Movitation Quantum simulation Speedups Trustworthy simulations

Quantum simulators, boson sampling and the quest for superpolynomial speedups

Jens Eisert, Freie Universität Berlin With Dom Hangleiter, Martin Schwarz, Robert Raussendorf, Juan Bermejo-Vega ICNFP 2017, Crete, July 2017 Movitation Quantum simulation Speedups Trustworthy simulations

Quantum simulators, boson sampling and the quest for superpolynomial speedups

! Recent years have seen rapid development of quantum devices Movitation Quantum simulation Speedups Trustworthy simulations

! Large-scale continuous variable systems Quantum simulators, boson sampling and the quest for superpolynomial speedups

Roslund, de Arujo, Jiang, Fabre, Treps, Nature Phot 8, 109 (2014)

Menicucci, Flammia, Pfister, Phys Rev Lett 13, 130501 (2008)

Yokohama, Ukai, Armstrong, Sornphiphatphong, Kaji, Suziki, Yoshikawa, Yonezawa, Menicucci, Furusawa, Nature Phot 7, 982 (2013) Movitation Quantum simulation Speedups Trustworthy simulations

! Linear optical devices and integrated optics Quantum simulators, boson sampling and the quest for superpolynomial speedups

Carolan, Harrold, Sparrow, Martín-López, Russell, Silverstone, Shadbolt, Matsuda, Oguma, Itoh, Marshall, Thompson, Matthews, Hashimoto, O’Brien, Laing, Science 349, 711 (2015)

Langford, Kundys, Gates, Smith, Smith, Walmsley, Nature Phot 8, 770 (2014) Movitation Quantum simulation Speedups Trustworthy simulations

! Trapped ions Quantum simulators, boson sampling and the quest for superpolynomial speedups

Schindler, Mueller, Nigg, Barreiro, Martinez, Hennrich, Monz, Diehl, Zoller, Blatt, Nature Phys 9, 361 (2013) Movitation Quantum simulation Speedups Trustworthy simulations

! Cold atomic quantum simulators Quantum simulators, boson sampling and the quest for superpolynomial speedups

Trotzky, Chen, Flesch, McCulloch, Schollwoeck, Eisert, Kaufman, Tai, Lukin, Rispoli, Schittko, Preiss, Greiner, Bloch, Nature Phys 8, 325 (2012) Science 353, 794 (2016) Movitation Quantum simulation Speedups Trustworthy simulations

! Industrial efforts, mostly on superconducting devices Quantum simulators, boson sampling and the quest for superpolynomial speedups Intel Google

D-wave IBM

! When can we expect quantum devices to computationally outperform classical computers? Movitation Quantum simulation Speedups Trustworthy simulations

Quantum simulators, boson sampling and the quest for superpolynomial speedups

! When can we expect quantum devices to computationally outperform classical computers? Movitation Quantum simulation Speedups Trustworthy simulations

Quantum simulators, boson sampling and the quest for superpolynomial speedups

! (Analog) quantum simulators

! Quantum simulators are promised to solve problems ! How! Howdo we can know they we outperform have done classical the right computers? thing? inaccessible to classical computers ! Not BQP-complete, what is computational power?

! Error correction/fault tolerance unavailable Movitation Quantum simulation Speedups Trustworthy simulations

Quantum simulators, boson sampling and the quest for superpolynomial speedups Movitation Quantum simulation Speedups Trustworthy simulations

Cold atomic quantum simulators Movitation Quantum simulation Speedups Trustworthy simulations Analog quantum simulators

! Cold atoms in optical lattices allow to probe condensed matter systems

4 5 ! Probe local Hamiltonians, 10 10 particles ⇠ ! Ground state problems itH itH ! ”Quenches” ⇢ ( t )= e ⇢ e (time evolution)

! Slow evolutions, driven and open settings

Bloch, Dalibard, Nascimbene, Nature Phys 8, 267 (2012) Movitation Quantum simulation Speedups Trustworthy simulations Analog quantum simulators A

! Equilibration and thermalisation of atoms in optical super-lattices (MPQ)

! Imbalance as function of time for (0) = 0 , 1 ,..., 0 , 1 under | i | i Bose-Hubbard Hamiltonian Movitation Quantum simulation Speedups Trustworthy simulations Analog quantum simulators A

! Equilibration and thermalisation of atoms in optical super-lattices (MPQ)

! Imbalance as function of time for (0) = 0 , 1 ,..., 0 , 1 under | i | i Bose-Hubbard Hamiltonian odd n

Best available classical matrix-product state simulation, bond dimension 5000 Trotzky, Chen, Flesch, McCulloch, Schollwoeck, Eisert, Bloch, Nature Phys 8, 325 (2012) Movitation Quantum simulation Speedups Trustworthy simulations Analog quantum simulators

Equilibration Kibble-Zurek mechanism Many-body localization

Short times can be 1D systems can be efficiently simulated, Schreiber, Hodgman, Bordia, Lüschen, Fischer, Vosk, efficiently simulated 2D systems not Altman, Schneider, Bloch, Science 349, 842 (2015) Trotzky, Chen, Flesch, McCulloch, Schollwoeck, Braun, Friesdorf, Hodgman, Schreiber, Eisert, Bloch, Nature Phys 8, 325 (2012) Ronzheimer, Riera, del Rey, Bloch, Eisert, Schneider, Proc Natl Acad Sci 112 3641 (2015) Movitation Quantum simulation Speedups Trustworthy simulations Analog quantum simulators

Equilibration Kibble-Zurek mechanism Many-body localization

Short times can be 1D systems can be efficiently simulated, Schreiber, Hodgman, Bordia, Lüschen, Fischer, Vosk, efficiently simulated 2D systems not Altman, Schneider, Bloch, Science 349, 842 (2015) Trotzky, Chen, Flesch, McCulloch, Schollwoeck, Braun, Friesdorf, Hodgman, Schreiber, Eisert, Bloch,! Nature Phys 8, 325 (2012) Ronzheimer, Riera, del Rey, Bloch, Eisert, Dynamical quantumSchneider, simulators Proc Natl Acad Sci 112 3641 (2015) Existing quantum simulators outperform state-of-the-art simulations on classical supercomputers

! Cleverer simulation method? Movitation Quantum simulation Speedups Trustworthy simulations Quest for intermediate problems

BQP

BPP

! Intermediate problems To be sure, we should prove the hardness of the task: Identify a (feasible) task that lies outside of BPP, but is not BQP hard Movitation Quantum simulation Speedups Trustworthy simulations

Super-polynomial computational speedups Movitation Quantum simulation Speedups Trustworthy simulations Super-polynomial speedups

! Aim: Find some problem with strong evidence for super-polynomial speedup

! Dubbed “quantum computational supremacy” Movitation Quantum simulation Speedups Trustworthy simulations Super-polynomial speedups

! Aim: Find some problem with strong evidence for super-polynomial speedup

! Boson sampling

1 1 0 0 0 ! n bosons in m optical modes

! Haar random mode transformation b Ub U U(m) T 7! 2 b =(b1,...,bm)

1 0 0 1 0 ! Photon detection Movitation Quantum simulation Speedups Trustworthy simulations Super-polynomial speedups

! Aim: Find some problem with strong evidence for super-polynomial speedup

! Boson sampling

1 1 0 0 0 ! n bosons in m optical modes

! Haar random mode transformation b Ub T 7! b =(b1,...,bm)

1 0 0 1 0 ! Photon detection Movitation Quantum simulation Speedups Trustworthy simulations Super-polynomial speedups

! Aim: Find some problem with strong evidence for super-polynomial speedup

! Boson sampling

1 1 0 0 0 ! n bosons in m optical modes

! Haar random mode transformation b Ub U U(m) T 7! 2 b =(b1,...,bm)

1 0 0 1 0 ! Photon detection

! Theorem: Sampling from a distribution close in l 1 norm to boson sampling distribution is "computationally hard" with high probability if the unitary U is chosen from Haar 5 measure and m increases sufficiently fast with n ( m ⌦ ( n ) ) 2 Aaronson, Arkhipov, Th Comp 9, 143 (2013) Movitation Quantum simulation Speedups Trustworthy simulations Super-polynomial speedups

! Aim: Find some problem with strong evidence for super-polynomial speedup

! Boson sampling

! 1B eautiful 1 experiments0 0 0 ! n bosons in m optical modes

! Haar random mode transformation b Ub U U(m) T 7! 2 b =(b1,...,bm)

1 0 0 1 0 ! Photon detection

! Theorem: Sampling from a distribution close in l 1 norm to boson sampling distribution is "computationally hard" with high probability if the unitary U is chosen from Haar 5 measure and m increases sufficiently fast Broomewith et n al, Science( m 339, ⌦ 794 ( (2012)n ) ) Spring et al, Science 339,2 798 (2012) Aaronson, Arkhipov, Th Comp 9, 143 (2013) Tillmann et al, Nature Photonics 7, 540 (2013) Crespi et al, Nature Photonics 7, 545 (2013) Bloch, Nature Phys 8, 325 (2012) Movitation Quantum simulation Speedups Trustworthy simulations Super-polynomial speedups

! Aim: Find some problem with strong evidence for super-polynomial speedup

! Boson sampling

1 1 0 0 0 ! n bosons in m optical modes

! Haar random mode transformation b Ub U U(m) T 7! 2 b =(b1,...,bm)

1 0 0 1 0 ! Photon detection Movitation Quantum simulation Speedups Trustworthy simulations Super-polynomial speedups! Certification using continuous-variables:

! Target fidelity F ( ⇢ t , ⇢ p ) F T with anticipated state ⇢t ! Aim: Find some problem with strong evidence for super-polynomial speedup

! Boson sampling

1 1 0 0 0 ? 1 FT %p ! n bosons in m optical modes

1 F %t ! Haar random mode transformation b Ub U U(m) T 7! 2 b =(b1,...,bm)

! Can perform robust fidelity certfication, with poly(m, 1/) O log(1/(1 ↵)) ✓ ◆ many preparations and homodyne measurements, with success probability

Aolita, Gogolin, Kliesch, Eisert, Nature Communications 6, 8498 (2015)

! Great tool, not quite good enough Movitation Quantum simulation Speedups Trustworthy simulations Super-polynomial speedups

! Aim: Find some problem with strong evidence for super-polynomial speedup

! Boson sampling

1 1 0 0 0 ! n bosons in m optical modes

! Haar random mode transformation U U(m) T 2 b =(b1,...,bm)

1 0 0 1 0 Movitation Quantum simulation Speedups Trustworthy simulations Super-polynomial speedups

! Aim: Find some problem with strong evidence for super-polynomial speedup

! Boson sampling

1 0 0 1 0

5.1 ! Let m n and let U U ( m ) be Haar random. Then with probability at least 2 1 , for every T and every ✏ > 0 , there exists a circuit of size T poly(n, 1/✏, 1/) that samples a distribution that is ✏ -indistinguishable from the boson sampling distribution by circuits of size at most T

Gogolin, Kliesch, Aolita, Eisert, arXiv:1306.3995 Trevisan, Tulsiani, Vadhan, Proc IEEE Conf Comp Complex, 126 (2009) Aaronson, Arkhipov, arXiv: 1309.7460 Brandao, private communication Movitation Quantum simulation Speedups Trustworthy simulations Super-polynomial speedups

! Aim: Find some problem with strong evidence for super-polynomial speedup

! Boson sampling

! Common predjudice: In order to be able to verify a quantum simulation, one needs to be able to efficiently simulate it Movitation Quantum simulation Speedups Trustworthy simulations Super-polynomial speedups

! Aim: Find some problem with strong evidence for super-polynomial speedup

! Boson sampling Aaronson, Arkhipov, Th Comp 9, 143 (2013)

! IQP circuits Bremner, Montanaro, Shepherd: Phys Rev Lett 117, 080501 (2016)

! Random universal circuits Boixo et al. (Google), arXiv:1608.00263 (2016)

! Ising-type interactions Gao, Wang, Duan, Phys Rev Lett, 118, 040502 (2017)

+ : Provable classical hardness with l 1 -errors (under reasonable assumptions) - : Very hard to implement: Either

! Hard to scale up with present technology

! Arbitrary gate choices necessary

! High (56) periodicity of Hamiltonian Movitation Quantum simulation Speedups Trustworthy simulations

Trustworthy dynamical simulators Movitation Quantum simulation Speedups Trustworthy simulations Super-polynomial speedups

! Aim: Find some problem with strong evidence for super-polynomial speedup

Here: Best of both worlds, bring speedups closer to experiment

! Hamiltonian quench architecture

! Low periodicity of the interaction Hamiltonian (NN or NNN)

! Hardness proofs with l 1 -norm error (under some assumptions)

Bermejo-Vega, Hangleiter, Schwarz, Raussendorf, Eisert, arXiv:1703.00466 Movitation Quantum simulation Speedups Trustworthy simulations Simple Ising models

! Aim: Find some problem with strong evidence for super-polynomial speedup

Here: Best of both worlds, bring speedups closer to experiment

Random Periodic Translationally invariant

Bermejo-Vega, Hangleiter, Schwarz, Raussendorf, Eisert, arXiv:1703.00466 Movitation Quantum simulation Speedups Trustworthy simulations Simple Ising models

! Prepare N qubits in n m square lattice in product ⇥ = n,m ( 0 + eii,j 1 ) | i ⌦i,j=1 | i | i with i,j 0 , ⇡ / 4 , , i.i.d. randomly 2 { } { }

Bermejo-Vega, Hangleiter, Schwarz, Raussendorf, Eisert, arXiv:1703.00466 Movitation Quantum simulation Speedups Trustworthy simulations Simple Ising models

! Prepare N qubits in n m square lattice in product ⇥ = n,m ( 0 + eii,j 1 ) | i ⌦i,j=1 | i | i with i,j 0 , ⇡ / 4 , , i.i.d. randomly 2 { } { } ! Reminscient of disordered optical lattices ⇡ iH ! Quench to H = Z i Z j + Z i and evolve under U = e 4 (i,j) E i V X2 X2

Schreiber, Hodgman, Bordia, Lüschen, Fischer, Vosk, Altman, Schneider, Bloch, Science 349, 842 (2015)

Bermejo-Vega, Hangleiter, Schwarz, Raussendorf, Eisert, arXiv:1703.00466 Movitation Quantum simulation Speedups Trustworthy simulations Simple Ising models

! Prepare N qubits in n m square lattice in product ⇥ = n,m ( 0 + eii,j 1 ) | i ⌦i,j=1 | i | i with i,j 0 , ⇡ / 4 , , i.i.d. randomly 2 { } { } ⇡ iH ! Quench to H = Z i Z j + Z i and evolve under U = e 4 (i,j) E i V X2 X2 ! Controlled coherent collisions long realized

Mandel, Greiner, Widera, Rom, Hänsch, Bloch, Nature, 425, 937 (2003)

Bermejo-Vega, Hangleiter, Schwarz, Raussendorf, Eisert, arXiv:1703.00466 Movitation Quantum simulation Speedups Trustworthy simulations Simple Ising models

! Prepare N qubits in n m square lattice in product ⇥ = n,m ( 0 + eii,j 1 ) | i ⌦i,j=1 | i | i with i,j 0 , ⇡ / 4 , , i.i.d. randomly 2 { } { } ⇡ iH ! Quench to H = Z i Z j + Z i and evolve under U = e 4 (i,j) E i V X2 X2

! Measure all qubits in X -basis

! Single-site addressing (most challenging)

Bakr, Gillen, Peng, Foelling, Greiner, Nature 462, 74–77 (2009) Weitenberg, Endres, Sherson, Cheneau, Schauß, Fukuhara, Bloch, Kuhr, Nature 471, 319 (2011)

Bermejo-Vega, Hangleiter, Schwarz, Raussendorf, Eisert, arXiv:1703.00466 Movitation Quantum simulation Speedups Trustworthy simulations Main result

! Theorem (Hardness of classical sampling): Assuming three highly plausible complexity-theoretic conjectures are true a classical computer cannot efficiently sample from the outcome distribution of our scheme up to constant error in l 1 distance

! Lemma (Approximation of outcome distribution in worst case): It is #P-hard to approximate the outcome distribution of an all- X measurement U to constant relative error | i

Bermejo-Vega, Hangleiter, Schwarz, Raussendorf, Eisert, arXiv:1703.00466 Movitation Quantum simulation Speedups Trustworthy simulations Equivalence to MBQC

! A hint at the argument: Universal architecture for postBQP

Logical Physical

Bermejo-Vega, Hangleiter, Schwarz, Raussendorf, Eisert, arXiv:1703.00466 Building upon Mantri, Demarie, Fitzsimons, arXiv:1607.00758 Raussendorf, Phys Rev A 72, 052301 (2005) Movitation Quantum simulation Speedups Trustworthy simulations Quantum simulators

! Ingredients ! A classical algorithm that samples from the output distribution of A U 0 such that p U pU l " k 0 k 1  ! A binary string x

Additive error✏ U U ! Approximate sampling pU pU 0 l1 < ✏ 0 k k

pU 0 (x) ! Stockmeyer error sU 0 (x) pU 0 (x) A | |  poly(n)

x Stockmeyer 1 error Multiplicative / poly(

n pU (x) ✏ ) s (x) p (x) + (1 + o(1)) | U 0 U |  poly 2n

sU 0 (x)

Bermejo-Vega, Hangleiter, Schwarz, Raussendorf, Eisert, arXiv:1703.00466 Movitation Quantum simulation Speedups Trustworthy simulations Complexity theoretic conjectures

! Polynomial hierarchy: The polynomial hierarchy is infinite (i.p. P = NP ) 6

! Average case complexity: For a constant fraction of the instances it is as hard to sample from the outcomes! Strong numerical of measurements evidence as in worst case

! Anti-concentration: The output probabilities of U anticoncentrate, i.e., | i 1 1 prob a U 2 U |h | | i| 2N e ✓ ◆

! Rigorous proofs for small variants

Hangleiter, Bermejo-Vega, Schwarz, Eisert, arXiv:1706.03786

Bermejo-Vega, Hangleiter, Schwarz, Raussendorf, Eisert, arXiv:1703.00466 Movitation Quantum simulation Speedups Trustworthy simulations Complexity theoretic conjectures

! Polynomial hierarchy: The polynomial hierarchy is infinite (i.p. P = NP ) ! The scheme provides a scalable quantum simulation producing 6outputs that are hard to sample from classically*

* In preparation: A continuous variable analog based on continuous-variable cluster states in frequency combs

Bermejo-Vega, Hangleiter, Eisert, in preparation Movitation Quantum simulation Speedups Trustworthy simulations Certification of trustworthy simulators

! One can with ✓ ( N ) many measurements detect closeness in l 1 -norm!

Bermejo-Vega, Hangleiter, Schwarz, Raussendorf, Eisert, arXiv:1703.00466 Hangleiter, M. Kliesch, M. Schwarz, J. Eisert, Quantum Sci Technol 2, 015004 (2017) Cramer et al, Nature Comm 1, 149 (2010) Movitation Quantum simulation Speedups Trustworthy simulations Summary

! TrustworthyCommon predjudice: quantum simulatorsIn order to can be ablebe verified, to verify even a quantum if the classical simulation, simulation one needs is beyond to be able reach to efficiently simulate it Movitation Quantum simulation Speedups Trustworthy simulations Summary

! Hope for feasible quantum devices with superpolynomial speedup Movitation Quantum simulation Speedups Trustworthy simulations Summary

! Hope for feasible quantum devices with superpolynomial speedup

! Is not fault tolerant (feature, not bug), but can be certified Movitation Quantum simulation Speedups Trustworthy simulations Summary

! Hope for feasible quantum devices with superpolynomial speedup

! Is not fault tolerant (feature, not bug), but can be certified

! One can efficiently assess correctness - even if simulators exhibit “quantum computational supremacy”

Thanks for your attention! http://www.physik.fu-berlin.de/en/einrichtungen/ag/ag-eisert