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Finite automata for caching in matrix product algorithms

Gregory M. Crosswhite, Dave Bacon

arXiv:0708.1221v3quant-ph

TL;DR

Finding matrix product factorizations for complicated effective Hamiltonians is nontrivial; the paper formalizes a connection between matrix product representations and complex-weighted finite state automata. It uses that connection to factor operators and reports O(N) expectation-value computation and amortized O(1) energy-minimization steps.

  • Problem

    Finding matrix product factorizations for complicated effective Hamiltonians is a nontrivial task.

    Source: §1 · p. 8

  • Method

    The paper formalizes an equivalence between matrix product representations and complex-weighted finite state automata, then uses automata to construct operator factorizations.

    Source: §I · p. 2, §IV · p. 16

  • Results

    Expectation values can be computed in O(N) time, and energy minimization takes amortized O(1) cost per step.

    Source: D. Calculation of expectations · p. 9, E. Energy minimization and caching · p. 10, §IV · p. 16

  • Takeaways & Limitations

    The formalism connects efficient physical-system simulation with automata hierarchy and formal language theory.

    Source: §IV · p. 16

    For grid states, one-directional information flow makes the representation depend on grid size and prevents cheap communication in both directions.

    Source: A. Motivation · p. 12

Abstract

from arXiv · show

A diagram is introduced for visualizing matrix product states which makes transparent a connection between matrix product factorizations of states and operators, and complex weighted finite state automata. It is then shown how one can proceed in the opposite direction: writing an automaton that ``generates'' an operator gives one an immediate matrix product factorization of it. Matrix product factorizations have the advantage of reducing the cost of computing expectation values by facilitating caching of intermediate calculations. Thus our connection to complex weighted finite state automata yields insight into what allows for efficient caching in matrix product algorithms. Finally, these techniques are generalized to the case of multiple dimensions.

I. MOTIVATION

Representing a general quantum state directly grows exponentially, so the paper motivates matrix product representations and their use in efficient expectation-value calculations. It connects matrix product states and operators to weighted finite automata, using that connection to clarify their construction and caching.

I. MOTIVATION

  • Direct representations of quantum systems grow exponentially, motivating representations that remain tractable while preserving states and supporting efficient expectation values.The paper identifies these as three requirements for a useful representation.Source: §I · p. 1
  • Matrix product operators can reduce expectation calculations from O(N^2) to O(N) when both the state and operator are factored.Most operators of interest require only O(N) tensor-product terms, while a general operator may require exponentially many.Source: §I · p. 1
  • Adjacent-site sweeps can cache earlier computations, reducing the cost to O(1) per site and O(N) per sweep.The naive cost is O(N^2) per site.Source: §I · p. 2
  • An operator factorization lets one caching implementation work across Hamiltonians, rather than requiring special caching code for each one.The input to the general caching code is a matrix product factorization of the operator.Source: §I · p. 2
  • Matrix product states and operators correspond to complex-weighted finite automata, enabling automata techniques to help construct operator factorizations.The paper presents this connection as a formalization of the memory and signal intuitions associated with tensor indices.Source: §I · p. 2

A. Background

  • A full state tensor grows exponentially, whereas a separable state has a linear-size representation but cannot model entanglement; matrix bonds allow entanglement to be represented.The paper describes matrix product states as using inner indices between observables and notes that tensor networks can extend to multiple dimensions and lattice structures.Source: A. Background · pp. 2–3

B. Matrix product diagram

The matrix product diagram represents the W state through nodes and weighted edges, with each allowed path corresponding to a term in the state. Missing edges rule out paths whose contributions vanish.

B. Matrix product diagram

  • The example factorizes the four-particle W state, a sum of configurations with exactly one spin-down particle.The factorization is presented using site tensors labeled by particle observables.Source: B. Matrix product diagram · p. 3
  • Diagram nodes represent indices and edges represent matrix elements; shared nodes mark indices summed over.Zero-valued matrix elements are omitted from the diagram.Source: B. Matrix product diagram · p. 4
  • Each allowed left-to-right walk generates a term in the state, while a missing edge makes that path contribute zero.Arrows impose an index ordering that makes the diagram readable as paths.Source: B. Matrix product diagram · p. 4

1. Extension to operators

The diagrammatic representation extends from states to operators by allowing site tensors to carry observable indices. Replacing the W-state labels with identity and Pauli Z matrices yields an operator for a z-direction magnetic field coupling to the particles.

1. Extension to operators

  • Site tensors may carry multiple observable indices, allowing matrix product diagrams to represent operators as well as states.The resulting operator representation is called a matrix product operator.Source: §1 · pp. 4–5
  • Replacing each up label in the W-state representation with the identity and each down label with Pauli Z produces the magnetic-field operator diagram.The operator describes a magnetic field in the z direction coupling to each particle.Source: Figure 2 · p. 4, §1 · p. 5

C. Weighted finite automata states

The paper treats matrix product states as functions on strings computed by complex-weighted finite automata, making their relationship explicit. It then uses automata to construct matrix product factorizations of operators, with different diagram choices for translationally invariant and position-dependent structure.

C. Weighted finite automata states

  • A matrix product state is equivalent to a function on alphabet strings computed by a complex-weighted finite automaton.For spin-1/2 chains, the alphabet is {0, 1}; the automaton output is defined by the initial weights, symbol-transition matrices, and final weights.Source: C. Weighted finite automata states · p. 5, §1 · p. 5

1. Q is a finite set of states

  • Matrix product states are a special case of weighted finite automata, with position-specific transition matrices; automaton states can be copied across particles to form a matrix product state.Shared states yield compact automaton representations, illustrated by the two-state W-state automaton and its matrix-product conversion.Source: §1 · p. 6, Figure 5 · p. 7
  • To factor an operator, construct an automaton for its pattern, translate it into a matrix product operator diagram, and read off the matrices.The paper identifies a nearest-neighbor coupling operator as an example of a factorization obtained this way.Source: §1 · p. 7
  • Automata suit translationally invariant behavior, while position-dependent structure is better placed in matrix product diagrams to avoid extra automaton states and larger factors.For an operator with a special-position Z, modifying the earlier diagram is preferred to encoding the position dependence through additional automaton states.Source: §1 · pp. 7–8
  • Automata operations such as unions, concatenations, and intersections can help construct increasingly complex matrix product operators, including effective Hamiltonians with nontrivial factorizations.The paper presents these operations as tools for obtaining factorizations of complicated effective Hamiltonians.Source: §1 · p. 8

D. Calculation of expectations

Factoring an operator into matrix product form makes its expectation with a matrix product state tractable through sitewise contractions. The resulting calculation takes O(N) time.

D. Calculation of expectations

  • Factoring O into matrix product form replaces the exponentially costly unfactored expectation network with a more tractable network.Without the factorization, the calculation would still require an exponential amount of work.Source: D. Calculation of expectations · p. 9
  • The factored network sums in stages by first combining the site and operator matrices into transfer matrices.The section frames this factorization as a way to compute expectations of matrix product states efficiently.Source: D. Calculation of expectations · pp. 8–9
  • O(3N) matrix multiplications suffice to contract the network, making matrix-product-operator expectation calculation an O(N) procedure.The largest matrices formed are the En matrices.Source: D. Calculation of expectations · p. 9

E. Energy minimization and caching

During variational energy minimization, matrix product operators and cached recursive contractions reduce the per-site cost of calculating optimization matrices. The procedure applies to any operator expressible in matrix product form.

E. Energy minimization and caching

  • Holding all but one site matrix constant lets the method optimize the remaining site by minimizing the normalized Hamiltonian expectation.The authors seek a reasonable approximation to the true ground state.Source: E. Energy minimization and caching · p. 9
  • The site-3 energy is a Rayleigh quotient whose minimization is equivalent to solving a generalized eigenvalue problem.Its contractions use H_3 for the Hamiltonian and N_3 for the identity operator.Source: E. Energy minimization and caching · p. 10
  • Caching left and right contractions across a sweep lets the method compute the matrices needed for each site from recursive updates.Moving left to right and then back reuses intermediate results; each subsequent update requires one additional matrix multiplication.Source: E. Energy minimization and caching · p. 10
  • Unlike the caching procedure described by Verstraete, Porras, and Cirac [3], this procedure works for any operator in matrix product form.The authors state that it uses the same process for all such operators rather than requiring separate procedures for different operator classes.Source: E. Energy minimization and caching · p. 10
  • Amortized O(1) time per site suffices to calculate the H_i and N_i matrices needed for site optimization using cached recursive contractions.The first step takes O(N) time to initialize the recursion; subsequent steps take O(1) on average.Source: E. Energy minimization and caching · pp. 10–11

A. Motivation

A total ordering lets matrix product states represent operators on a two-dimensional grid as finite-state automata. For square-arranged four-site X terms, however, the required automaton grows with the grid width because information flows in only one direction.

A. Motivation

  • For the four-site X operator on a grid, the automaton's memory states track how many sites have passed since the second X so it can place the final two Xs on the next row.The number of states required grows with the number of columns.Source: A. Motivation · pp. 11–12, Figure 13 · p. 11
  • The resulting matrix product representation depends on grid size because information can flow in only one direction.Ordering the sweep down columns instead would make communication across a row costly; matrix product states cannot make both directions easy simultaneously.Source: A. Motivation · pp. 11–12

B. Tensor network diagrams

The diagrams represent tensor networks with shared indices and branching walks, while weighted finite signaling agents provide a way to encode such networks and generate operator factorizations on a grid.

B. Tensor network diagrams

  • Shared indices let one tensor connect observables to multiple others, representing entanglement that need not be limited to adjacent observables.A single index may be shared by three tensors, while a tensor without a physical-observable index can coordinate communication between observables.Source: B. Tensor network diagrams · p. 12
  • Diagram walks may branch into simultaneous paths, which must rejoin at the same node because each summed index takes one value per term.The network imposes a partial ordering of steps, not the total ordering found in one-dimensional diagrams.Source: B. Tensor network diagrams · pp. 12–13

C. Weighted finite signaling agents

  • Ordinary automata do not directly fit these diagrams because branching can leave the system in multiple states without specifying which input symbol comes next.At the C transition, the system chooses states from both j and k, while the next order between E and F is unspecified.Source: C. Weighted finite signaling agents · p. 13
  • Weighted finite signaling agents replace states with channels carrying signals, letting each site make a nondeterministic, weighted decision from incoming signals and an input symbol.Agents then send signals through output channels, accommodating network structure without combining branched indices into a potentially larger tensor.Source: C. Weighted finite signaling agents · p. 13
  • For the 4-X Hamiltonian, a grid of signaling agents accepts placements of four X operators and immediately yields a matrix product-style operator factorization.The resulting factorization is a projected entangled-pair operator, with tensors located at grid points and links representing summed indices.Source: C. Weighted finite signaling agents · pp. 13–14

D. Calculation of expectations using recursion

The section expresses expectation calculations and local energy minimization through transfer-matrix contractions and recursion. Adjacent-site updates reuse contractions, but general multidimensional contraction remains intractable for arbitrarily large networks.

D. Calculation of expectations using recursion

  • When the state and operator share a network structure, the expectation reduces to contracting transfer matrices; the same contraction supplies the eigenvalue problem for minimizing energy at a site.The transfer-matrix viewpoint also connects to contour correlated states, whose map transports information across a lattice region.Source: D. Calculation of expectations using… · p. 15
  • Moving between adjacent sites allows O(1) amortized work to calculate O_ij; for O_43, only C_33 and then L_43 need calculation.The recursion rules update left, right, and intermediate contractions from neighboring sites.Source: D. Calculation of expectations using… · p. 15
  • Arbitrarily large multidimensional tensor networks are intractable to contract, and row-by-row contraction can grow exponentially with network size.A lossy row-compression technique has been used to model hard-core bosons in a two-dimensional optical lattice [24].Source: D. Calculation of expectations using… · p. 15

IV. CONCLUSION

The paper connects matrix product states and operators with complex-weighted finite state automata, using that correspondence to construct operator factorizations. These factorizations support efficient expectation and energy-minimization calculations, with a generalization to multiple spatial dimensions.

IV. CONCLUSION

  • The paper introduces diagrams for matrix product states and establishes their formal equivalence, along with operators, to complex-weighted finite state automata.The automata perspective provides a way to construct matrix product operator factorizations.Source: §IV · p. 16
  • With an operator factorization, expectation calculations take O(N) time and energy minimization costs amortized O(1) per step.Source: §IV · p. 16
  • The procedure is generalized to systems with more than one spatial dimension.The authors also connect efficient physical-system simulation to automata hierarchy and formal language theory, suggesting further insights may be possible.Source: §IV · p. 16

APPENDIX A: VISUALIZATION OF MATRIX DEGREE OF FREEDOM

Matrix product representations have a degree of freedom that permits equivalent diagram transformations by inserting inverse matrices. Periodic boundaries change the automata acceptance rule, and the idea extends to multiple dimensions.

APPENDIX A: VISUALIZATION OF MATRIX DEGREE OF FREEDOM

  • Inserting a matrix X and its inverse X^-1 into the product preserves the represented state and enables equivalent diagram forms.The appendix illustrates rearranging nodes, adding canceling paths, and inserting empty nodes when dimensions permit.Source: Appendix A · pp. 16–17

APPENDIX B: PERIODIC BOUNDARY CONDITIONS

  • Periodic boundary conditions add a bond between the first and last matrices.Source: Appendix B · p. 17
  • For periodic boundaries, the automaton may start in any state but accepts only strings that return it to its starting state, replacing initial and final distributions with a trace.The corresponding function is the trace of the product of the string's transition matrices.Source: Appendix B · p. 17
  • The periodic-boundary construction generalizes to multiple dimensions.Source: Appendix B · p. 17
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