Successive randomized compression: A randomized algorithm for the compressed MPO-MPS product

TL;DR

<3-5 sentence high-level summary> SRC introduces a single-pass, randomized algorithm for compressing MPO–MPS products by constructing the output MPS site-by-site from right to left using a common set of random test matrices. The method leverages randomized QB approximations with a Khatri–Rao constructed test matrix and reuses randomness to keep computations local and efficient, achieving near-optimal accuracy with a favorable time complexity and no iterative convergence. The paper provides extensive comparisons to existing contract-then-compress, optimization, and explicit construction methods, demonstrating substantial speedups and competitive accuracy, including an application to unitary time evolution via GSE-TDVP1. It also develops practical tools for adaptive bond-dimension selection, error estimation, and QR factorization updates, broadening the method’s applicability to large-scale tensor-network simulations.

Abstract

Tensor networks like matrix product states (MPSs) and matrix product operators (MPOs) are powerful tools for representing exponentially large states and operators, with applications in quantum many-body physics, machine learning, numerical analysis, and other areas. In these applications, computing a compressed representation of the MPO--MPS product is a fundamental computational primitive. For this operation, this paper introduces a new single-pass, randomized algorithm, called successive randomized compression (SRC), that improves on existing approaches in speed or in accuracy. The performance of the new algorithm is evaluated on synthetic problems and unitary time evolution problems for quantum spin systems.

Figures (10)

• Figure 1: (MPO--MPS contraction algorithm comparison). Comparison of relative error versus computation time of several algorithms for the MPO--MPS product of $n = 100$ tensor networks with bond dimension $D = \chi = 50$. The maximum bond dimension $\overline{\chi}$ was varied between $5$ and $100$, with the compressibility difficulty parameter $\alpha$ set to $-0.5$. The fitting (variational) method was run for a single left-to-right sweep; more sweeps led to higher computational cost at no appreciable gain in accuracy. Randomized C-T-C is the contract-then-compress method with randomized MPS rounding (see \ref{['sec:RCTC']}). Each data point represents the mean of five independent runs.
• Figure 2: (Fitting failure mode). Runtime (left) and relative error (right) of the fitting algorithm, SRC, and contract-then-compress on long range XY Hamiltonian given in \ref{['sec:TDVP_exp']}.
• Figure 3: (Fixed bond dimension). Runtime (left) and relative error (right) of several MPO--MPS multiplication algorithms for a synthetic problem as a function of the output bond dimension $\overline{\chi}$, as described in \ref{['sec:Fixed dimension']}.
• Figure 4: (Accuracy threshold experiment). Runtime (left), relative error (center), and final bond dimension (right) of several MPO--MPS multiplication algorithms for a synthetic problem as a specified approximation threshold.
• Figure 5: (TDVP unitary time evolution). Left: Magnetization difference \ref{['eq:magnetization_difference']} at each position $1\le i\le 101$ and time $0\le tJ\le 5$ computed by GSE-TDVP1 with randomized MPO--MPS multiplication, showing a light cone with power law leakage. Middle and right: Total runtime (middle) and runtime spent during Krylov vector construction (right) for different MPO--MPS methods per timestep.

Theorems & Definitions (5)

• Theorem 1: Randomized QB approximation: Gaussian
• Theorem 2: Randomized QB approximation: Khatri--Rao
• Theorem 3: Successive randomized compression: Exact recovery
• Theorem 4: Leave-one-out error estimation
• Proposition 5: Norm estimation