Permanents and boson sampling Stefan Scheel, Universität Rostock Summer School on Modern Quantum Technologies 13.09.2018 Permanents and boson sampling | Stefan Scheel, Universität Rostock 1 / 16 Menu Outline: • permanents in linear optics • computational complexity • boson sampling as intermediate model for quantum computation 13.09.2018 Permanents and boson sampling | Stefan Scheel, Universität Rostock 2 / 16 Permanents in linear optics lossless beam splitter transformation for photonic amplitude operators: h i b^ = T^a = U^ y^aU^ ; U^ = exp −i^ayΦ^a ; T = exp [−iΦ] • equivalent to discrete-time Heisenberg equation of motion • transform quantum states by discrete-time Schrödinger equation using inverse transformation %^0 = U^ %^U^ y hO^_ i = Tr[^%O^_ ] = Tr[^% U^ yO^ U^ ] = Tr[ U^ %^U^ y O^ ] = Tr[%^_O^ ] | {z } | {z } Heisenberg Schrödinger 13.09.2018 Permanents and boson sampling | Stefan Scheel, Universität Rostock 3 / 16 Permanents in linear optics transformation matrix T 2SU(2), but U^ in general is not: • n-photon Fock space is symmetric tensor product of single-photon spaces • quantum-state transformation %^0 = U^ %^U^ y according to a subgroup of SU(2n) example: matrix representation of U^ in basis fj0; 0i; j1; 0i; j0; 1i; j2; 0i; j1; 1i; j0; 2ig 01 0 0 0 0 0 1 ∗ B0 T −R 0 0 0 C B0 RT ∗ 0 0 0 C 1 U B p C M U = B0 0 0 T 2 2T ∗R∗ R∗2 C = n B p p C n 0 B0 0 0 2TR T 2 R 2 2T ∗R∗C = @ (j jp− j j ) − A 0 0 0 R2 − 2TR T ∗2 • U is block-diagonal with respect to Fock layers of total photon numbers (0; 1; 2) • U has direct product structure 13.09.2018 Permanents and boson sampling | Stefan Scheel, Universität Rostock 4 / 16 Permanents in linear optics matrix transforming quantum states acts on symmetric subspace ) can be constructed from permanents of transmission matrix T define set of all non-decreasing integer sequences ! as Gn;N = f! : 1 ≤ !1 ≤ ::: ≤ !n ≤ Ng matrix elements of U^ in the Fock basis are matrix permanents 1 2 −1=2 !− = 0 1 Y Y 0 hm1;:::; mN jU^ jn1;:::; nN i = ni ! @ mj !A per T[Ω jΩ] i j Ω = (1n1 ; 2n2 ;:::; NnN ), Ω0 = (1m1 ; 2m2 ;:::; NmN ) T[Ω0jΩ]: N × N-matrix with elements from T with row and column indices Ω0, Ω S. Scheel, quant-ph/0406127; S. Scheel and S.Y. Buhmann, Acta Phys. Slovaca 58, 675–810 (2008). 13.09.2018 Permanents and boson sampling | Stefan Scheel, Universität Rostock 5 / 16 Permanents in linear optics unitary transformation of N-mode Fock state with total n photons !−1=2 Y X 1 U^ jn1;:::; nN i = ni ! per T[!jΩ]jm1(!);:::; mN (!)i µ(!) i !2Gn;N mi (!): multiplicities of occurence of index i in non-increasing integer sequence ! µ(!) = Q mi (!)! i n per T P Q T T S = iσi matrix permanent of , n: (symmetric) group of permutations σ2Sn i immediate consequence: per T = h1; 1;:::; 1jU^ j1; 1;:::; 1i 2 2 e.g. h1; 1jU^ j1; 1i = per T = T11T22 + T12T21 = jT j − jRj S. Scheel, quant-ph/0406127; S. Scheel and S.Y. Buhmann, Acta Phys. Slovaca 58, 675–810 (2008). 13.09.2018 Permanents and boson sampling | Stefan Scheel, Universität Rostock 6 / 16 Permanents in linear optics unitary transformation of N-mode Fock state with total n photons !−1=2 Y X 1 U^ jn1;:::; nN i = ni ! per T[!jΩ]jm1(!);:::; mN (!)i µ(!) i !2Gn;N mi (!): multiplicities of occurence of index i in non-increasing integer sequence ! µ(!) = Q mi (!)! i n per T P Q T T S = iσi matrix permanent of , n: (symmetric) group of permutations σ2Sn i immediate consequence: per T = h1; 1;:::; 1jU^ j1; 1;:::; 1i 2 2 e.g. h1; 1jU^ j1; 1i = per T = T11T22 + T12T21 = jT j − jRj S. Scheel, quant-ph/0406127; S. Scheel and S.Y. Buhmann, Acta Phys. Slovaca 58, 675–810 (2008). 13.09.2018 Permanents and boson sampling | Stefan Scheel, Universität Rostock 6 / 16 Complexity of computing matrix permanents What is a permanent anyway, and why is it so special? n det M P 1 χ(σ) Q M matrix determinant: = (− ) iσi σ2Sn i • has its roots in linear algebra • volume of parallelepiped spanned by the column vectors of M • only defined for square matrices • similarity (principal axes) transformation 7! diagonal form • computational demand: O(n3) for LU/QR/Cholesky decomposition 13.09.2018 Permanents and boson sampling | Stefan Scheel, Universität Rostock 7 / 16 Complexity of computing matrix permanents What is a permanent anyway, and why is it so special? n det M P Q M matrix permanent: = iσi σ2Sn i • has its roots in combinatorics and graph theory • # permutations with restricted positions, weights of perfect matchings of a graph • also defined for rectangular matrices • only invariant under permutations • computational demand: O(n2n) for exact computation 13.09.2018 Permanents and boson sampling | Stefan Scheel, Universität Rostock 8 / 16 Complexity of computing matrix permanents Theorem (Valiant) The complexity of computing the permanent of n × n(0; 1)-matrices is NP-hard and, in fact, of at least as great difficulty (to within a polynomial factor) as that of counting the number of accepting computations of any nondeterministic polynomial time Turing machine. consequences: • computing permanents is really hard (#P-complete, i.e. not possible in polynomial time) • designing linear-optical networks for a specific task requires computing the permanent of the network ) designing LOQC is itself hard L.G. Valiant, Theor. Comp. Science 8, 189 (1979). 13.09.2018 Permanents and boson sampling | Stefan Scheel, Universität Rostock 9 / 16 Complexity of computing matrix permanents approximations to permanents: • Jerrum, Sinclair, and Vigoda: matrix elements nonnegative: approximations to per M can be made in probabilistic polynomial time • matrix elements complex: even approximating per M to within a constant factor is #P-complete! M. Jerrum, A. Sinclair, and E. Vigoda, J. ACM 51, 671 (2010). 13.09.2018 Permanents and boson sampling | Stefan Scheel, Universität Rostock 10 / 16 Boson sampling model Extended Church–Turing Thesis: all computational problems that are efficiently solvable by realistic physical device, are solvable by a probabilistic Turing machine. Shor: Predicting the (probabilistic) results of a given quantum-mechanical experiment, to finite accuracy, cannot be done by a classical computer in probabilistic polynomial time, unless factoring integers can as well. Shor’s argument is only valid if factoring is classically hard (not known)! Does one need a fully fledged universal quantum computer to disprove Extended Church–Turing Thesis? Aaronson and Arkhipov: no, linear optics is enough! ) quantum computation with noninteracting bosons (boson sampling) S. Aaronson and A. Arkhipov, The computational complexity of linear optics, Theory of Computing 9, 143–252 (2013). 13.09.2018 Permanents and boson sampling | Stefan Scheel, Universität Rostock 11 / 16 Boson sampling model Extended Church–Turing Thesis: all computational problems that are efficiently solvable by realistic physical device, are solvable by a probabilistic Turing machine. Shor: Predicting the (probabilistic) results of a given quantum-mechanical experiment, to finite accuracy, cannot be done by a classical computer in probabilistic polynomial time, unless factoring integers can as well. Shor’s argument is only valid if factoring is classically hard (not known)! Does one need a fully fledged universal quantum computer to disprove Extended Church–Turing Thesis? Aaronson and Arkhipov: no, linear optics is enough! ) quantum computation with noninteracting bosons (boson sampling) S. Aaronson and A. Arkhipov, The computational complexity of linear optics, Theory of Computing 9, 143–252 (2013). 13.09.2018 Permanents and boson sampling | Stefan Scheel, Universität Rostock 11 / 16 Boson sampling model Extended Church–Turing Thesis: all computational problems that are efficiently solvable by realistic physical device, are solvable by a probabilistic Turing machine. Shor: Predicting the (probabilistic) results of a given quantum-mechanical experiment, to finite accuracy, cannot be done by a classical computer in probabilistic polynomial time, unless factoring integers can as well. Shor’s argument is only valid if factoring is classically hard (not known)! Does one need a fully fledged universal quantum computer to disprove Extended Church–Turing Thesis? Aaronson and Arkhipov: no, linear optics is enough! ) quantum computation with noninteracting bosons (boson sampling) S. Aaronson and A. Arkhipov, The computational complexity of linear optics, Theory of Computing 9, 143–252 (2013). 13.09.2018 Permanents and boson sampling | Stefan Scheel, Universität Rostock 11 / 16 Boson sampling model Extended Church–Turing Thesis: all computational problems that are efficiently solvable by realistic physical device, are solvable by a probabilistic Turing machine. Shor: Predicting the (probabilistic) results of a given quantum-mechanical experiment, to finite accuracy, cannot be done by a classical computer in probabilistic polynomial time, unless factoring integers can as well. Shor’s argument is only valid if factoring is classically hard (not known)! Does one need a fully fledged universal quantum computer to disprove Extended Church–Turing Thesis? Aaronson and Arkhipov: no, linear optics is enough! ) quantum computation with noninteracting bosons (boson sampling) S. Aaronson and A. Arkhipov, The computational complexity of linear optics, Theory of Computing 9, 143–252 (2013). 13.09.2018 Permanents and boson sampling | Stefan Scheel, Universität Rostock 11 / 16 Boson sampling model Extended Church–Turing Thesis: all computational problems that are efficiently solvable by realistic physical device, are solvable by a probabilistic Turing machine. Shor: Predicting the (probabilistic) results of a given quantum-mechanical experiment, to finite accuracy, cannot be done by a classical computer in probabilistic polynomial time, unless factoring integers can as well. Shor’s argument is only valid if factoring is classically hard (not known)! Does one need a fully fledged universal quantum computer to disprove Extended Church–Turing Thesis? Aaronson and Arkhipov: no, linear optics is enough! ) quantum computation with noninteracting bosons (boson sampling) S.