Certifying almost all quantum states with few single-qubit measurements

TL;DR

This paper introduces a method to certify almost all n-qubit quantum states using only O(n^2) single-qubit measurements, enabling efficient verification and property prediction for complex quantum states.

Contribution

The authors develop a novel technique linking state certification to random walk mixing times, allowing certification of states with exponential circuit complexity using polynomial measurements.

Findings

Certification protocol works for almost all n-qubit states

Numerical experiments with up to 120 qubits demonstrate effectiveness

Outperforms existing methods like cross-entropy benchmarking

Abstract

Certifying that an n-qubit state synthesized in the lab is close to the target state is a fundamental task in quantum information science. However, existing rigorous protocols either require deep quantum circuits or exponentially many single-qubit measurements. In this work, we prove that almost all n-qubit target states, including those with exponential circuit complexity, can be certified from only O(n^2) single-qubit measurements. This result is established by a new technique that relates certification to the mixing time of a random walk. Our protocol has applications for benchmarking quantum systems, for optimizing quantum circuits to generate a desired target state, and for learning and verifying neural networks, tensor networks, and various other representations of quantum states using only single-qubit measurements. We show that such verified representations can be used to efficiently predict highly non-local properties that would otherwise require an exponential number of measurements. We demonstrate these applications in numerical experiments with up to 120 qubits, and observe advantage over existing methods such as cross-entropy benchmarking (XEB).