![1.2 The Space and the States 3
predicting, that is, in channeling reproducible knowledge. Yet the full theory,
the long-desired theory of everything, still eludes us.
Happily, in this introduction to quantum algorithms, we need only a sim-
ple model of how part of the universe works. We can avoid relativity, special
and general; we can avoid the complexities of the Standard Model of particle
physics, with its hundreds of particles; we can even avoid gravity and electro-
magnetism. We cannot quite go back to earth, water, wind, and fire, but we can
avoid having to know and understand much of modern physics. This avowal
of independence from physical qualities does not prevent us from imagining
nature’s workings. Instead, it speaks to our belief that algorithmic considera-
tions in information processing run deeper.
1.2 The Space and the States
So what do we need to know? We need to understand that the state of our
quantum systems will always be described by a single unit vector a that lies in
some fixed vector space of dimension N = 2n, for some n. That is, the state is
always a vector
a =
⎡
⎢
⎢
⎣
a0
.
.
.
aN−1
⎤
⎥
⎥
⎦,
where each entry ak is a real or complex number depending on whether the
space is real or complex. Each entry is called an amplitude. We will not need
to involve the full quantum theory of mixed states, which are formally the same
as classical probability distributions over states like a, which are called pure
states. That is, we consider only pure states in this text.
We must distinguish between general states and basis states, which form
a linear-algebra basis composed of configurations that we may observe. In the
standard basis, the basis states are denoted by the vectors ek whose entries are
0 except for a 1 in place k. We identify ek with the index k itself in [0,N − 1]
and then further identify k with the k-th string x in a fixed total ordering of
{0,1}n. That the basis states correspond to all the length-n binary strings is
why we have N = 2n. The interplays among basis vectors, numerical indices,
and binary strings encoding objects are drawn more formally in chapter 2.
Any vector that is not a basis state is a superposition. Two or more basis
states have nonzero amplitude in any superposition, and only one of them can](https://image.slidesharecdn.com/28200689-250518095010-3a177360/85/Quantum-Algorithms-Via-Linear-Algebra-A-Primer-Richard-J-Lipton-Kenneth-W-Regan-22-320.jpg)
![6 Chapter 1 Introduction
End: We get information out of the quantum computation by making a mea-
surement. If the final state is c, then k is seen with probability |ck|2. Note that
the output is just the index k, not the probability of the index. Often we will
have a distinguished set S of indices that stand for accept, with outcomes in
[0,N − 1] S standing for reject.
That is it.
1.4 Where Is the Input?
In any model of computation, we expect that there is some way to input infor-
mation into the model’s devices. That seems, at first glance, to be missing from
our model. The start state is fixed, and the output method is fixed, so where do
we put the input? The answer is that the input can be encoded by the choice
of the unitary transformation U, in particular by the first several unitary opera-
tions in U, so that for different inputs we will apply different transformations.
This can easily be done in the case of classical computations. We can dis-
pense with explicit input provided we are allowed to change the program each
time we want to solve a different problem. Consider a program of this form:
M = 21;
x = Factor(M);
procedure Factor(z) { ... }
Clearly, if we can access the value of the variable x, then we can determine
the factors of 21. If we wanted to factor a more interesting number such as 35
or 1,001 or 11,234,143, then we can simply execute the same program with M
set to that number.
This integration of inputs with programs is more characteristic of quantum
than classical computation. Think of the transformation U as the program, so
varying U is exactly the same as varying the above program in the classical
case. We will show general ways of handling inputs in chapter 6, while for
several famous algorithms, in chapters 8–10, the input is expressly given as a
transformation that is dropped into a larger program. Not to worry—our chap-
ter 7 in the middle also provides support for the classical notion of feeding a
binary string x directly as the input, while also describing the ingredients of
the “quantum power” that distinguish these programs from classical ones.](https://image.slidesharecdn.com/28200689-250518095010-3a177360/85/Quantum-Algorithms-Via-Linear-Algebra-A-Primer-Richard-J-Lipton-Kenneth-W-Regan-25-320.jpg)
![2 Numbers and Strings
Before we can start to present quantum algorithms, we need to discuss the
interchangeable roles of natural numbers and Boolean strings. The set N of
natural numbers consists of
0,1,2,3,...
as usual.
A Boolean string is a string that consists solely of bits: a bit is either 0 or 1.
In computer science, such strings play a critical role, as you probably already
know, because our computational devices are all based on being either “on” or
“off,” “charged” or “uncharged,” “magnetized” or “unmagnetized,” and so on.
The operations we use on natural numbers are the usual ones. For example,
x + y is the sum of x and y, and x · y is their product. There is nothing new
or surprising here. The operations we use on Boolean strings are also quite
simple: The length of a string is the number of bits in the string, and if x and
y are Boolean strings, then xy is their concatenation. Thus, if x = 0101 and
y = 111, then we have xy = 0101111. This is a kind of “product” operation on
strings, but we find it convenient not to use an explicit operator symbol. If you
see xy and both x and y are strings, then xy is the result of concatenating them
together.
What we need to do is switch from numbers to Boolean strings and back.
Sometimes it is best to use the number representation and other times the string
representation. This kind of dual nature is basic in computer science and will
be used often in describing the quantum algorithms. There are, however, some
hitches that must be regarded to make it work properly. Let’s look and see why.
If m is a natural number, then it can uniquely be written as a binary number:
Let
m = 2n−1
xn−1 + ··· + 2x1 + x0,
where each xi is a bit, and we insist that xn−1 is nonzero. Then we can use m to
denote the Boolean string xn−1 ...x1x0. For instance, 7 maps to the string 111.
In the reverse direction, we can use the string
xn−1,...,x0
to denote the natural number
2n−1
xn−1 + ··· + 2x1 + x0.
For example, the string 10010 is the number 16 + 2 = 18.
Often we will be concerned with numbers in a fixed range 0...N − 1 where
N = 2n. This range is denoted by [N]. Then it will be convenient to omit the](https://image.slidesharecdn.com/28200689-250518095010-3a177360/85/Quantum-Algorithms-Via-Linear-Algebra-A-Primer-Richard-J-Lipton-Kenneth-W-Regan-28-320.jpg)
![3.3 Matrices 17
obtained by concatenating vectors from the former with vectors from the latter.
Its vectors have the form a(i) with i ∈ [m + n].
Their tensor product H1 ⊗ H2, however, has vectors of the form a(k), where
1 ≤ k ≤ mn. Indeed, k is in 1–1 correspondence with pairs (i,j) of indices
where i ∈ [m] and j ∈ [n]. Because we regard indices as strings, we can write
them juxtaposed as a(ij).
The tensor product of two vectors a and b is the vector c = a ⊗ b defined by
c(ij) = a(i)b(j).
A vector denoting a pure quantum state is separable if it is the tensor product
of two other vectors; otherwise it is entangled. The vectors e00 and e11 are
separable, but their unit-scaled sum 1
√
2
(e00 + e11) is entangled. The standard
basis vectors of H1 ⊗ H2 are separable, but this is not true of many of their
linear combinations, all of which still belong to H1 ⊗ H2 because it is a Hilbert
space.
Often our quantum algorithms will operate on the product of two Hilbert
spaces, each using binary strings as indices, which gives us the space of vectors
of the form a(xy). Here x ranges over the indices of the first space and y over
those of the second space. Writing a(xy) does not entail that a is separable.
3.3 Matrices
Matrices represent linear operators on Hilbert spaces. We can add them
together, we can multiply them, and of course we can use them to operate on
vectors. We assume these notions are familiar to you; if not, please see sources
in this chapter’s end notes. A typical example is
UVa = b.
This means: apply the V transform to the vector a, then apply the U transform
to the resulting vector, and the answer is the vector b. The matrix IN denotes
the N × N identity matrix. We use square brackets for matrix entries to dis-
tinguish them further from vector amplitudes, so that the identity matrix has
entries IN[r,c] = 1 if r = c, IN[r,c] = 0 otherwise. One of the key properties
of matrices is that they define linear operations, namely:
U(a + b) = Ua + Ub.](https://image.slidesharecdn.com/28200689-250518095010-3a177360/85/Quantum-Algorithms-Via-Linear-Algebra-A-Primer-Richard-J-Lipton-Kenneth-W-Regan-36-320.jpg)
![18 Chapter 3 Basic Linear Algebra
The fact that all our transformations are linear is what makes quantum algo-
rithms so different from classical ones. This will be clearer as we give exam-
ples, but the linearity restriction both gives quantum algorithms great power
and curiously makes them so different from classical ones.
If U is a matrix, then we use Uk
to denote the k-th power of the matrix,
Uk
= UU ···U
k copies
.
DEFINITION 3.1 The transpose of a matrix U is the matrix V such that
V[r,c] = U[c,r].
We use UT
as usual to denote the transpose of a matrix. We also use trans-
pose for vectors but only when writing them after a matrix in lines of text. Gen-
erally, we minimize the distinction between row and column vectors, using the
latter as standard. The inner product of two real vectors a and b is given by
a,b =
m
k=0
a(k)b(k).
DEFINITION 3.2 A real matrix U is unitary provided UT
U = I.
Here are three unitary 2 × 2 real matrices. Note that the last, called the
Hadamard matrix, requires a constant multiplier to divide out a factor of
2 that comes from squaring it.
I =
1 0
0 1
, X =
0 1
1 0
, H =
1
√
2
1 1
1 −1
.
The first two are also permutation matrices, meaning square matrices each
of whose rows and columns has all zeros except for a single 1. All permutation
matrices are unitary.
Another definition of a unitary matrix is based on the notion of orthogonal-
ity. Call two vectors a and b orthogonal if their inner product is 0. A matrix
U is unitary proved each row is a unit vector and any two distinct rows are
orthogonal. The reason these matrices are so important is that they preserve
the Euclidean length.
LEMMA 3.3 If U is a unitary matrix and a is a vector, then ||Ua|| = ||a||.](https://image.slidesharecdn.com/28200689-250518095010-3a177360/85/Quantum-Algorithms-Via-Linear-Algebra-A-Primer-Richard-J-Lipton-Kenneth-W-Regan-37-320.jpg)
![3.4 Complex Spaces and Inner Products 19
Proof. Direct calculation gives:
||Ua||2
=
x
|Ua(x)|2
=
x
Ua(x) · Ua(x)
=
x
(
y
U[x,y]a(y))(
z
U[x,z]a(z))
=
x
(
y z
U[x,y]U[x,z]a(y)a(z))
=
y z
(
x
(U[x,y]U[x,z]))a(y)a(z))
=
y
a(y)a(y) = ||a||2
because the inner product of U[−,y] and U[−,z] is 1 or 0 according as y = z.
As far as we can, we try not to care about whether a Hilbert space is real or
complex. However, we do need notation for complex spaces.
3.4 Complex Spaces and Inner Products
To describe some algorithms, mainly Shor’s, we need to use complex vectors
and matrices. In this case, all definitions are the same except for the notion of
transpose and inner product. Both now need to use the conjugation operation:
if z = x + iy where x,y are real numbers and i =
√
−1 as usual, then the conju-
gate of z is x − iy and is denoted by z. We now define the adjoint of a matrix
U to be the matrix V = U∗
such that
V[r,c] = U[c,r].
A complex matrix U is unitary provided U∗
U = I. Furthermore, the inner
product of two complex vectors a and b is defined to be
a,b =
m
k=0
a(k)b(k).
Note that because r is the same as r for a real number r, these concepts
are the same as what we defined before when the entries of the vectors and](https://image.slidesharecdn.com/28200689-250518095010-3a177360/85/Quantum-Algorithms-Via-Linear-Algebra-A-Primer-Richard-J-Lipton-Kenneth-W-Regan-38-320.jpg)
![20 Chapter 3 Basic Linear Algebra
matrices are all real numbers. Rather then use special notation for the complex
case, we will use the same as in the real case—the context should make it clear
which is being used. The one caveat is that in the few cases where the entries
of a or U are complex, one needs to conjugate them. Upon observing this, the
proof of the following is much the same as for Lemma 3.3 above:
LEMMA 3.4 If U is a unitary matrix and a is a vector, then the length of Ua
is the same as the length of a.
We can also form tensor products of matrices having any dimensions. If U
is m × n and V is r × s, then W = U ⊗ V is the mr × ns matrix whose action
on product vectors c(ij) = a(i)b(j) is as follows:
(Wc)(ij) = (Ua)(i)(Vb)(j).
Because every vector d of dimension rs (whether entangled or not) can be
written as a linear combination of basis vectors, each of which is a product
vector, the action Wd is well defined via the same linear combination of the
outputs on the basis vectors. That is, if
d =
r
i=1
s
j=1
di,jei ⊗ ej,
then
Wd =
r
i=1
s
j=1
di,j (Uei) ⊗
Vej
.
Note that it does not matter whether the scalars di,j are regarded as multiplying
ei, ej, or the whole thing. This fact matters later in section 6.5. We mainly
use tensor products to combine operations that work on separate halves of an
overall index ij.
3.5 Matrices, Graphs, and Sums Over Paths
One rich source of real-valued matrices is graphs. A graph G consists of a
set V of vertices, also called nodes, together with a binary relation E on V
whose members are called edges. The adjacency matrix A = A(G) of a graph
G = (V,E) is defined for all u,v ∈ V by:
A[u,v] =
1 if (u,v) ∈ E
0 otherwise.](https://image.slidesharecdn.com/28200689-250518095010-3a177360/85/Quantum-Algorithms-Via-Linear-Algebra-A-Primer-Richard-J-Lipton-Kenneth-W-Regan-39-320.jpg)
![22 Chapter 3 Basic Linear Algebra
dot product in a single column, namely, column 5, so there is no substitution
of nonzero values for nonzero entries that will make them orthogonal.
Figure 3.2
3-regular prism graph G and stochastic adjacency matrix A.
1 2
3 4
5 6
A
=
1
3
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0 1 1 0 1 0
1 0 0 1 0 1
1 0 0 1 1 0
0 1 1 0 0 1
1 0 1 0 0 1
0 1 0 1 1 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
However, in chapter 14, we will see a general technique involving a tensor
product of AG with another matrix to create a unitary matrix, one representing
a quantum rather than a classical random walk on the graph G.
The square of an adjacency matrix has entries given by
A2
[i,j] =
k
A[i,k]A[k,j].
The sum counts the number of ways to go from vertex i through some vertex k
and end up at vertex j. That is, A2
[i,j] counts the number of paths of length 2
from i to j. Likewise, A3
[i,j] counts paths of length exactly 3, A4
[i,j] those of
length 4, and so on.
For directed graphs in which each edge goes from one of n sources to one
of m sinks, the adjacency matrices An,m lend themselves to a path-counting
composition. Consider
U = An,m1 Am1,m2 Am2,m3 ···Amk,r.](https://image.slidesharecdn.com/28200689-250518095010-3a177360/85/Quantum-Algorithms-Via-Linear-Algebra-A-Primer-Richard-J-Lipton-Kenneth-W-Regan-41-320.jpg)
![3.6 Problems 23
Note how the mi dimensions match like dominoes. This makes U a legal matrix
product. The corresponding graph identifies the sinks of the graph G repre-
sented by Am−1,m (here m0 = n) with the sources of G+1. Then U[i,j] counts
the number of paths in the product graph that begin at source node i of G1 and
end at sink node j of the last graph Gk+1.
What links quantum and classical concepts here is that weights can be put on
the edges of these graphs, even complex-number weights. When these are sub-
stituted for the “1” entries of the adjacency matrices, the value U[i,j] becomes
a sum of products over the possible paths. In the classical case, these weights
can be probabilities of individual choices along the paths. In the quantum case,
they can be complex amplitudes, and some of the products in the sum may can-
cel. Either way, the leading algorithmic structure is that of a sum over paths.
As a fundamental idea in quantum mechanics, it was advanced particularly by
Richard Feynman, and yet it needs no more than this much about graphs and
linear algebra to appreciate.
This discussion already conveys the basic flavor of quantum operations. The
matrices are unitary rather than stochastic; the entries are square-roots of prob-
abilities rather than probabilities; negative entries—and later imaginary num-
ber entries—are used to achieve cancellations. After defining feasible compu-
tations, we will examine some important unitary matrices in greater detail.
3.6 Problems
3.1. Show that the product of unitary matrices is unitary.
3.2. If U is a matrix, then what is Uek?
3.3. Show that the columns of a unitary matrix are unit vectors. Also show that
distinct columns are orthogonal.
3.4. Consider the matrix
w w
w −w
.
For what real values of w is it a unitary matrix?
3.5. Consider the matrix U equal to:
1
√
2
IN IN
IN −IN
.](https://image.slidesharecdn.com/28200689-250518095010-3a177360/85/Quantum-Algorithms-Via-Linear-Algebra-A-Primer-Richard-J-Lipton-Kenneth-W-Regan-42-320.jpg)
![30 Chapter 4 Boolean Functions, Quantum Bits, and Feasibility
Thus, representing Boolean functions by their truth tables is alway possible,
but is not always feasible. The tables will be large when there are thirty inputs,
and when there are over 100, the table would be impossible to write down.
The final technical concept we need is having not just a single Boolean func-
tion but rather a family [fn] of Boolean functions, each fn taking n inputs,
that are conceptually related. That is, the [fn] constitute a single function f
on strings of all lengths, so we write f : {0,1}∗ → {0,1}, or for general rather
than one-bit outputs, f : {0,1}∗ → {0,1}∗. Maybe it is confusing to write “f”
also for this kind of function with an infinite domain, but the intent is usually
transparent—as when letting AND, OR, and XOR above apply to any n. Now
we can finally define “feasible”:
DEFINITION 4.1 A Boolean function f = [fn] is feasible provided the indi-
vidual fn are computed by circuits of size nO(1).
4.2 An Example
Consider the Boolean function MAJ(x1,x2,x3,x4,x5), which takes the major-
ity of five Boolean inputs. A first idea is to compute it using applications
of OR and AND as follows: For every three-element subset S = {i,j,k} of
{1,2,3,4,5}, we compute yS = OR(xi,xj,xk). Define y to be the AND of each
of the ten subsets S. Then y = 1 ⇐⇒ no more than 2 bits of x1,...,x5 are 0
⇐⇒ MAJ(x1,x2,x3,x4,x5) is true. The complexity is counted as 11 operations
and, importantly, 35 total arguments of those operations.
On second thought, we can find a program of slightly lower complexity.
Consider the Boolean circuit diagram in figure 4.1.
Expressed as a sequence of operations, in one of many possible orders, the
circuit is equivalent to the following straight-line program:
v1 = OR(x1,x2,x3), v2 = OR(x4,x5),
w1 = AND(x1,x2), w2 = AND(x1,x3), w3 = AND(x2,x3),
w4 = AND(w1,w2), w5 = AND(v1,x4,x5)
u = OR(w1,w2,w3), t = AND(u,v2), y = OR(w4,t,w5).
This program has 10 operations and only 24 applications to arguments. To see
that it is correct, note that w4 is true if and only if x1 = x2 = x3 = 1, and w5 is
true if and only if x4 = x5 = 1 and one of x1,x2,x3 is 1. Finally, t is true if and](https://image.slidesharecdn.com/28200689-250518095010-3a177360/85/Quantum-Algorithms-Via-Linear-Algebra-A-Primer-Richard-J-Lipton-Kenneth-W-Regan-49-320.jpg)
![4.2 An Example 31
Figure 4.1
Monotone circuit computing MAJ(x1,x2,x3,x4,x5).
x
1 1
2 2
3
4 5
5
4
3
2
1
x x x x
v
v
w w
w
w
w
u
t
y
only if two of x1,x2,x3 and one of x4,x5 are true, which handles the remaining
six true cases.
Now clearly MAJ generalizes into a Boolean function MAJ of strings x of
any length n, such that MAJ(x) returns 1 if more than n/2 of the bits of x are
1. We ask the important question:
Is MAJ feasible?
The operational question about MAJ(x1,x2,x3,x4,x5) is, do the above pro-
grams scale when “5” is replaced by “n”? Technically, scalable is the same
idea as feasible but with the idea mentioned in chapter 2 that a polynomial
bound is the same as having a linear bound each time the size of the input
doubles.
The first idea, when generalized from “5” to “n,” says to take the AND
of every r-sized subset of [n], where r = n/2 + 1, and feed that to an OR.](https://image.slidesharecdn.com/28200689-250518095010-3a177360/85/Quantum-Algorithms-Via-Linear-Algebra-A-Primer-Richard-J-Lipton-Kenneth-W-Regan-50-320.jpg)
![4.3 Quantum Representation of Boolean Arguments 33
O(st) = O(t2) because we do at most one bit of scratchwork at each step. If the
machine time t(n) is polynomial, then so is t(n)2.
This relation also goes in the opposite direction if you think of using a
machine to verify the computation by the circuit—provided the circuits are
uniform in a technical sense that captures the conceptual sense we applied
above to Boolean function families.
Hence, the criterion of feasible is broad and is the same for any of the classi-
cal models of computation by machines, programs, or (uniform) circuits. There
is a huge literature on which functions are feasible and which are not. One
can encode anything via Boolean strings, including circuits themselves. The
problem of whether a Boolean circuit can ever output 1—even when it allows
applying NOT only to the original arguments xi and then has just one level of
ternary OR gates feeding into a big AND—is not known to be feasible. That is,
no feasible function is known to give the correct answer for every encoding X
of such a program: this is called the satisfiability problem and has a property
called NP-hardness. We define this with regard to an equivalent problem about
solving equations in chapter 16. For now we are happy with not only defining
classical feasible computation in detail but also showing that equivalent cri-
teria are reached from different models. Now we are ready for the quantum
challenge to this standard.
4.3 Quantum Representation of Boolean Arguments
Let N = 2n. Every coordinate in N-dimensional Hilbert space corresponds to
a binary string of length n. The standard encoding scheme assigns to each
index j ∈ [0,...,n − 1] the n-bit binary string that denotes j in binary nota-
tion, with leading 0’s if necessary. This produces the standard lexicographic
ordering on strings. For instance, with n = 2 and N = 4, we show the indexing
applied to a permutation matrix:
00 01 10 11
00 1 0 0 0
01 0 1 0 0
10 0 0 0 1
11 0 0 1 0
.](https://image.slidesharecdn.com/28200689-250518095010-3a177360/85/Quantum-Algorithms-Via-Linear-Algebra-A-Primer-Richard-J-Lipton-Kenneth-W-Regan-52-320.jpg)
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- 6. QUANTUM ALGORITHMS VIA LINEAR ALGEBRA
- 8. QUANTUM ALGORITHMS VIA LINEAR ALGEBRA A Primer Richard J. Lipton Kenneth W. Regan The MIT Press Cambridge, Massachusetts London, England
- 9. c 2014 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. MIT Press books may be purchased at special quantity discounts for business or sales promotional use. For information, please email special sales@mitpress.mit.edu. This book was set in Times Roman and Mathtime Pro 2 by the authors, and was printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Lipton, Richard J., 1946– Quantum algorithms via linear algebra: a primer / Richard J. Lipton and Kenneth W. Regan. p. cm. Includes bibliographical references and index. ISBN 978-0-262-02839-4 (hardcover : alk. paper) 1. Quantum computers. 2. Computer algorithms. 3. Algebra, Linear. I. Regan, Kenneth W., 1959– II. Title QA76.889.L57 2014 005.1–dc23 2014016946 10 9 8 7 6 5 4 3 2 1
- 10. We dedicate this book to all those who helped create and nourish the beautiful area of quantum algorithms, and to our families who helped create and nourish us. RJL and KWR
- 12. Contents Preface xi Acknowledgements xiii 1 Introduction 1 1.1 The Model 2 1.2 The Space and the States 3 1.3 The Operations 5 1.4 Where Is the Input? 6 1.5 What Exactly Is the Output? 7 1.6 Summary and Notes 8 2 Numbers and Strings 9 2.1 Asymptotic Notation 11 2.2 Problems 12 2.3 Summary and Notes 13 3 Basic Linear Algebra 15 3.1 Hilbert Spaces 16 3.2 Products and Tensor Products 16 3.3 Matrices 17 3.4 Complex Spaces and Inner Products 19 3.5 Matrices, Graphs, and Sums Over Paths 20 3.6 Problems 23 3.7 Summary and Notes 26 4 Boolean Functions, Quantum Bits, and Feasibility 27 4.1 Feasible Boolean Functions 28 4.2 An Example 30 4.3 Quantum Representation of Boolean Arguments 33 4.4 Quantum Feasibility 35 4.5 Problems 38 4.6 Summary and Notes 40 5 Special Matrices 41 5.1 Hadamard Matrices 41 5.2 Fourier Matrices 42 5.3 Reversible Computation and Permutation Matrices 43 5.4 Feasible Diagonal Matrices 44 5.5 Reflections 45 5.6 Problems 46
- 13. viii Contents 5.7 Summary and Notes 49 6 Tricks 51 6.1 Start Vectors 51 6.2 Controlling and Copying Base States 52 6.3 The Copy-Uncompute Trick 54 6.4 Superposition Tricks 55 6.5 Flipping a Switch 56 6.6 Measurement Tricks 58 6.7 Partial Transforms 59 6.8 Problems 60 6.9 Summary and Notes 62 7 Phil’s Algorithm 63 7.1 The Algorithm 63 7.2 The Analysis 63 7.3 An Example 64 7.4 A Two-Qubit Example 64 7.5 Phil Measures Up 66 7.6 Quantum Mazes versus Circuits versus Matrices 69 7.7 Problems 71 7.8 Summary and Notes 74 8 Deutsch’s Algorithm 77 8.1 The Algorithm 77 8.2 The Analysis 78 8.3 Superdense Coding and Teleportation 82 8.4 Problems 86 8.5 Summary and Notes 87 9 The Deutsch-Jozsa Algorithm 89 9.1 The Algorithm 89 9.2 The Analysis 90 9.3 Problems 92 9.4 Summary and Notes 92 10 Simon’s Algorithm 93 10.1 The Algorithm 93 10.2 The Analysis 94
- 14. Contents ix 10.3 Problems 95 10.4 Summary and Notes 96 11 Shor’s Algorithm 97 11.1 Strategy 97 11.2 Good Numbers 98 11.3 Quantum Part of the Algorithm 99 11.4 Analysis of the Quantum Part 100 11.5 Probability of a Good Number 102 11.6 Using a Good Number 105 11.7 Continued Fractions 106 11.8 Problems 107 11.9 Summary and Notes 108 12 Factoring Integers 109 12.1 Some Basic Number Theory 109 12.2 Periods Give the Order 110 12.3 Factoring 110 12.4 Problems 112 12.5 Summary and Notes 113 13 Grover’s Algorithm 115 13.1 Two Vectors 115 13.2 The Algorithm 117 13.3 The Analysis 117 13.4 The General Case, with k Unknown 118 13.5 Grover Approximate Counting 119 13.5.1 The Algorithm 122 13.5.2 The Analysis 122 13.6 Problems 126 13.7 Summary and Notes 128 14 Quantum Walks 129 14.1 Classical Random Walks 129 14.2 Random Walks and Matrices 130 14.3 An Encoding Nicety 132 14.4 Defining Quantum Walks 133 14.5 Interference and Diffusion 134
- 15. x Contents 14.6 The Big Factor 138 14.7 Problems 139 14.8 Summary and Notes 140 15 Quantum Walk Search Algorithms 143 15.1 Search in Big Graphs 143 15.2 General Quantum Walk for Graph Search 145 15.3 Specifying the Generic Walk 147 15.4 Adding the Data 149 15.5 Toolkit Theorem for Quantum Walk Search 150 15.5.1 The Generic Algorithm 151 15.5.2 The Generic Analysis 152 15.6 Grover Search as Generic Walk 152 15.7 Element Distinctness 153 15.8 Subgraph Triangle Incidence 154 15.9 Finding a Triangle 155 15.10Evaluating Formulas and Playing Chess 156 15.11Problems 157 15.12Summary and Notes 158 16 Quantum Computation and BQP 159 16.1 The Class BQP 159 16.2 Equations, Solutions, and Complexity 161 16.3 A Circuit Labeling Algorithm 163 16.4 Sum-Over-Paths and Polynomial Roots 165 16.5 The Additive Polynomial Simulation 168 16.6 Bounding BQP 169 16.7 Problems 170 16.8 Summary and Notes 173 17 Beyond 175 17.1 Reviewing the Algorithms 175 17.2 Some Further Topics 176 17.3 The “Quantum” in the Algorithms 179 Bibliography 183 Index 189
- 16. Preface This book is an introduction to quantum algorithms unlike any other. It is short, yet it is comprehensive and covers the most important and famous quantum algorithms; it assumes minimal background, yet is mathematically rigorous; it explains quantum algorithms, yet steers clear of the notorious philosophical problems and issues of quantum mechanics. We assume no background in quantum theory, quantum mechanics, or quan- tum anything. None. Quantum computation can be described in terms of ele- mentary linear algebra, so some familiarity with vectors, matrices, and their basic properties is required. However, we will review all that we need from linear algebra, which is surprisingly little. If you need a refresher, then our material should be enough; if you are new to linear algebra, then we suggest some places where you can find the required material. It is really not much, so do not worry. We do assume that you are comfortable with mathematical proofs; that is, we assume “mathematical maturity” in a way that is hard to define. Our proofs are short and straightforward, except for advanced topics in section 13.5 and chapters 15 and 16. This may be another surprise: for all the excitement about quantum algorithms, it is interesting that the mathematical tools and meth- ods used are elementary. The proofs are neat, clever, and interesting, but you should have little trouble following the arguments. If you do, it is our fault— we hope that our explanations are always clear. Our idea of a standard course runs through section 13.4, possibly including chapter 14. We strive for mathematical precision. There is always a fine line between being complete and clear and being pedantic—hopefully we stay on the right side of this. We started with the principle of supplying all the details—all of them—on all we present. We have compromised in three places, all having to do with approximations. The first is our using the quantum Fourier transform “as-is” rather than approximating it, and the others are in chapters 15 and 16. For better focus on the algorithms, we chose to de-emphasize quantum cir- cuits. In fact, we tried to avoid quantum circuits and particularities of quantum gates altogether. However, they are excellent to illuminate linear algebra, so we have provided a rich set of exercises in chapters 3 through 7, plus two pop- ular applications in section 8.3. These can in fact be used to support coverage of quantum circuits in a wider-scale course. The same goes for complexity classes. We prefer to speak operationally in terms of feasible computation, and we try to avoid being wedded to the “asymptotically polynomially bounded” definition of it. We avoid naming any complexity class until chapter 16. Never- theless, that chapter has ample complexity content anchored in computational
- 17. xii Preface problems rather than machine models and is self-contained enough to support a course that covers complexity theory. At the same stroke, it gives algebraic tools for analyzing quantum circuits. We featured tricks we regard as algorith- mic in the main text and delegated some tricks of implementation to exercises. What makes an algorithm a quantum algorithm? The answer should have nothing to do with how the algorithm is implemented in a physical quantum system. We regard this as really a question about how programming notation— mathematical notation—represents the feasibility of calculations in nature. Quantum algorithms use algebraic units called qubits that are richer than bits, by which they are allowed to count as feasible some operations that when writ- ten out in simple linear algebra use exponentially long notation. The rules for these allowed operations are specified in standard models of quantum compu- tation, which are all equivalent to the one presented in this book. It might seem ludicrous to believe that nature in any sense uses exponentially long notation, but some facet of this appears at hand because quantum algorithms can quickly solve problems that many researchers believe require exponential work by any “classical” algorithm. In this book, classical means an algorithm written in the notation for feasible operations used by every computer today. This leads to a word about our presentation. Almost all summaries, notes, and books on quantum algorithms use a special notation for vectors and matri- ces. This is the famous Dirac notation that was invented by Paul Dirac—who else. It has many advantages and is the de-facto standard in the study of quan- tum algorithms. It is a great notation for experts and instrumental to becom- ing an expert, but we suspect it is a barrier for those starting out who are not experts. Thus, we avoid using it, except for a few places toward the end to give a second view of some complicated states. Our thesis is that we can explain quantum algorithms without a single use of this notation. Essentially this book is a testament to that belief: if you find this book more accessible than oth- ers, then we believe it owes to this decision. Our notation follows certain ISO recommendations, including boldface italics for vectors and heavy slant for matrices and operators. We hope you will enjoy this little book. It can be used to gain an understand- ing of quantum algorithms by self-study, as a course or seminar text, or even as additional material in a general course on algorithms. Georgia Institute of Technology, Richard J. Lipton University at Buffalo (SUNY), Kenneth W. Regan
- 18. Acknowledgements We thank Aram Harrow, Gil Kalai, and John Preskill for contributions to a debate since 2012 on themes reflected in the opening and last chapters, Andrew Childs and Stacey Jeffery for suggestions on chapter 15, and several members of the IBM Thomas J. Watson Research Center for suggestions and illumi- nating discussions that stimulated extra coverage of quantum gates and circuit simulations in the exercises. We thank Marie Lufkin Lee and Marc Lowenthal and others at MIT Press for patiently shepherding this project to completion, and the anonymous reviewers of the manuscript in previous stages. We also thank colleagues and students for some helpful input. Quantum circuit dia- grams were typeset using version 2 of the Qcircuit.tex package by Steve Flammia and Bryan Eastin.
- 20. 1 Introduction One of the great scientific and engineering questions of our time is: Are quantum computers possible? We can build computers out of mechanical gears and levers, out of electric relays, out of vacuum tubes, out of discrete transistors, and finally today out of integrated circuits that contain thousands of millions of individual transis- tors. In the future, it may be possible to build computers out of other types of devices—who knows. All of these computers, from mechanical to integrated-circuit-based ones, are called classical. They are all classical in that they implement the same type of computer, albeit as the technology gets more sophisticated the computers become faster, smaller, and more reliable. But they all behave in the same way, and they all operate in a non-quantum regime. What distinguishes these devices is that information is manipulated as bits, which already have determinate values of 0 or 1. Ironically, the key compo- nents of today’s computers are quantum devices. Both the transistor and its potential replacement, the Josephson junction, won a Nobel Prize for the quan- tum theory of their operation. So why is their regime non-quantum? The reason is that the regime reckons information as bits. By contrast, quantum computation operates on qubits, which are based on complex-number values, not just 0 and 1. They can be read only by measuring, and the readout is in classical bits. To skirt the commonly bandied notion of observers interfering with quantum systems and postpone the discussion of measurement as an operation, we offer the metaphor that a bit is what you get by “cooking” a qubit. From this standpoint, doing a classical computation on bits is like cooking the ingredients of a pie individually before baking them together in the pie. The quantum argument is that it’s more expedient to let the filling bubble in its natural state while cooking everything at once. The engineering problem is whether the filling can stay coherent long enough for this to work. The central question is whether it is possible to build computers that are inherently quantum. Such computers would exploit the power and wonder of nature to create systems that can effectively be in multiple states at once. They open a world with apparent actions at a distance that the great Albert Ein- stein never believed but that actually happen—a world with other strange and counter-intuitive effects. To be sure, this is the world we live in, so the question becomes how much of this world our computers can enact.
- 21. 2 Chapter 1 Introduction This question is yet to be resolved. Many believe that such machines will be built one day. Some others have fundamental doubts and believe there are phys- ical limits that make quantum computers impossible. It is currently unclear who is right, but whatever happens will be interesting: a world with quantum computers would allow us to solve hard problems, while a barrier to them might shed light on deep questions of physics and information. Happily this question does not pose a barrier to us. We plan to study quan- tum algorithms, which are interesting whether quantum computers are built soon, in the next ten years, in the next fifty years, or never. The area of quantum algorithms contains some beautiful ideas that everyone interested in computa- tion should know. The rationale for this book is to supply a gentle introduction to quan- tum algorithms. We will say nothing more about quantum computers—about whether they will be built or how they may work—until the end. We will only discuss algorithms. Our goal is to explain quantum algorithms in a way that is accessible to almost anyone. Curiously, while quantum algorithms are quite different from classical ones, the mathematical tools needed to understand them are quite modest. The mathematics that is required to understand them is linear algebra: vectors, matrices, and their basic properties. That is all. So these are really linear-algebraic algorithms. 1.1 The Model The universe is complex and filled with strange and wonderful things. From lifeforms like viruses, bacteria, and people; to inanimate objects like comput- ers, airplanes, and bridges that span huge distances; from planets, to whole galaxies. There is mystery and wonder in them all. The goal of science in general, and physics specifically, is to explore and understand the universe by discovering the simplest laws possible that explain the multitude of phenomena. The method used by physics is the discovery of models that predict the behavior of all from the smallest to the largest objects. In ancient times, the models were crude: the earliest models “explained” all by reducing everything to earth, water, wind, and fire. Today, the models are much more refined—they replace earth and water by hundreds of particles and wind and fire by the four fundamental forces. Mainly, the models are better at
- 22. 1.2 The Space and the States 3 predicting, that is, in channeling reproducible knowledge. Yet the full theory, the long-desired theory of everything, still eludes us. Happily, in this introduction to quantum algorithms, we need only a sim- ple model of how part of the universe works. We can avoid relativity, special and general; we can avoid the complexities of the Standard Model of particle physics, with its hundreds of particles; we can even avoid gravity and electro- magnetism. We cannot quite go back to earth, water, wind, and fire, but we can avoid having to know and understand much of modern physics. This avowal of independence from physical qualities does not prevent us from imagining nature’s workings. Instead, it speaks to our belief that algorithmic considera- tions in information processing run deeper. 1.2 The Space and the States So what do we need to know? We need to understand that the state of our quantum systems will always be described by a single unit vector a that lies in some fixed vector space of dimension N = 2n, for some n. That is, the state is always a vector a = ⎡ ⎢ ⎢ ⎣ a0 . . . aN−1 ⎤ ⎥ ⎥ ⎦, where each entry ak is a real or complex number depending on whether the space is real or complex. Each entry is called an amplitude. We will not need to involve the full quantum theory of mixed states, which are formally the same as classical probability distributions over states like a, which are called pure states. That is, we consider only pure states in this text. We must distinguish between general states and basis states, which form a linear-algebra basis composed of configurations that we may observe. In the standard basis, the basis states are denoted by the vectors ek whose entries are 0 except for a 1 in place k. We identify ek with the index k itself in [0,N − 1] and then further identify k with the k-th string x in a fixed total ordering of {0,1}n. That the basis states correspond to all the length-n binary strings is why we have N = 2n. The interplays among basis vectors, numerical indices, and binary strings encoding objects are drawn more formally in chapter 2. Any vector that is not a basis state is a superposition. Two or more basis states have nonzero amplitude in any superposition, and only one of them can
- 23. 4 Chapter 1 Introduction be observed individually in any measurement. The amplitude ak is not directly meaningful for what we may expect to observe but rather its squared absolute value, |ak|2. This gives the probability of measuring the system to be in the basis state ek. The import of a being a unit vector is that these probabilities sum to 1, namely, N−1 k=0 |ak|2 = 1. For N = 2, the idea that the length of a diagonal line defined by the origin and a point in the plane involves the sum of two squares is older than Pythago- ras. The length is 1 precisely when the point lies on the unit circle. We may regard the basis state e0 as lying on the x-axis, while e1 lies on the y-axis. Then measurement projects the state a either onto the “x leg” of the triangle it makes with the x-axis or the “y leg” of the triangle along the y-axis. It may still seem odd that the probabilities are proportional not to the lengths of the legs but to their squares. But we know that if the angle is θ from the near part of the x-axis (so 0 ≤ θ ≤ π/2), then the lengths are cos(θ) and sin(θ), respectively, and it is cos2(θ) + sin2(θ), not cos(θ) + sin(θ), that sums to 1. If we wanted to use points whose legs sum directly to 1, we’d have to use the diamond that is inscribed inside the circle. In N-dimensional space, we’d have to use the N-dimensional simplex rather than the sphere. Well the simplex is spiky and was not really studied until the 20th century, whereas the sphere is smooth and nice and was appreciated by the ancients. Evidently nature agrees with ancient aesthetics. We may not know why the world works this way, but we can certainly say, why not? Once we agree, all we really need to know about the space is that it sup- ports the picture of Pythagoras, that is, ordinary Euclidean space. Both the real vector spaces RN and the complex ones CN do so. That is, they agree on how many components their vectors have and how distances are measured by taking squares of values from the vector components. They differ only on what kind of numbers these component values v can be, but the idea of the norm or absolute value |v| quickly reconciles this difference. This aspect was first formalized by Euclid’s great rigorizer of the late 19th and early 20th centuries, David Hilbert; in his honor, the common concept is called a Hilbert Space. Hilbert’s concept retains its vigor even if “N” can be infinite, or if the “vectors” and “compo- nents” are strange objects, but in this book, we need not worry: N will always be finite, and the space HN will be RN or CN. Allowing the latter is the reason that we say “Hilbert space” not “ordinary Euclidean space.”
- 24. 1.3 The Operations 5 1.3 The Operations In fact, our model embraces the circle and the sphere even more tightly. It throws out—it disallows—all other points. Every state a must be a point on the unit sphere. If you have heard the term “projective space,” then that is what we are restricting to. But happily we need not worry about restricting the points so long as we restrict the operations in our model. The operations must map any point on the unit sphere to some (other) point on the unit sphere. We will also restrict the operations to be linear. Apart from measurements, they must be onto the whole sphere—which makes them map all of HN onto all of HN. By the theory of vector subspaces, this means the operations must also be invertible. Operations with all these properties are called unitary. We will represent the operations by matrices, and we give several equivalent stipulations for unitary matrices in chapter 3, followed by examples in chap- ter 5 and tricks for working with them in chapter 6. But we can already under- stand that compositions of unitary operations are unitary, and their representa- tions and actions can be figured by the familiar idea of multiplying matrices. Thus, our model’s programs will simply be compositions of unitary matrices. The one catch is that the matrices themselves will be huge, out of proportion to the actual simplicity of the operation as we believe nature meters it. Hence, we will devote time in chapters 4 and 5 to ensuring these operations are feasible according to standards already well accepted in classical computation. Thus, we can comply with the requirements for any computational model that we must understand what state the computation starts in, how it moves from one state to another, and how we get information out of the computation. Start: We will ultimately be able to assume that the start state is always the elementary vector e0 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 1 0 . . . 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ of length N. Because the first binary string in our ordering of {0,1}n will be 0n, our usual start state will denote the binary string of n zeros. Move: If the system is in some state a, then we can move it by applying a unitary transformation U. Thus, a will move to b where b = Ua. Not all unitary transformations are allowed, but we will get to that later. Note that if a is a unit vector, then so is b.
- 25. 6 Chapter 1 Introduction End: We get information out of the quantum computation by making a mea- surement. If the final state is c, then k is seen with probability |ck|2. Note that the output is just the index k, not the probability of the index. Often we will have a distinguished set S of indices that stand for accept, with outcomes in [0,N − 1] S standing for reject. That is it. 1.4 Where Is the Input? In any model of computation, we expect that there is some way to input infor- mation into the model’s devices. That seems, at first glance, to be missing from our model. The start state is fixed, and the output method is fixed, so where do we put the input? The answer is that the input can be encoded by the choice of the unitary transformation U, in particular by the first several unitary opera- tions in U, so that for different inputs we will apply different transformations. This can easily be done in the case of classical computations. We can dis- pense with explicit input provided we are allowed to change the program each time we want to solve a different problem. Consider a program of this form: M = 21; x = Factor(M); procedure Factor(z) { ... } Clearly, if we can access the value of the variable x, then we can determine the factors of 21. If we wanted to factor a more interesting number such as 35 or 1,001 or 11,234,143, then we can simply execute the same program with M set to that number. This integration of inputs with programs is more characteristic of quantum than classical computation. Think of the transformation U as the program, so varying U is exactly the same as varying the above program in the classical case. We will show general ways of handling inputs in chapter 6, while for several famous algorithms, in chapters 8–10, the input is expressly given as a transformation that is dropped into a larger program. Not to worry—our chap- ter 7 in the middle also provides support for the classical notion of feeding a binary string x directly as the input, while also describing the ingredients of the “quantum power” that distinguish these programs from classical ones.
- 26. 1.5 What Exactly Is the Output? 7 1.5 What Exactly Is the Output? The problematic words in this section title are the two short ones. With apolo- gies to President Bill Clinton, the question is not so much what the meaning of “is” is, as what the meaning of “the” is. In a classical deterministic model of computation, every computation has one, definite, output. Even if the model is randomized, every random input still determines a single computation path, whose output is definite before its last step is executed. A quantum computation, however, presents the user at the end with a slot machine. Measuring is pulling the lever to see what output the wheels give you. Unlike in some old casinos where slot machines were rigged, you control how the machine is built, and hopefully you’ve designed it to make the prob- abilities work in your favor. As with the input, however, the focus shifts from a given binary string to the machinery itself, and to the action of sampling a distribution that pulling the measurement lever gives you. In chapter 6, we will also finesse the issue of having measurements in the middle of computations that continue from states projected down to a lower- dimensional space. This could be analogized to having a slot machine still spin- ning after one wheel has fixed its value. We show why measurements may gen- erally be postponed until the end, but the algorithms in this textbook already behave that way. This helps focus the idea that the ultimate goal of quantum computation is not a single output but rather a sampling device. Chapter 6 also lays groundwork for how those devices can be re-used to improve one’s chance of success. Chapter 7 lays out the format for how we present and analyze quan- tum algorithms and gives further attention to entanglement, interference, and measurement. All of this remains mere philosophy, however, unless we can show how the results of the sampling help solve concrete problems efficiently in distin- guished ways. These solutions are the ultimate outputs, as exemplified in chap- ters 8–10. In chapters 11 and 12, the outputs are factorizations of numbers, via the algorithm famously discovered by Peter Shor. In chapters 13–15, they are objects that we need to search for in a big search space. Chapter 16 branches out to topics in quantum complexity theory, defining the class BQP formally and proving upper and lower bounds on it in terms of classical complexity. Chapter 17 summarizes the algorithms and discusses some further topics and readings. Saying this completes the application of the model and the overview of this book.
- 27. 8 Chapter 1 Introduction 1.6 Summary and Notes In old Platonic fashion, we have tried to idealize quantum computation by pure thought apart from physical properties of the world around us. Of course one can make errors that way, as Aristotle showed by failing to climb a tower and drop equal-sized balls of unequal weights. It is vain to think that Pythagoras or Euclid or Archimedes could have come up with such a computational model but maybe not so vain to estimate it of Hilbert. Linear algebra and geometry were both deepened by Hilbert. Meanwhile, the quantum theory emerged and ideas of computation were extensively discussed long before Alan Turing gave his definitive classical answer (Turing, 1936). The main stumbling block may have been probability. Quantum physicists were forced to embrace probability from the get-go, in a time when Newtonian determinism was dominant and Einstein said “God”—that is, nature—“does not play dice.” Some of our algorithms will be deterministic—that is, we will encounter cases where the final points on the unit sphere coincide with standard basis vectors, whereupon all the probabilities are 0 or 1. However, coping with probabilistic output appears necessary to realize the full power of quantum computation. Another block, even for the physicists happy with dice, may have been the long time it took to realize the essence of computation. Turing’s paper reached full flower only with the emergence of computing machines during and after World War II. Even then, it took Richard Feynman, arguably the main visionary in the area after the passing of John von Neumann in 1956, until his last decade in the 1980s to set his vision down (Feynman, 1982, 1985). That is when it came to the attention of David Deutsch (1985) and some others. The theory still had several false starts—for instance, the second of us overlapped with Deutsch in 1984–1986 at Oxford’s Mathematical Institute and saw its fellows turn aside Deutsch’s initial claims to be able to compute classically uncomputable func- tions. It can take a long time for a great theory to mature, but a great theorem such as Peter Shor’s on quantum factoring can accelerate it a lot. We hope this book helps make the ascent to understanding it easier. We chose M = 21 in section 1.4 because it is currently the highest integer on which practical runs of Shor’s algorithm have been claimed, but even these are not definitively established (Smolin et al., 2013).
- 28. 2 Numbers and Strings Before we can start to present quantum algorithms, we need to discuss the interchangeable roles of natural numbers and Boolean strings. The set N of natural numbers consists of 0,1,2,3,... as usual. A Boolean string is a string that consists solely of bits: a bit is either 0 or 1. In computer science, such strings play a critical role, as you probably already know, because our computational devices are all based on being either “on” or “off,” “charged” or “uncharged,” “magnetized” or “unmagnetized,” and so on. The operations we use on natural numbers are the usual ones. For example, x + y is the sum of x and y, and x · y is their product. There is nothing new or surprising here. The operations we use on Boolean strings are also quite simple: The length of a string is the number of bits in the string, and if x and y are Boolean strings, then xy is their concatenation. Thus, if x = 0101 and y = 111, then we have xy = 0101111. This is a kind of “product” operation on strings, but we find it convenient not to use an explicit operator symbol. If you see xy and both x and y are strings, then xy is the result of concatenating them together. What we need to do is switch from numbers to Boolean strings and back. Sometimes it is best to use the number representation and other times the string representation. This kind of dual nature is basic in computer science and will be used often in describing the quantum algorithms. There are, however, some hitches that must be regarded to make it work properly. Let’s look and see why. If m is a natural number, then it can uniquely be written as a binary number: Let m = 2n−1 xn−1 + ··· + 2x1 + x0, where each xi is a bit, and we insist that xn−1 is nonzero. Then we can use m to denote the Boolean string xn−1 ...x1x0. For instance, 7 maps to the string 111. In the reverse direction, we can use the string xn−1,...,x0 to denote the natural number 2n−1 xn−1 + ··· + 2x1 + x0. For example, the string 10010 is the number 16 + 2 = 18. Often we will be concerned with numbers in a fixed range 0...N − 1 where N = 2n. This range is denoted by [N]. Then it will be convenient to omit the
- 29. 10 Chapter 2 Numbers and Strings leading 1 and use n bits for each number, so that zero is n-many 0s, one is 0n−11, and so on up to N − 1 = 1n, where 1n means n-many 1’s. We call this the canonical numbering of {0,1}n. For example, with n = 3: 000 = 0 100 = 4 001 = 1 101 = 5 010 = 2 110 = 6 011 = 3 111 = 7. The small but important issue is that, for the representation from numbers to strings to be unambiguous, we must know how long the strings are. Otherwise, what does the number 0 represent? Does it represent 0 or 00 or 000 and so on? This is why we said earlier that the mapping between numbers and strings is not exact. To make it precise, we need to know the length of the strings. A more technical way of saying this is that once we specify the mapping as being between the natural numbers 0,1,...,2n − 1 and the strings of length n (that is, {0,1}n), it is one-to-one. Note that 0 as a number now corresponds to the unique string 0...0. total of n zeros There is one more operation that we use on Boolean strings. If x and y are Boolean strings of length m, then x • y is their Boolean inner product, which is defined to be x1y1 ⊕ ··· ⊕ xmym. Here ⊕ means exclusive-or, which is the same as addition modulo 2. Hence, sometimes we may talk about Boolean strings as being members of an m- dimensional space with addition modulo 2. We must warn that the name inner product is also used when we talk about Hilbert spaces in chapter 3. Many sources use x · y to mean concatenation of strings, but we reserve the lighter dot for numerical multiplication. When x and y are single bits, x · y is the same as x • y, but using the lighter dot still helps remind us that they are single bits. Sometimes this type of overloading occurs in mathematics—we try to make clear which is used when. A further neat property of Boolean strings is that they can represent subsets of a set. If the set is {1,2,3}, in that order, then 000 corresponds to the empty set, 011 to {2,3}, 100 to {1}, 111 to the whole set {1,2,3}, and so on.
- 30. 2.1 Asymptotic Notation 11 2.1 Asymptotic Notation Suppose we run an algorithm that on problems of size n works in n “passes,” where the i-th pass takes i “steps.” How long does the whole algorithm take? If we want to be exact about the number s(n) of steps, then we can calculate: s(n) = n i=1 i = n(n + 1) 2 = 1 2 n2 + 1 2 n. If we view the process pictorially, then we see the passes are tracing out a trian- gular half of an n × n square along the main diagonal. This intuition says “1 2 n2” without worrying about whether the main diagonal is included or excluded or “halved.” The difference is a term 1 2 n whose added size is relatively tiny as n becomes moderately large, so we may ignore it. Formally, we define: DEFINITION 2.1 Two functions s(n) and t(n) on N are asymptotically equiv- alent, written s(n) ∼ t(n), if limn→∞ s(n) t(n) exists and equals 1. So s(n) ∼ 1 2 n2, which we can also encapsulate by saying s(n) is quadratic with “principal constant” 1 2 . But suppose now we don’t know or care about the actual time units for a “step,” only that the algorithm’s cost scales as n2. Another way of saying this is that as the data size n doubles, the time for the algorithm goes up by about a factor of 4. This idea doesn’t care what the constant multiplying n2 is, only that it is some constant. Hence, we define: DEFINITION 2.2 Given two functions s(n),t(n) on N, write: • s(n) = O(t(n)) if there are constants c,d such that for all n, s(n) ≤ c · t(n) + d. • s(n) = (t(n)) if t(n) = O(s(n)), and • s(n) = (t(n)) if s(n) = O(t(n)) and s(n) = (t(n)). In the first case, we say s(n) is “order-of” t(n) or “Big-Oh-of” t(n), whereas in the second, we might say t(n) is “asymptotically bounded below by” s(n), and in the third, we say s(n) and t(n) have the same “asymptotic order.” A sufficient condition for s(n) = (t(n)) is that the limit limn→∞ s(n) t(n) exists and is some positive number. If the limit is zero, then we write s(n) = o(t(n)) instead—this “little-oh” notation is stronger than writing s(n) = O(t(n)) here. Thus, we can say about our s(n) example above:
- 31. 12 Chapter 2 Numbers and Strings • s(n) = (n2); • s(n) = o(n3); • log(s(n)) = (logn). Indeed, the last gives log(s(n)) = (logn), but it does not give log(s(n)) ∼ log(n) because the exponent 2 in s(n) becomes a multiplier of 2 on the loga- rithm. The choice of base for the logarithm also affects the constant multiplier, but not the relation. The latter enables us not to care about what the base is or even whether two logarithms have the same base. This kind of freedom is important when analyzing the costs of algorithms and even in thinking what the goals are of designing them. One further important idea, which we will begin employing in chapter 13, uses logarithms in a different way. Write f(n) = Õ(g(n)) if there is some finite k such that f(n) = O(g(n)(logg(n))k). This is pronounced “f is Oh-tilde of g,” and carries the idea that sometimes logarithmic as well as constant factors can be effectively ignored. 2.2 Problems 2.1. Let x be a Boolean string. What type of number does the Boolean string x0 represent? 2.2. Let x be a Boolean string with exactly one bit a 1. What can you say about the number it represents? Does this identification depend on using the canonical numbering of {0,1}n, where n is the length of x? 2.3. Compute the 4 × 4 “times-table” of x • y for x,y ∈ {00,01,10,11}. Then write the entries in the form (−1)x•y. 2.4. Let x be a Boolean string of even length. Can the Boolean string xxx ever represent a prime number in binary notation? 2.5. Show that a function f : N → N is bounded by a constant if and only if f(n) = O(1), and is linear if and only if f(n) = (n). 2.6. Show that a function f : N → N is bounded by a polynomial in n, written f(n) = nO(1), if and only if there is a constant C such that for all sufficiently large n, f(2n) ≤ Cf(n). How does C relate to the exponent k of the polynomial? Thus, we can characterize algorithms that run in polynomial time as those for which the amount of work scales up only linearly as the size of the data
- 32. 2.3 Summary and Notes 13 grows linearly. Later we will use this criterion as a benchmark for feasible computation. 2.7. Let M be an n-bit integer, and let a M. Give a bound of the form O(s(n)) for the time needed to find the remainder when M is divided into a2. 2.8. Now suppose we want to compute a75 modulo M. Give a concrete bound on the number of squarings and divisions by M one needs to do, never allowing any number to become bigger than M2. 2.9. Use the ideas of problems 2.7 and 2.8 to show that given any a M, the function fa defined for all integers x M by fa(x) = ax mod M can be computed in nO(1) time. 2.3 Summary and Notes Numbers and strings are made out of the same “stuff,” which are characters over a finite alphabet. With numbers we call them “digits,” whereas with binary strings we call them “bits,” but to a computer they are really the same. Switch- ing mentally from one to the other is often a powerful way to promote one’s understanding of theoretical concepts. This is especially true in quantum com- putations, where the bits of a Boolean string—or rather their indexed loca- tions in the string—will be treated as quantum coordinates or qubits. The next chapter shows how we use both numbers and strings as indices to vectors and matrices. It is interesting that while the notion of natural numbers is ancient, the notion of Boolean strings is much more recent. Even more interesting is that it is only with the rise of computing that the importance of using just the two Boolean values 0,1 has become so clear. Asymptotic notation helped convince us that whether we operate in base 10 or 2 or 16 or 64, the difference is secondary compared to the top-level structure of the algorithm, which usually determines the asymptotic order of the running time.
- 34. 3 Basic Linear Algebra A vector a of dimension N is an ordered list of values, just as usual. Thus, a of dimension N stands for: ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ a0 a1 . . . aN−1. ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ Instead of the standard subscript notation to denote the k-th element, we will use a(k) to denote it. This has the advantage of making some of our equations a bit more readable, but just remember that a(k) is the same as ak. One rationale for this notation is that a vector can often be best viewed as being indexed by other sets rather than just the usual 0,...,N − 1. The functional notation a(k) seems to be more flexible in this regard. A fancier justification is that sometimes vector spaces are defined as func- tions over sets, so this notation is consistent with that. In any event, a(k) is just the element of the vector that is indexed by k. A concrete example is to consider a vector a of dimension 4. In this case, we may use the notation a(0),a(1),a(2),a(3), for its elements, or we may employ the notation a(00),a(01),a(10),a(11), using Boolean strings to index the elements. A philosophical justification for our functional notation is that vectors are pieces of code. We believe that nature computes with code—not with the graph of the code. For instance, each elementary standard basis vector ek is 0 except for the k-th coordinate, which is 1, and we use the subscript when thinking of it as an object. When N = 2n, we index the complex coordinates from 0 as 0,...,N − 1 and enumerate {0,1}n as x0,...,xN−1, but we index the binary string places from 1 as 1,...,n. Doing so helps tell them apart. When thinking of ek as a piece of code, we get the function ek(x) = 1 if x = xk and ek(x) = 0 otherwise. We can also replace k by a binary string as a subscript, for instance, writing the four standard basis vectors when n = 2 as e00,e01,e10,e11.
- 35. 16 Chapter 3 Basic Linear Algebra 3.1 Hilbert Spaces A real Hilbert space is nothing more than a fancy name for the usual Euclidean space. We will use HN to denote this space of dimension N. The elements are real vectors a of dimension N. They are added and multiplied by scalars in the usual manner: • If a,b are vectors in this space, then so is a + b, which is defined by ⎡ ⎢ ⎢ ⎣ a(0) . . . a(N − 1) ⎤ ⎥ ⎥ ⎦ + ⎡ ⎢ ⎢ ⎣ b(0) . . . b(N − 1) ⎤ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎣ a(0) + b(0) . . . a(N − 1) + b(N − 1) ⎤ ⎥ ⎥ ⎦. • If a is a vector again in this space and c is a real number, then b = ca is defined by c ⎡ ⎢ ⎢ ⎣ a(0) . . . a(N − 1) ⎤ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎣ ca(0) . . . ca(N − 1) ⎤ ⎥ ⎥ ⎦. The abstract essence of a Hilbert space is that each vector has a norm: The norm of a vector a, really just its length, is defined to be ||a|| = k a(k)2 1/2 . Note that in the case of two dimensions, the norm of the vector a = r s is its usual length in the plane, √ r2 + s2. A Hilbert space simply generalizes this to many dimensions. A unit vector is just a vector of norm 1. Unit vectors together comprise the unit sphere in any Hilbert space. 3.2 Products and Tensor Products The ordinary Cartesian product of an m-dimensional Hilbert space H1 and an n-dimensional Hilbert space H2 is the (m + n)-dimensional Hilbert space
- 36. 3.3 Matrices 17 obtained by concatenating vectors from the former with vectors from the latter. Its vectors have the form a(i) with i ∈ [m + n]. Their tensor product H1 ⊗ H2, however, has vectors of the form a(k), where 1 ≤ k ≤ mn. Indeed, k is in 1–1 correspondence with pairs (i,j) of indices where i ∈ [m] and j ∈ [n]. Because we regard indices as strings, we can write them juxtaposed as a(ij). The tensor product of two vectors a and b is the vector c = a ⊗ b defined by c(ij) = a(i)b(j). A vector denoting a pure quantum state is separable if it is the tensor product of two other vectors; otherwise it is entangled. The vectors e00 and e11 are separable, but their unit-scaled sum 1 √ 2 (e00 + e11) is entangled. The standard basis vectors of H1 ⊗ H2 are separable, but this is not true of many of their linear combinations, all of which still belong to H1 ⊗ H2 because it is a Hilbert space. Often our quantum algorithms will operate on the product of two Hilbert spaces, each using binary strings as indices, which gives us the space of vectors of the form a(xy). Here x ranges over the indices of the first space and y over those of the second space. Writing a(xy) does not entail that a is separable. 3.3 Matrices Matrices represent linear operators on Hilbert spaces. We can add them together, we can multiply them, and of course we can use them to operate on vectors. We assume these notions are familiar to you; if not, please see sources in this chapter’s end notes. A typical example is UVa = b. This means: apply the V transform to the vector a, then apply the U transform to the resulting vector, and the answer is the vector b. The matrix IN denotes the N × N identity matrix. We use square brackets for matrix entries to dis- tinguish them further from vector amplitudes, so that the identity matrix has entries IN[r,c] = 1 if r = c, IN[r,c] = 0 otherwise. One of the key properties of matrices is that they define linear operations, namely: U(a + b) = Ua + Ub.
- 37. 18 Chapter 3 Basic Linear Algebra The fact that all our transformations are linear is what makes quantum algo- rithms so different from classical ones. This will be clearer as we give exam- ples, but the linearity restriction both gives quantum algorithms great power and curiously makes them so different from classical ones. If U is a matrix, then we use Uk to denote the k-th power of the matrix, Uk = UU ···U k copies . DEFINITION 3.1 The transpose of a matrix U is the matrix V such that V[r,c] = U[c,r]. We use UT as usual to denote the transpose of a matrix. We also use trans- pose for vectors but only when writing them after a matrix in lines of text. Gen- erally, we minimize the distinction between row and column vectors, using the latter as standard. The inner product of two real vectors a and b is given by a,b = m k=0 a(k)b(k). DEFINITION 3.2 A real matrix U is unitary provided UT U = I. Here are three unitary 2 × 2 real matrices. Note that the last, called the Hadamard matrix, requires a constant multiplier to divide out a factor of 2 that comes from squaring it. I = 1 0 0 1 , X = 0 1 1 0 , H = 1 √ 2 1 1 1 −1 . The first two are also permutation matrices, meaning square matrices each of whose rows and columns has all zeros except for a single 1. All permutation matrices are unitary. Another definition of a unitary matrix is based on the notion of orthogonal- ity. Call two vectors a and b orthogonal if their inner product is 0. A matrix U is unitary proved each row is a unit vector and any two distinct rows are orthogonal. The reason these matrices are so important is that they preserve the Euclidean length. LEMMA 3.3 If U is a unitary matrix and a is a vector, then ||Ua|| = ||a||.
- 38. 3.4 Complex Spaces and Inner Products 19 Proof. Direct calculation gives: ||Ua||2 = x |Ua(x)|2 = x Ua(x) · Ua(x) = x ( y U[x,y]a(y))( z U[x,z]a(z)) = x ( y z U[x,y]U[x,z]a(y)a(z)) = y z ( x (U[x,y]U[x,z]))a(y)a(z)) = y a(y)a(y) = ||a||2 because the inner product of U[−,y] and U[−,z] is 1 or 0 according as y = z. As far as we can, we try not to care about whether a Hilbert space is real or complex. However, we do need notation for complex spaces. 3.4 Complex Spaces and Inner Products To describe some algorithms, mainly Shor’s, we need to use complex vectors and matrices. In this case, all definitions are the same except for the notion of transpose and inner product. Both now need to use the conjugation operation: if z = x + iy where x,y are real numbers and i = √ −1 as usual, then the conju- gate of z is x − iy and is denoted by z. We now define the adjoint of a matrix U to be the matrix V = U∗ such that V[r,c] = U[c,r]. A complex matrix U is unitary provided U∗ U = I. Furthermore, the inner product of two complex vectors a and b is defined to be a,b = m k=0 a(k)b(k). Note that because r is the same as r for a real number r, these concepts are the same as what we defined before when the entries of the vectors and
- 39. 20 Chapter 3 Basic Linear Algebra matrices are all real numbers. Rather then use special notation for the complex case, we will use the same as in the real case—the context should make it clear which is being used. The one caveat is that in the few cases where the entries of a or U are complex, one needs to conjugate them. Upon observing this, the proof of the following is much the same as for Lemma 3.3 above: LEMMA 3.4 If U is a unitary matrix and a is a vector, then the length of Ua is the same as the length of a. We can also form tensor products of matrices having any dimensions. If U is m × n and V is r × s, then W = U ⊗ V is the mr × ns matrix whose action on product vectors c(ij) = a(i)b(j) is as follows: (Wc)(ij) = (Ua)(i)(Vb)(j). Because every vector d of dimension rs (whether entangled or not) can be written as a linear combination of basis vectors, each of which is a product vector, the action Wd is well defined via the same linear combination of the outputs on the basis vectors. That is, if d = r i=1 s j=1 di,jei ⊗ ej, then Wd = r i=1 s j=1 di,j (Uei) ⊗ Vej . Note that it does not matter whether the scalars di,j are regarded as multiplying ei, ej, or the whole thing. This fact matters later in section 6.5. We mainly use tensor products to combine operations that work on separate halves of an overall index ij. 3.5 Matrices, Graphs, and Sums Over Paths One rich source of real-valued matrices is graphs. A graph G consists of a set V of vertices, also called nodes, together with a binary relation E on V whose members are called edges. The adjacency matrix A = A(G) of a graph G = (V,E) is defined for all u,v ∈ V by: A[u,v] = 1 if (u,v) ∈ E 0 otherwise.
- 40. 3.5 Matrices, Graphs, and Sums Over Paths 21 If E is a symmetric relation, then A is a symmetric matrix. In that case, G is an undirected graph; otherwise it is directed. The degree of a vertex u is the number of edges incident to u, which is the same as the number of 1s in row u of A. A graph is regular of degree d if every vertex has degree d. In that case, consider the matrix A = 1 d A. Figure 3.1 exemplifies this for a graph called the four-cycle, C4. This graph is bipartite, meaning that V can be partitioned into V1,V2 such that every edge connects a vertex in V1 and a vertex in V2. Figure 3.1 Four-cycle graph G = C4, stochastic adjacency matrix A G, and unitary matrix UG. 1 2 3 4 A G = 1 2 ⎡ ⎢ ⎢ ⎢ ⎣ 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 ⎤ ⎥ ⎥ ⎥ ⎦ , UG = 1 √ 2 ⎡ ⎢ ⎢ ⎢ ⎣ 0 1 0 1 1 0 1 0 0 1 0 −1 1 0 −1 0 ⎤ ⎥ ⎥ ⎥ ⎦ . Because every row has non-negative numbers summing to 1, A is a stochas- tic matrix. Because the columns also sum to 1, A is doubly stochastic. However, A is not unitary for two reasons. First, the Euclidean norm of each row and column is (1 2 )2(1 + 1) = 1 2 , not 1. Second, not all pairs of distinct rows or columns are orthogonal. To meet the first criterion, we multiply by 1 √ 2 instead of 1 2 . To meet the second, we can change the entries for the edge between nodes 3 and 4 from 1 to −1, creating the matrix UG, which is also shown in figure 3.1. Then UG is unitary—in fact, it is the tensor product H ⊗ X of two 2 × 2 unitary matrices given after definition 3.2. This fact is peculiar to the four-cycle. An example of a d-regular graph whose adjacency matrix cannot similarly be converted into a unitary matrix is the 3-regular prism graph shown in figure 3.2. Rows 1 and 3 have a nonzero
- 41. 22 Chapter 3 Basic Linear Algebra dot product in a single column, namely, column 5, so there is no substitution of nonzero values for nonzero entries that will make them orthogonal. Figure 3.2 3-regular prism graph G and stochastic adjacency matrix A. 1 2 3 4 5 6 A = 1 3 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 1 1 0 1 0 1 0 0 1 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 0 1 0 1 0 1 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ However, in chapter 14, we will see a general technique involving a tensor product of AG with another matrix to create a unitary matrix, one representing a quantum rather than a classical random walk on the graph G. The square of an adjacency matrix has entries given by A2 [i,j] = k A[i,k]A[k,j]. The sum counts the number of ways to go from vertex i through some vertex k and end up at vertex j. That is, A2 [i,j] counts the number of paths of length 2 from i to j. Likewise, A3 [i,j] counts paths of length exactly 3, A4 [i,j] those of length 4, and so on. For directed graphs in which each edge goes from one of n sources to one of m sinks, the adjacency matrices An,m lend themselves to a path-counting composition. Consider U = An,m1 Am1,m2 Am2,m3 ···Amk,r.
- 42. 3.6 Problems 23 Note how the mi dimensions match like dominoes. This makes U a legal matrix product. The corresponding graph identifies the sinks of the graph G repre- sented by Am−1,m (here m0 = n) with the sources of G+1. Then U[i,j] counts the number of paths in the product graph that begin at source node i of G1 and end at sink node j of the last graph Gk+1. What links quantum and classical concepts here is that weights can be put on the edges of these graphs, even complex-number weights. When these are sub- stituted for the “1” entries of the adjacency matrices, the value U[i,j] becomes a sum of products over the possible paths. In the classical case, these weights can be probabilities of individual choices along the paths. In the quantum case, they can be complex amplitudes, and some of the products in the sum may can- cel. Either way, the leading algorithmic structure is that of a sum over paths. As a fundamental idea in quantum mechanics, it was advanced particularly by Richard Feynman, and yet it needs no more than this much about graphs and linear algebra to appreciate. This discussion already conveys the basic flavor of quantum operations. The matrices are unitary rather than stochastic; the entries are square-roots of prob- abilities rather than probabilities; negative entries—and later imaginary num- ber entries—are used to achieve cancellations. After defining feasible compu- tations, we will examine some important unitary matrices in greater detail. 3.6 Problems 3.1. Show that the product of unitary matrices is unitary. 3.2. If U is a matrix, then what is Uek? 3.3. Show that the columns of a unitary matrix are unit vectors. Also show that distinct columns are orthogonal. 3.4. Consider the matrix w w w −w . For what real values of w is it a unitary matrix? 3.5. Consider the matrix U equal to: 1 √ 2 IN IN IN −IN .
- 43. 24 Chapter 3 Basic Linear Algebra Show that for any N, the matrix U is unitary. 3.6. For real vectors a and b, show that the inner product can be defined from the norm as: (a,b) = 1/2 ||a + b||2 − ||a||2 − ||b||2 . 3.7. For any complex N × N matrix U, we can uniquely write U = R + iQ, where Q and R have real entries. Show that if U is unitary, then so is the 2N × 2N matrix U given in block form by U = R Q −Q R . Thus, by doubling the dimension, we can remove the need for complex-number entries. 3.8. Apply the construction of the last problem to the matrix Y = 0 −i i 0 . This is the second of the so-called Pauli matrices, along with X above and Z defined below. 3.9. Consider the following matrix: V = 1 √ 2 eiπ/4 e−iπ/4 e−iπ/4 eiπ/4 = 1 2 1 + i 1 − i 1 − i 1 + i . What is V2 ? 3.10. Let Tα denote the 2 × 2 “twist” matrix 1 0 0 eiα , respectively. Show that it is unitary. Also find a complex scalar c such that cTα has determinant 1 and write out the resulting matrix, which often also goes under the name Tα. 3.11. The following cases of Tα for α = π, π 2 , π 4 have special names as shown: Z = 1 0 0 −1 , S = 1 0 0 i , T = 1 0 0 eiπ/4 .
- 44. 3.6 Problems 25 How are they related? Also find an equation relating the Pauli matrices X,Y,Z along with an appropriate “phase scalar” c as in the last problem. 3.12. In this and the next problem, consider the following commutators of the three matrices of the last problem with the Hadamard matrix H (noting also that H∗ = H−1 = H, i.e., the Hadamard is self-adjoint as a unitary matrix): Z = HZHZ∗ S = HSHS∗ T = HTHT∗ Show that Z and some multiple cS have nonzero entries that are powers of i. 3.13. Show, however, that no multiple cT has entries of this form by consid- ering the mutual angles of its entries. What is the lowest power 2r such that these angles are multiples of π/2r? 3.14. Define a matrix to be balanced if all of its nonzero entries have the same magnitude. Of all the 2 × 2 matrices in problem 3.12, say which are balanced. Is the property of being balanced closed under multiplication? 3.15. Show that the rotation matrix by an angle θ (in the real plane), Rx(θ) = cos(θ/2) sin(θ/2) −sin(θ/2) cos(θ/2) is unitary. Also, what does R2 x represent? 3.16. Show that for every 2 × 2 unitary matrix U, there are real numbers θ,α,β,δ such that U = eiδ TαRθ Tβ. Thus, every 2 × 2 unitary operation can be decomposed into a rotation flanked by two twists, multiplied by an arbitrary phase shift by δ. Write out the decom- position for the matrix V in problem 3.9. (It doesn’t matter which definition of “Tα” you use from problem 3.10.) 3.17. Show how to write V as a composition of Hadamard and T matrices. (All of these problems point out the special nature of the T-matrix and its close relative, V.) Either of these matrices is often called the “π/8 gate”; the confusing difference from π/4 owes to the constant c in problem 3.10. 3.18. Show that the four Pauli matrices, I, X, Y, and Z, form an orthonormal basis for the space of 2 × 2 matrices, regarded as a 4-dimensional complex Hilbert space.
- 45. 26 Chapter 3 Basic Linear Algebra 3.19. Define G to be the complete undirected graph on 4 vertices, whose 6 edges connect every pair (i,j) with i = j. Convert its adjacency matrix AG into a unitary matrix by making some entries negative and multiplying by an appro- priate constant. Can you do this in a way that preserves the symmetry on the matrix? 3.20. Show that a simple undirected graph G has a triangle if and only if A2 A has a nonzero entry, where A = AG and here means multiplying entry- wise. This means that triangle-detection in an n-vertex graph has complexity no higher than that of squaring a matrix, which is equivalent to that of multiplying two n × n matrices. Surprisingly, the obvious matrix multiplication algorithm and its O(n3) run- ning time are far from optimal, and the current best exponent on the “n” for matrix multiplication is about 2.372. This certainly improves on the O(n3) run- ning time of the simple algorithm that tries every set of three vertices to see if it forms a triangle. 3.7 Summary and Notes There are many good linear algebra texts and online materials. If this material is new or you took a class on it once and need a refresher, here are some suggested places to go: • The textbook Elementary Linear Algebra by Kuttler (2012). • Linear algebra video lectures by Gilbert Strang which are maintained at MITOPENCOURSEWARE: http://ocw.mit.edu/courses/mathematics/18- 06-linear-algebra-spring-2010/video-lectures/ • The textbook Graph Algorithms in the Language of Linear Algebra by Kep- ner and Gilbert (2011). The famous text by Nielsen and Chuang (2000) includes a full treatment of linear algebra needed for quantum computation. Feynman (1982, 1985) wrote the first two classic papers on quantum computation.
- 46. 4 Boolean Functions, Quantum Bits, and Feasibility A Boolean function f is a mapping from {0,1}n to {0,1}m, for some numbers n and m. When we define a Boolean function f(x1,...,xn) = (y1,...,ym), we think of the xi as inputs and the yj as outputs. We also regard the xi together as a binary string x and similarly write y for the output. When m = 1, there is some ambiguity between the output as a string or a single bit because we write just “y” not “(y)” in the latter case as well, but the difference does not matter in context. When m = 1, you can also think of f as a predicate: x satisfies the predicate if and only if f(x) = 1. Thus, Boolean functions give us all of the following: the basic truth values, binary strings, and, as seen in chapter 2, also numbers and other objects. The most basic have n = 1 or 2, such as the unary NOT function, and binary AND, OR, and XOR. We can also regard the following higher-arity versions as basic: • AND: This is the function f(x1,...,xn) defined as 1 if and only if every argument is 1. Thus, f(1,1,1) = 1 and f(1,0,1,1) = 0. • OR: This is the function f(x1,...,xn) defined as 1 if the number of 1’s in x1,...,xn is non-zero. Thus, f(0,1,1) = 1 and f(0,0,0,0) = 0. • XOR: This is the function f(x1,...,xn) defined as 1 if the number of 1’s in x1,...,xn is odd. Thus, f(0,1,1) = 0 and f(1,1,1,1,1) = 1. The latter is true because there are five 1’s. The binary operations can also be applied on pairs of strings bitwise. For instance, if x and y are both Boolean strings of length n, then x ⊕ y is equal to z = (x1 ⊕ y1,...,xn ⊕ yn). We could similarly define the bitwise-AND and the bitwise-OR of two equal- length binary strings. These are not the same as the above n-ary operations but are instead n applications of binary operations. Each operation connects the i-th bit of x with then i-th bit of y, for some i, and they intuitively run “in parallel.” The Boolean inner product, which we defined in chapter 2, is computed by feeding the bitwise binary AND into the n-ary XOR, that is: x • y = XOR(x1 ∧ y1,...,xn ∧ yn).
- 47. 28 Chapter 4 Boolean Functions, Quantum Bits, and Feasibility In cases like x ⊕ y, we say we have a circuit of n Boolean gates that collec- tively compute the Boolean function f : {0,1}r → {0,1}n, with r = 2n, defined by f(x,y) = x ⊕ y. Technically, we need to specify whether x and y are given sequentially as (x1,...,xn,y1,...,yn) or shuffled as (x1,y1,...,xn,yn), and this matters to how we would draw the circuit as a picture. But either way we have a 2n-input function that represents the same function of two n-bit strings. The number of gates is identified with the amount of work or effort expended by the circuit, and this in turn is regarded as the sequential time for the circuit to execute. It does not matter too much whether one counts gates or wires between gates. What is critical is that only basic operations can be used, and that they can only apply to previously computed values. In the following sketch, the NOT of a ∨ b is allowed because a ∨ b has already been computed: ...(a ∨ b)... ...¬(a ∨ b)... Two Boolean functions that we should not regard as basic are: • PRIME: This is the function f(x1,...,xn) defined as 1 if the Boolean string x = x1,...,xn represents a number that is a prime number. Recall a prime number is a natural number p greater than 1 with only 1 and p as divisors. • FACTOR: This is the function f(x1,...,xn,w1,...,wn) regarded as having two integers x and w as arguments—note that we can pad w as well as x by leading 0’s. It returns 1 if and only if x has no divisor greater than w, aside from x itself. The game is, how efficiently can we build a circuit to compute these func- tions? They are related by PRIME(x) = FACTOR(x,1) for all x. This implies that a circuit for FACTOR immediately gives one for solving the predicate PRIME because one can simply fix the “w” inputs to be the padded version of the number 1. This does not imply the converse relation, however. Although both of these functions have been studied for 3,000 years, PRIME was shown only a dozen years ago to be feasible in a sense we describe next, while many believe that FACTOR is not feasible at all. Unless you are allowed a quantum circuit, that is. 4.1 Feasible Boolean Functions Not all Boolean functions are created equal; some are more complex than oth- ers. In the above examples, which n-bit function would you like to compute if
- 48. 4.1 Feasible Boolean Functions 29 you had to? I think we all will agree that the easiest is the OR function: just glance at the input bits and check whether one is a 1. If you only see 0’s, then clearly the OR function is 0. AND is similar. Next harder it would seem is the XOR function. The intuitive reason is that now you have to count the number of bits, and this count has to be exact. If there are 45 inputs that are 1 and you miscount and think there are 44, then you will get the wrong value for the function. Indeed, one can argue that n-ary XOR is harder than the bitwise-XOR function because each of the n binary XOR operations is “local” on its own pair of bits. More difficult is the PRIME function. There is no known algorithm that we can use and just glance at the bits. Is 1010101101101011010111110101 a prime number or not? Of course you first might convert it to a decimal num- ber: it represents the number 11234143. This still now requires some work to see if it has any nontrivial divisors, but it does: 23 and 488441. One of the achievements of computer science is that we can define the clas- sical complexity of a Boolean function. Thus, AND and XOR are computable in a linear number of steps, that is, O(n). Known circuits for PRIME take more than linearly many steps, but the time is still polynomial, that is, nO(1). But there are also Boolean functions that require time exponential in n. Many people believe that FACTOR is one of them, but nobody knows for sure. To see the issue, consider that any Boolean function can be defined by its truth table. Here is the truth table for the exclusive-or function XOR: x y x ⊕ y 0 0 0 0 1 1 1 0 1 1 1 0 . Each row of the table tells you what the function, in this case, the exclusive- or function, does on the inputs of that row. In general, a Boolean function f(x1,...,xn) is defined by a truth table that has 2n rows—one for each possible input. Thus, if n = 3, there are eight possible rows: 000,001,010,011,100,101,110, and 111. The difficulty is that as the number of inputs grows, the truth table increases exponentially in size.
- 49. 30 Chapter 4 Boolean Functions, Quantum Bits, and Feasibility Thus, representing Boolean functions by their truth tables is alway possible, but is not always feasible. The tables will be large when there are thirty inputs, and when there are over 100, the table would be impossible to write down. The final technical concept we need is having not just a single Boolean func- tion but rather a family [fn] of Boolean functions, each fn taking n inputs, that are conceptually related. That is, the [fn] constitute a single function f on strings of all lengths, so we write f : {0,1}∗ → {0,1}, or for general rather than one-bit outputs, f : {0,1}∗ → {0,1}∗. Maybe it is confusing to write “f” also for this kind of function with an infinite domain, but the intent is usually transparent—as when letting AND, OR, and XOR above apply to any n. Now we can finally define “feasible”: DEFINITION 4.1 A Boolean function f = [fn] is feasible provided the indi- vidual fn are computed by circuits of size nO(1). 4.2 An Example Consider the Boolean function MAJ(x1,x2,x3,x4,x5), which takes the major- ity of five Boolean inputs. A first idea is to compute it using applications of OR and AND as follows: For every three-element subset S = {i,j,k} of {1,2,3,4,5}, we compute yS = OR(xi,xj,xk). Define y to be the AND of each of the ten subsets S. Then y = 1 ⇐⇒ no more than 2 bits of x1,...,x5 are 0 ⇐⇒ MAJ(x1,x2,x3,x4,x5) is true. The complexity is counted as 11 operations and, importantly, 35 total arguments of those operations. On second thought, we can find a program of slightly lower complexity. Consider the Boolean circuit diagram in figure 4.1. Expressed as a sequence of operations, in one of many possible orders, the circuit is equivalent to the following straight-line program: v1 = OR(x1,x2,x3), v2 = OR(x4,x5), w1 = AND(x1,x2), w2 = AND(x1,x3), w3 = AND(x2,x3), w4 = AND(w1,w2), w5 = AND(v1,x4,x5) u = OR(w1,w2,w3), t = AND(u,v2), y = OR(w4,t,w5). This program has 10 operations and only 24 applications to arguments. To see that it is correct, note that w4 is true if and only if x1 = x2 = x3 = 1, and w5 is true if and only if x4 = x5 = 1 and one of x1,x2,x3 is 1. Finally, t is true if and
- 50. 4.2 An Example 31 Figure 4.1 Monotone circuit computing MAJ(x1,x2,x3,x4,x5). x 1 1 2 2 3 4 5 5 4 3 2 1 x x x x v v w w w w w u t y only if two of x1,x2,x3 and one of x4,x5 are true, which handles the remaining six true cases. Now clearly MAJ generalizes into a Boolean function MAJ of strings x of any length n, such that MAJ(x) returns 1 if more than n/2 of the bits of x are 1. We ask the important question: Is MAJ feasible? The operational question about MAJ(x1,x2,x3,x4,x5) is, do the above pro- grams scale when “5” is replaced by “n”? Technically, scalable is the same idea as feasible but with the idea mentioned in chapter 2 that a polynomial bound is the same as having a linear bound each time the size of the input doubles. The first idea, when generalized from “5” to “n,” says to take the AND of every r-sized subset of [n], where r = n/2 + 1, and feed that to an OR.
- 51. 32 Chapter 4 Boolean Functions, Quantum Bits, and Feasibility However, there are n r such subsets, which is exponential when r ∼ n/2. So the first idea definitely does not scale. The trouble with the second, shorter program is its being rather ad hoc for n = 5. The question of whether there are programs like it with only AND and OR gates that scale for all n is a famous historical problem in complexity the- ory. The answer is known to be yes, but no convenient recipe for constructing the programs for each n is known, and their size O(n5.3) is comparatively high. However, if we think in terms of numbers, we can build circuits that easily scale. Take k = log2(n + 1). We can hold the total count of 1’s in an x of length n in a k-bit register. So let us mentally draw x going down rather than across and draw to its right an n × k grid, whose rows will represent successive values of this register. It is nicest to put the least significant bit leftmost, i.e., closest to x, but this is not critical. The row to the right of x1 has value 0 (that is, 0k as a Boolean string) if x1 = 0 and value 1 (that is, 10k−1) if x1 = 1. As we scan x downward, if xi = 0, then the row of k bits has the same value as the previous one, but if xi = 1, then the register is incremented by 1. We might not regard the increment operation as basic—instead, we might use extra “helper bits” and gates to compute binary addition with carries. But we can certainly tell that we will get a Boolean circuit of size O(kn) = O(nlogn), which is certainly feasible. At the end, we need only compare the final value v with n/2, and this is also feasible. This idea broadens to any computation that we can imagine doing with paper and pencil and some kind of grid, such as multiplication, long division, and other arithmetic. To avoid the scratchwork and carries impinging on our basic grid, we can insist that they occupy h-many “helper rows” below the row with xn, stretching wires down to those rows as needed. The final idea that helps in progressing to quantum computation is instead of saying we have k(n + h) bits, say that we have n + h bits that “evolve” going left to right. This also fits the classical picture articulated by Turing. The cells directly below xn in the column with the input can be regarded as a segment of “tape” for scratchwork. The change to this tape at each step of a Turing Machine computation on x, including changes to the scratchwork, can be recorded in an adjacent new column. If the machine computation lasts t steps, then with s = n + h as the measure of “space,” we have an s × t grid for the whole computation. To finish touring machine and circuit models, we can next imagine that every cell in this grid depends on its neighbors in the preceding column by a fixed finite combination of basic Boolean operations. This gives us a circuit of size
- 52. 4.3 Quantum Representation of Boolean Arguments 33 O(st) = O(t2) because we do at most one bit of scratchwork at each step. If the machine time t(n) is polynomial, then so is t(n)2. This relation also goes in the opposite direction if you think of using a machine to verify the computation by the circuit—provided the circuits are uniform in a technical sense that captures the conceptual sense we applied above to Boolean function families. Hence, the criterion of feasible is broad and is the same for any of the classi- cal models of computation by machines, programs, or (uniform) circuits. There is a huge literature on which functions are feasible and which are not. One can encode anything via Boolean strings, including circuits themselves. The problem of whether a Boolean circuit can ever output 1—even when it allows applying NOT only to the original arguments xi and then has just one level of ternary OR gates feeding into a big AND—is not known to be feasible. That is, no feasible function is known to give the correct answer for every encoding X of such a program: this is called the satisfiability problem and has a property called NP-hardness. We define this with regard to an equivalent problem about solving equations in chapter 16. For now we are happy with not only defining classical feasible computation in detail but also showing that equivalent cri- teria are reached from different models. Now we are ready for the quantum challenge to this standard. 4.3 Quantum Representation of Boolean Arguments Let N = 2n. Every coordinate in N-dimensional Hilbert space corresponds to a binary string of length n. The standard encoding scheme assigns to each index j ∈ [0,...,n − 1] the n-bit binary string that denotes j in binary nota- tion, with leading 0’s if necessary. This produces the standard lexicographic ordering on strings. For instance, with n = 2 and N = 4, we show the indexing applied to a permutation matrix: 00 01 10 11 00 1 0 0 0 01 0 1 0 0 10 0 0 0 1 11 0 0 1 0 .
- 53. 34 Chapter 4 Boolean Functions, Quantum Bits, and Feasibility The mapping is f(00) = 00, f(01) = 01, f(10) = 11, f(11) = 10, and in gen- eral f(x1,x2) = (x1,x1 ⊕ x2). Thus, the operator writes the XOR into the sec- ond bit while leaving the first the same. Once can also say that it negates the second bit if-and-only-if the first bit is 1. This negation itself is represented on one bit by a matrix we have seen before—now with the indexing scheme, it is: X = 0 1 0 0 1 1 1 0 . Thus, the negation is controlled by the first bit, which explains the name “Controlled-NOT” (CNOT) for the whole 4 × 4 operation. To get a general Boolean function y = f(x1,...,xn), we need n + 1 Boolean coordinates, which entails 2N = 2n+1 matrix coordinates. What we really com- pute is the function F(x1,...,xn,z) = (x1,...,xn,z ⊕ f(x1,...,xn)). Formally, F is a Boolean function with outputs in {0,1}n+1 rather than just {0,1}. Its first virtue, which is necessary to the underlying quantum physics, is that it is invertible—in fact, F is its own inverse: F(F(x1,...,xn,z)) = F(x1,...,xn,z ⊕ y) = (x1,...,xn,(z ⊕ y) ⊕ y) = (x1,...,xn,z). Its second virtue is having a 2N × 2N permutation matrix Pf that is easy to describe: the lone 1 in each row x1x2 ···xnz is in column x1x2 ···xnb, where b = z ⊕ f(x1,...,xn). If f is a Boolean function with m outputs (y1,...,ym) rather than a single bit, then we have the same idea with F(x1,...,xn,z1,...,zm) = (x1,...,xn,z1 ⊕ y1,...,zm ⊕ ym) instead. The matrix Pf is still a permutation matrix, although of even larger dimensions 2n+m × 2n+m. Often left unsaid is what happens if we need h-many “helper bits” to compute the original f. The simple answer is that we can treat them all as extra outputs of the function, allocating extra zj variables as dummy inputs so that the ⊕ trick preserves invertibility. Because h is generally poly- nomial in n, this does not upset feasibility. In this scheme, adopting the “rows-and-columns” picture we gave for classi- cal computation above, everything is laid out in n = n + m + h rows, with the
- 54. 4.4 Quantum Feasibility 35 input x laid out in the first column. Each row is said to represent a qubit, which is short for quantum bit. In order to distinguish the row from the idea of a qubit as a physically observable object, we often prefer to say qubit line for the row itself in the circuit. The h-many helper rows even have their own fancy name as ancilla qubits, using the Latin word for “chambermaid” or, more simply, helper. Writing out a big 2n × 2n matrix, just for a permutation, is of course not feasible. This is a chief reason we prefer to think of operators Pf as pieces of code. The qubit lines are really coordinates of binary strings that represent indices to these programs. These strings have size n, and their own indices 1,...,n are what we call quantum coordinates, when trying to be more care- ful than saying “qubits.” As long as we confine ourselves to linear algebra operations that are efficiently expressible via these n quantum indices, we can hope to keep things feasible. The rest of the game with quantum computation is, which operations are feasible? 4.4 Quantum Feasibility A quantum algorithm applies a series of unitary matrices to its start vector. Can we apply any unitary matrix we wish? The answer is no, of course not. If the quantum algorithms are to be efficient, then there must be a restriction on the matrices allowed. If we look at the matrices Pf in section 4.3, we see several issues. First, the design of Pf seems to take no heed of the complexity of the Boolean func- tion f but merely creates a permutation out of its exponential-sized truth table. Because infeasible (families of) Boolean functions exist, there is no way this alone could scale. Second, even for simple functions like AND(x1,x2,...,xn), the matrix still has to be huge—even larger than 2n on the side. How do we distinguish “basic feasible operations”? Third, what do we use for variables? If we have a 2n-sized vector, do we need exponentially many variables? The answer is to note that if we keep the number k of arguments for any operation to a constant, then 2k stays constant. We can therefore use 2k × 2k matrices that apply to just a few arguments. But what are the arguments? They are not the same as the Hilbert space coordinates 0,...,N − 1, which would involve us in exponentially many. The quantum coordinates start off being labeled x1,x2,...,xn as for Boolean input strings and extend to places for out- puts and for ancillae, which is the plural of ancilla.
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- 60. “Viyella” FOR THE MOTORIST For Shirt-Blouses, Skirts, Costumes, etc. For Nightdresses, Pyjamas, etc. “The acme of comfort.” To be obtained in the latest designs and colourings from the leading Drapers, or name of nearest sent, on application, by “VIYELLA” (D.L.) 25 26 Newgate Street, London, E.C. “Viyella” Hosiery Underwear, and “Viyella” Gloves, made from the same yarns as the celebrated cloth. Specially suitable for sensitive skins. THE COUNTRY HANDBOOKS A Series of Illustrated Practical Handbooks dealing with Country Life. Suitable for the Pocket or Knapsack. Under the General Editorship of HARRY ROBERTS. Foolscap 8vo (6½ by 4 inches). Price, bound in Limp Cloth, 3s. net. THE TRAMP’S HANDBOOK. By H. Roberts. THE STILL ROOM. By Mrs. Roundell. THE BIRD BOOK. By A. J. R. Roberts. THE LITTLE FARM. By “Home Counties.” THE FISHERMAN’S HANDBOOK. By Edgar S. Shrubsole. THE SAILING HANDBOOK. By Clove Hitch. THE KENNEL HANDBOOK. By C. J. Davies. THE GUN ROOM. By Alex Innes Shand. THE COUNTRY COTTAGE. By G. H. Morris and Esther Wood. THE MOTOR BOOK. By R. J. Mecredy. THE STABLE HANDBOOK. By T. F. Dale. THE TREE BOOK. By M. R. Jarvis and Harry Roberts. THE INSECT BOOK. By W. Percival Westell. THE PHOTOGRAPHER’S HANDBOOK. By Charles Harrison and John C. Douglas. THE VET. BOOK. By F. Townend Barton. THE SMALL HOLDING. By F. E. Green. John Lane, The Bodley Head, Vigo Street, London
- 61. “It is giving us every satisfaction.” OF THE THIRD De Dion Bouton owned by Mrs. Chester, Ashurst, Haslemere, her chauffeur, M. E. J. James, wrote in January 1908 as follows: “We have sold our 12-h.p. De Dion and have now a 24- h.p. car. It is giving us every satisfaction, as we have now done 5000 miles without any trouble at all. “I should like to mention that I have driven this 24-h.p. car now for seven months, before that the 12-h.p. for two years, and before I had an 8-h.p. for two years also, and during that time of just on five years I have never been hung up on the road, except, of course, for tyres, which I think speaks very well for De Dion cars.” Models from 8 h.p. to 30 h.p. Catalogue gratis on application. De Dion Bouton (1907), Ltd. Sole Authorised Representatives of Messrs. De Dion Bouton et Cie., of Puteaux, France, for the United Kingdom and all British Colonies and Dependencies. 10 Great Marlborough St., Regent St., W. Telegrams—“Andesite, London.” Telephone—Nos. 8160 8161 Central THE MOTOR BOOK By R. J. MECREDY
- 62. With Numerous Illustrations (“The County Handbooks.”) Fcap. 8vo. 3s. net. PRESS OPINIONS Scotsman.—“An admirable, succinct and clear account of the mechanism of a typical petrol car. Contains as much information as the ordinary owner of a motor is likely to want.” Aberdeen Free Press.—“An exceedingly exhaustive account of how the motor works. The machinery is described with the utmost clearness. It should prove of the utmost value to all motorists who are not practical mechanics.” Birmingham Post.—“His work is very valuable. In addition it is a very dainty volume, nicely printed, illustrated, and bound.” Morning Post.—“In any case the book will help inexperienced enthusiasts to run their cars straight and to keep them in order.” Motor-Car World.—“The book is the most satisfactory work on motor-cars which we have perused. In the last two of these chapters Mr. Mecredy gives many valuable hints on the care of cars, how to prevent trouble, and how to remedy defects when they do occur. The book is profusely illustrated by means of both photographic reproductions and diagrams.” John Lane, The Bodley Head, Vigo St., London
- 63. NEW FICTION GALAHAD JONES. A Tragic Farce. By Arthur H. Adams. Crown 8vo, 6s. ⁂ Galahad Jones is a middle-aged bank clerk, with a family. One day, on his way home, a letter falls to his feet from the balcony of a house he is passing. It is addressed “To You,” and on reading it he discovers that he is requested to meet the writer in the garden of the house at 10 o’clock that night. In a spirit of knight-errantry he decides to do so, and learns that the writer—a young girl—is kept practically in prison by her father, because of her affection for a man of
- 64. whom he does not approve. The chivalry of Galahad Jones plunges him into many difficulties and leads to some very awkward and extremely amusing situations. JOAN OF THE HILLS. A Novel by T. B. Clegg, author of “The Love Child,” “The Wilderness,” “The Bishop’s Scapegoat.” Crown 8vo, 6s. ⁂ Mr. Clegg’s previous novels have given him a position as an Australian novelist to be reckoned with. The present story opens in London, but Mr. Clegg is soon back in Australia, describing the life on a remote Australian station with its refreshing bush atmosphere. “Joan of the Hills” should increase the reputation that Mr. Clegg has already achieved. THE MEASURE OF OUR YOUTH. A Novel. By Alice Herbert. Crown 8vo, 6s. ⁂ A brilliant novel of modern life, by a new author. Its leading interest is the eternal one of sex; but the treatment is particularly fresh and fearless, and there is a sense of humour and of style that will please the fastidious. The realism of the writing will be forgiven for the sake of the delicate and poetic vein of thought that underlies the story, which is full of interest for the psychologist. SIXPENNY PIECES. By A. Neil Lyons. Uniform with “Arthur’s.” Crown 8vo. 6s. ⁂ Mr. Lyons’ new book has for its central figure a “Sixpenny Doctor” in the east end of London. The sketches are connected by a thread of continuous interest as in Mr. Lyons’ former book, the now famous “Arthur’s.” The volume is instinct with a realism that differs altogether from the so- called realism of the accepted “gutter” novels, for it is the realism of life as it is, and not as imagined. THE PRINCE’S PRANKS. A Novel. By Charles Lowe. Crown 8vo, 6s.
- 65. John Lane, The Bodley Head, Vigo St., London NEW FICTION CHIP: A Novel. By F. E. Mills Young, author of “A Mistaken Marriage.” Crown 8vo, 6s. ⁂ This is a story of the veld, of the lives of a small community of Europeans dwelling far from civilisation amid the silence and solitude of the swamps of East Africa. To the unhealthiness of the climate is added another danger—the
- 66. disaffection of the natives upon the farm, caused by their fear and dislike of their employer, Mordaunt, the hero of the tale. Reckless, holding life cheaply, and with a scorn of fear, Mordaunt, a man of great strength of character, yet one who allows an early disappointment to embitter his life, courts danger as he has for years vainly courted death. Then across his path comes Chip, the heroine of the tale, who, disguised as a boy, seeks and obtains the post of overseer on the ranch. The story describes their daily life, the dangers which they face together, and the great influence which the mysterious boy- overseer exercises over the dissipated misogynist, his employer. LITTLE DINNERS WITH THE SPHINX. By Richard LeGallienne. Crown 8vo, 6s. DIANA DETHRONED: A Novel. By W. M. Letts. Crown 8vo, 6s. ⁂ Phœbe Lankester, unconsciously to herself, is pledged to the old pagan ideal represented by Diana the Huntress. Healthy in body and mind, Phœbe stands aloof from the troubles and desires of humanity, until in her own wrecked happiness she awakes slowly to the need of some power greater and kindlier than ever Diana knew. It is only after the absolute surrender of self and after the awakening of a greater, more maternal love than she has as yet known that she finds peace. Love and death and pity have conquered Diana, and the statue of the goddess that once adorned the Lankesters’ hall is banished to a lumber-room. SOMEONE PAYS: A Novel. By Noel Barwell. Crown 8vo, 6s. ⁂ “Someone Pays,” though exemplifying a subtle train of cause and effect, is not a novel with a problem or a purpose. The story is told by means of the correspondence passing between a number of persons. We are first introduced to the post-bag at a country Vicarage where Sir Bernard Orr’s son is
- 67. being coached. Later the scene changes to Cambridge, and we watch the developments of a romance and an awkward entanglement which arise at the Vicarage. Everything is smoothed out and ends happily for all parties, especially for an unscrupulous triumphant cleric. THE ODD MAN. A Novel. By Arnold Holcombe. Crown 8vo, 6s. John Lane, The Bodley Head, Vigo St., London
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