Classically estimating observables of noiseless quantum circuits

2024-09-03

arXiv.org

AI Summary

This work addresses the classical efficient estimation of expectation values of arbitrary observables under noiseless random quantum circuits. Prior methods were restricted to noisy circuits and struggled with fully connected or deep architectures. We propose the first universal classical algorithm, based on Pauli operator path expansion in the Heisenberg picture, integrated with single-qubit rotation-invariant measures and classical shadow techniques. Our algorithm is provably efficient for arbitrary circuit geometries—including all-to-all connectivity—and arbitrary depth, achieving polynomial-time high-precision estimation for constant accuracy (ε, δ), and quasi-polynomial time for inverse-polynomial precision. It succeeds on the overwhelming majority of circuit instances with controllable failure probability. Crucially, we provide the first rigorous proof that noiseless quantum circuits dominated by chaos and local shuffling remain classically estimable—thereby breaking the prior reliance on noise for classical simulability.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We present a classical algorithm for estimating expectation values of arbitrary observables on most quantum circuits across all circuit architectures and depths, including those with all-to-all connectivity. We prove that for any architecture where each circuit layer is equipped with a measure invariant under single-qubit rotations, our algorithm achieves a small error $varepsilon$ on all circuits except for a small fraction $delta$. The computational time is polynomial in qubit count and circuit depth for any small constant $varepsilon, delta$, and quasi-polynomial for inverse-polynomially small $varepsilon, delta$. For non-classically-simulable input states or observables, the expectation values can be estimated by augmenting our algorithm with classical shadows of the relevant state or observable. Our approach leverages a Pauli-path method under Heisenberg evolution. While prior works are limited to noisy quantum circuits, we establish classical simulability in noiseless regimes. Given that most quantum circuits in an architecture exhibit chaotic and locally scrambling behavior, our work demonstrates that estimating observables of such quantum dynamics is classically tractable across all geometries.

Problem

• Estimating observables in noiseless quantum circuits classically
• Analyzing random unstructured circuits across all architectures
• Proving classical tractability for chaotic quantum behaviors

Innovation

• Classical algorithm using Pauli propagation
• Polynomial time for small error rates
• Effective across all circuit geometries