Sub-Universal Models of Quantum Computation in Continuous Variables

Transcription

1 Sub-Universal Models of Quantum Computation in Continuous Variables Giulia Ferrini Chalmers University of Technology Genova, 8th June 2018

2 OVERVIEW Sub-Universal Models of Quantum Computation Continuous Variables (CV) Sub-Universal Models of Quantum Computation in CV 2

3 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...

4 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... A step back, the new goal: to demonstrate quantum advantage for simple problems, e.g. sampling Classical Computer Calculator Universal Quantum Computer Sub-universal model of quantum computation B. Terhal, D. DiVicenzo, Quant. Inf. Comp. 4, 134 (2004); S. Aaronson, A.Arkhipov, Theory Comput. 9, 143 (2013) Nat. Phot. (Bristol, O'Brien) & (Sciarrino, Rome), Science (Wamsley, Oxford) & (White, Queensland),

5 BOSON SAMPLING m single photons passive linear optics evolution U single photon detection 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 S. Aaronson, A.Arkhipov, Theory Comput. 9, 143 (2013) 5

6 RELEVANT COMPLEXITY CLASSES: 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

7 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, coherent state Discrete basis : Finite-dimensional Hilbert space Continuous basis Infinite-dimensional Hilbert space

8 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, squeezed state Discrete basis : Finite-dimensional Hilbert space Continuous basis Infinite-dimensional Hilbert space

9 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, squeezed state Discrete basis : Finite-dimensional Hilbert space Continuous basis Infinite-dimensional Hilbert space CV Universal gate set :

10 AN INCREASING INTEREST TOWARDS CONTINUOUS VARIABLES = measurement of quadratures (e.g. )

11 GAUSSIAN VS NON-GAUSSIAN RESOURCES Wigner function : Quasi-probability distribution allowing to represent quantum states, evolutions and measurements in phase space Coherent state Squeezed state Photon subtracted squeezed state Gaussian resources : non-gaussian resources : Positive Wigner function Can have negative Wigner function Easy to produce experimentally Hard to produce experimentally

12 QUANTUM CIRCUITS Input state Evolution Measurement 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, (2002); A. Mari, J. Eisert, PRL 109, (2012) But non-gaussian resources are hard to achieve experimentally! Minimal extensions of Gaussian models that yield to non-trivial sampling Sub-Universal Quantum Circuits in CV!!!

13 CV SUB-UNIVERSAL MODELS The non-gaussian element can be either......the input state...the unitary evolution...the detection on-off on-off on-off CV Non-Gaussian input circuit CV Instantaneous Quantum Computing CV Boson Sampling on-off Chabaud et al, PRA (2017) Chakhmakhchyan, PRA (2017) Lund et al, PRA (2017) Douce et al, PRL (2017) Douce et al, in preparation Hamilton et al, PRL 119, (2017) Efficient sampling that is hard for classical computers (like in Boson Sampling)

14 CV SUB-UNIVERSAL MODELS The non-gaussian element can be either......the input state...the unitary evolution...the detection on-off on-off on-off CV Non-Gaussian input circuit CV Instantaneous Quantum Computing CV Boson Sampling on-off Chabaud et al, PRA (2017) Chakhmakhchyan, PRA (2017) Lund et al, PRA (2017) Douce et al, PRL (2017) Douce et al, in preparation Hamilton et al, PRL 119, (2017) Efficient sampling that is hard for classical computers (like in Boson Sampling)

15 CONTINUOUS VARIABLE SAMPLING, MAIN RESULT (1) Continuous-Variable Sampling (CVS) circuits: total number of modes m photon subtracted squeezed states passive linear optics evolution heterodyne detection = projection onto real orthogonal; symmetric real orthogonal; m even, Sampling from the output exact probability distribution is classically hard, or the polynomial hierarchy collapses (to the third level) U. Chabaud, T. Douce, D. Markham, P. van Loock, E. Kashefi and G. Ferrini, PRA (2017)

16 CONTINUOUS VARIABLE SAMPLING, MAIN RESULT (2) Limit of zero input squeezing: m single photons passive linear optics evolution heterodyne detection real orthogonal; symmetric real orthogonal; m even, Boson Sampling with heterodyne detection is classically hard, or the polynomial hierarchy collapses (to the third level) U. Chabaud, T. Douce, D. Markham, P. van Loock, E. Kashefi and G. Ferrini, PRA (2017)

17 SKETCH OF THE PROOF: STRUCTURE a) 1) Map input state: photon subtracted squeezed states = squeezed single photons heterodyne detection Boson Sampling obtained for s = 0 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).

18 CV SUB-UNIVERSAL MODELS The non-gaussian element can be either......the input state...the unitary evolution...the detection on-off on-off on-off CV Non-Gaussian input circuit CV Instantaneous Quantum Computing CV Boson Sampling on-off Chabaud et al, PRA (2017) Chakhmakhchyan, PRA (2017) Lund et al, PRA (2017) Douce et al, PRL (2017) Douce et al, in preparation Hamilton et al, PRL 119, (2017) Efficient sampling that is hard for classical computers (like in Boson Sampling)

19 INSTANTANEOUS QUANTUM COMPUTING (IQP) Input : X eigenstates Evolution : Diagonal in Z Measurement: X Gates commute, hence they can be performed simultaneously («Instantaneous») 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, (2016)

20 CV INSTANTANEOUS QUANTUM COMPUTING, MAIN RESULT 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 IQP is classically hard, or the polynomial hierarchy collapses (to the third level)

21 SKETCH OF THE PROOF: STRUCTURE b) Adding post-selection to the model makes it universal: Then, if it were possible to efficiently simulate CVrIQP on a classical computer, a postselected classical computer would be at least as powerful as PostBQP. This violates important conjectures in computer science! Hence it must not be possible to efficiently classically simulate IQP circuits We need to show : adding post-selection promotes CVrIQP to Universal QC

22 (1) FOURIER TRANSFORM Fourier gadget: infinitely p-squeezed state Post-selection allows to recover the Fourier transform Universal set of CV gates With finite resolution and finite squeezing, the gadget yields to first order in a noisy version of the Fourier transform :

23 (2) GKP ENCODING Ideal GKP states: Allow to encode qubits in CV: Finitely-squeezed GKP states: D. Gottesman, A. Kitaev, and J. Preskill, Phys. Rev. A 64, (2001) GKP encoding and ancillae make CV quantum computation Fault-Tolerant N. Menicucci, PRL 112, (2014)

24 SUMMARY OF CV IQP HARDNESS PROOF 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 (2017).

25 FURTHER STEP, SOON ON ARXIV: Probabilistic GKP state generation can be given in terms of elementary gates and subsumed in the definition of the circuit itself hardness of: T. Douce, D. Markham, E. Kashefi, P. van Loock, and G. Ferrini, to be submitted!

26 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! on-off on-off on-off on-off Next: approximate sampling and study the origin of quantum advantage (resource theory) Experimental implementation: optics (Treps, Paris) or microwaves (Delsing, Gothenburg) 26

27 Thank you for your attention! «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, (2017) «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, (2017) PhD and Postdoc positions open at Chalmers! see also WACQT website

28 Thank you for your attention! «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, (2017) «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, (2017) AQC and Quantum Annealing experts wanted for collaboration!

29 Thank you for your attention! Göran Johansson, Per Delsing, Jonas Bylander, Göran Wendin MPQ (Paris) Thomas Coudreau Pérola Milman LKB (Paris) Valentina Parigi Claude Fabre LIP6 (Paris) Damian Markham Elham Kashefi Nicolas Treps Tom Douce Francesco Arzani JGU (Mainz) Peter van Loock RMIT (Melbourne) Macquire University (Sydney) SUTD (Singapore) Nicolas Menicucci Gavin Brennen Tommaso Demarie

30 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 with Toda s theorem, this yields a collapse of the polynomial hierarchy! (Toda theorem: PH included in P #P ) We conclude that it must not be possible to sample efficiently from CVS circuits

31 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

32 Technical details of the proof: average case With two additional conjectures, we have an average case result (still exact): (1) Real version of Permanent of Gaussian Estimation (RGPE) conjecture in AA: Estimating Perm(X) for X random Gaussian (real) matrix is #P-hard (2) Real version of the permanent anti-concentration conjecture in AA: 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

33 Technical details of the proof: With two additional conjectures, we have an average case result (still exact): Real version of Permanent of Gaussian Estimation (RGPE) conjecture in AA Real version of the permanent anticoncentration conjecture in AA

34 Heterodyne detection

35 and PostBQP = PP (Aaronson)