Enabling large-depth simulation of noisy quantum circuits with positive tensor networks

TL;DR

This paper introduces a method to reduce the complexity of tensor network representations of noisy quantum states, enabling large-depth simulations by optimizing the environment basis and maintaining bounded bond dimensions.

Contribution

The authors develop a basis optimization technique for MPDOs that reduces bond dimensions, facilitating efficient simulation of noisy quantum circuits with controlled memory.

Findings

Bond dimension reduction through environment basis optimization.

Enables simulation of deeper noisy quantum circuits with bounded memory.

Maintains reasonable fidelity in noisy circuit emulation.

Abstract

Matrix product density operators (MPDOs) are tensor network representations of locally purified density matrices where each physical degree of freedom is associated to an environment degree of freedom. MPDOs have interesting properties for mixed state representations: guaranteed positivity by construction, efficient conservation of the trace and computation of local observables. However, they have been challenging to use for noisy quantum circuit simulation, as the application of noise increases the dimension of the environment Hilbert space, leading to an exponential growth of bond dimensions. MPDOs also lack a unique canonical form, due to the freedom in the choice of basis for the environment Hilbert space, which leads to a vast variation of bond dimensions. In this work, we present a systematic way to reduce the bond dimensions of MPDOs by disentangling the purified state. We optimize the basis for the environment Hilbert space by performing density matrix renormalization group (DMRG)-like sweeps of local 2-qubit basis optimization. Interestingly, we find that targeting only the disentanglement of the purified state leads to a reduction of the environment dimension. In other words, a compact MPDO representation requires a low-entanglement purified state. We apply our compression method to the emulation of noisy random quantum circuits. Our technique allows us to keep bounded bond dimensions, and thus bounded memory, contrary to previous works on MPDOs, while keeping reasonable truncation fidelities.