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Particle number conservation and block structures in matrix product states

Markus Bachmayr, M Götte, Max J. Pfeffer

Published May 25, 2022• 6 citations
Publisher

Abstract

Abstract The eigenvectors of the particle number operator in second quantization are characterized by the block sparsity of their matrix product state representations. This is shown to generalize to other classes of operators. Imposing block sparsity yields a scheme for conserving the particle number that is commonly used in applications in physics. Operations on such block structures, their rank truncation, and implications for numerical algorithms are discussed. Explicit and rank-reduced matrix product operator representations of one- and two-particle operators are constructed that operate only on the non-zero blocks of matrix product states.

DOI
10.1007/s10092-022-00462-9

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Abstract The eigenvectors of the particle number operator in second quantization are characterized by the block sparsity of their matrix product state representations. This is shown to generalize to other classes of operators. Imposing block sparsity yields a scheme for conserving the particle number that is commonly used in applications in physics. Operations on such block structures, their rank truncation, and implications for numerical algorithms are discussed. Explicit and rank-reduced matrix product operator representations of one- and two-particle operators are constructed that operate only on the non-zero blocks of matrix product states.
The eigenvectors of the particle number operator in second quantization are characterized by the block sparsity of their matrix product state representations. This is shown to generalize to other classes of operators. Imposing block sparsity yields a scheme for conserving the particle number that is commonly used in applications in physics. Operations on such block structures, their rank truncation, and implications for numerical algorithms are discussed. Explicit and rank-reduced matrix product operator representations of one- and two-particle operators are constructed that operate only on the non-zero blocks of matrix product states.
We consider an iterative eigensolver for Schrödinger equations that constructs low-rank approximations of eigenfunctions with accuracy-adapted ranks, with particular focus on fermionic Schrödinger equations in second-quantized form and on matrix product state approximations enforcing particle number conservation. We provide a complete analysis of a solver based on preconditioned inverse iteration combined with rank truncation and propose a generalization to subspace iteration for the joint approximation of several eigenspaces. The practical performance of the method is illustrated by numerical tests for several model problems.
Incorporating conservation laws explicitly into matrix product states (MPSs) has proven to make numerical simulations of quantum many-body systems much less resource consuming. We will discuss here to what extent this concept can be used in simulation where the dynamically evolving entities are matrix product operators (MPOs). Quite counter-intuitively the expectation of gaining in speed by sacrificing information about all but a single symmetry sector is not in all cases fulfilled. It turns out that often the entanglement imposed by the global constraint of fixed particle number is the limiting factor.
While quantum algorithms for solving large scale systems of linear equations offer potentially exponential speedups, their application has largely been confined to sparse matrices. This work extends the scope of these algorithms to a broad class of structured dense matrices arise in potential theory, covariance modeling, and computational physics, namely, hierarchically block separable (HBS) matrices. We develop two distinct methods to make these systems amenable to quantum solvers. The first is a pre-processing approach that transforms the dense matrix into a larger but sparse format. The second is a direct block encoding scheme that recursively constructs the necessary oracles from the HBS structure. We provide a detailed complexity analysis and rigorous error bounds for both methods. Numerical experiments are presented to validate the effectiveness of our approaches.
Matrix product states (MPS) serve as a key tool for studying quantum systems from chemistry and condensed-matter physics, making their preparation on quantum computers an important task in interfacing classical and quantum simulation. Many systems of interest have U(1)U(1)U(1)-symmetries induced by particle number and spin projection conservation, allowing to restrict the MPS tensors to be of block-sparse form, a property widely used in the implementation of classical algorithms such as the density matrix renormalization group. We reduce the cost of fault-tolerantly preparing block-sparse MPS within the standard ancilla-assisted linear-depth approach by implementing row and column permutations that transform the block-sparse matrices into block-diagonal form. These block-diagonal unitaries are then implemented via unitary synthesis, with the cost being determined by the size of the largest block. In this context, we modify the unitary synthesis approach of Berry et al. in order to reduce the Toffoli cost for real-valued unitaries by a factor of 2\sqrt{2}2​. In numerical benchmarks, we achieve Toffoli cost improvement factors of 10−3010 - 3010−30 compared to the state-of-the-art for MPS of various molecular systems.
We reconstruct a matrix product state (MPS) in reduced spaces using density matrix. This scheme applies to a MPS built on a blocked quantum lattice. Each block contains NNN physical sites that have a local space of rank RRR. The simulation in the original spaces of rank RNR^NRN is used to construct density matrices for every block. They are diagonalized and only the eigenvectors corresponding to significant diagonal elements are used to transform the original spaces to smaller ones and to reconstruct the MPS in those smaller spaces accordingly. Simulations in the reduced spaces are used to reliably extrapolate the result in unreduced spaces. Moreover, to obtain a required accuracy, the ratio of the reduced space rank over the original decreases quickly with NNN. The reduced space has a saturated rank to obtain a demanded accuracy when N→∞N\rightarrow \inftyN→∞.