Umesh V. Vazirani U. C. Berkeley Quantum Algorithms: a survey

Exponential Superposition  Superposition of all 2 n classical states:  Measurement: See |x i with probability |  x | Quantum Algorithms: tension between these two phenomena 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 P 1 (x) = |  1 (x)| 2 P 2 (x) = |  2 (x)| 2  1,2 =  1 (x) +  2 (x) P 1,2 (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: aa b a © b - Single Bit Gates (Rotations)  |1 i |0 i |1’ i |0’ i |0 i ! cos  |0 i + sin  |1 i |1 i ! sin  |0 i - cos  |1 i..\..\clipart\atomphoton\index1.html

Quantum Fourier Transform Quantum: Input: |  i =  j=0 m  j |ji O(logm) qubits Fourier transform: F|  i =  j=0 m-1  j |ji O(log 2 m) gates Limited Access: Measure: see |ji with probability |  j | 2 Classical : FFT O(m logm)

Quantum Fourier Transform One Qubit or Z 2  |1 i |0 i |1’ i |0’ i |0 i ! |0 i + |1 i |1 i ! |0 i - |1 i Two Qubits or Z 2 2   |00 i ! (|0 i + |1 i ) ­ (|0 i + |1 i ) = 1/2(|00 i + |01 i + |10 i + |11 i ) |01 i ! (|0 i + |1 i ) ­ (|0 i - |1 i ) = 1/2(|00 i - |01 i + |10 i - |11 i ) |10 i ! (|0 i - |1 i ) ­ (|0 i + |1 i ) = 1/2(|00 i + |01 i - |10 i - |11 i ) |11 i ! (|0 i - |1 i ) ­ (|0 i - |1 i ) = 1/2(|00 i - |01 i - |10 i + |11 i )

n Qubits or Z 2 n     |u i x1x1 xnxn

Outline of Shor’s Factoring Algorithm N ! exponential superposition  x  x |x i 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 q0q

Outline of Shor’s Factoring Algorithm (Example) 0 q0q N = 15 = p ¢ q Randomly choose a = 7 (mod 15) Consider sequence a x (mod 15) for x =0,1,2,… 1, 7, 4, 13, 1, 7, 4, 13, 1, 7, 4, 13, … Period r = 4. N | (a r -1)=(a r/2 +1)(a r/2 -1) = (7 2 +1)(7 2 -1) = 50 ¢ 48 p = gcd(15, 50), q = gcd(15, 48). Create superposition  x |x i |a x i

Outline of Shor’s Factoring Algorithm (Example) N = 15 = p ¢ q Randomly choose a = 7 (mod 15) Consider sequence a x (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 |x i |a x i 0 q0q

The Hidden Subgroup Problem Given f : G ! S, constant and distinct on cosets of subgroup H. Find H. Examples Factoring N: G = Discrete log: G = G:G:

1. State Preparation G:G: Given f : G ! S, constant and distinct on cosets of subgroup H. Find H. FGFG f Measure 1. Create Random Coset state  h 2 H |g + h i :

2. Fourier Sampling G:G: F.T. G: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: x r = 1 mod N. Let f: a ! x a mod N. Underlying group = Z M, where M =  (N) = (P-1) ¢ (Q- 1) Hidden subgroup = H = h r i = {0,r, 2r, …, M/r} H ? = h M/r i = {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 x 2 – d ¢ y 2 = 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 v 1, …, v n vectors in R n. The lattice is a 1 v 1 + … + a n v n for all integers a 1, … a n. Find shortest vector in lattice. Short vector in Lattice:

Two Challenges Finding short vector not easy! 1. Short vector in Lattice: Regev: D N Dihedral group 2. Graph Isomorphism S N 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 D N : Group of symmetries of a regular N-gon. Generated by x, y: x N = 1, y 2 =1, xyxy = 1. Assume N = 2 n. D N has 4 1-d irreps and (N-1)/2 2-d irreps. [Kuperberg ’03] algorithm for dihedral HSP.

Dihedral HSP algorithm Assume wlog H = {1, x h y} 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.

Dihedral HSP algorithm Claim:  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.

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’03 Adiabatic Computation ≈ Quantum Computation

Classical Simulation of Quantum Systems Vidal ’03 Polynomial time simulation of one dimensional spin chains with O(log n) entanglement length. ABC  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.