Locally purified density operators for noisy quantum circuits

TL;DR

This paper investigates the effectiveness of locally purified density operators (LPDOs) in simulating noisy quantum circuits, revealing their strengths in the quantum region and limitations at critical points, with MPOs offering an alternative for entanglement analysis.

Contribution

It introduces a unified method for managing LPDOs in noisy quantum circuit simulation and compares their performance with MPOs, highlighting their respective advantages and limitations.

Findings

LPDOs effectively describe mixed states in quantum regions.

LPDOs face challenges at the quantum-classical critical point.

MPOs successfully track entanglement trends during simulation.

Abstract

Simulating open quantum systems is essential for exploring novel quantum phenomena and evaluating noisy quantum circuits. In this Letter, we address the problem of whether mixed states generated from noisy quantum circuits can be efficiently represented by locally purified density operators (LPDOs). We map an LPDO of $N$ qubits to a pure state of size $2\times N$ defined on a ladder and introduce a unified method for managing virtual and Kraus bonds. We numerically simulate noisy random quantum circuits with depths up to $d=40$ using fidelity and entanglement entropy as accuracy measures. LPDO representation proves to be effective in describing mixed states in both quantum and classical regions but encounters significant challenges at the quantum-classical critical point, limiting its applicability to the quantum region exclusively. In contrast, the matrix product operator (MPO) successfully characterizes the entanglement trend throughout the simulation, while truncation in MPOs breaks the positivity condition required for a physical density matrix. This work advances our understanding of efficient mixed-state representation in open quantum systems and provides insights into the entanglement structure of noisy quantum circuits.