Umesh V. VaziraniU. C. Berkeley

Quantum Algorithms: a survey

Exponential Superposition

Superposition of all 2n classical states:

Measurement:

See |xi with probability |x|2

+-

+

-

+

-

+-

Quantum Algorithms: tension between these two phenomena

x

x x 1|| 2x

x

all n-bit

strings

Why limit computers to electronics implemented on

silicon?

• Extended Church-Turing thesis - Any “reasonable” model of computation can be efficiently simulated by a probabilistic Turing Machine. - Circuits, Random access machines, cellular

automata. - “Reasonable” = physically realizable in principle

• Quantum computers only model that violate this thesis

• Shor’s quantum factoring algorithm. Breaks modern cryptography.

• Simulating quantum mechanics.

• Symmetry - Discrete logarithm - Pell’s equation - Shifted Legendre Symbol - Gauss sums - Elliptic curve cryptography

Quantum Algorithms – Exponential Speedups

Young’s Double slit experiment

P1(x) = |1(x)|2

P2(x) = |2(x)|2

1,2 = 1(x) + 2(x)

P1,2(x) = |1,2(x)|2

n-slits• Etch n slits in a pattern based on the input.

• Send photon through and measure.

• Location at which photon detected gives provides information about solution to input.

Input-based slit pattern

photon

screen

Quantum Circuits

- Each Wire Carries a qubit of information

- Controlled Not:

a a

b a © b

- Single Bit Gates (Rotations)

|1i

|0i

|1’i

|0’i

|0i ! cos |0i + sin |1i

|1i ! sin |0i - cos |1i

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U

Quantum Fourier Transform

Quantum: Input: |i = j=0

m j |ji O(logm) qubits

Fourier transform: F|i = j=0m-1 j |ji O(log2m) gates

Limited Access:

Measure: see |ji with probability |j|2

1

1

0

)1)(1()1(21

12

1

1

0

.

.

.1

.....

.....

.1

1.111

.

.

mmmmm

m

m

Classical: FFT O(m logm)

Quantum Fourier TransformOne Qubit or Z2

|1i

|0i

|1’i

|0’i

|0i ! |0i + |1i

|1i ! |0i - |1i

Two Qubits or Z22

|00i ! (|0i + |1i) (|0i + |1i) = 1/2(|00i + |01i + |10i + |11i)

2

1

2

1

2

1

2

1

|01i ! (|0i + |1i) (|0i - |1i) = 1/2(|00i - |01i + |10i - |11i) 2

1

2

1

|10i ! (|0i - |1i) (|0i + |1i) = 1/2(|00i + |01i - |10i - |11i) 2

1

2

1

|11i ! (|0i - |1i) (|0i - |1i) = 1/2(|00i - |01i - |10i + |11i) 2

1

2

1

2

1

2

1

n Qubits or Z2n

|ui

x1

xn

xunx

n

xu

|2

)1(|

}1,0{2/

Outline of Shor’s Factoring Algorithm

• N ! exponential superposition x x |xi

• Factors of N encoded in global property of superposition – its period.

• quantum fourier transform and measure to extract period.

• Reconstruct factors of N from the period.

0 q 0 q

Outline of Shor’s Factoring Algorithm (Example)

0 q 0 q

• N = 15 = p¢ q• Randomly choose a = 7 (mod 15)• Consider sequence ax (mod 15) for x =0,1,2,… 1, 7, 4, 13, 1, 7, 4, 13, 1, 7, 4, 13, …• Period r = 4. • N | (ar-1)=(ar/2+1)(ar/2 -1) = (72 +1)(72 -1) = 50¢ 48• p = gcd(15, 50), q = gcd(15, 48).• Create superposition x |xi |axi

Outline of Shor’s Factoring Algorithm (Example)

• N = 15 = p¢ q• Randomly choose a = 7 (mod 15)• Consider sequence ax (mod 15) for x =0,1,2,… 1, 7, 4, 13, 1, 7, 4, 13, 1, 7, 4, 13, …• Period r = 4. N | (r+1)(r-1) = (4+1)(4-1) = 5¢ 3• p = gcd(15, 5), q = gcd(15, 3).

• Create superposition x |xi |axi

0 q 0 q

The Hidden Subgroup ProblemGiven f : G ! S, constant and distinct on cosets of subgroup H. Find H.

Examples• Factoring N: G = • Discrete log: G =

G:

1. State Preparation

G:

Given f : G ! S, constant and distinct on cosets of subgroup H. Find H.

FG

f

Measure

1. Create Random Coset state h2 H |g + hi:

2. Fourier Sampling

G:

F.T.

G:

Measure a random element of H?.

3) (Classically) reconstruct H from polynomialy many samples.

Example: Factoring

To factorize N = P¢Q, sufficient to compute order of randomly chosen x mod N. i.e. smallest positive r: xr = 1 mod N.

Let f: a ! xa mod N. Underlying group = ZM, where M = (N) = (P-1)¢(Q-

1)Hidden subgroup = H = h r i = {0,r, 2r, …, M/r}

H? = h M/ri = {0,M/r, 2M/r, …, M}Fourier sampling gives kM/r for random k: 0 · k · r-1

gcd(M, kM/r) = M/r if k,r relatively prime.

• Given a positive non-square integer d, find integer solutions x,y of x2 – d¢ y2 = 1.

Pell’s Equation

• [Hallgren 2002]: 1) Quantum algorithm for Pell’s Equation

2) Breaks Buchman-Williams cyptosystem

• One of the oldest studied problem in algorithmic number theory.• Appears harder than factoring • [1989] Buchman-Williams cryptosystem

Abelian Hidden subgroup problem – but the group is not finitely generated

Two Challenges

• Basis v1, …, vn vectors in Rn.

• The lattice is a1 v1 + … + an vn for all integers a1, … an.

• Find shortest vector in lattice.

Short vector in Lattice:

Two Challenges

Finding short vector not easy!

1. Short vector in Lattice:

Regev: DN Dihedral group

2. Graph Isomorphism

SN Symmetric group

Ajtai-Dwork Cryptosystem.

[GSVV] For random choice of basis, for sufficiently non-abelian groups (e.g. S_n), exponentially many samples necessary to distinguish |H|=2 from |H| =1.

Dihedral HSP

• Dihedral Group DN : Group of

symmetries of a regular N-gon. • Generated by x, y: xN = 1, y2=1, xyxy = 1.• Assume N = 2n.

• DN has 4 1-d irreps and (N-1)/2 2-d irreps.

jl

jll

j x

0

0)(

0

0)(

jl

jll

j yx

[Kuperberg ’03] )(2 nO algorithm for dihedral HSP.

Dihedral HSP algorithm• Assume wlog H = {1, xhy}• Set up random coset state, fourier transform and

sample irrep to get random j,

• Can sample column superposition and do phase estimation to get coin flip of bias

• O(log N) samples sufficient to determine h, but reconstruction problem hard.

• Would like to sample particular irreps j.

jhkhj

khjjh

)(

)(

N

jh2cos

Dihedral HSP algorithmClaim: i j = i+j © i-j

Algorithm: • Start with registers in random coset states, FT, sample irreps.• Sort irrep names, pair up successive registers• Apply above transformation, and retain iff i,j ! i-j• Number of bits reduced by per iteration. iterations

• Number of irreps reduced by 4 per iteration.

n22

n2

2

n

Non-abelian Hidden Subgroup Problem

• Abelian quantum algorithm doesn’t generalize - Prepare random coset state - Measure in Fourier basis

• Ettinger, Hoyer, Knill ’98:

- Prepare several registers with random coset states - Perform appropriate joint measurement

• Ip ’03:

- Fourier transform & Measure irrep (character) for each register - Perform appropriate joint measurement on residual state.

Adiabatic Quantum State Generation

Aharonov, Ta-Shma ‘02

AharonovvanDamKempeLandauLloydRegev’03AharonovvanDamKempeLandauLloydRegev’03

Adiabatic Computation ≈ Quantum ComputationAdiabatic Computation ≈ Quantum Computation

Classical Simulation of Quantum Systems

Vidal ’03 Polynomial time simulation of one dimensional spin chains with O(log n) entanglement length.

A B C

AC = A C

ABC = AB1 B2C

1 2

Summary

• Quantum computation only model that violates extended Church-Turing thesis.

• Exponential superposition vs limited access.

• Exponential speedups appear to require symmetry.

• Fast quantum algorithm for abelian hidden subgroup problem

• Non-abelian case open.