Efficient benchmarking of logical magic state

TL;DR

This paper introduces efficient methods for benchmarking high-fidelity logical magic states in quantum computing, reducing sample complexity from quadratic to linear in the inverse of infidelity, and validates their robustness through simulations.

Contribution

It proposes two novel benchmarking schemes that achieve optimal linear sample complexity for magic states, overcoming previous quadratic limitations.

Findings

Proves that single-copy benchmarking requires at least O(1/ε^2) samples.

Introduces two schemes achieving O(1/ε) sample complexity, proven to be optimal.

Demonstrates robustness of protocols via numerical simulations under realistic noise.

Abstract

High-fidelity logical magic states are a critical resource for fault-tolerant quantum computation, enabling non-Clifford logical operations through state injection. However, benchmarking these states presents significant challenges: one must estimate the infidelity $\epsilon$ with multiplicative precision, while many quantum error-correcting codes only permit Clifford operations to be implemented fault-tolerantly. Consequently, conventional state tomography requires $\sim1/\epsilon^2$ samples, making benchmarking impractical for high-fidelity states. In this work, we show that any benchmarking scheme measuring one copy of the magic state per round necessarily requires $\Omega(1/\epsilon^2)$ samples for single-qubit magic states. We then propose two approaches to overcome this limitation: (i) Bell measurements on two copies of the twirled state and (ii) single-copy schemes leveraging twirled multi-qubit magic states. Both benchmarking schemes utilize measurements with stabilizer states orthogonal to the ideal magic state and we show that $O(1/\epsilon)$ sample complexity is achieved, which we prove to be optimal. Finally, we demonstrate the robustness of our protocols through numerical simulations under realistic noise models, confirming that their advantage persists even at moderate error rates currently achievable in state-of-the-art experiments.