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Boson sampling, where quantum optics meet computer science Raul Garcia-Patron QuIC Université Libre de Bruxelles (Brussels) DIQIP & Qalgo Joint meeting Brussels, May 13, 2014 Motivation New look at “old stuff”: regime of large-size (complex) liner-optics Test-bed for protocols of certification of large-size quantum systems Better understanding of quantum optics Evidence for the separation between quantum and classical computation Inspire new algorithms for permanent? Boson-sampling Boson Sampling “Gedankenexperiment” RELATIVELY “easy” to implement Single-photons detectors Single-photons input Linear optics interferometer ● Beamsplitter ● Phase-shift Statistic are given by PRMANENTS: HARD to calculate ( #P-complete) Efficient weak simulation (sampling) on a classical computer HARD Linear optics representations 1 PHOTON=1 QUDIT Dimension D = # Modes Path encoding |x ⟩ =|0,0,1,0,0,. .. , 0 ⟩ Interferometer is a unitary Preserving the # photons | y ⟩ =U D|x ⟩ Creation operator notation ● |0,0,1,0,0,. .. ,0 ⟩ =a↑i |0 ⟩ Evolution through linear interferometer ̄b =Λ ā b̂i =∑ j Λi , j â j Linear optics and Symmetric subspace N PHOTON=N QUDIT Dimension D = # Modes √ 1 |1,1,2 ⟩ |ψ ⟩ 0 =|2,1,0 ⟩ = ∑ 3−tuples 3 Mode 1 Photon 1 Linear optics and Symmetric subspace N PHOTON=N QUDIT Dimension D = # Modes √ 1 |ψ ⟩ 0 =|t 1, t 2, t 3,. .. , t D ⟩= ∑ BT X BT = N! t 1 !t 2 ! ... t D ! T ∈B T |x1 , x2 ,... , x N ⟩ Basis of the symmetric subspace of dimension ( ) (N + D−1)! Λ DN = N + D−1 = N N !( D−1)! Evolution by interferometer (non-interacting) Interferometer is a unitary Preserving the # photons |ψ ⟩ f =U | y ⟩ =U D|x ⟩ ∘N D | ψ⟩f Boson sampling permanent rule INPUT: N single-photons distributed over D modes |T ⟩=|1,1,1,0,. .. ,0 ⟩ = √ 1 |π(1), π (2) ,... , π (N ) ⟩ ∑ π∈S N! P (S∥T )=|⟨ s 1 , s2 , ... , s M|U D ∘N D 2 |t 1 ,t 2 ,... ,t M ⟩| Modes with detected photons 2 P(S∥T )=|Per (Λ S ,T )| S. Scheel quant-ph/0406127 Per ( A)=∑σ ∈S n n ∏i A i ,σ (i ) Input modes ( U 1,1 U 1,2 U 2,1 U 2,2 ⋮ ⋮ U D ,1 U D ,2 … U 1, D … U 2, D ⋱ ⋮ … U D, D Because D=O(N^2): no-collision regime, random Gaussian matrix, Sub-matrix of UD of dimension NxN As opposed to Determinant, Permanent is hard to calculate: #P-Hard (Valiant 1979) ) Hardness of simulating boson-sampling Hardness of Exact Sampling KLM proof Single photons+linear optics +post-selection Exact classical algorithm for BOSON-SAMPLING Universal quantum computation Collapse PH to 3rd order An artist vision on the collapses of the PH to the 3rd order... An the FULL collapse N=NP Hardness of Exact Sampling KLM proof Single photons+linear optics +post-selection Universal quantum computation PP PH ⊆ P =P Post −BQP Toda 91 Exact classical algorithm for BOSON-SAMPLING P Post −BPP ⊆Δ3 Han 96 Collapse PH to 3rd order Hardness of Exact Sampling χ + post-selection Universal quantum computation Non-universal set of gates and states Exact classical algorithm for χ-SAMPLING Collapse PH to 3rd order Approximate Sampling BOSON-SAMPLING(N,M,Λ) 1 ‖P (S)−~ P(S)‖= ∑ s |P (s)−~ P(s)|<ϵ 2 Appr. classical algorithm for BOSON-SAMPLING ● ● Conjectured Collapse PH to 3rd order Approximate Permanent of matrix of independent Gaussian entries hard (#P-hard) Permanent anti-concentration conjecture √ n! ∣Per (Λ S , T )∣≥ poly (n) Experiments Experiments 3 to 4 photons over 8 modes “Our results demonstrate that boson sampling is related to the computation of matrix permanents, a problem believed to be classically hard” Spring et al. Science 339, 798 (2013) "The bottom line is that we have built boson sampling machines, that they work and that the errors are not fatal" Ian Walmsley We want the simplest and most robust experimental setup that can be proven to be hard to sample Open Questions Can we prove “supremacy”? No experiment will provide a “mathematical proof” of “quantum supremacy” What shall we aim to? In practice it will look more like a competition between Classical Team Quantum Team VS Classical Sampler Bosonic Sampler Verifier Can we prove “supremacy”? No experiment will provide a “mathematical proof” of “quantum supremacy” What shall we aim to? Build a Boson Sampler big enough that a classical computer can not efficiently simulate But that it can be certified in practice (regime ~ 30 photons) In practice it will look more like a competition between... Quantum Team Classical Team VS Classical Sampler Bosonic Sampler Can we certify a Boson-sampling device? Is BS far from uniform distribution? ● ● ● Instead of building a device that implements Boson-Sampling, program a classical computer to efficiently sample from the uniform distribution. If one chooses U at random, the chances of being caught cheating becomes large only after exponentially many samples. The findings of any experimental realization of Boson-Sampling have to be interpreted with great care, as far as the notion “quantum supremacy” is concerned. Is BS far from uniform distribution? ● ● ● Instead of building a device that implements Boson-Sampling, program a classical computer to efficiently sample from the uniform distribution. If one chooses U at random, the chances of being caught cheating becomes large only after exponentially many samples. The findings of any experimental realization of Boson-Sampling have to be interpreted with great care, as far as the notion “quantum supremacy” is concerned. BS is far from uniform distribution ● ● ● Instead of building a device that implements Boson-Sampling, program a classical computer to efficiently sample from the uniform distribution. If one chooses U at random, the chances of being caught cheating becomes large only after exponentially many samples. The findings of any experimental realization of Boson-Sampling have to be interpreted with great care, as far as the notion “quantum supremacy” is concerned. ● ● 1 October 2013 We prove that the BosonSampling distribution is far from the uniform distribution in total variation distance. We give an efficient algorithm that distinguishes these two distributions with constant bias.” Nonetheless, certifying a BosonSampling distribution might be classically hard “BosonSampling device faster than its fastest classical simulation, but without classical certification being practically impossible (regime n = 30 photons)” Verification Verification of output data is Hard For the same reason sampling is hard... Build a Boson Sampler big enough that a classical computer can not efficiently simulate But that it can be certified in practice (regime ~ 30 photons) SOLUTIONS? Find a set of matrices that are hard to sample, but easy to verify. Probably not... Use RUV Classical Command Nature 496, 456–460 (25 April 2013) Yes, but..... What about an honest experimentalist? What about if we assume the circuit to be linear optics? Tomography of UD Self-Testing (Mosca, ...) + random circuits + two boxes Boson-sampling variations Quantum harmonic oscillators Intermezzo Fock vs Phase-space Fock Representation Phase-space Representation P X ρ=∑ ρn ,m|n ⟩ ⟨ m| [ a,a ↓] =1 a|n ⟩ = √ n|n−1 ⟩ x= ( a↓ +a ) / √ 2 p=i ( a↓−a ) / √ 2 [ x,p ] =i Wigner Function: quasi-probability distribution N |n ⟩=a a|n ⟩=n|n ⟩ a|0 ⟩ =0 1 H^ = N^ + ↓ 2 a |n ⟩= √ n+1|n+1 ⟩ ↓ ↓ D (α)=exp (α a −ᾱ a) Characteristic function:χ ( ξ ) =Tr [ ρD ( ξ ) ] COHERENT STATE ∞ ∣α 〉 =e−∣α ∣/2 ∑n= 0 2 α 〉 =D (α)∣0 〉 ∣ ( ) αn ∣n 〉 √n ! Coherent State Vacuum p Poisson Distribution Quadratures first moments d i=Tr [ ρ x̂ i ] ̄d =√ 2(ℜ α , ℑ α ) Quadratures second moments γi , j =Tr [ ρ { x̂ i−d i , x̂ j−d j } ] ( ) 1 0 γ= 0 1 x SQUEEZED STATE Vacuum ∣r 〉 =S (r )∣0 〉 S (r )=exp ( r (a↓ a↓−aa ) ) ∞ |r ⟩ =∑n=0 (√ √(2n)! cosh (r )2n n ! ) Quadratures second moments γ= ( −2r e 0 0 2r e n (tanh (r)) |2 n ⟩ ) Displaced Squeezed State ∣α , r 〉 =D(α)S (r )∣0 〉 Squeezed Vacuum State ENTANGLEMENT Two-Mode Squeezing (OPA) ↓ G ↓ A( τ)=exp ( τ( a b −ab) ) Two-mode Squeezed Vacuum State ∞ 1 n (tanh (r)) |n ⟩|n ⟩ |ψ(r) ⟩= ∑ n=0 cosh (r ) Beam-splitter operator ↓ ↓ B(τ )=exp ( τ (a b −a b) ) THERMAL STATES Two-mode Squeezed Vacuum State ∞ 1 n (tanh (r)) |n ⟩|n ⟩ |ψ(r) ⟩= ∑ n=0 cosh (r ) Thermal State ( ) ∞ ̄ 1 N ρN̄ = ∣n 〉 〈 n∣ ∑ n=0 ̄ +1 ̄ +1 N N Partial Trace Phase-space P ρN̄ 2 ̄ N =sinh (r ) n α〉 ∣ Maximally mixed state for fixed energy X g ( N̄ )=( N̄ +1)log ( N̄ +1)− N̄ log ( N̄ ) MEASUREMENT Homodyne: measurement of x Heterodyne: measurement of x and p Gaussian POVM H ∣0 〉 Variations Measurement Coherent States Gaussian States Single-photon Adaptive QND BQP BQP BQP Adaptive NOT-QND P BQP BQP [KLM] NOT-Adaptive QND Sampling Hard Sampling Hard Sampling Hard NOT-Adaptive NOT-QND P Sampling Hard Sampling Hard Homodyne (Gaussian) P P ? Gaussian Fock-basis sampling Gaussian Boson Sampling Sample on the photon-number basis a n-mode Gaussian state Pure states ∣0 〉 ∣G 〉=U G∣0 〉 =e −i τ H ∣0 〉 ∣0 〉 Quadratic Hamiltonian in creation and annihilation operators. H=i(α a+ a F a↓ + a G a)+h.c. H ∣0 〉 Euler decomposition: S(r1) H K S(r2) S(r3) S(r1) L S(r2) S(r3) L Gaussian Boson Sampling Sample on the photon-number basis a n-mode Gaussian state I Hardness of Exact Gaussian BS χ + post-selection Universal quantum computation + Complexity Theory tricks Non-universal set of gates and states Exact classical algorithm for χ-SAMPLING Collapse PH to 3rd order Gaussian states Post-selection + Two-mode squeezer + Linear optics ∣1×1∣ ∣0 〉 TMS ∣0 〉 ∣1 〉 Approximate Gaussian BS ? Scattershot boson-sampling Scattershot boson-sampling Two-mode squeezer SCATTERSHOT-BS(N,M,Λ) G Lund et al. arXiv:1305.4346 Scattershot boson-sampling permanent rule INPUT: N single-photons distributed over D modes |T ⟩=|1,1,1,0,. .. ,0 ⟩ = √ 1 |π(1), π (2) ,... , π (N ) ⟩ ∑ π∈S N! P (S∥T )=|⟨ s 1 , s2 , ... , s M|U D ∘N D 2 |t 1 ,t 2 ,... ,t M ⟩| Modes with detected photons 2 P(S∥T )=|Per (Λ S ,T )| S. Scheel quant-ph/0406127 Per ( A)=∑σ ∈S n n ∏i A i ,σ (i ) Input modes ( U 1,1 U 1,2 U 2,1 U 2,2 ⋮ ⋮ U D ,1 U D ,2 Because D=O(N^2): no-collision regime, random Gaussian matrix, Sub-matrix of UD of dimension NxN … U 1, D … U 2, D ⋱ ⋮ … U D, D ) Approximate Scattershot-BS SCATTERSHOT-BS(N,M,Λ) Appr. classical algorithm for SCATTERSHOT-BS 1 ̃ ̃ s)∣<ϵ ∥P(S )− P(S)∥= ∣P( s)− P( 2 ∑s Hardness of approximate Sampling of Gaussian-sampling Conjectured Collapse PH to 3rd order What if we pump more photons in? More photons per input? Can we get rid of the “collision-free” condition? 2 P( S∥T )= ∣ Per (Λ S ,T )∣ ∏i (si !)∏ j (t j !) S. Scheel quant-ph/0406127 Much easier for experimentalists! Example: Hong Ou Mandel 2 P( S∥T )= ∣ Per (Λ S ,T )∣ ∏i ( si !)∏ j (t j !) One photon per output ( 1 1 1 Λ S ,T =Λ= √ 2 −1 1 Per (Λ)=0 ) 2 P((1,1)∥(1,1))= ∣ Per (Λ )∣ 2 2 (1!) (1!) =0 Example: Hong Ou Mandel 2 P( S∥T )= ∣ Per (Λ S ,T )∣ ∏i ( si !)∏ j (t j !) Two photons on one output ( 1 1 √ 2 Λ= −1 1 ) ( 1 1 −1 1 ) ( 1 1 −1 1 ) ( ) 1 1 = 2Λ √ S ,T 1 1 2 ∣Per (Λ S , T )∣ 1 P((2,0)∥(1,1))= = 2 2 2 ! 0 ! (1 !) Per (Λ S , T )=1 General permanent rule Boson Sampling “Gedankenexperiment” M ∣T 〉 =∣t 1 , t2 , ... ,t M 〉 =∏ i=1 ● N photons @ input of M modes ● Evolution through linear interferometer ● Photon counting @ output P (S∥T )=|⟨ s 1 , s2 , ... , s M|U ∘N ( ) (a↑ )t ∣0 〉 √ ti i b̂i =∑ j Λi , j â j ̄b =Λ ā 2 |t 1 ,t 2 ,... ,t M ⟩| 2 P (S∥T )= |Per (Λ S ,T )| ∏i (si !)∏ j (t j !) S. Scheel quant-ph/0406127 Per ( A)=∑σ ∈S n n ∏i Ai , σ (i) As opposed to Determinant, Permanent is hard to calculate: #P-Hard (Valiant 1979) Homodyne detection sampling Homodyne sampling Gaussian states+Gaussian operations+homodyne measurement Is classically simulatable what about non-Gaussian states + homodyne? Full quantum modeling of homodyne measurement (Fock basis)... Implementations and tolerance to noise Effect of errors Showing results 1 through 7 (of 7 total) for (au:Rohde AND all:(boson AND sampling)) 1. arXiv:1403.4007 [pdf, other] Scalable boson-sampling with time-bin encoding using a loop-based architecture Keith R. Motes, Alexei Gilchrist, Jonathan P. Dowling, Peter P. Rohde Comments: 7 pages, 7 fgures Subjects: Quantum Physics (quant-ph) 2. arXiv:1402.0531 [pdf, other] Boson sampling with photon-added coherent states Kaushik P. Seshadreesan, Jonathan P. Olson, Keith R. Motes, Peter P. Rohde, Jonathan P. Dowling Comments: 5 pages, 2 fgures, Comments are welcome Subjects: Quantum Physics (quant-ph) 3. arXiv:1401.2199 [pdf, other] Will boson-sampling ever disprove the Extended Church-Turing thesis? Peter P. Rohde, Keith R. Motes, Paul A. Knott, William J. Munro Comments: 3 pages, 0 fgures Subjects: Quantum Physics (quant-ph) 4. arXiv:1310.0297 [pdf, other] Sampling generalized cat states with linear optics is probably hard Peter P. Rohde, Keith R. Motes, Paul Knott, William Munro, Jonathan P. Dowling Comments: 8 pages, 2 fgures Subjects: Quantum Physics (quant-ph) 5. arXiv:1307.8238 [pdf, other] Spontaneous parametric down-conversion photon sources are scalable in the asymptotic limit for boson-sampling Keith R. Motes, Jonathan P. Dowling, Peter P. Rohde Comments: 6 pages, 5 fgures; added reference to Meany et al Journal-ref: Phys. Rev. A 88, 063822 (2013) Subjects: Quantum Physics (quant-ph) 6. arXiv:1208.2475 [pdf, other] Optical quantum computing with photons of arbitrarily low fdelity and purity Peter P. Rohde s Comments: Version submitted to Phys. Rev. A Journal-ref: Phys. Rev. A 86, 052321 (2012) Subjects: Quantum Physics (quant-ph) ̃ S)∥= 1 ∑ ∣P(s)− P(s) ̃ ∣<ϵ ∥P(S )− P( 2 Anthony Leverrier, R. G.-P., arXiv:1309.4687 Future directions Future Directions Improve the Hardness proof ● ● ● Hardness for approximated Boson Sampling (proof the conjecture) Hardness for larger input photons (collisions) Hardness for realistic experimental conditions (losses, detector efficiency,...) Verification of Boson Sampling Variants of Boson Sampling Qudit repr of active transformations Classical Simulation Boson Sampling ● ● Optimize classical sampling algorithm Does exist families of states with efficient classical sampling? (bosonic “stabilizer states”) THANK YOU! @QuantumCrypto