Single-Shot Decoding and Fault-tolerant Gates with Trivariate Tricycle Codes

TL;DR

The paper introduces trivariate tricycle (TT) codes, a CSS qLDPC family defined by a length-3 chain complex that unifies high circuit-level thresholds, partial single-shot decodability, and a broad set of fault-tolerant gates. TT codes generalize BB codes and the 3D toric code, enabling large data-qubit savings for equivalent-distance encodings and offering constant-depth non-Clifford CCZ implementations via cup-product constructions. Demonstrations include threshold analyses under phenomenological and circuit-level noise, single-shot decoding evidence, and multiple code instances that outperform the 3D toric code in X- and Z-memory scenarios; TT codes also support transversal CZ gates between blocks and shift automorphisms for Clifford operations. The work suggests TT codes as a promising path toward scalable, fault-tolerant quantum computation with favorable resource overheads and new avenues for lattice-surgery and non-Clifford gate protocols in qLDPC frameworks.

Abstract

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$\times$ fewer data qubits as equivalent 3DTC encodings. We construct syndrome-extraction circuits for these codes and numerically demonstrate single-shot decoding in the X error channel in both phenomenological and circuit-level noise models. Under circuit-level noise, TT codes have a threshold of $0.3\%$ in the Z error channel and $1\%$ in the X error channel (with single-shot decoding). All TT codes possess several transversal $CZ$ gates that can partially address logical qubits between two code blocks. Additionally, 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 allows for a constant-depth implementation of logical $CCZ$ gates. We find examples of error-correcting and error-detecting codes using these constructions whose parameters out-perform those of the 3DTC, using up to $4\times$ fewer data qubits for equivalent-distance 3DTC encodings.

Figures (7)

• Figure 1: Layout of the 3D toric code on a cubic lattice. The code's defining polynomials are $A = 1+x$, $B=1+y$, $C=1+z$. Qubits are placed on edges, $X$-checks on vertices and $Z$-checks on faces. (a) $X$-checks are connected to adjacent data qubits by terms in $A$ (for Left qubits, shaded grey), $B$ (for Centre qubits, shaded black), or $C$ (for Right qubits, shaded white). Example $X$-check $x1z$ and its connected qubits is shown. (b) $Z$-checks fall into three families, coloured red ($Z_a$), blue ($Z_b$), and yellow $(Z_c)$, respectively. Each family corresponds to an orthogonal orientation of a plane in the lattice. $Z$-checks are connected to data qubits on the edges surrounding the face by terms from (transposes of) two of the three polynomials. As an example, the red $Z$-checks are connected to Centre qubits by terms in $C^\top$ and Right qubits by terms in $B^\top$. Three example $Z$-checks and their connected qubits are shown. TT codes can be locally laid out on the same 3D cubic lattice, with additional long-range connections. For TT codes built from weight-3 polynomials, the $X$ checks gain 3 long-range connections, and the $Z$ checks gain 2 long-range connections.
• Figure 2: Performance of the TT codes under a phenomenological noise model. We present logical error rates (LERs) in memory experiments performed over $2d_Z$ rounds of syndrome measurements. (a) TT and 3DTC performance in $X$ memory experiments, for small instances of TT codes from Table \ref{['tab:mycodes']}. For all instances considered, the TT codes perform better, while encoding more logical qubits than the $[[81,3,3]]$ 3DTC (yellow). Decoding was performed with BP+OSD-CS0 over the full syndrome history. (b) TT code LERs in $Z$ memory experiments with a (2,1) windowing strategy. The asymmetric distances of the TT codes lead to strong error suppression in the $X$ error channel ($Z$ memory), resulting in order of magnitude reductions in the LER, compared to the $Z$ error channel ($X$ memory).
• Figure 3: Total logical error probability for $Z$ memory vs window size used in the overlapping window strategy, for several TT codes, listed in \ref{['tab:codes_2']}. The plateaus demonstrates that optimal decoding performance can be obtained with a fixed window and commit size, indicating full syndrome histories do not need to considered. Simulations were performed under a phenomenological noise model with physical noise rate $p=3\times 10^{-2}$ and $N=14$ rounds. The decoding windows had commit regions of size 1.
• Figure 4: Performance of the TT codes under circuit level noise. (a) Logical error rate per round in $X$ memory experiments, decoded using BP+OSD-CS30, over the full syndrome history. A crossing between instances of TT codes is found at a physical error rate of $2.9 \times 10^{-3}$. (b) Logical error rate per round in $Z$ memory experiments. Decoding was performed using a $(2,1)$ overlapping window strategy (Sec. \ref{['sec: single-shot']}) with BP+OSD-CS0. We find a crossing at $1.1 \times 10^{-2}$. Arrowheads show the upper limit of the logical error rate.
• Figure 5: Logical error plateaus, for a commit size of 1 and varying window size, under a circuit level noise model for small TT codes. The $[[72,6,6]]$ and $[[81,6,6]]$ codes both exhibit single-shot decodability in the $X$ error channel, at error rates of $p=0.003$ and $p=0.006$. At $p=0.003$, the plateau begins around $w=3$, however at $p=0.006$ the plateau begins around $w=5$.

Theorems & Definitions (14)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7