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arXiv:2601.22939

Fast magic state preparation by gauging higher-form transversal gates in parallel

Dominic J. Williamson

quant-phSubmitted Jan 30, 2026• Announced Feb 2, 2026• 0 citations
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Abstract

Magic states are a foundational resource for universal quantum computation. To survive in a realistic noisy environment, magic states must be prepared fault-tolerantly and protected by a quantum error-correcting code. The recent discovery of highly efficient quantum low-density parity-check codes, together with efficient logic gates, lays the groundwork for low-overhead fault-tolerant quantum computation. This motivates the search for fast and parallel protocols for logical magic state preparation to enable universal quantum computation. Here, we introduce a fast code surgery procedure that performs a fault-tolerant measurement of many transversal logic gates in parallel. This is achieved by performing a generalized gauging measurement on a quantum code that supports a higher-form transversal gate. The time overhead of our procedure is constant, and the qubit overhead is linear. The procedure inherits fault-tolerance properties from the base code and the structure of the higher-form transversal gate. When applied to codes that support higher-form Clifford gates our procedure achieves fast and fault-tolerant preparation of many magic states in parallel. This motivates the search for good quantum low-density parity-check codes that support higher-form Clifford gates.

Comments
16 pages, 5 figures

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Magic states are a foundational resource for universal quantum computation. To survive in a realistic noisy environment, magic states must be prepared fault-tolerantly and protected by a quantum error-correcting code. The recent discovery of highly efficient quantum low-density parity-check codes, together with efficient logic gates, lays the groundwork for low-overhead fault-tolerant quantum computation. This motivates the search for fast and parallel protocols for logical magic state preparation to enable universal quantum computation. Here, we introduce a fast code surgery procedure that performs a fault-tolerant measurement of many transversal logic gates in parallel. This is achieved by performing a generalized gauging measurement on a quantum code that supports a higher-form transversal gate. The time overhead of our procedure is constant, and the qubit overhead is linear. The procedure inherits fault-tolerance properties from the base code and the structure of the higher-form transversal gate. When applied to codes that support higher-form Clifford gates our procedure achieves fast and fault-tolerant preparation of many magic states in parallel. This motivates the search for good quantum low-density parity-check codes that support higher-form Clifford gates.
Non-Clifford operations required for universal quantum computation can be implemented by measuring logical Clifford operators. We present a generalized framework to fault-tolerantly measure transversal Clifford logical operators in quantum low-density parity check (QLDPC) codes. Our protocol takes any QLDPC code and deforms the code across additional ancillary qubits such that a product of stabilizers measures the desired logical Clifford operator. We prove that the deformed, Clifford-stabilized code remains LDPC and that its distance is preserved up to a constant factor independent of the number of physical qubits. We further prove that the Clifford measurement has a fault distance against data qubit and measurement errors that grows linearly with the distance of the original code. We apply our methods to implement logical non-Clifford gates in algorithmically relevant subroutines and to prepare magic states in quantum stabilizer codes.
Quantum low-density parity-check (qLDPC) codes offer a promising route to reducing the qubit overhead of fault tolerance, with recent ultra-high-rate constructions achieving encoding rates above one half. In this work, we develop fast and parallel logical computation methods for ultra-high-rate codes. We first utilize the group symmetries of the codes to partition the logical qubits into orbits, and then apply parallel surgery methods to implement group-invariant collections of logical Pauli-product measurements. These measurements are highly programmable, as we may arbitrarily couple logical orbits as long as each measurement is repeated across the whole orbit(s). To reduce the space and time overheads, we develop a technique called chain-map convolution, which uses the group structure to boost both the spatial and temporal distances of our protocol in a highly efficient manner. As case studies, we construct distance-preserving parallel measurement gadgets for the [[256, 68, 12]], [[576, 148, 18]] pair-partition codes, and a [[1152, 580, 12]] rate-1/21/21/2 code, with only around 2.5×2.5\times2.5× space overhead and 3-4 rounds of syndrome extraction each. The same methods also enable parallel injection of cultivated magic states into high-rate codes. Finally, we extend programmability through subgroup restrictions, revealing a tradeoff between logical control and the spacetime resources required for fault tolerance and highlighting the crucial role structured parallelism plays in reducing overhead. These results provide new methods for designing qLDPC logical operations with low spacetime overhead, opening a route toward more efficient fault-tolerant quantum architectures.
Magic states are essential yet resource-intensive components for realizing universal fault-tolerant quantum computation. Preparing magic states within emerging quantum low-density parity-check (qLDPC) codes poses additional challenges, due to the complex encoding structures. Here, we introduce a generic and scalable method for magic state injection into arbitrarily selected logical qubits encoded using qLDPC codes. Our approach, based on parallelized code surgery, supports the injection from either physical qubits or low-distance logical qubits. For qLDPC code families with asymptotically constant encoding rates, the method achieves injection into Θ(k)Θ(k)Θ(k) logical qubits -- where kkk denotes the logical qubit number of the code -- with only constant qubit overhead and a time complexity of O~(d2)\tilde{O}(d^2)O~(d2), where ddd is the code distance. A central contribution of this work is a rigorous proof that errors affecting the injected magic states remain independent throughout the procedure. This independence ensures the resilience of logical qubits against interactions with noisy ancillae and preserves the presumption of subsequent magic state distillation protocols. We further support our theoretical results with numerical validation through circuit-level simulations. These findings advance the feasibility of scalable, fault-tolerant universal quantum computing using qLDPC codes, offering a pathway to significantly reduced qubit resource requirements in magic state injection.
In this work, we introduce constructions for non-Abelian qLDPC codes obtained by gauging transversal Clifford gates using measurement and feedback. In particular, we identify two qualitatively different approaches to gauging qLDPC codes to obtain their non-Abelian counterparts. The first approach applies to codes that exhibit a generalized form of Poincaré duality and leads to a qLDPC non-Abelian Clifford stabilizer code, whose stabilizers are reminiscent of the action of a Type-III twisted quantum double. Our second approach applies to general qLDPC codes, and uses a graph of ancilla qubits which may be tailored to properties of the input codes to gauge a single transversal gate. For both constructions, the resulting gauged codes are shown to have properties analogous to 2D non-Abelian topological order -- e.g. the analog of a single anyon on a torus. We conclude by demonstrating that our gauging procedures enable magic state preparation via the measurement of logical Clifford gates. Consequently, our gauging constructions offer a protocol for performing non-Clifford operations on any qLDPC code.
Generalized code surgery is a versatile and low-overhead technique for performing fault-tolerant computation on quantum low-density parity-check (qLDPC) codes. In many settings, surgery exhibits practical space overheads, while its time overhead remains a bottleneck at O(d)O(d)O(d) syndrome rounds per operation. In this work, we construct surgery gadgets that perform parallel logical measurements on 2D hypergraph product codes in constant time overhead (O(1)O(1)O(1)) and near-constant space overhead (O~(1)\tilde{O}(1)O~(1)). The reduced time overhead is a result of amortization, as we show, following the formulation by Cowtan et al. (arXiv:2510.14895), that performing ddd surgery operations in O(d)O(d)O(d) time is fault tolerant. Our gadgets combine the strengths of different approaches to fault-tolerant logical operations: they partially retain the flexibility of surgery while achieving overheads comparable to transversal gates. Consequently, they are well-suited for near-term experimental realization and demonstrate new possibilities in the design of gadgets for fast logical computation.
We propose a fault-tolerant quantum computation scheme that is broadly applicable to quantum low-density parity-check (qLDPC) codes. The scheme achieves constant qubit overhead and a time overhead of O ( d a + o ( 1 ) ) for any [ [ n , k , d ] ] qLDPC code with constant encoding rate and distance d = Ω ( n 1 / a ) . For good qLDPC codes, the time overhead is minimized and reaches O ( d 1 + o ( 1 ) ) . In contrast, code surgery based on gauging measurement and brute-force branching requires a time overhead of O ( d w 1 + o ( 1 ) ) , where d ≤ w ≤ n . Thus, our scheme is asymptotically faster for all codes with a < 2 . This speedup is achieved by developing techniques that enable parallelized code surgery under constant qubit overhead and leverage classical locally testable codes for efficient resource state preparation. These results establish a new paradigm for accelerating fault-tolerant quantum computation on qLDPC codes, while maintaining low overhead and broad applicability.