Quantum Computers

Presentation on
Quantum Computers
Prepared by-
Khan Saad Bin Hasan
Saurav Yadav
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Contents
1. Wha qu m o p s a w ar y e d?
2. Pre is Qu t Me h s
3. In e n W ki s
4. Qu n Log
5. Qu n Al o t s
6. Cha n an S r mi s, Rec Ad a c d t e
Ro d A a

What are Quantum
Computers?
● A Quantum Computer is any device for
computation that makes direct use of
distinctively quantum mechanical
phenomena, such as superposition and
entanglement, to perform operations on
data.
● Quantum computing takes advantage of
the strange ability of subatomic particles
to exist in more than one state at any time.
● Due to the way the tiniest of particles
behave, operations can be done much
more quickly and use less energy than
classical computers.
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Why are they needed?
● With the ever shrinking chip
size, transistors will hit a
physical limit where
electrons will just tunnel
through them.
● Quants can solve BQP or
Bounded-Error Quantum
Polynomial-Time problems.
● E.g, Integer Factorisation
(Shor’s Algorithm), Discrete
Logarithm, Simulation of
Quantum Systems etc.[2]
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Qubits
● A qubit[3] is the basic unit of
quantum information.
● Qubits can be 0/1 or a
coherent superposition of
these states.
● Can be realized by using the
spins of an electron or
polarization of a photon
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Superposition
Principle
● Consider the superposition
of two states, A and B,
which on observation give
results a and b.[4]
● The observation made on
system in superposed state
will be sometimes a and
sometimes b, according to a
probability law depending
on the relative weights of A
and B in the superposition
process.[4]
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Entanglement
● Quantum entanglement
occurs when pairs or groups
of particles are generated,
interact, or share spatial
proximity in ways such that
the quantum state of each
particle cannot be described
independently of the state of
the other(s), even when the
particles are separated by a
large distance
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How it works?
● The qubits are cooled to about
0.1K to cancel the effect of
thermal energy provided by the
environment.
● Qubits are now in a low energy
environment so their spins can
be measured reliably and can
be altered effectively(using
microwaves) and the results
can be converted to a string of
1’s and 0’s.
● Observing superposed states
will destroy the informative
states.
8

Why Quantum
Computers
● To define a classical system with N-bits
we only need N-bits, To define a
quantum system with N-qubits we need
2^N states.
● N-qubits can be in 2^N states, resulting
in exponential increase in computation
power.
● Hence, Quantum Computers can have
exponentially more power than a
classical computer.
● But when we measure a superposition it
results in loss of information for all other
states, hence we need algorithms that
converges quantum states to the right
answer.
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● In quantum mechanics, bra–ket notation is a standard notation for describing quantum states.
● |ket⟩ is a vector and <bra| is the hermitian conjugate of ket with same label.
● Remember:
○ |0⟩ stands for qubit being zero.
○ |1⟩ stands for qubit being one.
○ ( |0⟩ + |1⟩ ) and ( |0⟩ - |1⟩ ) represent superposition of |0⟩ and |1⟩
<Bra|ket> Notation
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Quantum Logic Gates
● A quantum logic gate is a basic quantum circuit operating on a small number of qubits. They are the building blocks
of quantum circuits, like classical logic gates are for conventional digital circuits.
● Measurement of a qubit can also change the state of the system and functions very similar to gates but it is not an
actual quantum gate.
● Quantum logic gates are reversible. This is because quantum mechanics require a quantum system to never lose
information over time and it must always possible to reconstruct the past.
● Few important logic gates are- CNOT Gate, Hadamard Gate, Toffoli Gate etc.
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CNOT Gate
● Any quantum circuit can be simulated to an
arbitrary degree of accuracy using a combination of
CNOT gates and single qubit rotations.
● It can be used to entangle and disentangle EPR
states.
● The CNOT gate operates on a quantum register
consisting of 2 qubits. The CNOT gate flips the
second qubit (the target qubit) if and only if the first
qubit (the control qubit) is |1⟩.
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● Toffoli gate is a universal reversible logic
gate, which means that any reversible
circuit can be constructed from Toffoli
gates.
● It is also known as the
"controlled-controlled-not" gate.
● It has 3-bit inputs and outputs; if the first
two bits are both set to 1, it inverts the
third bit, otherwise all bits stay the same.
Toffoli Gate
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Hadamard gate
● The Hadamard gate is a single-qubit
operation
● It maps the basis state:
∣0⟩ to (∣0⟩+∣1⟩)/√2 and
∣1⟩ to (∣0⟩−∣1⟩)/√2,
thus creating an equal superposition of
the two basis states.
● For Real numbers, it amounts to
reflection.
● The Hadamard gate can also be
expressed as a 90º rotation around the
Y-axis, followed by a 180º rotation
around the X-axis.
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Quantum Algorithms
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● Shor’s Algorithm can be used for integer factorization, It uses number theory to achieve this, It can theoretically
break RSA encryption
● Deutsch-Jozsa problem takes n-digit binary values as input and produces either a 0 or a 1 as output for each
such value. We are promised that the function is either constant (0 on all outputs or 1 on all outputs) or balanced,
the task then is to determine if f is constant or balanced by using the oracle(black box quantum computer).
● Grover's algorithm is a quantum algorithm that finds with high probability the unique input to a black box
function that produces a particular output value, using just O ( N ) evaluations of the function, where N is the size
of the function's domain. It was devised by Lov Grover in 1996.
● Other Quantum Algorithms- Simon’s Algorithm, Hidden subgroup problem, Boson sampling problem etc.

Grover’s Algorithm (Unstructured Search)
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Consider a function which is always equal 0 except a single value u. How are we going to find
u?

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● To find the pink box – the marked item – using classical computation, one would have to
check on average N/2 of these boxes, and in the worst case, all N of them.
● On a QC, however this can be achieved in √N steps using Grover’s amplitude
amplification trick.
● A quadratic speedup is indeed a substantial time-saver for finding marked items in long
lists.
● Additionally, the algorithm does not use the list’s internal structure, which makes it
generic.

The Oracle
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● A common way to encode such a list is in terms of a function f which returns f(x)=0 for all
unmarked items x and f(w)=1 for the winner.
● First we choose a binary encoding of the items x,w∈{0,1}n
so that N = 2n
; now we can
represent it in terms of qubits on a quantum computer.
● Then we define the oracle matrix Uf
to act on any of the simple, standard basis states |x⟩ by

Amplitude amplification
19
If at this point we were to measure in the standard basis {|x⟩}, the probability would collapse to
1/N = 1/2n
Step 0: The amplitude amplification procedure starts out in the uniform superposition |s⟩.

Amplitude amplification (contd.)
20
Step 1: We apply the oracle reflection Uf
to the state Uf
|ψt
⟩ = |ψt′
⟩. Geometrically this corresponds to
a reflection of the state |ψt
⟩ about −|w⟩.
Step 2: We now apply an additional reflection Us
about the state |s⟩ represented as Us
= 2|s⟩⟨s|−1.
|ψt+1
⟩ = Us
Uf
|ψt
⟩
Step1 Step2

Amplitude amplification(contd.)
21
Repeating Step 1 and Step 2 sufficient time enhances the probability of the winner.
But how many times? - After t steps the state will have transformed to |ψt
⟩ = (Us
Uf
)t
|ψ0
⟩. It turns out
that roughly √N rotations suffice.

Challenges and shortcomings
● Quantum Decoherence - the decay of quantum state, either due to environmental noise of
simply the nature of quantum mechanics and the particular terminology use to implement
qubits.
● Errors in results - Since Quantum information is fragile, quantum computers suffers from
errors and these errors seem to be fundamental.
● If there are fundamental corrections to the laws of quantum mechanics for a large system,
we would be unable to discover them because of our inability to tell what exactly quantum
mechanics would predict.
22

Recent Advances
● Intel recently announced a 49 bit quantum computer named ‘Tangle Lake’.
● Google has reportedly built a 72-bit quantum computer chip called ‘Bristlecone’.
● IBM unveils world’s first commercial quantum computer named the IBM Q System One.
● At the end of January, led by physicists at the University of Science and Technology of China
(USTC) as part of the Quantum Experiments at Space Scale project, a 75-minute
quantum-encrypted video conference call was conducted between Asia and Europe. [4]
● China sent its “Micius” satellite into space for intercontinental quantum communications.
● The race to quantum supremacy is on, with researchers claiming their machines manage tasks
beyond any modern rival.
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The Road Ahead
● Seth Lloyd believes that there will eventually
be quantum microchips in future smartphones.
● Communities and emerging companies are
striving to build even computationally intense
and power efficient systems with more
number of qubits.
● Quantum Internet, Quantum Cryptography,
Quantum Teleportation satellites, Quantum AI
etc.
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References
[1] S. Abramsky, High-level methods for quantum computation and information, in: Proceedings of the 19th Annual
IEEE Symposium on Logic in Computer Science, pp. 410–414
[2] S.L. Andresen, John McCarthy: Father of AI, IEEE Intelligent Systems (September/October 2002) 84–85
[3] Schumacher, B. (1995). Quantum coding. Physical Review A, 51(4), 2738.
[4] P.A.M. Dirac (1947). The Principles of Quantum Mechanics (2nd edition). Clarendon Press. p. 12.
[5] E. Bernstein, U. Vazirani, Quantum complexity theory, SIAM Journal on Computing 26 (1997) 1411–1473
[6] A. Baltag, S. Smets, LQP: Quantum information, Mathematical Structures in Computer Science 16 (2006) 491–525
[7] D. Aerts, M. Czachor, Quantum aspects of semantic analysis and symbolic artificial intelligence, Journal of Physics
A: Mathematical and General 37 (2004) L123–L132
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