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Overview. | Le-Shin Wu >. History and background The quantum computation model. Example: Shor’s algorithm. Quantum World…. | Le-Shin Wu >. Introduction. | Le-Shin Wu >. The important milestones: In the early 1980s, the idea of quantum computer was introduced by Benioff and Feynman.

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Overview

Overview

| Le-Shin Wu >

  • History and background

  • The quantum computation model.

  • Example: Shor’s algorithm


Quantum world

Quantum World…

| Le-Shin Wu >


Introduction

Introduction

| Le-Shin Wu >

The important milestones:

  • In the early 1980s, the idea of quantum computer was introduced by Benioff and Feynman.

  • In 1994, an explosion of interest in quantum computation was caused by Shor’s discovery of the first quantum algorithm.

  • In 2001, IBM's Almaden Research Center have performed the world's most complicated quantum-computer calculation to get the prime factors of 15.


What is quantum computation

What is quantum computation?

| Le-Shin Wu >

  • A new mode of information processing by utilizing quantum mechanics.

  • So…What is quantum mechanics?

  • Quantum mechanics describes the motion, mechanics and dynamics of particles on extremely small scales. The rules are very different to those in the classical world!


From bits to qubits

From Bits to Qubits…

| Le-Shin Wu >

  • What makes quantum computers so different from their classical counterparts?

    The answer is BIT.

  • From the physical point of view a bit is a two state system: it can be prepared in one of two distinguishable states representing two logical values : no or yes, false or true, or simply 0 or 1.


From bits to qubits cont

From Bits to Qubits…(Cont.)

| Le-Shin Wu >

  • From the quantum mechanics, if a bit can exist in either of two distinguishable states, it can also exist in coherent superposition of them. These are further states, which in general have no classical analogues, in which the atom represents both values, 0 and 1, simultaneously.


Quantum mechanics experiment 1

Quantum Mechanics Experiment (1)

| Le-Shin Wu >

  • half-silvered mirror reflects half the light that impinges upon it.

  • Place a photodetector behind the mirror in each of the two possible exit beams, the photon is detected with equal probability at either detector.

  • The photon takes both paths at once


Quantum mechanics experiment 2

Quantum Mechanics Experiment (2)

| Le-Shin Wu >

  • Add two fully silvered mirrors and placing another half-silvered mirror at their meeting point, with two photodectors in direct lines of the two beams

  • There is a 100% probability that the photon reaches the detector 1 and 0% probability that it reaches the other detector 2


Quantum mechanics experiment 3

Quantum Mechanics Experiment (3)

| Le-Shin Wu >

  • An absorbing screen is placed in the way of either of the two routes

  • Then it becomes equally probable that detector 1 or 2 is reached


The conclusion of the experiment

The Conclusion of The Experiment

| Le-Shin Wu >

  • The photon is in a coherent superposition of being in the transmitted beam and in the reflected beam.

  • A quantum two state system, called a quantum bit or a qubit, can be prepared in a superposition of its two logical states 0 and 1.

  • Thus one qubit can encode at a given moment of time both 0 and 1.


Push the idea of superposition further

Push The Idea of Superposition Further

| Le-Shin Wu >

  • A register composed of three physical bits can store in a given moment of time only one out of eight different numbers.

  • A quantum register composed of three qubits can store in a given moment of time all eight numbers in a quantum superposition.

  • In general L qubits can store 2L numbers at once.


Push the idea of superposition further1

Push The Idea of Superposition Further

| Le-Shin Wu >

  • A register composed of three physical bits can store in a given moment of time only one out of eight different numbers.

  • A quantum register composed of three qubits can store in a given moment of time all eight numbers in a quantum superposition.

  • In general L qubits can store 2L numbers at once.


Push the idea of superposition further cont

Push The Idea of Superposition Further(cont.)

| Le-Shin Wu >

  • A quantum computer can in only one computational step perform the same mathematical operation on 2L different input numbers encoded in coherent superpositions of L qubits.

  • Any classical computer has to repeat the same computation 2L times or one has to use 2L different processors working in parallel.


Push the idea of superposition further cont1

Push The Idea of Superposition Further(cont.)

| Le-Shin Wu >

  • A quantum computer offers an enormous gain in the use of computational resources such as time and memory.


Factorisation problem

Factorisation Problem

| Le-Shin Wu >

  • 127 x 229 = ? is a very easy problem.

  • ? x ? = 29083 is a harder problem.

  • We know fast algorithms for multiplication but we do not know equally fast ones for factorisation.


Factorisation problem cont

Factorisation Problem (Cont.)

| Le-Shin Wu >

  • Multiplication requires little extra work when we switch from two three digit numbers to two thirty digits numbers. But factoring a thirty digit number using the simplest trial divison method is about 1013 times more time or memory consuming than factoring a three digit number.


Factoring on quantum computers

Factoring on Quantum Computers

| Le-Shin Wu >

Problem: to find the prime factors of N = 15.

  • Pick a random number a smaller than N, for instance a = 7.

  • Define a function f(x) = 7x mod 15. For instance, if x = 3, f(x) = 13 because 73 = 343 = 15 times 22 +13.


Factoring on quantum computers cont

Factoring on Quantum Computers (Cont.)

| Le-Shin Wu >

3. Mathematics shows that f(x) is periodic and that its period r can be related to the factors of 15. we can check easily that f(x) evaluates to 1, 7, 4, 13, 1, 7, 4... for the values of x = 0, 1, 2, 3, 4, 5, 6.... and conclude that the period is r = 4.


Factoring on quantum computers cont1

Factoring on Quantum Computers (Cont.)

| Le-Shin Wu >

4. Evaluate the greatest common divisor of N and ar/2 +/- 1.

5. In our example computing the greatest common divisor of 15 and 50 =7 4/2 + 1 (or 48=7 4/2 - 1) returns indeed the values 5 (or 3), the factors of 15.


Factoring on quantum computers cont2

Factoring on Quantum Computers (Cont.)

| Le-Shin Wu >

  • Classical computers cannot make much of this new method: finding the period of f(x) requires to evaluate the function f(x) many times.

  • With a quantum computer: by setting a quantum register in a superposition of states representing 0, 1, 2, 3, 4... it is possible to compute in a single go the values f(0), f(1), f(2) ... .

  • This algorithm can be performed very efficiently on a quantum computer.


Quantum computer model

Quantum Computer Model

| Je-Luen Tzeng >

  • Quantum logic gates

    • Perform unitary operations

    • Must be reversible

  • Quantum networks

    • Consisting of quantum logic gates

    • Computational steps synchronized in time


Unitary matrices

[ ]

0 i

i 0

M =

= MT

[ ]

0 -i

-i 0

MTC =

[ ]

1 0

0 1

M ´ MTC =

= I

Unitary Matrices

| Je-Luen Tzeng >

  • Definition:A unitary matrix is a matrix whose conjugate transpose is its own inverse.

  • Example:


Why unitary

Why Unitary?

| Je-Luen Tzeng >

  • Want to compute some function f:

  • Time evolution operator U:

  • Solve the Hamiltonian H:


Dirac s bra ket notation

Dirac’s Bra-ket Notation

| Je-Luen Tzeng >

  • Primary objects of QCs are vectors and matrices of a Hilbert space over the complex numbers.

  • Bras (áf |) and kets (|y ñ).

  • |f ñ : a normal (vertical) vector.

  • áf | : the conjugate transpose (horizontal) vector of |f ñ.


Notations for qubits

Notations for Qubits

| Je-Luen Tzeng >

  • 2-dimensional, normalized, complex in a Hilbert space with base vectors |0ñ and |1ñ .

  • |f ñ = a |0ñ + b |1ñ

  • State vectors of the qubits:

    e.g., |0ñ Ä |1ñ = |01ñ


Tensor product

Tensor Product

| Je-Luen Tzeng >

  • Definition:

  • Example:


Entanglement

Entanglement

| Je-Luen Tzeng >

  • A state of three particles:

  • Can we find the superposition describing the first qubit?

NO!


Quantum logic gates

Quantum Logic Gates

| Je-Luen Tzeng >

  • Definition: A quantum gate on k qubits is a unitary matrix U of dimensions 2k x 2k.


Quantum logic gates1

Quantum Logic Gates

| Je-Luen Tzeng >

  • A quantum gate:

  • Quantum computation is reversible

  • U: |001> -> |010>

  • U-1: |010> -> |001>


Quantum logic gates cont

Quantum Logic Gates (cont.)

| Je-Luen Tzeng >

  • An example: C-NOT (Controlled-NOT) is a logic gate in which one bit is flipped conditional on the state of the other bit.

|00> -> |00>

|01> -> |01>

|10> -> |11>

|11> -> |10>


Universality problem

Universality Problem

| Je-Luen Tzeng >

  • Deutsch (1989) identified three-qubit universal quantum gates.


More universal quantum gates

More Universal Quantum Gates

| Je-Luen Tzeng >

  • Several researchers (1995) independently announced two-qubit gates.


Implementation of two qubit gates

Implementation of Two-Qubit Gates

| Je-Luen Tzeng >

  • Linear ion-trap by Cirac and Zoller in 1995.


Rsa encryption

RSA Encryption

| Yu-Chun Wang >

  • Encryptography

    How to transmit data from computer A to computer B safely?

    Public key v.s. Private key


How does rsa work

How does RSA work?

| Yu-Chun Wang >

  • Step 1. Find P and Q, two large (e.g., 1024-bit) prime numbers.

    e.g. P = 11, Q = 2, PQ(one public key) = 22.


How does rsa work1

How does RSA work?

| Yu-Chun Wang >

  • Step 2. Choose E such that 1 < E < PQ, and E and (P-1)(Q-1) are coprime.

    e.g. E(another public key) = 7.


How does rsa work2

How does RSA work?

| Yu-Chun Wang >

  • Step 3. Compute D such that DE = 1 mod (P-1)(Q-1). This is easy to do -- simply find an integer X which causes D = (X(P-1)(Q-1) + 1)/E to be an integer, then use that value of D.

    e.g. D(private key) = 3 = (2(P-1)(Q-1)+1)/E


Rsa encryption1

RSA Encryption

| Yu-Chun Wang >

  • Encryption

    C = TE mod PQ. (C = Cipher, T = Text)

    e.g. 37 = 9 mod 22

  • Decryption

    T = CD mod PQ

    e.g. 93 = 3 mod 22


Prove rsa encryption

Prove RSA Encryption

| Yu-Chun Wang >

Let P, Q be 2 prime numbers, DE = 1 mod (P-1)(Q-1), b = aE mod PQ, c = bD, then c = a mod PQ.

  • Fermat’s little theorem

    Let p be a prime number which does not divide the integer a, then ap-1 = 1 mod p.


Why rsa works

Why RSA works

| Yu-Chun Wang >

  • PQ and E are available, but D is not.

  • In order to obtain D, you have to factor PQ.

  • Sounds easy? Keep in mind that P and Q are extremely large.


Why rsa works1

Why RSA works

| Yu-Chun Wang >

  • Currently difficult for classical computer to factor large number

  • Running time:

    O(exp((64/9)1/3N1/3(ln N)2/3)) N:bits

  • Using Shor’s Algorithm on QC:

    O((log N)2log(log N)) on QC, O(log N) steps on classical computer


Shor s algorithm

Shor’s Algorithm

| Yu-Chun Wang >

  • The function f(a) = xa mod n is periodic, where x is coprime to n.

  • Calculating this function would take exponential time on a classical computer. Shor's algorithm utilizes quantum parallelism to perform the exponential number of operations in one step


Shor s algorithm1

Shor’s Algorithm

| Yu-Chun Wang >

  • Assume the periodic number of f(a) is r, and r is even:

  • x0 = 1 mod n, so xr = 1 mod n. (n is the number we want to factor)

  • xr – 1 = 0 mod n


Shor s algorithm2

Shor’s Algorithm

| Yu-Chun Wang >

  • r is even, so (xr/2)2 – 1 = 0 mod n

  • (xr/2 + 1)(xr/2 - 1) = 0 mod n

  • (xr/2 + 1)(xr/2 - 1) = kn, which means either (xr/2 + 1) or (xr/2 - 1) has a factor in common with n


Shor s algorithm3

Shor’s Algorithm

| Yu-Chun Wang >

  • So calculate gcd(xr/2 + 1, n) and gcd(xr/2 – 1, n), we can obtain at least a factor of n.

  • Example:


Shor s algorithm4

Shor’s Algorithm

| Yu-Chun Wang >

  • Pick x = 2, n = 15.

    20 = 1 mod 15

    21 = 2 mod 15

    22 = 4 mod 15

    23 = 8 mod 15

    24 = 1 mod 15

  • So r = 4.


Shor s algorithm5

Shor’s Algorithm

| Yu-Chun Wang >

  • Calculate gcd(24/2 + 1, 15) and gcd((24/2 - 1, 15)15 = 5 x 3.


Implementation

Implementation

| Yu-Chun Wang >

  • To do this, create a quantum memory register with 2 parts.

    • Part 1 stores the superposition of a, and part 2 stores the result of xa mod n

    • Choose a to be from 0 to q–1, where q is the power of 2 such that n2 <= q < 2n2


Implementation1

Implementation

| Yu-Chun Wang >

  • Do calculation and measurement

  • After the measurement, part 2 contains a value k, and part 1 contains a superposition of the base states which make xa = k mod n, say s, s+r, s+2r…, where s is the smallest integer such that

    xs = k mod n


Implementation2

Implementation

| Yu-Chun Wang >

  • Perform a discrete Fourier transform on the contents of part 1, which has the effect of peaking the probability amplitudes of part 1 at integer multiples of the quantity q/r

  • Now measuring part 1 will generate the multiples of q/r. Using classical computer can analyze this value and guess the actual value of r

  • Problem solved!


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