Statistical mechanical mapping and maximum-likelihood thresholds for the surface code under generic single-qubit coherent errors

TL;DR

This work addresses the impact of generic single-qubit coherent errors on the surface code by introducing a statistical-mechanics mapping to a complex, inter-layer RBIM and a nonunitary (1+1)D transfer-matrix circuit. It demonstrates the existence of an error-correcting phase with a maximum-likelihood threshold $α_ ext{th}$ that, for coherent errors, exceeds the corresponding incoherent Pauli-twirl bound and MWPM thresholds. The authors develop an efficient MPS-based method to simulate the transfer-matrix circuit and introduce a syndrome-sampling algorithm that handles interference between error strings, enabling quantitative threshold estimates across rotation axes. They show a phase diagram featuring area-law below threshold and a slower, (approximately) logarithmic entanglement growth above threshold, with entanglement fluctuations serving as a practical threshold indicator. These results advance understanding of coherent errors in QEC and suggest pathways to decoders that approach the maximum-likelihood limit for realistic, coherent noise models.

Abstract

The surface code, one of the leading candidates for quantum error correction, is known to protect encoded quantum information against stochastic, i.e., incoherent errors. The protection against coherent errors, such as from unwanted gate rotations, is however understood only for special cases, such as rotations about the $X$ or $Z$ axes. Here we consider generic single-qubit coherent errors in the surface code, i.e., rotations by angle $α$ about an axis that can be chosen arbitrarily. We develop a statistical mechanical mapping for such errors and perform entanglement analysis in transfer matrix space to numerically establish the existence of an error-correcting phase, which we chart in a subspace of rotation axes to estimate the corresponding maximum-likelihood thresholds $α_\mathrm{th}$. The classical statistical mechanics model we derive is a random-bond Ising model with complex couplings and four-spin interactions (i.e., a complex-coupled Ashkin-Teller model). The error correcting phase, $α<α_\mathrm{th}$, where the logical error rate decreases exponentially with code distance, is shown to correspond in transfer matrix space to a gapped one-dimensional quantum Hamiltonian exhibiting spontaneous breaking of a $\mathbb{Z}_2$ symmetry. Our numerical results rest on two key ingredients: (i) we show that the state evolution under the transfer matrix -- a non-unitary (1+1)-dimensional quantum circuit -- can be efficiently numerically simulated using matrix product states. Based on this approach, (ii) we also develop an algorithm to (approximately) sample syndromes based on their Born probability. The $α_\mathrm{th}$ values we find show that the maximum likelihood thresholds for coherent errors are larger than those for the corresponding incoherent errors (from the Pauli twirl), and significantly exceed the values found using minimum weight perfect matching.

Figures (7)

• Figure 1: Summary of results. (a) Bloch sphere visualizing the coherent error from a unitary rotation of single qubits [Eq. \ref{['eq:rotation']}]. We numerically simulate rotations by the angle $\alpha$ about three different axes in the red shaded plane with $\varphi=\pi/4$ and $\vartheta = 0.05\pi,0.15\pi,0.304\pi$. (b) Approximate phase diagram for coherent errors and $\varphi=\pi/4$. In the phase with $\alpha$ below threshold, the logical error rate decays exponentially with distance (QEC✓); the dual quantum circuit yields a 1D area law for entanglement. Above threshold, the logical error rate decays slowly (QEC✗) and the dual quantum circuit yields a logarithmic entanglement phase. Black dots denote conservative threshold estimates based on maximal entanglement fluctuations Kjall:2014bd, and the error bars arise from the uncertainty of this maximum. The circuit is Gaussian at $\vartheta=0$, which enables the simulation of larger system sizes, yielding a more accurate threshold (value at $\vartheta=0$ from Ref. Venn:2023fp). The orange crosses mark the corresponding incoherent Pauli twirl Emerson:2007ejSilva:2008ij threshold Dennis:2002dsBombin:2012km. (c) The surface code geometry we study, illustrated for $L=M=4$. The direct lattice is shown in black, with physical qubits on its links (black dots), stabilizers $S_v^X$ (white disks) on its vertices, and $S_p^Z$ (gray disks) on its faces; the dual lattice is shown in gray. The blue dashed line shows a representative for the logical $\overline{X}$ and the red dashed line one for the logical $\overline{Z}$. We map the surface code to a RBIM (Sec. \ref{['sec:stat_mech']}) with complex vertex-vertex couplings $J^x$ (wiggly light green line), plaquette-plaquette couplings $J^z$ (wiggly dark green line), and four-spin interactions $J^y$ (light green square) that couple direct and dual lattice. (d) Transfer matrix representation (Sec. \ref{['sec:quantum_circuit']}): The three coupling terms act as three-qubit gates $\hat{H}_{\mu,s}^{(l,m)}$ and $\hat{V}_{\mu,s}^{(l,m)}$. To sample error strings (Sec. \ref{['sec:algorithm']}), for each site we successively (visualized by the red line) apply one of four possible three-qubit gates, sampled based on its probability.
• Figure 2: (a) Example of a syndrome that is characterized by $S_v^X =-1$ and $S_p^Z=-1$, marked by red and blue crosses, respectively (the white and gray disks without crosses denote $S_v^X=+1$ and $S_p^Z=+1$ eigenvalues). A reference string $\overline{\mathsf{O}}^\mu C_s$ connects pairs of crosses of the same type: The blue and red lines denote Pauli strings of $X_j$ operators and $Z_j$ operators, respectively. In the RBIM, reference strings translate to signs $\eta_{\mu,s}^{x,j}=-1$ for sites with $X_j$ (blue), and $\eta_{\mu,s}^{z,j}=-1$ for sites with $Z_j$ (red); cf. Eq. \ref{['eq:reference_string']}. (b) Pauli string $P_\varsigma$ that equals $\overline{\mathsf{O}}^\mu C_s$ up to closed loops. In the RBIM, closed loops that include $S_v^X$ and $S_p^Z$ translate to the spins $\sigma_v=-1$ (blue vertices) and $\tilde{\sigma}_p=-1$ (red plaquettes), respectively. In this example, the Pauli strings $\overline{\mathsf{O}}^\mu C_s$ [panel (a)] and $P_\varsigma$ [panel (b)] anticommute: An odd number of vertices [here: one vertex marked by a dark blue arrow in (b)] is simultaneously a $S_v^x=-1$ syndrome and part of a loop characterized by $\sigma_v=-1$.
• Figure 3: (a) Quantum circuit $\hat{\mathcal{M}}_{\mu,s}$ consisting of individual gates $\hat{H}_{\mu,s}^{(l,m)}$ and $\hat{V}_{\mu,s}^{(l,m)}$, shown for $L=3$, $M=4$. The boundary conditions are encoded by the product state $\ket{\phi_0}$ [Eq. \ref{['eq:boundary1']}] shown to the left and right of $\hat{\mathcal{M}}_{\mu,s}$, where rectangles represent $\ket{0}$ (to pick $\sigma = 1$ in the Ising model) and diamonds $\sqrt{2} \ket{+}$ (to ensure summation over $\sigma=\pm1$). (b) Visualization of the sampling algorithm: For convenience, we label the individual gates from panel (a) by $\hat{T}_j^{(\eta_j)}$; dark gray gates have a fixed error string [labeled by superscript signs $(\eta_j^x,\eta_j^z)$], light gray gates are not sampled yet, and the red gate is the currently sampled gate. Gates are successively applied to the initial $\ket{\phi_0}$, resulting in the state $\ket{\phi_j^{\{\eta\}}}$ for the $j^\text{th}$ step (here: $j=6$), and $(\eta_j^x,\eta_j^z)$ are sampled according to the marginal distribution $P_j$ (which equals $\lVert \ket{ \phi_{j}^{\{\eta\}} }\rVert^2$ for all but the last layer of the circuit).
• Figure 4: (a)--(c) Syndrome-averaged minimum infidelity and (d)--(f) half-system entanglement entropy as a function of the coherent rotation angle $\alpha$, averaged over 1000 to 10000 error strings. The different colors denote the code distance $d$, and the angle $\vartheta$ of the rotation axis is shown in the panels. The inset in (c) shows the exponential decay of $P_L$ with code distance below threshold ($\alpha=0.07\pi$), where the colors denote different $\vartheta$. Error bars indicating the standard error are imperceptible. The insets in (d)--(f) show $\sigma_S$, the standard deviation of the entanglement entropy, whose maximum we use to identify the error threshold.
• Figure 5: Logical error from MWPM $P_L^\text{(MWPM)}$ as a function of the coherent rotation angle $\alpha$, averaged over 1000 to 10000 error strings; error bars are imperceptible. The different colors denote the code distance $d$, and the angle $\vartheta$ of the rotation axis is shown in the panels. The inset in (a) shows the exponential decay of $P_L^\text{(MWPM)}$ with code distance below threshold ($\alpha=0.07\pi$).