Color code off-the-hook: avoiding hook errors with a single auxiliary per plaquette

Abstract

Syndrome extraction in the planar color code is complicated by high weight stabilizers and hook errors that can reduce the circuit-level distance. With a single auxiliary qubit per plaquette, any spatially uniform circuit halves the circuit-level distance. We propose a single-auxiliary syndrome extraction circuit with color-dependent gate schedules that avoids all malign hook errors in the bulk, thereby preserving the full circuit-level distance. The circuit has minimal depth: all stabilizers of the same Pauli type are measured in parallel in six time steps. Furthermore, this schedule can be readily applied to the XYZ color code circuit, yielding an improved temporal distance. We find that at the boundary, no single hook error alone reduces the distance; instead, only certain combinations of hook errors do, which we call fractional hook errors. We demonstrate through Monte Carlo simulations over a range of circuit-level noise models and physical error rates that our circuit outperforms the previous state of the art.

Figures (3)

• Figure 1: Malign hook errors and our schedule.(a) Illustration of all malign hook error pairs in plaquettes of each color (triplets not shown as they are benign). Each highlighted pair would reduce the distance by one if it could be generated by a hook error on the plaquette hosting that pair. The pairs themselves are colored according to the type of anyon for which they would provide a shortcut. Malign hook errors on trapezoidal boundary plaquettes differ slightly from those in the bulk. On the corner plaquettes, all hook errors are malign (not shown). (b) Our syndrome extraction circuit. All malign hook errors are avoided in the bulk, and the depth is minimal. On the boundary certain combinations of hook errors can reduce the distance slightly; we call these "fractional hook errors." Three hook errors (orange lines) and two data-qubit errors (orange dots) are equivalent to a sequence of six data-qubit errors (purple dots) along the boundary, forming part of a logical operator.
• Figure 2: Hook error propagation. In each plaquette, there are three possible hook errors that can act on the auxiliary qubit during the measurement of each of the two Pauli stabilizers (dark blue/red). Each of these errors propagates through the syndrome extraction circuit to form a distinct correlated data-qubit error (light blue/red). (a) Errors occurring after the second CNOT during the measurement of each stabilizer. (b) Errors between the third and fourth CNOTs. (c) Errors between the fourth and fifth CNOTs.
• Figure 3: Logical error rate vs. total qubit count across noise models and error rates. Top row: SI1000 noise model. Middle row: uniform depolarizing noise model. Bottom row: noisy CNOT noise model. The physical error rate increases from left to right across the panels. Solid lines show our circuits, and dashed lines show previous work. Highlights indicate hypotheses with likelihoods within a factor of 1000 of the maximum-likelihood hypothesis, given the sampled data.