Model realiza+on and model reduc+on for quantum systems

Transcription

1 Model realiza+on and model reduc+on for quantum systems Mohan Sarovar Scalable and Secure Systems Research Sandia Na+onal Laboratories, Livermore, USA Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy s National Nuclear Security Administration under contract DE-AC04-94AL PRACQSYS 2014, Cambridge

2 Outline & Acknowledgements! Model realization and system identification! Estimating unknown Hamiltonian parameters Quantum Hamiltonian identification from measurement time traces Jun Zhang, MS arxiv: , to appear in PRL 2014! Model reduction Jun Zhang Shanghai Jiao Tong University! Reducing simulation cost for certain many-body quantum systems On model reduction for quantum dynamics: symmetries and invariant subspaces Akshat Kumar, MS arxiv: Akshat Kumar Sandia National Laboratories

3 System Iden+fica+on Identify system from input-output behavior e.g. Process tomography: identify process (CP-map, unitary) at a particular time Alternative: identify generator (e.g. Hamiltonian) of system Advantages: physical constraints (locality, connectivity) more naturally introduced intro generator of dynamics e.g. n qubits: arbitrary unitary parameterized by 4 n -1 real parameters, arbitrary 2-local Hamiltonian parameterized by 3n+9n(n-1)/2 real parameters

4 System Iden+fica+on How powerful are time traces? ho 1 (t 1 )i, ho 1 (t 2 )i,... ho 2 (t 1 )i, ho 2 (t 2 )i,... ho n (t 1 )i, ho n (t 2 )i,... Additional considerations Measurements could be restricted May have partial information about system Assumptions: 1. system is finite dimensional 2. Hamiltonian dynamics (closed system) See also: Cole, J. H., Schirmer, S. G., Greentree, A. D., Wellard, C. J., Oi, D. K. L., & Hollenberg, L. C. L. Phys. Rev. A, 71, (2005) Burgarth, D., Maruyama, K., & Nori, F. Phys. Rev. A, 79, (2009) Burgarth, D., & Maruyama, K. New J. Phys., 11, (2009) Di Franco, C., Paternostro, M., & Kim, M. Phys. Rev. Lett., 102, Burgarth, D., & Yuasa, K. Phys. Rev. Lett., 108, (2012) Granade, C. E., Ferrie, C., Wiebe, N., & Cory, D. G. New J. Phys., 14(10), (2012)

5 An example Measurable spins Parametric Hamiltonian H( 1, 2,..., M ) h 1 z(t 0 )i, h 1 z(t 1 )i,...,h 1 z(t n )i Time trace of some accessible observable Can we identify the parameters in the Hamiltonian from just this?

6 The setup Choose an orthogonal operator basis for the linear operator space (e.g. generalized Paulis) [ix j,ix k ]= X N 2 1 C jkl (ix l ), j,k =1,,N 2 1, l=1 Hamiltonian can be expanded in this basis H = X M m=1 a m( )X m Goal: to identify a m Leads to a linear, autonomous equation for state x(t) d dt x k = X N 2 1 XM C mkla m x l l=1 m=1 d dt x = Ax, x k(t) =h (t) X k (t)i i2c N dimh = N N x 2 R (N 2 1) dima =(N 2 1) (N 2 1)

7 The setup Similarly, each directly measured observable can be expanded in the same basis. Resulting in a AC system d dt x = Ax y(t) =Cx(t) But this may be too complex a description. E.g. C = A = a b 0 0 c d a 0 b c 0 d

8 Filtra+on to find minimal descrip+on O i = X j o (i) j X j M = {X 1,X 2,...,X p } Directly measured set H = X M m=1 a m( )X m = {X m } M m=1 Hamiltonian set Filtration recursively constructed as: G 0 = M, and where G i =[G i 1, ] [ G i 1 [G i 1, ] {X j :tr(x j [g, h]) 6= 0, where g 2 G i 1,h2 } Finite algebra => procedure terminates, resulting in filtration Ḡ Accessible set Results in minimal description d dt x a = Ãx a y(t) = Cx a (t)

9 Example XX spin chain Fukuhara et al. Nature Physics, (2013) H = nx k=1! k 2 nx 1 k z + k=1 k k + k+1 + k k+1 + M = Measure end spin 1 x Ḡ ={2 n/2 1 x, 2 n/2 1 y} [{2 n/2 z 1 k 1 k z x, 2 n/2 z 1 k 1 k z y} n k= ! 1 0 1! ! Ã = 1 0! n n n 1 0! n 5 n 1 0! n 0 C =[1, 0,...,0] Filtration (generalized Pauli basis) Filtered system matrix is 2n x 2n (as opposed to 2 n x 2 n ) x a =[ x 1, ȳ 1,..., x n, ȳ n ] T x 1 = 1 x, ȳ1 = 1 y x k 1 z ȳ k 1 z k z 1 k z 1 k x, k y, k 2

10 Sampling and discre+za+on x a (j + 1) = Ãdx a (j) Ã d = eã t y(j) = Cx a (j) Explicit solution Note: x a(j) x a (j t) y(j) y(j t) y(j) = CÃj d x a(0) Goal: Use {y(j)} J j=0 to estimate {a m } M m=1 Strategy: 1. Find the minimal linear model that generates the collected data 2. Back out the unknown parameters from this model

11 Eigenstate realiza+on algorithm [Juang, J. N., & Pappa, R. S. Journal of Guidance, Control, and Dynamics, 8, 620 (1985)] Step 1: Form Hankel matrix from data 2 3 y(k) y(k + 1) y(k +(s 1)) y(1 + k) y(1 + k + 1) y(1 + k +(s 1)) H rs (k) = y(r 1+k) y(r 1+k + 1) y(r 1+k +(s 1)) Step 2: Take SVD of Hankel matrix at k=0 H rs (0) = P apple QT = apple 0 P 1 P apple Q T 1 Q T 2 O P C 1 2 Q T 1 Step 3: Form realizations of linear model from SVD components  d = 1 2 P T 1 H rs (1)Q Ĉ = e T 1 P e T 1 =[1, 0,, 0] ˆx(0) 1 2 Q T 1 e 1

12 Eigenstate realiza+on algorithm Step 3: Form realizations of linear model from SVD components  d = 1 2 P T 1 H rs (1)Q Ĉ = e T 1 P ˆx(0) 1 2 Q T 1 e 1 The triple (Âd, Ĉ, ˆx(0)) is a realization of the triple (Ãd, C, x a (0)) Certain quantities are model realization invariants, e.g. the Markov parameters CA j x(0) 8j (A, C, x(0)) for any LTI model Therefore, y(j) = CÃj d x a(0) = ĈÂj dˆx(0), for all j 0,

13 Realiza+on to parameter es+ma+on y(j) = CÃj d x a(0) = ĈÂj dˆx(0), for all j 0, Define  = log Âd/ t CÃj x a (0) = ĈÂj ˆx(0), for all j 0 Polynomial equation in unknown parameters Determined by the data Solving these equations yields estimates of parameters Notes: 1. Parameter estimates can be non-unique (gauge freedom/symmetries) 2. Δt must be satisfy Nyquist relation to one-over-fastest-frequency in system

14 Example XX spin chain (n=3 qubits) H = 3X k=1! k 2 k z + 2X k=1 k k + k+1 + k + k+1! 1 =1.3,! 2 =2.4,! 3 =1.7, 1 =4.3, 2 = (0)i = 0i + i 1i p 2 00i <σ x1 (t)> x Time (s) time

15 Example XX spin chain (n=3 qubits) H = 3X k=1! k 2 k z + 2X k=1 k k + k+1 + k + k+1! 1 =1.3,! 2 =2.4,! 3 =1.7, 1 =4.3, 2 =5.2 Construct Hankel matrix with r=100, s= (3! 1 +2! 2 )+ 2 1! 1 = Ĉˆx(0) = 1.3! (2! 1 +! 2 )= ĈÂ3ˆx(0) = (2! 1 +2! 2 +! 3 )+4! ! 2! ! 2 2! 1 +! 3 2 +! 5 1 = ĈÂ5ˆx(0) = Coupling parameters only occur up to even order (symmetry) => can only be determined up to sign

16 Summary System identification through model realization Most useful when measurements are restricted prior information about process is available Continuing work: Noisy measurements Use a different model realization invariant (transfer function) ω 1 ω 2 ω 3 δ 1 δ Markovian open-system evolution Continuous time weak measurement

17 Outline! Model realization and system identification! Estimating unknown Hamiltonian parameters! Model reduction! Reducing simulation cost for certain many-body quantum systems

18 Outline Quantum state space: exponential Full-scale simulation of quantum systems very difficult Formal state is exponentially large in the number of particles 1 H 1 dimh 1 = n 1 2 H 2 dimh 2 = n 2 c H 1 H 2 dimh 1 H 2 = n 1 n 2 = n 1 + n 2

19 Outline Quantum state space: not really exponential? For most practical systems, this exponential scaling is only formal Many-body Hilbert space Physically relevant set of states Identifying this set of relevant states is difficult Dynamics within this relevant set of states is a reduced order model (ROM)

20 Iden+fying reduced order models Existing techniques/results: - static: DMRG, MPS, etc. - dynamic: Nakajima-Zwanzig (statistical), Bloch equations, adiabatic elimination

21 Model reduc+on Desired Output Resources available State snapshots Input-output map Dynamical model Full state vector want to reproduce (t)i Input-output map want to reproduce y(t) Proper orthogonal decomposition (POD), manifold learning Empirical balanced truncation (BPOD)? Identify or approximate invariant subspaces Minimal model realization algorithms (e.g. ERA) Balanced truncation - In the context of continuous measurement / quantum filtering: Mabuchi, PRA 78, (2008), Nielsen, Hopkins, Mabuchi, NJP 11, (2009) - In the context of coherent feedback control / quantum optical networks: Nurdin, arxiv: , Nurdin, Gough arxiv:

22 Compressible dynamics (t)i = e iht (0)i (0)i H( )= X i ih i Problem: Identify subspace of Hilbert space that contains (0)i and is invariant under Hamiltonian for all choices of Assumption: 1. system is finite dimensional e.g. quantum H = B X Ising model: i x J X i <i,j> i z j z

23 Compressible dynamics H( )= X i ih i (0)i Problem: Identify subspace of Hilbert space that contains (0)i and is invariant under Hamiltonian for all choices of Solutions: 1. Certificates Is this dynamics compressible? 2. Computing reduced order models What is the invariant subspace and moreover, what is the reduced order dynamical model? Projective linear model reduction: columns of P are basis vectors in this invariant subspace d dt P (t)i = P HP P (t)i q x q compressed description dimp = N q, q N

24 Cer+ficate H( )= X i ih i H 2 L(H) Coe (H) {h i } dim H = N Theorem: (algebraic certificate) The Hamiltonian acting on H keeps invariant a non-trivial proper subspace iff the subalgebra generated by Coe (H) is a proper subalgebra of L(H). Intuition: (t)i =exp{i( 1 h h 2 )t} (0)i = X n (it) n ( 1 h h 2 ) n (0)i n! Kumar, MS. arxiv: Products of h i generate an algebra. If the full operator algebra is not generated, there are directions in state space that are not explored

25 Cer+ficate H( )= X i ih i H 2 L(H) Coe (H) {h i } dim H = N Theorem: (algebraic certificate) The Hamiltonian acting on H keeps invariant a non-trivial proper subspace iff the subalgebra generated by Coe (H) is a proper subalgebra of L(H). Identifies a symmetry: Certifies the existence of a unitary matrix that simultaneously diagonalizes all h i UHU = m

26 Cer+ficate H( )= X i ih i H 2 L(H) Coe (H) {h i } dim H = N Theorem: (algebraic certificate) The Hamiltonian acting on H keeps invariant a non-trivial proper subspace iff the subalgebra generated by Coe (H) is a proper subalgebra of L(H). dim A(Coe (H)) < dim L(H)? To answer this: generate linear basis for count dimension. A(Coe (H)) (the Burnside basis) and Algorithm for basis generation: repeatedly multiply out h i and keep linearly independent results. This generates: {B 1,B 2,...B K }

27 Construc+ng reduced order models d dt P (t)i = P HP P (t)i dimp = N q, q N q x q compressed description Two methods: 1. Sampling from full-order model 1. Fix H( ) 2. Generate samples (t 1 )i, (t 2 )i,... (t k )i 3. The columns of P are formed from the (orthonormalized) minimal linearly independent set of these state samples (Krylov/cyclic subspace) Careful: cyclic subspace invariant subspace for model 2. Use Burnside basis generated in algebraic characterization (certificate) 1. Find 1 i = B 1 (0)i, 2 i = B 2 (0)i,... K i = B K (0)i 2. The columns of P are formed from the (orthonormalized) minimal linearly independent set of these states Generates true reduced subspace for model Kumar, MS. arxiv:

28 Example: quench dynamics Quantum Ising model Fukuhara et al. Nature Physics, (2013) H = B X i i x J X <i,j> i z j z Basic model for magnetism in crystalline material Competition between B and J results in phase transition behavior Can be emulated using cold atoms As a result: intense interest in dynamical phase transitions, quenching dynamics Quenching dynamics: 1. Prepare ground state of H 0 = B 0 X i 2. Rapidly change B and evolve system under 3. The resulting dynamics is very informative; e.g. contains information about static phases of system i x J X <i,j> H 1 = i z j z B 1 X i i x J X <i,j> i z j z

29 Example Simulation of quench dynamics in quantum Ising model (Circles: full model, lines: reduced order model) Quenches to different parameters are indicated by different colors Total transverse magnetization B = 0.1 B = 0.5 B = 1.0 B = 1.4 B = 1.8 N=8 qubits Full order model: = 255 complex numbers Reduced order model: 23 complex numbers time

30 Expense of compu+ng ROM d dt P (t)i = P HP P (t)i H( )= X i ih i Certificate: Multiply d x d matrices entering the Hamiltonian Compute linear dependency of the results Construction of model reduction matrix P Time sampling: Need samples from full order system Compute linear dependency of samples Burnside basis construction Multiply d x d matrices entering the Hamiltonian Compute linear dependency of the results Multiply Burnside basis elements by (0)i Rational computations, but exponential complexity in general However, complexity simplifies greatly in special cases (e.g. Pure Pauli spin models). See Kumar, MS. arxiv:

31 Cer+ficate Special case: Pauli Hamiltonian H( )= X i i i H 2 L(H) H = C 2n dim H =2 n e.g. i : (1) x 1... y (n) Theorem: (Pauli algebraic certificate) Any Pauli Hamiltonian acting on n qubits with fewer than 2n terms has a non-trivial proper invariant subspace. Note: a sufficient condition Kumar, MS. arxiv: e.g. Random quantum (transverse field) Ising model with open boundary conditions

32 Con+nuing work Approximations to invariant subspaces? True model reduction UHU = m Methods for approximate simultaneous block diagonalization Reduce computational complexity of ROM computation for special cases Algebraic approach to observability-based model reduction (i.e. only care to reproduce some observables)

33 Thanks!