Certifying almost all quantum states with few single-qubit measurements (2404.07281v1)

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).

• The paper introduces a novel protocol using single-qubit Pauli measurements to approximate fidelity via the shadow overlap.
• The method certifies almost all quantum states with O(n²) measurements, demonstrating scalability in systems up to 120 qubits.
• The protocol offers practical benefits for quantum device benchmarking, machine learning tomography, and quantum circuit optimization.

Certifying Quantum States with Few Single-Qubit Measurements

Introduction to the Certification Challenge

Quantum state certification is an essential task within quantum information science, aimed at verifying whether a synthesized quantum state in the laboratory aligns closely with a desired target state. Standard approaches for quantum state certification typically necessitate either deep quantum circuits or a prohibitive number of measurements, challenging their practical application. The research presented here introduces a novel protocol enabling the certification of quantum states—including those of exponential circuit complexity—with only $\mathcal{O}(n^2)$ single-qubit measurements, significantly reducing the experimental burden.

Main Results

The core achievement of this work is a certification protocol that estimates a quantity referred to as the shadow overlap by performing single-qubit Pauli measurements. This protocol establishes that, under favorable conditions, the shadow overlap effectively approximates the fidelity between the target and the synthesized quantum states. Importantly, this approach circumvents the need for complex measurement strategies, leveraging instead the concept of shadow overlap, which offers a practical surrogate for fidelity. Numerical demonstrations of this protocol have validated its efficacy across simulated quantum systems up to $120$ qubits, showcasing a clear advantage over current methods like cross-entropy benchmarking.

Two primary theoretical results underpin this certification method:

1. Theoretical Certification for General States: For any target pure state with a polynomial relaxation time, a certification procedure using single-qubit Pauli measurements can accurately verify the fidelity of an unknown state to the target. The procedure requires $\mathcal{O}(\tau^2 / \epsilon^2)$ samples to ascertain whether fidelity lies above a specified threshold, establishing a pragmatic balance between resource expenditure and certification accuracy.
2. Certification of Almost All Quantum States: A significant fraction of quantum states can be efficiently certified with $\mathcal{O}(n^2 / \epsilon)$ single-qubit measurements. This result suggests the broad applicability of the proposed certification method, encompassing a wide variety of quantum states with potentially complex entanglements.

The Certification Procedure

The certification protocol employs single-qubit measurements to evaluate a derived quantity—the shadow overlap—acting as a fidelity surrogate. This step involves randomly selecting qubits for measurement in one of the Pauli bases and correlating these measurements with queries to a model representing the target state. The expectation of the computed overlaps offers an estimate of the shadow overlap, which in turn provides insights into the fidelity between the synthesized and target states.

Analytical Foundation and Implications

Theoretical analysis reveals that the proposed certification method is rooted in the dynamics of certain random walks associated with the quantum states under investigation. The relaxation time of these random walks, quantitatively linked to the concept of mixing time, serves as a key factor determining the efficiency of the certification process. Specifically, the analysis leverages results from the paper of Markov chains to bound the relaxation time for almost all quantum states, offering a theoretical foundation for the protocol's broad applicability.

Practical Applications

Beyond state certification, the shadow overlap concept finds utility in several other areas:

• Machine Learning Tomography: This framework facilitates the learning and verification of neural network models of quantum states, potentially enhancing our ability to simulate and understand complex quantum systems.
• Quantum Device Benchmarking: By providing a statistically rigorous yet practical measure for evaluating quantum state preparation, the shadow overlap serves as a valuable tool for benchmarking quantum hardware.
• Optimization of Quantum Circuits: The certification approach also aids in the optimization of quantum circuits for state preparation, showing promise for overcoming challenges like barren plateaus encountered in fidelity-based optimization.

Conclusion and Future Directions

The introduction of a practical method for certifying quantum states with minimal experimental requirements marks a significant advance in the field of quantum computing. By effectively leveraging single-qubit measurements to approximate state fidelity, the presented protocol offers a scalable solution applicable to a wide range of quantum systems. Future research will likely explore further applications of this method, potentially extending its utility in quantum simulation, machine learning, and hardware benchmarking, as well as refining its theoretical underpinnings to enhance its efficiency and applicability.