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arXiv:2608.25797

Fault-tolerant $|\sqrt{ \mathrm{T} }\rangle$ state preparation and injection for more efficient fine-grained quantum circuit synthesis

Berat Yenilen, Markus Müller, Manuel Rispler

quant-phSubmitted Aug 26, 2026• Announced Aug 27, 2026• 0 citations
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Abstract

Magic-state injection is a standard route to realize universal fault-tolerant quantum computation. Whereas the set of Clifford gates in combination with the non-Clifford T gate is a widely used universal gate set, extending the available set of non-Clifford primitives can reduce compilation overhead, provided that the additional primitives can be prepared fault-tolerantly with competitive resource costs and at sufficiently low logical noise rates. In this work, we introduce flag fault-tolerant protocols for preparing logical $|\sqrt{\mathrm{T}} \rangle $ magic states on the 3D tetrahedral color code and its smaller morphed variant. Our simulations under circuit-level noise verify fault tolerance, quantify acceptance and logical error rates, and we reconstruct the effective logical channels of the corresponding circuits for gate injection via logical process tomography. We find that access to $\sqrt{\mathrm{T}}$ reduces the average space-time cost of synthesizing Haar-random single-qubit unitaries by approximately $20$-$30$% relative to the Clifford+T gate set across practically relevant approximation regimes. These results demonstrate how expanding the set of fault-tolerant non-Clifford primitives can improve computational efficiency and broaden the design space for universal quantum computation in the early fault-tolerant era.

Comments
18 pages, 10 figures

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Fault-tolerant ∣T⟩|\sqrt{ \mathrm{T} }\rangle∣T​⟩ state preparation and injection for more efficient fine-grained quantum circuit synthesis
Magic-state injection is a standard route to realize universal fault-tolerant quantum computation. Whereas the set of Clifford gates in combination with the non-Clifford T gate is a widely used universal gate set, extending the available set of non-Clifford primitives can reduce compilation overhead, provided that the additional primitives can be prepared fault-tolerantly with competitive resource costs and at sufficiently low logical noise rates. In this work, we introduce flag fault-tolerant protocols for preparing logical ∣T⟩|\sqrt{\mathrm{T}} \rangle ∣T​⟩ magic states on the 3D tetrahedral color code and its smaller morphed variant. Our simulations under circuit-level noise verify fault tolerance, quantify acceptance and logical error rates, and we reconstruct the effective logical channels of the corresponding circuits for gate injection via logical process tomography. We find that access to T\sqrt{\mathrm{T}}T​ reduces the average space-time cost of synthesizing Haar-random single-qubit unitaries by approximately 202020-303030% relative to the Clifford+T gate set across practically relevant approximation regimes. These results demonstrate how expanding the set of fault-tolerant non-Clifford primitives can improve computational efficiency and broaden the design space for universal quantum computation in the early fault-tolerant era.
Fault-tolerant preparation of non-Clifford resource states is a major contributor to the overhead of quantum computation, motivating protocols that achieve high output fidelity with minimal qubit and circuit costs. We introduce a low-cost magic-state preparation protocol in which the choice of stabilizer generators is co-designed with the flag gadgets, allowing the syndrome-extraction circuit itself to filter correlated faults across a non-Clifford layer. The protocol prepares a logical plus state in the 15-qubit quantum Reed-Muller code, applies a transversal T gate, and gauge-fixes the same register into the seven-qubit Steane code. By reorganizing equivalent Z-type stabilizer generators into jointly flagged measurement groups, the protocol eliminates all accepted logical-error contributions arising from one or two circuit faults under destructive error detection. Under a uniform circuit-level depolarizing noise model, the postselected infidelity is 210.2p3+O(p4)210.2p^3+O(p^4)210.2p3+O(p4). At p=10−3p=10^{-3}p=10−3, exact low-order enumeration combined with stratified sampling bounds the infidelity by 2.2×10−72.2\times10^{-7}2.2×10−7 at 99.9% joint confidence, while retaining an acceptance probability of 86.9%. The complete circuit requires only 19 qubits and 82 CNOT gates. These results demonstrate that stabilizer-generator design can substantially reduce the cost of postselected magic-state preparation, although corrected operation and the fidelity of an unmeasured output block require separate analysis.
Quantum low-density parity check (qLDPC) codes are among the leading candidates to realize error-corrected quantum memories with low qubit overhead. Potentially high encoding rates and large distance relative to their block size make them appealing for practical suppression of noise in near-term quantum computers. In addition to increased qubit-connectivity requirements compared to more conventional topological quantum error correcting codes, qLDPC codes remain notoriously hard to compute with. In this work, we introduce a construction to implement all Clifford quantum gate operations on the recently introduced lift-connected surface (LCS) codes (Old et al. 2024). These codes can be implemented in a 3D-local architecture and achieve asymptotic scaling [[n,O(n1/3),O(n1/3)]][[n, \mathcal{O}(n^{1/3}), \mathcal{O}(n^{1/3})]][[n,O(n1/3),O(n1/3)]]. In particular, LCS codes realize favorable instances with small numbers of qubits: For the [[15,3,3]][[15,3,3]][[15,3,3]] LCS code, we provide deterministic fault-tolerant (FT) circuits of the logical gate set {H‾i,H‾i,CiXj‾}i,j∈(0,1,2)\{\overline{H}_i, \overline{H}_i, \overline{C_i X_j}\}_{i,j \in (0,1,2)}{Hi​,Hi​,Ci​Xj​​}i,j∈(0,1,2)​ based on flag qubits. By adding a procedure for FT magic state preparation, we show quantitatively how to realize an FT universal gate set in d=3d=3d=3 LCS codes. Numerical simulations indicate that our gate constructions can attain pseudothresholds in the range pth≈4.8⋅10−3−1.2⋅10−2p_{\mathrm{th}} \approx 4.8\cdot 10^{-3}-1.2\cdot 10^{-2}pth​≈4.8⋅10−3−1.2⋅10−2 for circuit-level noise. The schemes use a moderate number of qubits and are therefore feasible for near-term experiments, facilitating progress for fault-tolerant error corrected logic in high-rate qLPDC codes.
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.
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.
As quantum hardware enters the fault-tolerant regime, algorithms are beginning to be executed on encoded qubits, yet scalable certification of how well they run remains an open problem. Here we introduce magic-free certification, a certification protocol for universal fault-tolerant circuits that consumes no magic, the resource that generally places them beyond efficient classical simulation and is by far the most expensive element of a fault-tolerant computation. The protocol is code- and decoder-agnostic, runs interleaved with the target circuit on the same device, and has a sample complexity that grows linearly with the number of logical qubits and non-Clifford gates while remaining decoupled from the logical error rate itself. Given a predefined error tolerance pL∗p_L^*pL∗​, the protocol determines at a chosen confidence level whether the device executes the target circuit at a logical error rate below pL∗p_L^*pL∗​. The method relies on trap circuits, copies of the target with every magic state replaced by a stabilizer state, whose outcomes are deterministic and which, being indistinguishable from the target to stochastic Pauli noise, fail at the same rate. We illustrate the protocol in Steane-code simulations of a magic state injection gadget and a quantum Fourier transform.
We propose a scheme for the fault-tolerant implementation of arbitrary Clifford circuits. To achieve this, we extend previous work on flag gadgets for syndrome extraction to a general framework that flags any Clifford circuit. This framework opens new pathways toward universal fault tolerance by allowing transversal implementation of TTT gates alongside fault-tolerant realization of selected non-transversal Clifford gates using flags. The construction we present allows a Clifford circuit consisting of nnn two-qubit gates and O(n)O(n)O(n) single-qubit gates acting upon physical qubits in a code of distance ddd to be made fault tolerant to distance ddd using O(d2log⁡(nd2log⁡n))O(d^2 \log(nd^2\log n))O(d2log(nd2logn)) ancilla qubits and O(nd2log⁡(nd2log⁡n))O(nd^2 \log(nd^2 \log n))O(nd2log(nd2logn)) extra CNOTs. Beyond asymptotic analysis, we demonstrate our construction by implementing the non-transversal logical Hadamard gate for the [[15,1,3]] code, which has transversal T, and compare to alternative approaches for universality using this code. We also apply our construction to magic-state preparation, general state preparation using Clifford circuits, and data-syndrome codes.