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Quantum Simulators, Boson Sampling and the Quest for Superpolynomial Speedups 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
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