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Overview

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| Le-Shin Wu >

- History and background
- The quantum computation model.
- Example: Shor’s algorithm

| Le-Shin Wu >

| 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.

| 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!

| 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.

| 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.

| 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

| 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

| 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

| 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.

| 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.

| 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.

| 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.

| Le-Shin Wu >

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

| 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.

| 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.

| 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.

| 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.

| 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.

| 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.

| Je-Luen Tzeng >

- Quantum logic gates
- Perform unitary operations
- Must be reversible

- Quantum networks
- Consisting of quantum logic gates
- Computational steps synchronized in time

[ ]

0 i

i 0

M =

= MT

[ ]

0 -i

-i 0

MTC =

[ ]

1 0

0 1

M ´ MTC =

= I

| Je-Luen Tzeng >

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

- Example:

| Je-Luen Tzeng >

- Want to compute some function f:

- Time evolution operator U:

- Solve the Hamiltonian H:

| 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.

| 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

| Je-Luen Tzeng >

- Definition:

- Example:

| Je-Luen Tzeng >

- A state of three particles:

- Can we find the superposition describing the first qubit?

NO!

| Je-Luen Tzeng >

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

| Je-Luen Tzeng >

- A quantum gate:

- Quantum computation is reversible
- U: |001> -> |010>
- U-1: |010> -> |001>

| 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>

| Je-Luen Tzeng >

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

| Je-Luen Tzeng >

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

| Je-Luen Tzeng >

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

| Yu-Chun Wang >

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

Public key v.s. Private key

| 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.

| 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.

| 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

| 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

| 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.

| 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.

| 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

| 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

| 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

| 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

| 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:

| 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.

| Yu-Chun Wang >

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

| 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

| 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

| 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!