Sub-Universal Models of Quantum Computation in Continuous Variables

Giulia Ferrini

Chalmers University of Technology

Genova, 8th June 2018

OVERVIEW

➤ Sub-Universal Models of Quantum Computation

➤ Continuous Variables (CV)

➤ Sub-Universal Models of Quantum Computation in CV

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QUANTUM ADVANTAGE➤ Quantum computers expected to solve efficiently certain problems that are hard to solve

on a classical computer (e.g. : factorization, Shor alhorithm) Efficient = polynomial time Hard = exponential time

➤ Millions of qubits required for factoring, we know how to build a few tenth nowadays...

QUANTUM ADVANTAGE➤ Quantum computers expected to solve efficiently certain problems that are hard to solve

on a classical computer (e.g. : factorization, Shor alhorithm) Efficient = polynomial time Hard = exponential time

➤ Millions of qubits required for factoring, we know how to build a few tenth nowadays...

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ClassicalComputer Calculator

Sub-universal model of quantum computation

Nat. Phot. (Bristol, O'Brien) & (Sciarrino, Rome), Science (Wamsley, Oxford) & (White, Queensland), 2013

Universal Quantum Computer

B. Terhal, D. DiVicenzo, Quant. Inf. Comp. 4, 134 (2004); S. Aaronson, A.Arkhipov, Theory Comput. 9, 143 (2013)

A step back, the new goal: to demonstrate quantum advantage for simple problems, e.g. sampling

BOSON SAMPLING

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S. Aaronson, A.Arkhipov, Theory Comput. 9, 143 (2013)

passive linear optics evolution

single photon detectionsingle photonsm

U

Sampling from this output probability distribution is classically hard, or the polynomial hierarchy collapses (to the third level)

Hardness proof a): based on the fact that approximating is #P-hard

Hardness proof b): adding post-selection makes the circuit universal

➤ NP = set of decision problems for which the solutions can be verified in polynomial time

➤ ♯P = class of function problems that counts the number of solutions of NP problems

➤ A problem is ♯P-hard if its solution allows solving all other problems in ♯P

➤ P = set of decision problems solvable in polynomial time by a Turing machine;

➤ BPP = set of problems solvable efficiently (poly time) by a probabilistic Turing machine

➤ Polinomial Hierarchy: = P

➤ BQP = set of problems solvable efficiently (poly number of gates) by a quantum computer

RELEVANT COMPLEXITY CLASSES:

DV : information encoded in qubits CV : information encoded in continuous states e.g. eigenstates of e.m. field quadratures ,

Discrete basis :Finite-dimensional Hilbert space

Continuous basisInfinite-dimensional Hilbert space

coherent state

DV VS CV ENCODING OF QUANTUM INFORMATION

DV : information encoded in qubits CV : information encoded in continuous states e.g. eigenstates of e.m. field quadratures ,

Discrete basis :Finite-dimensional Hilbert space

Continuous basisInfinite-dimensional Hilbert space

squeezed state

DV VS CV ENCODING OF QUANTUM INFORMATION

DV : information encoded in qubits CV : information encoded in continuous states e.g. eigenstates of e.m. field quadratures ,

Discrete basis :Finite-dimensional Hilbert space

Continuous basisInfinite-dimensional Hilbert space

CV Universal gate set :

squeezed state

DV VS CV ENCODING OF QUANTUM INFORMATION

= measurement of quadratures (e.g. )

AN INCREASING INTEREST TOWARDS CONTINUOUS VARIABLES

● Positive Wigner function ● Easy to produce experimentally

● Can have negative Wigner function ● Hard to produce experimentally

Coherent state Squeezed state Photon subtractedsqueezed state

Wigner function : Quasi-probability distribution allowing to represent quantum states, evolutions and measurements in phase space

Gaussian resources : non-Gaussian resources :

GAUSSIAN VS NON-GAUSSIAN RESOURCES

Theorem : if all the elements of a quantum circuits have positive W, then the output can be efficiently simulated by a classical computer

S. D. Bartlett et al, PRL 88, 097904 (2002); A. Mari, J. Eisert, PRL 109, 230503 (2012)

But non-Gaussian resources are hard to achieve experimentally!

Sub-Universal Quantum Circuits in CV !!!

Minimal extensions of Gaussian models that yield to non-trivial sampling

Input state Evolution Measurement

QUANTUM CIRCUITS

on-off

Douce et al, PRL 118 070503 (2017)

The non-Gaussian element can be either...

CV Instantaneous Quantum Computing

CV Boson SamplingCV Non-Gaussian input circuit

...the unitary evolution...the input state ...the detection

Hamilton et al, PRL 119, 170501 (2017)

on-offon-off

on-off

Chabaud et al, PRA 062307 (2017)

Efficient sampling that is hard for classical computers (like in Boson Sampling)

Chakhmakhchyan, PRA 032326 (2017)

Lund et al, PRA 022301 (2017)

Douce et al, in preparation

CV SUB-UNIVERSAL MODELS

on-off

Douce et al, PRL 118 070503 (2017)

The non-Gaussian element can be either...

CV Instantaneous Quantum Computing

CV Boson SamplingCV Non-Gaussian input circuit

...the unitary evolution...the input state ...the detection

Hamilton et al, PRL 119, 170501 (2017)

on-offon-off

on-off

Chabaud et al, PRA 062307 (2017)

Efficient sampling that is hard for classical computers (like in Boson Sampling)

Chakhmakhchyan, PRA 032326 (2017)

Lund et al, PRA 022301 (2017)

Douce et al, in preparation

CV SUB-UNIVERSAL MODELS

U. Chabaud, T. Douce, D. Markham, P. van Loock, E. Kashefi and G. Ferrini, PRA 062307 (2017)

Continuous-Variable Sampling (CVS) circuits:

photon subtracted squeezed states

Sampling from the output exact probability distribution is classically hard, or the polynomial hierarchy collapses (to the third level)

passive linear optics evolution

heterodyne detection

real orthogonal; symmetric real orthogonal;

= projection onto

m even,

⎨m

total number of modes

CONTINUOUS VARIABLE SAMPLING, MAIN RESULT (1)

U. Chabaud, T. Douce, D. Markham, P. van Loock, E. Kashefi and G. Ferrini, PRA 062307 (2017)

Limit of zero input squeezing:

Boson Sampling with heterodyne detection is classically hard, or the polynomial hierarchy collapses (to the third level)

passive linear optics evolution

heterodyne detectionsingle photons

real orthogonal; symmetric real orthogonal; m even,

m

CONTINUOUS VARIABLE SAMPLING, MAIN RESULT (2)

SKETCH OF THE PROOF: STRUCTURE a)

1) Map input state: photon subtracted squeezed states = squeezed single photons

Boson Sampling obtained for s = 0

heterodyne detection

2) Map to Time-Reversed CVS circuit using symmetry of Born rule

real square matrix

3) multiplicative approximation of is #P-hard Aaronson & Arkhipov, Theor. Comput. 9, 143 (2013).

on-off

Douce et al, PRL 118 070503 (2017)

The non-Gaussian element can be either...

CV Instantaneous Quantum Computing

CV Boson SamplingCV Non-Gaussian input circuit

...the unitary evolution...the input state ...the detection

Hamilton et al, PRL 119, 170501 (2017)

on-offon-off

on-off

Chabaud et al, PRA 062307 (2017)

Efficient sampling that is hard for classical computers (like in Boson Sampling)

Chakhmakhchyan, PRA 032326 (2017)

Lund et al, PRA 022301 (2017)

Douce et al, in preparation

CV SUB-UNIVERSAL MODELS

● Input : X eigenstates

● Evolution : Diagonal in Z

The probability distribution of the measurement outcomes is hard to sample

M. J. Bremner, R. Josza, and D. Shepherd, Proc. R. Soc. A 459, 459 (2010).M. J. Bremner, A. Montanaro, and D. J. Shepherd, Phys. Rev. Lett. 117, 080501 (2016)

Gates commute, hence they can be performed simultaneously (« Instantaneous »)

● Measurement: X

INSTANTANEOUS QUANTUM COMPUTING (IQP)

● Input : p-squeezed states

● Evolution : Diagonal in q

● Measurement: p homodyne detection (finite resolution)

For instance, one could take a uniform combination of gates from the set

CV INSTANTANEOUS QUANTUM COMPUTING, MAIN RESULT

CV IQP is classically hard, or the polynomial hierarchy collapses (to the third level)

Then, if it were possible to efficiently simulate CVrIQP on a classical computer, a post-selected classical computer would be at least as powerful as PostBQP.

This violates important conjectures in computer science !

We need to show : adding post-selection promotes CVrIQP to Universal QC

Hence it must not be possible to efficiently classically simulate IQP circuits

Adding post-selection to the model makes it universal:

SKETCH OF THE PROOF: STRUCTURE b)

With finite resolution and finite squeezing, the gadget yields to first order in a noisy version of the Fourier transform :

Post-selection allows to recover the Fourier transform

infinitely p-squeezed state

→ Universal set of CV gates

Fourier gadget:

(1) FOURIER TRANSFORM

D. Gottesman, A. Kitaev, and J. Preskill, Phys. Rev. A 64, 012310 (2001)

Allow to encode qubits in CV:

N. Menicucci, PRL 112, 120504 (2014)

GKP encoding and ancillae make CV quantum computation Fault-Tolerant

Finitely-squeezed GKP states:

Ideal GKP states:

(2) GKP ENCODING

The discrete set of gates in our model, plus the Fourier (Hadamard) gate obtained by post-selection yield a universal gate set within GKP encoding

For each computation in PostBQP it exists a circuit in our circuit family that, augmented with post-selection, yields the same computation

Continuous Variable Instantaneous Quantum Computing is hard to sample

T. Douce, D. Markham, E. Kashefi, T. Coudreau, P. Milman, P. van Loock, and G. Ferrini, Phys. Rev. Lett. 118 070503 (2017).

SUMMARY OF CV IQP HARDNESS PROOF

FURTHER STEP, SOON ON ARXIV:

T. Douce, D. Markham, E. Kashefi, P. van Loock, and G. Ferrini, to be submitted!

Probabilistic GKP state generation can be given in terms of elementary gates and subsumed in the definition of the circuit itself

hardness of:

CONCLUSIONS AND PERSPECTIVES➤ Proven hardness of two families of quantum circuits: CVS and CV IQP

➤ CVS, limit of zero squeezing: Boson Sampling with heterodyne detection is classically hard

➤ CV are promising for investigating quantum advantage!

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on-offon-offon-off

on-off

➤ Next: approximate sampling and study the origin of quantum advantage (resource theory)

➤ Experimental implementation: optics (Treps, Paris) or microwaves (Delsing, Gothenburg)

« Continuous-variable sampling from photon-added or photon-subtracted squeezed states » U. Chabaud, T. Douce, D. Markham, P. van Loock, E. Kashefi and G. Ferrini, Phys. Rev. A 96, 062307 (2017)

Thank you for your attention !

« Continuous-Variable Instantaneous Quantum Computing is hard to sample » T. Douce, D. Markham, E. Kashefi, E. Diamanti, T. Coudreau, P. Milman, P. van Loock and G. Ferrini, Phys. Rev. Lett. 118, 070503 (2017) 

PhD and Post-doc positions

open at Chalmers!

see also WACQT website

« Continuous-variable sampling from photon-added or photon-subtracted squeezed states » U. Chabaud, T. Douce, D. Markham, P. van Loock, E. Kashefi and G. Ferrini, Phys. Rev. A 96, 062307 (2017)

Thank you for your attention !

« Continuous-Variable Instantaneous Quantum Computing is hard to sample » T. Douce, D. Markham, E. Kashefi, E. Diamanti, T. Coudreau, P. Milman, P. van Loock and G. Ferrini, Phys. Rev. Lett. 118, 070503 (2017) 

AQC and Quantum Annealing

experts wanted for

collaboration!

MPQ (Paris)

● Thomas Coudreau ● Pérola Milman

LKB (Paris)

● Valentina Parigi ● Claude Fabre ● Nicolas Treps ● Francesco Arzani

● Damian Markham ● Elham Kashefi ● Tom Douce

COMB

JGU (Mainz)

● Peter van Loock

SUTD (Singapore)

● Tommaso Demarie

Macquire University (Sydney)RMIT (Melbourne)● Nicolas Menicucci ● Gavin Brennen

Thank you for your attention !

LIP6 (Paris)

Göran Johansson, Per Delsing, Jonas Bylander, Göran Wendin…

Sketch of the proof 4/4

4) Stockmeyer counting algorithm allows to approximate the value in zero from samples of CVS circuits in the third level of the polynomial hierarchy

Therefore, if efficient sampling from CVS circuits were possible, one could solve a #P-hard problem in the third level of the polynomial hierarchy

We conclude that it must not be possible to sample efficiently from CVS circuits

with Toda’s theorem, this yields a collapse of the polynomial hierarchy!(Toda theorem: PH included in P#P)

Technical details: discretization of the probability

• We actually sample from:

discrete boxes with resolution

The value for the box at zero relates to via a Taylor expansion:

Stockmeyer allows to approximate from which I can approximate

Technical details of the proof: average case

• With two additional conjectures, we have an average case result (still exact):

(2) Real version of the permanent anti-concentration conjecture in AA:

(1) Real version of Permanent of Gaussian Estimation (RGPE) conjecture in AA:Estimating Perm(X) for X random Gaussian (real) matrix is #P-hard

The probability of the value Perm(X) for X random Gaussian matrix is bounded

Picking X randomly, with high probability the CVS is hard

heterodyne detection

Technical details of the proof:

• With two additional conjectures, we have an average case result (still exact):

Real version of the permanent anti-concentration conjecture in AA

Real version of Permanent of Gaussian Estimation (RGPE) conjecture in AA

Heterodyne detection

and PostBQP = PP (Aaronson)