Efficient benchmarking of logical magic state

TL;DR

This work addresses the challenge of efficiently benchmarking high-fidelity logical magic states under fault-tolerant Clifford-only operations, where multiplicative precision in the infidelity $ε$ is essential. It proves a no-go bound for single-copy benchmarking, $N = Ω\left(1/(r^2 ε^2)\right)$, and introduces two efficient strategies that achieve $N = O\left(\log(1/δ)/(r^2 ε)\right)$: (i) Bell measurements on two copies of a twirled magic state, and (ii) single-copy schemes on twirled multi-qubit magic states using stabilizer components from the twirl decomposition. These results are underpinned by representation theory of twirling groups, Schur’s lemma, and a SWAP-test–style analysis of purity, with proofs of optimality for general joint measurements. Numerical simulations under realistic depolarizing noise in color-code architectures (e.g., [[7,1,3]] color code for T-state distillation and [[8,3,2]] for CCZ) show substantial sampling overhead reductions (over two orders of magnitude at moderate noise), supporting near-term experimental feasibility of efficient logical magic-state benchmarking.

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 $ε$ with multiplicative precision, while many quantum error-correcting codes only permit Clifford operations to be implemented fault-tolerantly. Consequently, conventional state tomography requires $\sim1/ε^2$ 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 $Ω(1/ε^2)$ 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 $O(1/ε)$ 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.

Figures (11)

• Figure 1: (a) Single-copy benchmarking scheme for $\ket{T}$ state with corresponding measurement output distribution, and (b) Bell measurement benchmarking scheme for $\ket{T}$ state with corresponding measurement output distribution.
• Figure 2: (a) Circuit for encoding a physical $\ket{T}$ state into the [[7, 1, 3]] color code, generating a noisy logical magic state. (b) Circuit for performing 5-to-1 magic-state distillation (MSD) at the logical level, resulting in a higher-fidelity logical state $\rho$. Both circuits in (a) and (b) follow the methodology of Ref. rodriguezExperimentalDemonstrationLogical2024. (c) Infidelity $\epsilon$ of the state $\rho$ and the logical error rates from benchmarking circuits. The dashed lines (black, blue, and orange) indicate power-law fits, given by $y=8.11x^{2.07}$, $y=2.52x^{3.01}$, and $y=16.8x^{3.01}$, respectively. (d) Sampling overhead $N$ for the standard state tomography and the Bell measurement scheme with multiplicative accuracies $r = 0.1$ and $r = 0.5$, with $68\%$ confidence.
• Figure 3: The SWAP test circuit for measuring the Hilbert-Schmidt inner product $\Tr[\rho_1 \rho_2]$.
• Figure 4: Equivalence of the SWAP test and the Bell measurement circuits. (a) The original SWAP test circuit with additional measurements at the end. (b) The circuit equivalent to the SWAP test circuit. (c) The Bell measurement circuit followed by feedforward channels.
• Figure 5: (a) Graphical description of the [[7, 1, 3]] color code. Each generator acts $X$ or $Z$ on red, blue, or green patches. (b) The non-fault-tolerant circuit on the physical level encodes a physical $\ket{T}$ state into [[7, 1, 3]] color code. (c) The 5-to-1 MSD circuit on the logical level. (d) The state tomography circuit on the logical level. (e) The Bell measurement circuit on the logical level.

Theorems & Definitions (26)

• Theorem 1
• Definition 1
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 2
• Definition 3
• Proposition 1: Adapted from Prop. 1.5 in fultonRepresentationTheory2004
• proof
• Definition 4