Error Interference in Quantum Simulation

TL;DR

A novel method is introduced that directly estimates the long-time algorithmic errors with multiple segments and introduces the concept of approximate error interference, which is more broadly applicable to scenarios such as power-law interaction models, the Fermi-Hubbard model, and higher-order Trotter formulas.

Abstract

Understanding algorithmic error accumulation in quantum simulation is crucial due to its fundamental significance and practical applications in simulating quantum many-body system dynamics. Conventional theories typically apply the triangle inequality to provide an upper bound for the error. However, these often yield overly conservative and inaccurate estimates as they neglect error interference -- a phenomenon where errors in different segments can destructively interfere. Here, we introduce a novel method that directly estimates the long-time algorithmic errors with multiple segments, thereby establishing a comprehensive framework for characterizing algorithmic error interference. We identify the sufficient and necessary condition for strict error interference and introduce the concept of approximate error interference, which is more broadly applicable to scenarios such as power-law interaction models, the Fermi-Hubbard model, and higher-order Trotter formulas. Our work demonstrates significant improvements over prior ones and opens new avenues for error analysis in quantum simulation, offering potential advancements in both theoretical algorithm design and experimental implementation of Hamiltonian simulation.

Figures (4)

• Figure 1: The total error of the simulation can be illustrated by a walk on the plane (a) When the error does not interfere, i.e., $\delta \Lambda$ is the major error term, then the error accumulates linearly as the trotter step grows, the triangle inequality gives a tight bound. (b) When the error interferes, i.e., $\delta P$ is the major error term, then only the first error contributes to the error, the error accumulates sublinearly, the triangle inequality gives a loose bound.
• Figure 2: Trotter error VS evolution time $t$ with fixed Trotter step $r=10000$. (a) The 1D nearest-neighbor Heisenberg model \ref{['eq:heisenberg']} of 8 qubits with parameters $J_x=J_y=J_z=2$, $h=0.5$. The red dots are the empirical PF1 Trotter error with the XYZ-group and the red solid line is our interference bound. The grey dashed line is plotted proportionally to $t$ as a guide for the eye. The empirical error and the triangle bound of PF2 with the even-odd EO-group are also plotted for reference. (b) The 1D Fermi-Hubbard Hamiltonian \ref{['eq:hubbard']} naturally partitioned into three groups \ref{['eq:hubbard_group']}. We choose the Hamiltonian of 4 sites (represented by 8 qubits) and set the interaction parameters as $u=-v=1$. The PF1 empirical error and bounds are plotted in red similar to (a), while the PF2 empirical error and the triangle bound are plotted for comparison.
• Figure 3: Numerical evaluation of a number of Trotter steps $r$ needed to achieve the fixed error threshold $\epsilon=0.01$ according to different bounds. The simulated quantum system is the 1D lattice power-law Heisenberg model without external magnetic field and the decaying coefficient $\alpha = 4$. The simulation time is 30 times the system size. The empirical curve suggests an $\mathcal{O}(n^{2.58})$ scaling of Trotter steps, which is close to our theoretical interference bound $\mathcal{O}(n^{2.78})$. While the estimate derived from the triangle inequality suggests that the required number of Trotter steps is $\mathcal{O}(n^{3.17})$.
• Figure 4: Error interference of both PF1 and PF2 with the XZ-group when the transverse-field Ising (TFI) model with parameters $J=2$ and $h=0.001$. The empirical error of PF2 with the even-odd EO-group is plotted for reference.

Theorems & Definitions (27)

• Definition 1: Error interference, informal
• Definition 2: Orthogonality condition
• Theorem 1: Necessary and sufficient condition for error interference
• Corollary 1: Error lower bound
• Corollary 2
• Corollary 3
• Theorem 2
• Theorem 3: Approximate error interference of PF2 of two terms when one term is the major term
• Theorem 4: Tight upper bound for general error interference
• Theorem 5: Tight upper bound for PF1