Faster randomized partial trace estimation

We develop randomized matrix-free algorithms for estimating partial traces. Our algorithm improves on the typicality-based approach used in [T. Chen and Y-C. Cheng, Numerical computation of the equilibrium-reduced density matrix for strongly coupled open quantum systems, J. Chem. Phys. 157, 064106 (2022)] by deflating important subspaces (e.g. corresponding to the low-energy eigenstates) explicitly. This results in a significant variance reduction for matrices with quickly decaying singular values. We then apply our algorithm to study the thermodynamics of several Heisenberg spin systems, particularly the entanglement spectrum and ergotropy

Estimating the trace of matrix functions with application to complex networks

The approximation of trace(f(Ω)), where f is a function of a symmetric matrix Ω, can be challenging when Ω is exceedingly large. In such a case even the partial Lanczos decomposition of Ω is computationally demanding and the stochastic method investigated by Bai et al. (J. Comput. Appl. Math. 74:71–89, 1996) is preferred. Moreover, in the last years, a partial global Lanczos method has been shown to reduce CPU time with respect to partial Lanczos decomposition. In this paper we review these techniques, treating them under the unifying theory of measure theory and Gaussian integration. This allows generalizing the stochastic approach, proposing a block version that collects a set of random vectors in a rectangular matrix, in a similar fashion to the partial global Lanczos method. We show that the results of this technique converge quickly to the same approximation provided by Bai et al. (J. Comput. Appl. Math. 74:71–89, 1996), while the block approach can leverage the same computational advantages as the partial global Lanczos. Numerical results for the computation of the Von Neumann entropy of complex networks prove the robustness and efficiency of the proposed block stochastic method

Computation of the von Neumann entropy of large matrices via trace estimators and rational Krylov methods

We consider the problem of approximating the von Neumann entropy of a large, sparse, symmetric positive semidefinite matrix $A$, defined as $\operatorname{tr}(f(A))$ where $f(x)=-x\log x$. After establishing some useful properties of this matrix function, we consider the use of both polynomial and rational Krylov subspace algorithms within two types of approximations methods, namely, randomized trace estimators and probing techniques based on graph colorings. We develop error bounds and heuristics which are employed in the implementation of the algorithms. Numerical experiments on density matrices of different types of networks illustrate the performance of the methods.Comment: 32 pages, 10 figure

Enhanced estimation of quantum properties with common randomized measurements

We present a technique for enhancing the estimation of quantum state properties by incorporating approximate prior knowledge about the quantum state of interest. This method involves performing randomized measurements on a quantum processor and comparing the results with those obtained from a classical computer that stores an approximation of the quantum state. We provide unbiased estimators for expectation values of multi-copy observables and present performance guarantees in terms of variance bounds which depend on the prior knowledge accuracy. We demonstrate the effectiveness of our approach through numerical experiments estimating polynomial approximations of the von Neumann entropy and quantum state fidelities

Just SLaQ When You Approximate: Accurate Spectral Distances for Web-Scale Graphs

Graph comparison is a fundamental operation in data mining and information retrieval. Due to the combinatorial nature of graphs, it is hard to balance the expressiveness of the similarity measure and its scalability. Spectral analysis provides quintessential tools for studying the multi-scale structure of graphs and is a well-suited foundation for reasoning about differences between graphs. However, computing full spectrum of large graphs is computationally prohibitive; thus, spectral graph comparison methods often rely on rough approximation techniques with weak error guarantees. In this work, we propose SLaQ, an efficient and effective approximation technique for computing spectral distances between graphs with billions of nodes and edges. We derive the corresponding error bounds and demonstrate that accurate computation is possible in time linear in the number of graph edges. In a thorough experimental evaluation, we show that SLaQ outperforms existing methods, oftentimes by several orders of magnitude in approximation accuracy, and maintains comparable performance, allowing to compare million-scale graphs in a matter of minutes on a single machine.Comment: To appear at TheWebConf (WWW) 202

Numerical computation of the equilibrium-reduced density matrix for strongly coupled open quantum systems

We describe a numerical algorithm for approximating the equilibrium-reduced density matrix and the effective (mean force) Hamiltonian for a set of system spins coupled strongly to a set of bath spins when the total system (system+bath) is held in canonical thermal equilibrium by weak coupling with a "super-bath". Our approach is a generalization of now standard typicality algorithms for computing the quantum expectation value of observables of bare quantum systems via trace estimators and Krylov subspace methods. In particular, our algorithm makes use of the fact that the reduced system density, when the bath is measured in a given random state, tends to concentrate about the corresponding thermodynamic averaged reduced system density. Theoretical error analysis and numerical experiments are given to validate the accuracy of our algorithm. Further numerical experiments demonstrate the potential of our approach for applications including the study of quantum phase transitions and entanglement entropy for long-range interaction systems