anthony leverrier (inria paris) 29 march 2018 orap 41st...

Quantum algorithms

Anthony Leverrier (Inria Paris)

29 March 2018

ORAP 41st Forum: Quantum Computing

Anthony Leverrier (INRIA) ORAP Forum, 29 March 2018 1 / 13

Outline

I 1982 - 1993: the genesis of quantum computing, a scientific curiosity

I 1994 - 2009: Shor, Grover and generalizations

I 2010 - : solving linear equations, quantum machine learning

Genesis of quantum computingHow can we simulate the behaviour/evolution of quantum systems, since we need tokeep track of an exponential number of parameters?

Feynman 1981

“Can quantum systems be probabilistically simulated by aclassical computer? [. . . ]The answer is almost certainly, No!”=⇒ use quantum systems to simulate quantum systems!

Deutsch 1985

I quantum Turing machine

I existence of a universal machine

Bernstein, Vazirani 1993

I efficient quantum Turing machine (complexity class BQP)

I Bernstein-Vazirani problem: f : {0, 1}n → {0, 1} such that f (x) = a · xFind a. =⇒ ok with 1 quantum query vs n classically

Feynman 1981

Deutsch 1985

Feynman 1981

Deutsch 1985

The first algorithms

Simon, Shor 1994exponential speedups for

I period finding

I factoring

I discrete logarithm

=⇒ exploits Quantum Fourier Transform=⇒ consequences for public-key cryptography: a main argument to obtain

funding in quantum computation, seen as a real threat today by crypto agencies(NIST competition for post-quantum cryptography)

Grover 1996I search an n-item list with O(

√n) queries

I Lots of applications (find collisions, approximate counting,shortest path)

but only quadratic improvement

Hidden subgroup problem (≈ exponential speedup)

The problem for group G

I input: function f “periodic” on subgroup H ≤ G :

f (x) = f (y) ⇐⇒ y ∈ xH

I output: generator set for H

Applications

I Simon problem: G = (Z2)n, H = {0, s}I factoring: G = Z, H = rZ

I discrete logarithm: G = Z2, H = {(rx , x) : x ∈ Z}

I Pell’s equation: G = R

I graph isomorphism: G = Sn

Polynomial-time algorithms (in log |G |)ok for Abelian groups G (Shor 1995), for normal subgroups H (2000)

Polynomial speedup

Generalizations of Grover’s search: “blackbox tools”

I amplitude amplification: unstructured problems

I quantum walks (Szegedy 2004)

I span programs (Reichardt, Spalek 2008): AND-OR tree evaluations

I learning graphs (Belovs 2012): triangle finding, etc

Applications to symmetric cryptography

I find a key: quadratic speedup =⇒ double key size

I break hash functions: find collisions x , y such that h(x) = h(y)

Lower bounds

If one models complexity as the number of queries to the function/oracle, thenquantum speedup for many problems is at most polynomial.

This was the situation until not so long ago.

Main algorithms: Shor and Grover (+ generalizations)

(also quantum chemistry . . . )

then came HHL and the prospect of quantum machine learning

This was the situation until not so long ago.

Main algorithms: Shor and Grover (+ generalizations)

(also quantum chemistry . . . )

then came HHL and the prospect of quantum machine learning

The HHL algorithm, “exponential speedup”

the problem

Given an n × n matrix A and a vector b, solve the linear system:

Ax = b

I classically, complexity at least n2

I quantumly, O(log n)!!

How is that even possible??by cheating a little bit... Read the fine prints!

the problem

Ax = b

the problem

Ax = b

How is that even possible??

by cheating a little bit... Read the fine prints!

the problem

Ax = b

the problem

Ax = b

the algorithm (Harrow, Hassidim, Lloyd 2009)

Given access to a matrix A and a vector b such that

1. b can be efficiently loaded in a quantum memory

2. A has sparsity s per row and condition number κ

there exists a Q algo that

I outputs |x〉 =∑n

i=1 xi |i〉 where Ax = b

I runs in time polynomial in (s, κ) and logarithmic in the dimension

Requires A to be sparse and well-conditioned

Is this algorithm useful?? (does not return the complete answer)

the problem

Ax = b

the algorithm (Harrow, Hassidim, Lloyd 2009)

Given access to a matrix A and a vector b such that

1. b can be efficiently loaded in a quantum memory

2. A has sparsity s per row and condition number κ

there exists a Q algo that

I outputs |x〉 =∑n

i=1 xi |i〉 where Ax = b

I runs in time polynomial in (s, κ) and logarithmic in the dimension

Requires A to be sparse and well-conditioned

Is this algorithm useful?? (does not return the complete answer)

Quantum machine learningHHL initiated the field of quantum machine learning (subroutine for many algos).

Least Square Fitting [Wiebe, Braun, Lloyd 12]

I input: N labelled points (xi , yi )

I output: A fit function f (x , λ) =∑

j fj(x)λj that minimizes

err =N∑i=1

|f (xi , λ)− yi |2

Using HHL, the algorithm returns |λ〉.

Support Vector Machine [Lloyd, Mohseni, Rebentrost 13]I input: M labelled N-dimensional points

(xi, yi ), xi ∈ RN , yi ∈ {−1, 1}I output: A maximum margin hyperplane that separates the

classes

SVM can be recast as a system of linear equations=⇒ the quantum algorithm returns |w〉, the normal vector to the hyperplane

Application: classify data by estimating the inner product with w (l2-norm)

Quantum machine learningHHL initiated the field of quantum machine learning (subroutine for many algos).

Least Square Fitting [Wiebe, Braun, Lloyd 12]

I input: N labelled points (xi , yi )

I output: A fit function f (x , λ) =∑

j fj(x)λj that minimizes

err =N∑i=1

|f (xi , λ)− yi |2

Using HHL, the algorithm returns |λ〉.

Support Vector Machine [Lloyd, Mohseni, Rebentrost 13]I input: M labelled N-dimensional points

(xi, yi ), xi ∈ RN , yi ∈ {−1, 1}I output: A maximum margin hyperplane that separates the

classes

SVM can be recast as a system of linear equations=⇒ the quantum algorithm returns |w〉, the normal vector to the hyperplane

Application: classify data by estimating the inner product with w (l2-norm)

Quantum machine learning

Singular Value Estimation

I Sampling eigenvalues/vectors [Lloyd, Mohseni, Rebentrost 13]For a psd matrix with trace 1, quantum algorithm that efficiently samples aneigenvector with corresponding eigenvalue

I Singular Value Estimation [LMR 13, Prakash 15]Given a matrix and a singular vector, outputs an estimate of the singular value

Not clear whether these algorithms offer a true speedup

I to obtain a good complexity, the data should be “nice”, b should be efficientlyaccessible. Maybe there are efficient classical algorithms in such cases?

=⇒ HHL should be seen as a template for a generic algorithm: one needs to providea setup where

I b can be loaded efficiently in the quantum memory

I the data is sufficiently nice

I we’re not interested in x but in quantities efficiently computable from |x〉

Quantum machine learning

Singular Value Estimation

I Sampling eigenvalues/vectors [Lloyd, Mohseni, Rebentrost 13]For a psd matrix with trace 1, quantum algorithm that efficiently samples aneigenvector with corresponding eigenvalue

I Singular Value Estimation [LMR 13, Prakash 15]Given a matrix and a singular vector, outputs an estimate of the singular value

Not clear whether these algorithms offer a true speedup

I to obtain a good complexity, the data should be “nice”, b should be efficientlyaccessible. Maybe there are efficient classical algorithms in such cases?

=⇒ HHL should be seen as a template for a generic algorithm: one needs to providea setup where

I b can be loaded efficiently in the quantum memory

I the data is sufficiently nice

I we’re not interested in x but in quantities efficiently computable from |x〉

Quantum machine learning: speedup for a real-life problem

Quantum recommendation systems [Kerenidis, Prakash 16]

I input: a hidden preference matrix T with Ti,j ∈ {0, 1}, depending on whetherproduct j is “good” for user i

I output: a high value of row i (i.e. a recommendation for user i)

I complexity: polylog(mn) (= exponential speedup)

Conclusion

Summary

I there are quantum algorithms beyond Shor and Grover

I e.g., generalization of these techniques

I lower bounds in the query complexity model

“Hot topics”

I applications of HHL =⇒ quantum machine learningreal applications?

I cryptanalysis of quantum-resistant (“postquantum”) cryptography

I not mentioned but probably relevant in the next few years:algorithms for quantum chemistry

thanks!

Conclusion

Summary

I there are quantum algorithms beyond Shor and Grover

I e.g., generalization of these techniques

I lower bounds in the query complexity model

“Hot topics”

I applications of HHL =⇒ quantum machine learningreal applications?

I cryptanalysis of quantum-resistant (“postquantum”) cryptography

I not mentioned but probably relevant in the next few years:algorithms for quantum chemistry

thanks!

anthony leverrier (inria paris) 29 march 2018 orap 41st...

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