[Paper Review] Efficient Computation of the Permanent of Block Factorizable Matrices

Kristan Temme, Paweł Wocjan|arXiv (Cornell University)|Aug 31, 2012

Matrix Theory and Algorithms24 references3 citations

TL;DR

This paper presents a polynomial-time algorithm for computing the permanent of block factorizable matrices—matrices expressible as a product of L block-diagonal matrices with 1×1 or 2×2 blocks. By mapping the permanent to a quantum circuit transition amplitude and leveraging matrix product states (MPS), the method achieves time complexity O(N·2^{3L²}) and space O(N·2^{2L²}), enabling efficient exact computation for matrices with banded structure and banded inverses.

ABSTRACT

We present an efficient algorithm for computing the permanent for matrices of size N that can written as a product of L block diagonal matrices with blocks of size at most 2. For fixed L, the time and space resources scale linearly in N, with a prefactor that scales exponentially in L. This class of matrices contains banded matrices with banded inverse. We show that such a factorization into a product of block diagonal matrices gives rise to a circuit acting on a Hilbert space with a tensor product structure and that the permanent is equal to the transition amplitude of this circuit and a product basis state. In this correspondence, a block diagonal matrix gives rise to one layer of the circuit, where each block to a gate acting either on a single tensor component or on two adjacent tensor components. This observation allows us to adopt matrix product states, a computational method from condensed matter physics and quantum information theory used to simulate quantum systems, to evaluate the transition amplitude.

Motivation & Objective

• To develop an efficient exact algorithm for computing the permanent of a class of structured matrices beyond the #P-hard general case.
• To address the computational intractability of the permanent by exploiting matrix factorization into block-diagonal components with small blocks.
• To establish a connection between the permanent and quantum circuit amplitudes, enabling the use of tensor network methods from quantum many-body physics.
• To demonstrate that matrices with banded structure and banded inverses admit such factorizations, enabling efficient permanent computation.

Proposed method

• The permanent is expressed as the transition amplitude of a quantum circuit, where each block-diagonal factor corresponds to a circuit layer with single- or two-qubit gates.
• Matrix product states (MPS) are used to represent and efficiently update the quantum state after each circuit layer, tracking the evolution of the system.
• The algorithm applies a sequential update of MPS tensors, increasing bond dimension at each layer, with the maximum bond dimension bounded by O(2^{L²}).
• The final permanent is computed as the scalar product of the evolved MPS with the computational basis state |1⟩^⊗N, evaluated via a contraction of MPS tensors.
• The method leverages commutativity of blocks within each factor to allow parallel gate application, reducing circuit depth and improving efficiency.
• The runtime and space bounds are derived from the growth of the bond dimension and the cost of tensor contractions, yielding O(N·2^{3L²}) time and O(N·2^{2L²}) space.

Experimental results

Research questions

• RQ1 Can the permanent of matrices with special structure be computed exactly in polynomial time, despite its #P-hardness?
• RQ2 Does a factorization into small block-diagonal matrices enable efficient permanent computation via quantum-inspired tensor network methods?
• RQ3 Can matrix product states be used to evaluate quantum circuit amplitudes that correspond to the permanent of a matrix?
• RQ4 What is the computational complexity of computing the permanent for matrices that are products of block-diagonal matrices with blocks of size at most 2?
• RQ5 How does the bond dimension growth in the MPS representation scale with the number of factorization layers L?

Key findings

• The algorithm computes the permanent of block factorizable matrices in time O(N·2^{3L²}) and space O(N·2^{2L²}), with linear scaling in matrix size N.
• The method applies matrix product states to efficiently evaluate the transition amplitude of a quantum circuit whose evolution corresponds to the matrix product A = F₁F₂⋯Fₗ.
• The bond dimension D_max grows at most as O(2^{L²}), which determines the overall runtime and space complexity.
• Any matrix A that admits such a factorization must be banded with bandwidth at most L, and if invertible, its inverse is also banded with the same bandwidth.
• The algorithm is applicable to matrices with banded structure and banded inverses, including tridiagonal and banded Toeplitz matrices, as such matrices can be decomposed into O(L) block-diagonal factors.
• The bounds are worst-case estimates; in practice, performance may be better when many singular values in the MPS truncation are negligible.