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arXiv:2605.01494

Completely Positive and Trace Preserving Schemes with Tensor Train Compression for the Lindblad Equation

Peter DelMastro, Daniel Appelö, Yingda Cheng

math.NAquant-phSubmitted May 2, 2026• Announced May 5, 2026• 0 citations
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Abstract

We propose a family of low-rank, completely positive and trace preserving schemes for the Lindblad equation, a common model for open quantum systems. Low-rank representation is employed at two levels: the density matrix is factorized into the product of tall-skinny matrices, and the columns of these matrices are further represented using the tensor train (TT) format, also know as matrix product states (MPS). This two-level low-rank format fits naturally into our existing Kraus is King scheme (arXiv:2409.08898v2 [math.NA]) for the Lindblad equation, whose underlying operations are arithmetic on the columns of the tall-skinny matrices. We show how these operations can be performed efficiently in the TT/MPS format, with particular emphasis on density matrix rank-truncation. We conclude with extensive numerical experiments demonstrating the convergence of this scheme and its efficiency in simulating systems with up to $10^{19}$ degrees of freedom using only modest compute resources.

AI Summary

The paper addresses deterministic Lindblad simulation when density matrices have low rank but their state vectors are too large to store densely. It extends the existing “Kraus is King” integrator using a two-level representation: $\rho=VV^\dagger$, with each column of $V$ stored independently as a tensor train/matrix product state (TT/MPS). The proposed schemes are described as completely positive and trace preserving; positivity follows from the factorized representation, and trace is enforced by normalization. The central construction compresses the density-matrix rank using the Gram matrix $X^\dagger X$, avoiding TT-based QR factorization. Retained factors are reconstructed through approximate MPS linear combinations. An error bound separates discarded singular-value weight from tensor-arithmetic error: $\|XX^\dagger-\tilde X\tilde X^\dagger\|_F\leq\sum_{i>r}\sigma_i^2+\sum_{i\leq r}(\|e_i\|_2^2+2\sigma_i\|e_i\|_2)$. Norm screening, shared-core contractions for local jump operators, and reused randomized sketches reduce compression costs. Numerical tests demonstrate second- and fourth-order convergence against exact evolution for six spins and step-refinement convergence for 64 spins. The latter example has Hilbert-space dimension $2^{64}>10^{19}$, starts with two excitations, and maintains MPS bond dimensions at most five. Further demonstrations simulate a 25-qubit heavy-hex mock SWAP circuit and six four-level qudits coupled through five ten-level resonators; the latter retains density-matrix rank below 30. Efficiency depends on low density-matrix rank and compressible states and propagators. Compression remains the main bottleneck. Error-budget allocation and randomized sketch sizes use heuristics, with randomization error excluded from the stated rounding estimate. Time-dependent controls are approximated piecewise constantly.

No sign of AI-generated textIncrementalMissing comparisonOverclaimsPossible error

Central claim may not hold: The compression rule is generally nonlinear, and constructing an input-dependent Kraus operator does not prove that it is a completely positive linear map. Dividing each output by its computed trace generally makes the update nonlinear, so positive, normalized outputs do not establish the advertised CPTP-channel property. The randomized compression algorithm advertises an unconditional error bound while its tolerance allocation explicitly omits randomization error without bounding or verifying it. The CP proof incorrectly assumes X^+X = I for potentially rank-deficient factor matrices, and its proposed Kraus operator can therefore fail even to reproduce the specified output. The stated Frobenius truncation-error equality is incorrect: the exact error is sqrt(sum_i sigma_i^4), although the stated sum_i sigma_i^2 is a valid conservative bound.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

Comments
33 pages, 9 figures

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We propose a family of low-rank, completely positive and trace preserving schemes for the Lindblad equation, a common model for open quantum systems. Low-rank representation is employed at two levels: the density matrix is factorized into the product of tall-skinny matrices, and the columns of these matrices are further represented using the tensor train (TT) format, also know as matrix product states (MPS). This two-level low-rank format fits naturally into our existing Kraus is King scheme (arXiv:2409.08898v2 [math.NA]) for the Lindblad equation, whose underlying operations are arithmetic on the columns of the tall-skinny matrices. We show how these operations can be performed efficiently in the TT/MPS format, with particular emphasis on density matrix rank-truncation. We conclude with extensive numerical experiments demonstrating the convergence of this scheme and its efficiency in simulating systems with up to 101910^{19}1019 degrees of freedom using only modest compute resources.
The paper addresses deterministic Lindblad simulation when density matrices have low rank but their state vectors are too large to store densely. It extends the existing “Kraus is King” integrator using a two-level representation: ρ=VV†\rho=VV^\daggerρ=VV†, with each column of VVV stored independently as a tensor train/matrix product state (TT/MPS). The proposed schemes are described as completely positive and trace preserving; positivity follows from the factorized representation, and trace is enforced by normalization.
The central construction compresses the density-matrix rank using the Gram matrix X†XX^\dagger XX†X, avoiding TT-based QR factorization. Retained factors are reconstructed through approximate MPS linear combinations. An error bound separates discarded singular-value weight from tensor-arithmetic error: ∥XX†−X~X~†∥F≤∑i>rσi2+∑i≤r(∥ei∥22+2σi∥ei∥2)\|XX^\dagger-\tilde X\tilde X^\dagger\|_F\leq\sum_{i>r}\sigma_i^2+\sum_{i\leq r}(\|e_i\|_2^2+2\sigma_i\|e_i\|_2)∥XX†−X~X~†∥F​≤∑i>r​σi2​+∑i≤r​(∥ei​∥22​+2σi​∥ei​∥2​). Norm screening, shared-core contractions for local jump operators, and reused randomized sketches reduce compression costs.
Numerical tests demonstrate second- and fourth-order convergence against exact evolution for six spins and step-refinement convergence for 64 spins. The latter example has Hilbert-space dimension 264>10192^{64}>10^{19}264>1019, starts with two excitations, and maintains MPS bond dimensions at most five. Further demonstrations simulate a 25-qubit heavy-hex mock SWAP circuit and six four-level qudits coupled through five ten-level resonators; the latter retains density-matrix rank below 30.
Efficiency depends on low density-matrix rank and compressible states and propagators. Compression remains the main bottleneck. Error-budget allocation and randomized sketch sizes use heuristics, with randomization error excluded from the stated rounding estimate. Time-dependent controls are approximated piecewise constantly.
Numerical simulation of quantum computing hardware and open quantum systems governed by the Lindblad equation is challenging due to the high dimensionality of the density matrix and the need to preserve fundamental physical properties. In our previous work, we developed an arbitrary-order, low-rank, completely positive and trace preserving (CPTP) method for the Lindblad equation with time-dependent Hamiltonians by nested Picard iteration (NPI). In this work, we develop Gregory NPI schemes, which are CPTP schemes constructed by Gregory-type quadrature on equispaced nodes. The methods, which are of order up to nine, substantially reduce the computational cost compared to our previously proposed NPI schemes with Gaussian quadrature rules, while retaining high-order accuracy and structure preservation. We analyze the stability of the resulting scheme for a physics-based test equation. Numerical experiments verify the convergence of the method and demonstrate the effectiveness of the low-rank approximation. We study the performance of a previously constructed CNOT gate for both closed and open quantum systems.
Abstract. The Lindblad equation is a widely used quantum master equation to model the dynamical evolution of open quantum systems whose states are described by density matrices. These solution matrices are characterized by semipositiveness and trace preserving properties, which must be guaranteed in any physically meaningful numerical simulation. In this paper, novel full- and low-rank exponential Euler integrators are developed for approximating the Lindblad equation that preserve positivity and trace unconditionally. Theoretical results are presented that provide sharp error estimates for the two classes of exponential integration methods. Results of numerical experiments are discussed that illustrate the effectiveness of the proposed schemes, beyond present state-of-the-art capabilities.
Quantum state tomography is a fundamental technique for estimating the state of a quantum system from measured data and plays a crucial role in evaluating the performance of quantum devices. However, standard estimation methods become computationally prohibitive as the system size increases due to the exponential growth of the density matrix, describing a quantum state, with the number of qubits. We propose a low-rank tensor-network framework for mixed-state quantum state tomography based on a block tensor train (Block-TT) factorization. Specifically, the density matrix is represented as the contraction of a Block-TT with its Hermitian transpose, yielding a TT analogue of the Burer-Monteiro factorization. This parameterization guarantees Hermiticity and positive semidefiniteness by construction while compressing the number of optimization variables from exponential to linear in the number of qubits. Building on this representation, we develop single-site and two-site density matrix renormalization group (DMRG) algorithms for estimating quantum states from compressed measurements. The resulting methods operate directly on the compressed parameterization, support adaptive rank refinement, and exploit efficient tensor-network contractions for expectation-value evaluation. The framework is applicable to a broad class of low-rank quantum states, including pure states, nearly pure states, and ground states that admit accurate tensor-network approximations. Numerical experiments demonstrate accurate state reconstruction from limited measurements together with substantial reductions in memory requirements and computational cost compared with conventional low-rank tomography methods.
We investigate quantum-inspired tensor networks (QTNs) for approximating flow maps of hydrodynamic partial differential equations (PDEs). Motivated by the effective low-rank structure that emerges after tensorization of discretized transport and diffusion dynamics, we encode PDE states as matrix product states (MPS) and represent the evolution operator as a structured low-rank matrix product operator (MPO) in tensor-train form (e.g., arising from finite-difference discretizations assembled in MPO form). The MPO is applied directly in MPS form, and rank growth is controlled via canonicalization and SVD-based truncation after each step. We provide theoretical context through standard matrix product properties, including exact MPS representability bounds, local optimality of SVD truncation, and a Lipschitz-type multi-step error propagation estimate. Experiments on one- and two-dimensional linear advection-diffusion and nonlinear viscous Burgers equations demonstrate accurate short-horizon prediction, favorable scaling in smooth diffusive regimes, and error growth in nonlinear multi-step predictions.
Matrix Product States (MPS), also known as Tensor Train (TT) decomposition in mathematics, has been proposed originally for describing an (especially one-dimensional) quantum system, and recently has found applications in various applications such as compressing high-dimensional data, supervised kernel linear classifier, and unsupervised generative modeling. However, when applied to systems which are not defined on one-dimensional lattices, a serious drawback of the MPS is the exponential decay of the correlations, which limits its power in capturing long-range dependences among variables in the system. To alleviate this problem, we propose to introduce long-range interactions, which act as shortcuts, to MPS, resulting in a new model \textit{ Shortcut Matrix Product States} (SMPS). When chosen properly, the shortcuts can decrease significantly the correlation length of the MPS, while preserving the computational efficiency. We develop efficient training methods of SMPS for various tasks, establish some of their mathematical properties, and show how to find a good location to add shortcuts. Finally, using extensive numerical experiments we evaluate its performance in a variety of applications, including function fitting, partition function calculation of 2−2-2−d Ising model, and unsupervised generative modeling of handwritten digits, to illustrate its advantages over vanilla matrix product states.