DQC1-hardness of estimating correlation functions

TL;DR

This paper proves that estimating out-of-time-order correlation functions and N-time correlation functions over all eigenstates is computationally complete for the DQC1 model, revealing a complexity dichotomy in quantum measurements.

Contribution

It establishes DQC1-hardness of estimating correlation functions over all eigenstates, extending previous results and highlighting a complexity gap between query and circuit complexities.

Findings

Estimating OTOCs is DQC1-Complete.

N-time correlation functions are DQC1-Complete.

A complexity dichotomy exists between query and circuit complexities.

Abstract

Out-of-Time-Order Correlation function measures transport properties of dynamical systems. They are ubiquitously used to measure quantum mechanical quantities, such as scrambling times, criticality in phase transitions, and detect onset of thermalisation. We characterise the computational complexity of estimating OTOCs over all eigenstates and show it is Complete for the One Clean Qubit model (DQC1). We then generalise our setup to establish DQC1-Completeness of N-time Correlation functions over all eigenstates. Building on previous results, the DQC1-Completeness of OTOCs and N-time Correlation functions then allows us to highlight a dichotomy between query complexity and circuit complexity of estimating correlation functions.