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Quantum Algorithms & Complexity. Umesh Vazirani U.C. Berkeley. One does not, by knowing all the physical laws as we know them today, immediately obtain an understanding of anything much. (Richard Feynman, 1918-1988) . One does not, by knowing all the physical laws as we know

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Quantum Algorithms & Complexity

Umesh Vazirani

U.C. Berkeley


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One does not, by knowing all the physical laws as we know

them today, immediately obtain an understanding of anything

much. (Richard Feynman, 1918-1988)


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One does not, by knowing all the physical laws as we know

them today, immediately obtain an understanding of anything

much. (Richard Feynman, 1918-1988)

Quantum computers are the only known model of

Computation that violate the Extended Church-Turing

thesis.


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Goals of Quantum Algorithms/Complexity

  • Find exponential speedups for a range of natural

  • computational problems.

  • Establish the limits of quantum algorithms.

  • Relate quantum complexity classes, such as BQP and

  • QMA, to classical complexity classes, such as

  • BPP, MA, PH.


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Goals of Quantum Algorithms/Complexity

  • Find exponential speedups for a range of natural

  • computational problems.

  • Establish the limits of quantum algorithms.

  • Relate quantum complexity classes, such as BQP and

  • QMA, to classical complexity classes, such as

  • BPP, MA, PH.

Far reaching implications for cryptography,

computational complexity, physics, … Each of these

gives its own unique flavor to the questions.


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Quantum resistant cryptography

  • Quantum computers break much of modern cryptography.

  • RSA (factoring), Diffie-Helman (discrete log),

  • Elliptic curve crypto, Buchmann-Williams (Pell eqn)…

  • Suppose we had a classical cryptosystem that was

  • as efficient and convenient as RSA, but was provably

  • not breakable even on a quantum computer.

  • Then there would be an incentive to switch to the

  • new cryptosystem, well before a large scale quantum

  • computer were experimentally realized.


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  • The answer relies crucially on our understanding of

  • the power and limitations of quantum computers.


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Hidden Subgroup Problem

G finite group. H subgroup of G.

Given black box that evaluates f: G -> S:

f is constant on cosets of H.

Determine H.

G:

  • G abelian: lens = fourier transform over G.

  • polynomial time quantum algorithm.

  • Shor: factoring. G = ZN. Period finding.

  • discrete log. G = Zp x Zp

  • [Hallgren] Pell’s equation

  • [van Dam, Hallgren, Ip] Hidden shift problems,

  • Breaking homomorphic encryption

  • [van Dam, Seroussi] Gauss sums


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Quantum Algorithm for Abelian HSP

Random coset state: use f to set up state

G:

gH

=

FT over G

FT over G:

FT + measurement gives uniformly random element of

Think of this as a random linear constraint on H …


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Graph Isomorphism

SN Symmetric group

Non-abelian hidden subgroup problem

Lens = (non-abelian) fourier transform over G.

Short vector in Lattice:

Finding short vector not easy!

DNDihedral group

[Regev]


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Lattice Problems

  • Finding short lattice vectors closely related to

  • Dihedral HSP.

  • Random coset state preparation + Fourier sampling

  • gives sufficient info to reconstruct subgroup.

  • But classically reconstructing subgroup appears to be

  • very difficult. Related to subset sum.

  • Kuperberg’s quantum reconstruction algorithm.


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Public-key cryptosystems based on Quantum

hardness of Shortest Lattice Vector.

  • [Ajtai-Dwork] cryptosystem.

  • [Regev]

  • Improved efficiency based on assumption that finding

  • short lattice vectors is hard for quantum algorithms.

  • New cryptosystem resembles hardness of solving noisy

  • linear equations mod p.

  • Worst-case to average case reduction.


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Learning with errors

Linear equations in n variables over Zp for p prime,

where n2 < p < 2n2

m noisy equations:

where

and is gaussian with mean 0 and standard

deviation n1.5

Theorem [Regev]: LWE is as hard as approximating

the shortest vector in a lattice to within n1.5


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Worst-case to average-case reduction

  • LWE specifies an average-case problem. Inputs

  • sampled from a fixed distribution.

  • Quantum reduction showing that an arbitrary lattice

  • problem (worst-case) can be mapped to LWE.

  • Example of the quantum method. Prove a purely

  • classical statement by quantum methods.

  • [Kerenidis, deWolf] lower bounds for locally

  • decodable codes.


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LWE and Lattices

  • Lattice L = {integer linear combinations of u1, …, un }

  • Dual lattice L* = {v: <v,u> integer for all u in L}

  • L* is the fourier transform of L.


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LWE and Lattices

  • Lattice L = {integer linear combinations of u1, …, un }

  • Dual lattice L* = {v: <v,u> integer for all u in L}

  • L* is the fourier transform of L.

D*L

DL


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D*L

DL

  • Sampling from DL with small width Gaussian implies

  • good approximation of shortest lattice vector.

  • Polynomially large samples from DL yield an unbiased

  • estimator for D*L . If the width of the Gaussian

  • is large, this gives a way of, given x, approximating

  • the closest lattice vector to x in L*.

  • Quantum reduction, given algorithm for approximating

  • closest vector in L*, to sampling from DL .


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D*L

DL

  • Sampling from DL with small width Gaussian implies good approximation

  • of shortest lattice vector.

  • Polynomially large samples from DL yield an unbiased estimator for D*L .

  • If the width of the Gaussian is large, this gives a way of, given z,

  • approximating the closest lattice to z.

  • Quantum reduction, given algorithm for approximating

  • closest vector in L*, to sampling from DL .

To erase x, compute x given z=x+y:


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Improving the Efficiency

  • Based on cyclic lattices:

  • Lattices where the basis consists of vector v, and

  • all its cyclic shifts.

  • Much more succinct. Key size n2 -> n

  • Faster computation – use Fourier transforms.

  • [Piekart, Rosen] collision resistant hash functions.

  • [Gentry] Homomorphic encryption.


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Open Questions

  • Is there a quantum algorithm to find a short

  • vector in a cyclic lattice?

  • Does the van Dam, Hallgren, Ip quantum algorithm for

  • breaking homomorphic encryption extend to

  • Gentry’s scheme?

  • Is it possible to speed up Kuperberg’s quantum

  • reconstruction algorithm for the dihedral HSP?

  • Is it possible to design a public-key cryptosystem

  • based on cyclic lattices?


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Greater Security?

[Hallgren, Moore, Roettler, Russell, Sen 06] provide

very strong evidence of quantum hardness:

Hg1

Hg2

Hgk

k < poly(n) implies exponentially many measurements

For sufficiently non-abelian groups. Eg Sn, GLn

in particular: graph isomorphism.

Sufficiently non-abelian ~ exponential sized irreps + …

Can one base public-key cryptography on these stronger

impossibility results?

[Moore, Russell, V] One-way function, related to McEliese

Cryptosystem, based on hardness of HSP over


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Goals of Quantum Algorithms/Complexity

  • Find exponential speedups for a range of natural

  • computational problems.

  • Establish the limits of quantum algorithms.

  • Relate quantum complexity classes, such as BQP and

  • QMA, to classical complexity classes, such as

  • BPP, MA, PH.


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An Old Question in Quantum Complexity Theory

  • Is BQP C PH?

  • [Bernstein, V ‘93] There is an oracle A: BQPA C MAA

  • Conjectured that same holds for PH – that recursive

  • fourier sampling is in BQP but not in PH.

  • [Aaronson ‘09] Conjecture: Fourier checking is in

  • BQP, but not in PH.

  • Proof that this is true under the generalized Linial-Nisan

  • conjecture.

  • The original Linial-Nisan conjecture states that

  • logn-wise independent distributions fool AC0 circuits.

  • Resolved by Braverman. Generalized = almost logn-wise.


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Hamiltonian Complexity

Computational complexity <--> condensed matter physics

  • H = H1 + … + Hm , each Hi k-local.

  • [Kitaev] Computing ground energy of H is QMA-hard.

  • [Aharonov, et. al.] Adiabatic quantum computation is

  • universal.

  • [Hastings] Area law for 1-D local Hamiltonians.

  • Efficient simulation of gapped Hamiltonians.

  • [Aharonov, Gottesman, Irani, Kempe] Computing

  • ground states of 1-D local Hamiltonians QMA-hard.


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Quantum PCP theorem?

  • Given a promise that k-local hamiltonian H has

  • either ground energy 0 or cm for constant c,

  • determine which.

  • Classical PCP theorem is a cornerstone of classical

  • complexity theory.

  • Theory of inapproximability, room temperature QC

  • [Aharonov, Arad, Landau, V] quantum gap amplification.


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  • How do you verify a theory where you require

  • exponential resources to calculate the predicted

  • outcome of the experiment?

  • One-way function. Start with P, Q primes.

  • Multiply N = PQ. See if quantum computer can

  • Factor.

  • How do you verify the claims of a company

  • New-Wave, that claims to have built a quantum

  • Computer?

  • [Aharonov, et. Al.], [Broadbent, et. Al.]

  • Quantum interactive proofs.


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Conclusions

Quantum algorithms and complexity theory explore

fundamental questions with profound implications:

  • Quantum resistant cryptography.

  • Probabilistic method <--> quantum method

  • Quantum complexity <--> classical complexity

  • quantum complexity theory <--> condensed matter physics

  • Verifying quantum computations.


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