Quantum simulation in the Heisenberg picture via vectorization

TL;DR

This paper introduces a comprehensive framework for simulating quantum systems in the Heisenberg picture using quantum hardware, leveraging vectorization to map operator tasks to Schrödinger-picture tasks for improved accessibility and implementation.

Contribution

It presents a novel vectorization-based method for Heisenberg picture quantum simulation, enabling efficient computation of operator-related quantities on quantum devices.

Findings

Framework enables simulation of Heisenberg operators on quantum hardware

Allows computation of OTOCs, correlators, and entanglement measures

Proposes implementations for 2D quantum simulators considering device constraints

Abstract

We present a general framework for simulating quantum systems in the Heisenberg picture on quantum hardware. Based on the vectorization map, our framework fully exploits the mapping between operators and quantum states, allowing any task defined on Heisenberg operators to be mapped to standard Schr\"odinger-picture tasks that are naturally accessible via quantum computers and simulators. This yields new or improved protocols for tasks such as operator sampling, the computation of OTOCs/superoperator expectation values and their higher order moments, two-point correlators, and operator stabilizer and entanglement entropies. Our approach is also amenable to implementation, as it inherits the structure and resource requirements of the (forward and time-reversed) Schr\"odinger-picture quantum simulation problem. We demonstrate this by proposing implementations of our framework for a 2D problem on digital and analog quantum simulators, taking into account device connectivity constraints.