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arXiv:2505.09687 · Physical Review Letters

Efficient Benchmarking of Logical Magic State

Su-un Lee, Ming Yuan, Senrui Chen, Kento Tsubouchi, Liang Jiang

Published Jan 9, 2026• 1 citation
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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 ∼1/ε^{2} samples, making benchmarking impractical for high-fidelity states. In this Letter, 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 multiqubit 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 the moderate error rates currently achievable in state-of-the-art experiments.

DOI
10.1103/fwjt-mw2c

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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 ∼1/ε^{2} samples, making benchmarking impractical for high-fidelity states. In this Letter, 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 multiqubit 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 the moderate error rates currently achievable in state-of-the-art experiments.
The preparation of high-fidelity logical magic states is a crucial subroutine for universal fault-tolerant quantum computation (FTQC). Predicting the performance of FTQC and developing improved protocols rely on numerical methods to classically simulate logical magic state preparation in the presence of noise. Clifford logic on Pauli-stabilizer codes with circuit-level Pauli errors can be efficiently simulated using Pauli-stabilizer formalism, but the non-Clifford operations required to prepare logical magic states render generic simulation inefficient. We introduce Clifford-stabilizer simulation, an exact and efficient algorithm based on updating a Clifford-stabilizer group to simulate noisy preparation protocols for a broad class of logical magic states used to implement non-Clifford gates in the third level of the Clifford hierarchy under circuit-level Pauli errors. Clifford-stabilizer simulation applies to a range of operations that commonly appear in preparation protocols for such logical magic states, including Pauli-stabilizer measurements, logical Clifford measurements, and transversal non-Clifford gates. Our algorithm for Clifford-stabilizer simulation maps a non-Clifford circuit with sampled circuit-level Pauli errors to a Clifford circuit that exactly reproduces its measurement outcome distribution, achieving time and space complexities polynomial in relevant protocol parameters. We perform exact simulation of magic state cultivation up to fault distance 7 by Clifford-stabilizer simulation. Our method provides a route to perform exact benchmarking of large-scale logical magic state preparation protocols required for useful FTQC.
Non-Clifford gates are essential for avoiding classical simulability under the Gottesman-Knill theorem. An error-corrected circuit with non-Clifford gates will therefore be an important element for realizing useful fault-tolerant quantum algorithms. However, accurately understanding the fault-tolerant performance of such circuits faces challenges due to the classical hardness of simulating these circuits. We avoid simulating logical measurements and develop exact and efficient classical methods for benchmarking circuits composed of X, CNOT, and diagonal gates in the Clifford hierarchy, including state and gate teleportation gadgets with feedforward. For circuits with third-level diagonal gates, our benchmarking algorithm efficiently samples syndrome statistics and estimates the output logical fidelity and expectation values of certain Clifford observables, including all Pauli observables. The evaluation of syndrome expectation values also extends to circuits with fourth-level diagonal gates. Our algorithm scales polynomially in the number of physical qubits n and circuit depth T, independent of the number of logical qubits. The benchmarking applications include IQP sampling and magic state preparation/cultivation. We demonstrate the practical utility of our simulation algorithms by using them to optimize high-rate magic state distillation factories based on three copies of a [[27,3,3]] tricycle code.
Encoding quantum information to protect it from errors is essential for performing large-scale quantum computations. Performing a universal set of quantum gates on encoded states demands a potentially large resource overhead and minimizing this overhead is key for the practical development of large-scale fault-tolerant quantum computers. We propose and experimentally implement a magic-state preparation protocol to fault-tolerantly prepare a pair of logical magic states in a [[6,2,2]] quantum error-detecting code using only eight physical qubits. Implementing this protocol on H1-1, a 20 qubit trapped-ion quantum processor, we prepare magic states with experimental infidelity 7−1+3×10−57^{+3}_{-1}\times 10^{-5}7−1+3​×10−5 with a 14.8−1+1%14.8^{+1}_{-1}\%14.8−1+1​% discard rate and use these to perform a fault-tolerant non-Clifford gate, the controlled-Hadamard (CH), with logical infidelity ≤2.3−9+9×10−4\leq 2.3^{+9}_{-9}\times 10^{-4}≤2.3−9+9​×10−4. Notably, this significantly outperforms the unencoded physical CH infidelity of 10−310^{-3}10−3. Through circuit-level stabilizer simulations, we show that this protocol can be self-concatenated to produce extremely high-fidelity magic states with low space-time overhead in a [[36,4,4]] quantum error correcting code, with logical error rates of 6×10−106\times 10^{-10}6×10−10 (5×10−145\times 10^{-14}5×10−14) at two-qubit error rate of 10−310^{-3}10−3 (10−410^{-4}10−4) respectively.
Developing space- and time-efficient logical magic state preparation protocols will likely be an essential step towards building a large-scale fault-tolerant quantum computer. Motivated by this need, we introduce a scalable method for simulating logical magic state preparation protocols under the standard circuit-level noise model. When applied to protocols based on code switching, magic state cultivation, and magic state distillation, our method yields a complexity polynomial in (i) the number of qubits and (ii) the non-stabilizerness, e.g., stabilizer rank or Pauli rank, of the target encoded magic state. The efficiency of our simulation method is rooted in a curious fact: every circuit-level Pauli error in these protocols propagates to a Clifford error at the end. This property is satisfied by a large family of protocols, including those that repeatedly measure a transversal Clifford that squares to a Pauli. We provide a proof-of-principle numerical simulation that prepares a magic state using such logical Clifford measurements. Our work enables practical simulation of logical magic state preparation protocols without resorting to approximations or resource-intensive state-vector simulations.
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Preparation of high-fidelity logical magic states has remained as a necessary but daunting step towards building a large-scale fault-tolerant quantum computer. One approach is to fault-tolerantly prepare a magic state in one code and then switch to another, a method known as code switching. We experimentally demonstrate this protocol on an ion-trap quantum processor, yielding a logical magic state encoded in an error-correcting code with state-of-the-art logical fidelity. Our experiment is based on the first demonstration of code switching between color codes, from the fifteen-qubit quantum Reed-Muller code to the seven-qubit Steane code. We prepare an encoded magic state in the Steane code with 82.58% probability, with an infidelity of at most 5.1 ( 2.7 ) × 10 − 4 . The reported infidelity is lower than the leading infidelity of the physical operations utilized in the protocol by a factor of at least 2.7, indicating the quantum processor is below the pseudothreshold. Furthermore, we create two copies of the magic state in the same quantum processor and perform a logical Bell basis measurement for a sample-efficient certification of the encoded magic state. The high-fidelity magic state can be combined with the already-demonstrated fault-tolerant Clifford gates, state preparation, and measurement of the 2D color code, completing a universal set of fault-tolerant computational primitives with logical error rates equal or better than the physical two-qubit error rate.