The preparation of high-fidelity non-Clifford (magic) states is an essential subroutine for universal quantum computation but imposes substantial space-time overhead. Magic-state factories based on high-rate and -distance quantum low-density parity-check (LDPC) codes equipped with transversal non-Clifford gates can potentially reduce these overheads significantly, by circumventing the need for multiple rounds of distillation and by producing a large number of magic states in a single code block. As a step toward realizing efficient, fault-tolerant magic-state production, we introduce a class of finite block-length quantum LDPC codes which we name tricycle codes, generalizing the well-known bicycle codes to three homological dimensions. These codes can support constant-depth physical circuits that implement logical C C Z gates between three code blocks. To construct these constant-depth C C Z circuits, we develop analytical and numerical techniques that apply to a broad class of three-dimensional homological and balanced-product codes. We further show that tricycle codes enable single-shot state preparation and error correction, leading to a highly efficient magic-state generation protocol. Numerical simulations of specific codes confirm robust performance under circuit-level noise, demonstrating a high circuit-noise threshold of > 0.5 % . With modest postselection, certain tricycle codes of block lengths of only approximately 50–100 qubits are shown to produce | C C Z ¯ ⟩ magic states with logical error rates ranging from 2 × 10 − 8 to below 3 × 10 − 11 . Finally, we construct optimal depth syndrome extraction circuits for tricycle codes and present a protocol for implementing them efficiently on a reconfigurable neutral-atom platform.
Quantum low-density parity-check (qLDPC) codes can substantially reduce the qubit overhead of fault-tolerant quantum computation, but any useful architecture based on such codes also requires an efficient mechanism to implement non-Clifford gates in a fault-tolerant manner. We introduce a space-time-efficient magic state factory for high-rate qLDPC codes by combining the transversal non-Clifford gates of recently developed tricycle codes with single-shot four-dimensional balanced product quadcycle codes. The key ingredient is an efficient "escape" mechanism that teleports postselected logical magic states from tricycle codes to larger distance quadcycle codes. Under a standard circuit-level noise model at physical error rate
10−3, our factories produce
CCZ states at logical error rates as low as
10−9 with 10-40 times lower space-time cost than state-of-the-art magic state generation methods. The resulting states are directly encoded in high-rate blocks suitable for computation. We identify the quadcycle codes by an AI-assisted search and show that they can serve as excellent memories capable of single-shot error correction. Finally, we present a concrete implementation of the entire factory with highly efficient atom transport on reconfigurable neutral-atom quantum computers. Together, these results provide an end-to-end, practical protocol for producing and utilizing high-fidelity magic states within a high-rate qLDPC architecture, with the lowest space-time cost to date.
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.
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.
While quantum low-density parity check (qLDPC) codes are a low-overhead means of quantum information storage, it is valuable for quantum codes to possess fault-tolerant features beyond this resource efficiency. In this work, we introduce trivariate tricycle (TT) codes, qLDPC codes that combine several desirable features: high thresholds under a circuit-level noise model, partial single-shot decodability for low-time-overhead decoding, a large set of transversal Clifford gates and automorphisms within and between code blocks, and (for several sub-constructions) constant-depth implementations of a (non-Clifford)
CCZ gate. TT codes are CSS codes based on a length-3 chain complex, and are defined from three trivariate polynomials, with the 3D toric code (3DTC) belonging to this construction. We numerically search for TT codes and find several candidates with improved parameters relative to the 3DTC, using up to 48
× fewer data qubits as equivalent 3DTC encodings. We construct syndrome-extraction circuits for these codes and numerically demonstrate single-shot decoding in
Z memory, in both phenomenological and circuit-level noise models. We also demonstrate that the codes have good performance under single-shot decoding even in
X memory. Under circuit-level noise, TT codes have a crossing point of
0.3% in
X memory and
1% in
Z memory (with single-shot decoding). All TT codes possess several transversal
CZ gates that can partially address logical qubits between two code blocks. The codes possess a large set of automorphisms that can perform Clifford gates within a code block. Finally, we establish several TT code polynomial constructions that allow for a constant-depth implementation of logical
CCZ gates. We find codes with
CCZ gates that use up to
5.7× fewer data qubits as equivalent-distance 3DTC encodings, and others with encoding rates as high as
1/4.
Magic State Distillation is considered to be one of the promising methods for supplying the non-Clifford resources required to achieve universal fault tolerance. Conventional MSD protocols implemented in surface codes often require multiple code blocks and lattice surgery rounds, resulting in substantial qubit overhead, especially at low target error rates.
In this work, we present practical magic state distillation factories on Bivariate Bicycle (BB) codes that execute Pauli-measurement-based Clifford circuits inside a single BB code block. We formulate distillation circuit design as a joint optimization of logical qubit mapping, gate scheduling, measurement nativization, and protocol compression via qubit recycling. Based on detailed resource analysis and simulations, our BB factories have space-time volume comparable to that of leading distillation factories while delivering lower target error at a smaller qubit footprint, and are particularly compelling as second-round distillers following magic state cultivations.
Magic state distillation is a key component of fault-tolerant quantum computation, as it enables the implementation of non-Clifford gates such as the
T gate and the
CCZ gate via gate teleportation. However, conventional distillation protocols require a large number of logical qubits and introduce substantial spatial and temporal overhead, posing a significant bottleneck for scalable fault-tolerant quantum computation. In this work, we propose a zero-level distillation protocol that efficiently generates a high-fidelity logical
CCZ magic state using only physical qubits on a two-dimensional square lattice with nearest-neighbor interactions. Our method leverages the transversal
T/T† operation of the
[[8,3,2]] code to fault-tolerantly encode the state
CCZ∣+++⟩, which is subsequently teleported to three surface-code logical qubits via lattice surgery. To enable teleportation between codes with different distances, we introduce adaptively initialized teleportation (AIT), a tailored initialization procedure for the surface code. Numerical simulations demonstrate that the logical error rate scales as
pL≃300×p2 with respect to the physical error rate
p. For example, the proposed method improves the logical error rate by approximately one and two orders of magnitude at
p=10−3 and
p=10−4, respectively, compared to conventional seven-
T-gate approaches. The distillation circuit requires only 22 physical qubits, 3 logical qubits, and a circuit depth of 24, reducing the space-time overhead by a factor of approximately 5-10 compared to previous methods. This result highlights the practicality of
CCZ-state distillation in early fault-tolerant quantum computation and offers a new direction toward resource-efficient physical-level magic state distillation beyond conventional
T-state generation.