usingqcomp.pptx

Using Quantum Computers
Barry C. Sanders
Quantum Information Summer School
15-18 July 2019
Institute of Business Administration Karachi

What is computation?
 A [universal] computer is a [programmable]
machine that process input information (bits) into
output information (bits) via a logical sequence of
hardware instructions.
 Computation is the physical process in this
machine, determined by an algorithm, whose
purpose is to solve a computational problem.
 Computational problems arise as different types,
including ”decision”, “function”, “search”, “promise”
and ”optimization”.

Decision problems
 Map bits to a single bit (Yes/No).
 Boolean function f:{0,1}* {0,1}.
 Boolean function decomposable into NAND gates.
 NAND:{0,1}2 → {0,1}:
(x,y)↦ ¬(x ⋀ y).
 Problem complexity
concerns how space
and time costs scale
with size of problem
instance (#bits).

Complexity
 Space cost S
 Time cost T
 Given problem,
complexity
characterizes how
S and T grow with
respect to instance
n, which is the
number of bits
required to specify
the problem

Reversible computing
 Motivated 1960s to avoid heat generation
 Every logical mapping is reversible:
for each 𝐹: 0,1 *→ 0,1 * whose input and
output string are the same size, 𝐹−1 exists.
Fredkin Gate

Quantum computer
 Accepts classical information (bits) as input and
yields classical information (bits) as output
 Convert classical to quantum information
 Process quantum information in machine
 Measure quantum information
 Classical out as bits
 |Ψ = 𝜳𝒃 |𝒃 for |𝒃 qubit strings, eg |010 … 11 ,
which are vectors in Hilbert space.
 Processing by unitary maps (isometries on Hilbert
space: preserve inner product).

Universal quantum instructions

Simulating Schrödinger’s cat

Physical representation of bit
Classical
 Mechanical: Position of
a gear or lever
 Dynamic RAM: charge
buildup in capacitor
 Read-only memory:
presence/absence of
conducting pathway
 Bar code
Quantum
 Magnetic: electron spin
or flux quantum
 Energy: dipole in atom
 Charge: Cooper pair left
or right of junction
 Light: Photon path or
polarization or arrival
time

Dealing with imperfection
Q Error Correction
 Block coding: a logical
qubit is represented by
a few physical qubits in
an entangled state
 Encode to overcome
exponential sensitivity
to decoherence
 Decode to extract
answer
Fault tolerance
 Each new instrument
brings new errors
 Fault tolerance ensures
that complicating the
set-up has the net
effect of reducing the
overall error.
 Convergence from q
threshold theorem

Using the q computer
Q algorithms
 Proven quadratic
speed-up such as
amplitude
amplification
 Believed
subexponential
speedup for hidden
abelian subgroup
Q simulation
 Hamiltonian simulation
for q-state input to q-
state output under
generic, restricted
Hamiltonian (H)
 Another definition
refers to analogue q
machine for dynamical
simulation trusting H

Hamiltonian H
 A q “program” is a unitary (isometric) map U of
states (Hilbert-space vectors): 𝑈|𝜓 = |𝜓 ,
|𝜓 = 𝜓 𝜓
 Stone’s theorem on one-parameter unitary groups:
∀𝑈(𝑡)∃𝐻(= 𝐻†
):𝑈 𝑡 = e−i𝐻𝑡
 Hamiltonian generates dynamics of a system; fits
with physicists’ concept of quantum mechanics

H as an adjacency graph
y1
yd
x :
α1
αd 2
2
2
2
2
1
1
1
2
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
1
1
1
1 1
1
3
3
3
3
3
3
3
3
3
3
3
3
2
1
2
2
2
2
2
2
3
3
3
3
3
3

Solve Schrödinger’s Equation
 Solve state |𝜓 as a function of t.
 Determine the spectrum of H
 Find eigenvectors of H, e.g. ground state.
 Estimate mean of some operator 𝜓 Ω 𝜓
 Applications: Chemistry, Physics, coupled linear
equations, differential equations, machine learning,
….

H 2
Äntotal
˜
Y0
˜
Yt '
H
Y0
Yt
exp -it ˆ
H
{ }
exp -i
t
r
ˆ
˜
H ji
#
%
&
¢
(
i=1
N
Õ ˜
Yt
e
Physical
Space
QComp
Space

Q circuit for Pauli evolution
H H
S6
H H S2

Linear equations for learning
 Fit model to data set
 Linear regression
 Regularization (regression with small coefficients).
 Principal-component analysis (dimensional
reduction via matrix factorization e.g. SVD*)
 Latent semantic analysis for natural language
processing using matrix factorization
 Recommender systems
 Deep learning

Harrow-Hassidim-Lloyd strategy
 Modify Kitaev’s eigenvalue-estimation algorithm for
Hermitian A
 Prepare and inject |𝒃 as input
 Then approximate |𝒃 in A-eigenbasis {|uj>} with
corresponding eigenvalues {lj}
 Requires O(nlogn) steps
 Kitaev q algorithm output: eigenvalues and
corresponding eigenvectors of A
 Then controlled rotations on |𝒃 and then undo to
get |𝒙

Amplitude amplification
 Given output
determines
corresponding
black-box input
with high prob that
yields given output
 Uses √N queries for
N the domain size
 Quadratic q
improvement

Hidden Abelian Subgroup
Problem
 Consider group G, subgroup H, and set X
 For O(log|G|+log|X|)-bit oracular hiding function
f:G→X st f(g)=f(g’)⟺gH=g’H, determine generating
set for H from oracular evaluations of G
 Hard classically (subexponential); easy quantumly
(polynomial)
 Includes factoring, discrete logarithm, graph
isomorphism, shortest-vector

Quantum simulation
 For sparse Hamiltonian H on n degrees of freedom,
exp(-iHt) can be simulated with poly(n,t) gates
whereas superpolynomial classical unless BPP=BQP
 Can tractably approximate ground states for some
systems and some tensor-network states
 Applications to chemistry, condensed matter,
relativistic q dynamics, q field theory

Optimization
 Minimize 𝑓 𝑥 : 𝑔𝑖 𝑥 ≤ 0 , ℎ𝑗 𝑥 = 0 .

Combinatorial optimization
 Find optimum from a finite set of feasible solutions
 Approaches are generally search algorithms
 Not guaranteed to solve efficiently
 Applications: machine learning, auctions,
operations research, algorithms

Simulated annealing
 Metaheuristic* to approximate global optimum
 Goal: minimize energy of a system
 Agent moves probabilistically from current state s to
neighbor s’ or stays at s
 Accept worse neighbours with low probability as a way
to escape local optimal that are not global
 Acceptance prob depends on energy & “temperature”
*Problem-independent strategy to guide search
procedures for approximating global optima, including
sim annealing, evolutionary algorithms & local searches

Cooling schedule
 Temperature T corresponds to probability of
accepting jumps to worse neighbouring states
 Cooling must be slow to allow near-equilibrium
dynamics; depends on topography and present T
 Strategies for cooling include adaptive and
thermodynamic approaches
 Sometimes restart rather than continue from
current state and temperature

Quantum annealing
 Augments temperature-
driven jumps to worse
neighbours with
quantum tunneling
through bad-neighbour
regions
 Typically used for
combinatorial
optimization problems

Hamiltonian
 Construct problem Hamiltonian Hp whose ground
state encodes the problem.
 Start with ground state of simple Hamiltonian H0
 Cooling schedule slow compared to spectral gap

Boolean Satisfiability problem
 Boolean expression built from binary variables,
operation AND (∧), OR (∨), NOT and parentheses
 Literals include variables and their negation
 Conjunctive normal form:
• (x1 ∨ ¬x2) ∧ (¬x1 ∨ x2 ∨ x3) ∧ ¬x1
 Conjunctive normal form:
 3SAT:
• (l1 ∨ l2 ∨ x2) ∧ (¬x2 ∨ l3 ∨ x3) ∧ (¬x3 ∨ l4 ∨ x4) ∧ ⋯ ∧ (¬xn −
3 ∨ ln − 2 ∨ xn − 2) ∧ (¬xn − 2 ∨ ln − 1 ∨ ln)

MAX-SAT
 Generalizes SAT
 What is maximum number of clauses that can be
satisfied in a Boolean expression?
 Efficiently solved approximately but NP-Hard
exactly.

usingqcomp.pptx

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