Spacetime Spins: Statistical mechanics for error correction with stabilizer circuits

TL;DR

The paper tackles the challenge of analyzing circuit-level quantum error correction with dynamic syndrome extraction by mapping stabilizer circuits to spacetime subsystem codes and expressing error coset probabilities as partition functions of classical spin Hamiltonians. It introduces spin diagrams as a modular, circuit-based method to build these Hamiltonians and demonstrates the approach on the repetition and toric codes, deriving analytic and Monte Carlo estimates for maximum-likelihood decoding thresholds and revealing how circuit compilations and logical operations affect decodability. A key insight is that decoding success corresponds to minimizing a free-energy cost, with $\,\mathbb{P}(\overline{E})=\mathcal{Z}_E\,$ and $F_E=-\ln\mathcal{Z}_E$, while the ML success probability is a softmax over coset free energies. The framework exposes connections between dynamical quantum systems and noise-resilient phases of matter, providing a universal prescription to analyze, simulate, and compare decoding across stabilizer circuits and enabling targeted optimizations for fault-tolerant quantum computation.

Abstract

A powerful method for analyzing quantum error-correcting codes is to map them onto classical statistical mechanics models. Such mappings have thus far mostly focused on static codes, possibly subject to repeated syndrome measurements. Recent progress in quantum error correction, however, has prompted new paradigms where codes emerge from stabilizer circuits in spacetime -- a unifying perspective encompassing syndrome extraction circuits of static codes, dynamically generated codes, and logical operations. We show how to construct statistical mechanical models for stabilizer circuits subject to independent Pauli errors, by mapping logical equivalence class probabilities of errors to partition functions using the spacetime subsystem code formalism. We also introduce a modular language of spin diagrams for constructing the spin Hamiltonians, which we describe in detail focusing on independent circuit-level X-Z error channels. With the repetition and toric codes as examples, we use our approach to analytically rank logical error rates and thresholds between code implementations with standard and dynamic syndrome extraction circuits, describe the effect of transversal logical Clifford gates on logical error rates, and perform Monte Carlo simulations to estimate maximum likelihood thresholds. Our framework offers a universal prescription to analyze, simulate, and compare the decoding properties of any stabilizer circuit, while revealing the innate connections between dynamical quantum systems and noise-resilient phases of matter.

Figures (16)

• Figure 1: The gauge generators (black labels) introduced by various Clifford gates and corresponding spin model ingredients. Each generator is associated with a spin (white square); interactions (colored lines) connect spins according to Eq. \ref{['eq:H_ES_gauge_ind']}. Red and blue lines are interactions on $H_X$ and $H_Z$ respectively. Gridlines show the spacetime locations $(i,\tau)$, with time going up the page. A qubit $i$ idling at integer time $t$, introduces two gauge generators: $[X]_{i,t-0.5}[X]_{i,t+0.5}$ and $[Z]_{i,t-0.5}[Z]_{i,t+0.5}$. A CNOT has four gauge generators; $g_1 = [X]_{i,t-0.5}[XX]_{\{i,j\},t+0.5}$ acts on three spacetime qubits and hence its spin has three legs (interactions). A single-qubit measurement $M_Z$ (a single-qubit reset, $R_Z$, produces the same spins and interactions) has gauge generators with only one interaction in $H_Z$ so we draw its spins atop the corresponding spacetime qubit. $M_Z$ introduces no spins to $H_X$ (and similarly for $M_X$ and $H_Z$). Two-qubit measurements $M_{ZZ}$ introduce a spin with four interactions and three spins with two-interactions. Shown here is one basis choice; $[ZZ]_{\{i,j\},t-0.5}$ could also have been used instead of $[ZZ]_{\{i,j\},t+0.5}$.
• Figure 2: (Top) An example circuit for a two-qubit $ZZ$ measurement via an ancilla qubit, with associated spin diagrams on the right, using the conventions in Fig. \ref{['fig:spins_intro']}. Endcaps mark the corresponding termination of the interaction lines (open lines may join up with interactions at earlier/later times.) Errors occur in-between circuit operations; an example error realization is shown. These flip the sign of bonds on the spin model (dashed lines). $Y$ errors flip signs on both $H_X$ and $H_Z$. (Bottom) Spins with one interaction are integrated out and the interaction removed from $H_X$ (marked by crosses). Spins with two remaining interactions are integrated out and their interactions grouped together. The combined interactions are weighted by the number of spacetime locations grouped into that interaction (unlabeled lines are weight one).
• Figure 3: Common circuit elements and their spin diagrams after integrating out spins with one or two interactions. $M_\alpha$ are measurements of Pauli $\alpha$. The SWAP gate has two weight-$2$ interactions for each $H_\alpha$ that cross over each other, swapping physical qubits. The two-qubit measurements each introduce three two-qubit gauge generators (cf. Fig. \ref{['fig:spins_intro']}); when all three are integrated out, they create a weight-$4$ grouped interaction on all four spacetime qubits adjacent to the measurement.
• Figure 4: Measurement circuits for repetition code experiments, where errors occur in-between each circuit operation. We show representatives of error cosets that lead to decoding failures. The detector cells triggered by these errors are highlighted in yellow; some untriggered detector cells are shown in pink. Both circuits map onto a random-bond Ising model on a rotated square lattice. The errors flip the sign of bonds, indicated by dashed lines, and form a chain on the dual lattice. Detector cells form plaquettes of the lattice, and triggered detector cells host Ising vortices at the endpoints of the chains. The spatial and temporal (open) boundary conditions are interchanged between the two experiments.
• Figure 5: Error rates for memory and stability experiments corresponding to the repetition code measurement circuits in Figs. \ref{['fig:rep_memory']} and \ref{['fig:rep_stability']} respectively, using MWPM decoders with circuits of size $N=d$, $T=2d+1$. Both exhibit the same threshold, the noise rate $p_X$ below which increasing $d$ decreases the experimental error rate: approximately $10.0\%$.