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Efficient time-evolution of matrix product states using average Hamiltonians

Belal Abouraya, Jirawat Saiphet, Fedor Jelezko, Ressa S. Said

TL;DR

This work proposes a simple, yet efficient, method to augment the already available MPS algorithms to simulate the dynamics of time-dependent Hamiltonians with better accuracy and a faster convergence rate, giving a second-order convergence compared to the first-order convergence of the standard method.

Abstract

Simulating quantum many-body systems (QMBS) is one of the long-standing, highly non-trivial challenges in condensed matter physics and quantum information due to the exponentially growing size of the system's Hilbert space. To date, tensor networks have been an essential tool for studying such quantum systems, owing to their ability to efficiently capture the entanglement properties of the systems they represent. One of the well-known tensor network architectures, namely matrix product states (MPS), is the standard method for simulating one-dimensional QMBS. Here, we propose a simple, yet efficient, method to augment the already available MPS algorithms to simulate the dynamics of time-dependent Hamiltonians with better accuracy and a faster convergence rate, giving a second-order convergence compared to the first-order convergence of the standard method. We apply our proposed method to simulate the dynamics of a chain of single spins associated with nitrogen-vacancy color centers in diamonds, which has potential applications for practical and scalable quantum technologies, and find that our method improves the average error for a system of few NV centers by a factor of about 1000 for moderate step sizes. Our work paves the way for efficient simulation of QMBS under the influence of time-dependent Hamiltonians.

Efficient time-evolution of matrix product states using average Hamiltonians

TL;DR

This work proposes a simple, yet efficient, method to augment the already available MPS algorithms to simulate the dynamics of time-dependent Hamiltonians with better accuracy and a faster convergence rate, giving a second-order convergence compared to the first-order convergence of the standard method.

Abstract

Simulating quantum many-body systems (QMBS) is one of the long-standing, highly non-trivial challenges in condensed matter physics and quantum information due to the exponentially growing size of the system's Hilbert space. To date, tensor networks have been an essential tool for studying such quantum systems, owing to their ability to efficiently capture the entanglement properties of the systems they represent. One of the well-known tensor network architectures, namely matrix product states (MPS), is the standard method for simulating one-dimensional QMBS. Here, we propose a simple, yet efficient, method to augment the already available MPS algorithms to simulate the dynamics of time-dependent Hamiltonians with better accuracy and a faster convergence rate, giving a second-order convergence compared to the first-order convergence of the standard method. We apply our proposed method to simulate the dynamics of a chain of single spins associated with nitrogen-vacancy color centers in diamonds, which has potential applications for practical and scalable quantum technologies, and find that our method improves the average error for a system of few NV centers by a factor of about 1000 for moderate step sizes. Our work paves the way for efficient simulation of QMBS under the influence of time-dependent Hamiltonians.
Paper Structure (1 section, 20 equations, 3 figures, 2 tables)

This paper contains 1 section, 20 equations, 3 figures, 2 tables.

Table of Contents

  1. Results for two NV centers

Figures (3)

  • Figure 1: (a) One-dimensional tensor networks of a quantum state, and (b) the corresponding Hamiltonian representation. (c) A chain of nitrogen-vacancy color centers in diamond. The blue and red spheres represent a pair of nitrogen and vacancy (NV), tetrahedrally coordinated in the carbon lattice having an axial trigonal $C_{3V}$ symmetry. The negatively-charged NV (denoted simply as the $NV$ throughout the paper), has an extra electron located at the vacancy site forming a spin $(S=1)$ pair with one of the vacancy electrons, and therefore inducing spin triplet ground states. The spin chain is due to nearest-neighbor interactions of dipolar coupling (not shown in the figure), where each spin can have one out of four different quantization axes.
  • Figure 2: Numerical performances for various system sizes of the NV spin chain. The upper panel represents the average (a) error, (b) error ratio, and (c) runtime of Riemann and Simpson steppers versus the number of integration steps $N_s$, for a system of three NV centers. The lower panel amounts to the corresponding quantities, as the number of NV centers $N$ increases: (d) the average error, (e) the average error ratio, and (f) the average runtime for both steppers. The average error ratio is defined as the ratio of the average error resulting from the Riemann stepper ($E_r$) divided by the average error resulting from the Simpson stepper ($E_s)$.
  • Figure 3: Simulation results for two NV centers. (a) Average error $E$ of the Riemann and Simpson steppers as a function of the number of steps $N_s$. (b) Average error ratio $E_r/E_s$ between the two steppers versus $N_s$. (c) Average runtime $T$ of both steppers as $N_s$ increases.