Certifying almost all quantum states with few single-qubit measurements
This addresses a fundamental bottleneck in quantum information science for benchmarking and optimizing quantum systems, offering a scalable solution with broad applications.
The paper tackles the problem of certifying n-qubit quantum states, which previously required deep circuits or exponential measurements, by proving that almost all such states can be certified with only O(n^2) single-qubit measurements, as demonstrated in numerical experiments up to 120 qubits.
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).