Magic tricycles: Efficient magic state generation with finite block-length quantum LDPC codes

TL;DR

This work introduces tricycle codes, a finite-block-length quantum LDPC construction in three dimensions that enables transversal CCZ across three code blocks, enabling single-shot magic-state distillation with constant-depth syndrome extraction. By combining a three-block Abelian group-algebra framework with symmetric triple cup-product (and a numerical Leibniz-rule variant), the authors demonstrate high-rate, distance-robust codes at modest block-lengths (examples up to $N=480$, $K$ up to 24, distances up to $\,\leq 17$) and provide concrete CCZ-depths from 8 to 128. Circuit-noise simulations show a threshold around $0.5\\%$ with promising post-selection improvements yielding logical errors as low as $6\times 10^{-10}$ on small codes and $<10^{-12}$ for larger ones, with optimized, low-depth syndrome-extraction circuits suitable for neutral-atom platforms. The paper thus offers a viable path toward LDPC-based magic-state factories, combining analytic code-construction (via cup-product) with numerical search (NLR) for short-depth gates, potentially reducing the resource overhead for universal fault-tolerant quantum computation.

Abstract

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 towards 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 $CCZ$ gates between three code blocks. To construct these constant-depth $CCZ$ circuits, we develop new 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 post-selection, certain tricycle codes of block-lengths of only $50-100$ qubits are shown to achieve logical error-rates of $6\times 10^{-10}$ or lower. 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.

Figures (9)

• Figure 1: Structure of transversal $CCZ$ gates of tricycle codes.a) Schematic of a transversal $CCZ$ circuit on a 12-qubit code. Each code block is partitioned into three sectors of 4 qubits, labeled $\alpha, \beta, \gamma, \delta$ with subscripts and superscripts indicating the code block and sector. Colored curves denote $CCZ$ gates between triples of qubits where $f_{CCZ}$ is non-zero. b) Structure of transversal $CCZ$ circuits: all sectors participate via two disjoint sets of circuit layers denoted by orange and black edges that can individually be parallelized across qubits. Each qubit undergoes a maximum of $l$ black and a maximum of $m$ orange $CCZ$ gates, leading to a maximum degree of $l+m$. c) Logical $CCZ$ connectivity after basis optimization for a $K=3$ code . Circles denote logical qubits; rows correspond to separate code blocks. Thick black lines indicate usable $\overline{CCZ}$ gates for magic state distillation ($K_{CCZ}=2$ shown), while thin lines involve gauge qubits (pink), initialized in $\overline{\ket{0}}$. Blue circles represent logical qubits in disjoint triples connected only to other qubits in the triple or to gauge qubits.
• Figure 2: Single-shot distillation with tricycle codes. The logical $\ket{\overline{+}}^{\otimes K}$ state of the tricycle code can be prepared fault-tolerantly in constant depth by harnessing the code’s intrinsic resilience in one basis—namely, by preparing the physical qubits such that the associated stabilizer checks are deterministic—together with single-shot error correction in the complementary, non-deterministic, basis. The logical non-Clifford operation is implemented via a constant-depth circuit composed of physical $CCZ$ gates. The output is a logical hypergraph magic state which embeds $K_{CCZ} \leq K$ disjoint logical $\ket{\overline{CCZ}}$ resource states.
• Figure 3: Phenomenological noise simulation of single-shot state preparation in the $Z$ basis for $4-2-2$ tricycle codes. Our method follows Ref. hong2024single and assumes that the initial $Z$ syndrome is trivial. For each code, we simulate one round of syndrome measurement in which measurement errors occur with probability $p$, though we expect performance to improve with a larger decoding window (see \ref{['sec::noise_sim']}). A most likely error (MLE) decoder applies a minimum weight correction to both the data and measurement qubits. Then, we simulate a noisy transversal $Z$ basis measurement of the data qubits, decode the reconstructed syndrome with the MLE decoder, and apply the corresponding correction. A logical failure is said to occur if the residual $X$ operator is a logical operator of the tricycle code, and the logical error rate is normalized per logical qubit. The observed phenomenological threshold is $\gtrsim 13\%$. Logical error rates are determined via Monte Carlo simulations; error bars indicate standard errors, computed as $\sqrt{p_L(1-p_L)/M}$, where $M$ is the number of samples.
• Figure 4: Circuit-level noise simulation results for tricycle codes. (a) Logical error rate for the $[[48,6,(d_z=4,d_x=8)]]$ code as a function of abort rate under cluster postselection (blue) and full error detection (orange). The cluster postselection data show the tradeoff between logical error rate and postselection (abort) probability using a BP+LSD decoder, while full error detection corresponds to strictly accepting only trials with no detected stabilizer flips. (b) Logical error rate versus two-qubit physical gate error rate $p_{2q}$ for $d$-rounds, fault-tolerant error correction in the $X$ basis, for tricycle codes of increasing size and distance using a most-likely error (MLE) decoder. In both panels, errors are sampled according to a standard two-qubit depolarizing circuit-level noise model and the logical error rate corresponds to the total logical error rate normalized by the number of QEC rounds and by the number of logical qubits. Logical error rates are determined via Monte Carlo simulations, with each data point corresponding to $M$ samples; error bars indicate the standard error as $\sqrt{p_L(1-p_L)/M}$.
• Figure 5: Implementation of tricycle codes on neutral atom arrays. (a) Syndrome extraction circuit. Each line denotes a sector. CNOTs are applied on pairs of qubits across two sectors, with the pairing determined by the permutation matrices. (b) Within each sector, physical qubits sharing the same $x$ index form tiles arranged in a row; within each tile, qubits are ordered by their $y$ and $z$ indices. Two AOD movements can realize $x$-, $y$-, or $z$-cycling. (c) Parallel CNOTs can be performed by Rydberg interaction on two overlaying sectors. Data sectors reside in the entangling zone and have larger spacing to avoid intra-sector interactions. After permutation in the workspace below, check sectors are fetched to overlay with the corresponding data sectors and perform CNOTs. Then, check sectors are put back to their original positions, and perform permutation for the next layer.

Theorems & Definitions (48)

• Definition 1: Tricycle codes
• Definition 2: Support and intersection subgroups
• Definition 3: Connected codes
• Proposition 1: Non-binary 3D HGP
• proof
• Theorem A.1
• proof
• Theorem A.2
• proof
• Lemma A.3