Error thresholds of toric codes with transversal logical gates

TL;DR

This work provides a decoder-agnostic, rigorous bound on how transversal Clifford gates affect the error threshold in toric codes by mapping a two-block tCNOT circuit to classical stat-mech models. For persistent bit-flip errors with perfect syndromes, the problem reduces to a 2D random Ashkin–Teller model, yielding thresholds $p_c^c \approx 0.099$ and $p_c^t \approx 0.080$, both below the memory threshold $p_{th} \approx 0.109$. When syndrome errors are present, the mapping becomes a 3D random 4-body Ising model with a plane defect, conservatively placing the target threshold at $p_c^t \approx 0.028$, still below the memory-case $p^* \approx 0.033$. Overall, the results show that error spreading due to transversal gates modestly lowers decoding thresholds but does not catastrophically degrade fault-tolerance, and establish a general statistical-mechanics framework for analyzing transversal quantum circuits through decoder-agnostic thresholds. The approach offers insights for designing FTQC primitives and suggests potential connections between quantum error thresholds and topological phase transitions.

Abstract

The threshold theorem promises a path to fault-tolerant quantum computation by suppressing logical errors, provided the physical error rate is below a critical threshold. While transversal gates offer an efficient method for implementing logical operations, they risk spreading errors and potentially lowering this threshold compared to a static quantum memory. Available threshold estimates for transversal circuits are empirically obtained and limited to specific, sub-optimal decoders. To establish rigorous bounds on the negative impact of error spreading by the transversal gates, we generalize the statistical mechanical (stat-mech) mapping from quantum memories to logical circuits. We establish a mapping for two toric code blocks that undergo a transversal CNOT (tCNOT) gate. Using this mapping, we quantify the impact of two independent error-spreading mechanisms: the spread of physical bit-flip errors and the spread of syndrome errors. In the former case, the stat-mech model is a 2D random Ashkin-Teller model. We use numerical simulation to show that the tCNOT gate reduces the optimal bit-flip error threshold to $p=0.080$, a $26\%$ decrease from the toric code memory threshold $p=0.109$. The case of syndrome error coexisting with bit-flip errors is mapped to a 3D random 4-body Ising model with a plane defect. There, we obtain a conservative estimate error threshold of $p=0.028$, implying an even more modest reduction due to the spread of the syndrome error compared to the memory threshold $p=0.033$. Our work establishes that an arbitrary transversal Clifford logical circuit can be mapped to a stat-mech model, and a rigorous threshold can be obtained correspondingly.

Figures (10)

• Figure 1: Stat-mech mapping of toric code under errors. (a) Sketch of trivial $C\in S$ and non-trivial cycle $C\in L$ on top of an error $E$ in one toric code block, which triggers a pair of syndromes denoted by the two red dots. (b) Stat-mech model of toric code under bit-flip errors. Here the physical qubits of the toric code are marked by circles, and the weight-4 Z stabilizers are defined on the shaded plaquettes. A trivial cycle $C\in S$ can be parameterized by a domain wall of Ising spins $\{\sigma\}$ (marked by the arrows) on a 2D square lattice that is rotated by $45^\circ$, marked by dashed lines. Two ends of a link $l$ on this lattice are denoted by $l_{1,2}$. (c) The 3D stat-mech model in the case of syndrome errors for a single code block. The Ising spins live at the links of thd 3D cubic lattice, marked by the arrows. Some of the 4-body interaction terms in the Hamiltonian in Eq. \ref{['eq:R4bIM']} are marked by grey stars. The dashed lines at $t\in\mathbb{Z}+\frac{1}{2}$ form a 2D square lattice where Ising spins live on the sites, which is the same 2D square lattice in (b).
• Figure 2: (a) The implementation of the tCNOT gate between two rounds of syndrome extractions. The black dots at the centers of shaded plaquettes represent the measurement of the $Z$ stabilizers defined on these plaquettes. The choices of spacetime detectors $d^c$ and $d^t$ for the control and target code blocks between the two rounds of SE are marked in blue and green, repectively. Due to the tCNOT gate, the spacetime detector of the target block, $d^t$, extends to the control block, whose value is obtained from multiplying three $Z$ stabilizer measurement outcomes $d^t=M^c_1M^t_1M^t_2$. To avoid visual overlap, we sketch the spacetime detectors $d^c$ and $d^t$ in two adjacent plaquettes that host $Z$ stabilizers. (b) The logical circuit with tCNOT gate, persistent bit-flip noise channels $\mathcal{N}_{\tilde{p}}$ and two rounds of perfect syndrome extractions (SE), during which the weight-4 $Z$ stabilizers are measured.
• Figure 3: Stat-mech mapping of two toric code blocks undergoing a tCNOT gate, where the probability distribution of $E^c$ and $E^t$ are correlated due to the tCNOT gate. Here the physical qubits are denoted by gray circles living on the lattice makred by solid lines. Bit-flip errors $E^{c/t}$ and $E^{c/t}\cdot C^{c/t}$ of the physical qubits in the control/target code blocks are marked by gray circles with colored fillings. Syndromes triggered by $E^{c/t}$ are marked by dots at the center of shaded plaquettes which host $Z$ stabilizers. Trivial cycles $C^{c/t}$ in control/target blocks are parameterized by two sets of Ising spins $\{\sigma\}$ and $\{\tau\}$, which are located at lattice sites of the dotted-line lattice, and are denoted by arrows in blue/green.
• Figure 4: Coupling constants $K_2$ and $K_4$ in the random AT model Hamiltonian in Eq. \ref{['eq:atmodel']}, whose relations with $\tilde{p}$ are given in the Appendix Eq.\ref{['eq:abc']}. Coupling of RBIM, $J=\frac{1}{2}\ln\frac{1-2\tilde{p}(1-\tilde{p})}{2\tilde{p}(1-\tilde{p})}$, is shown as a reference.
• Figure 5: Magnetization of (a) $\tau$ and (b) $\sigma$ ($|M_\tau|$ and $|M_\sigma|$) versus persistent noise strength $\tilde{p}$ for system sizes $L=8, 12, 16$.