Direct measurement of Bacon-Shor code stabilizers

A Bacon-Shor code is a subsystem quantum error-correcting code on an $L\times L$ lattice where the $2(L-1)$ weight-$2L$ stabilizers are usually inferred from the measurements of $2L(L-1)$ weight-2 gauge operators. Here, we show that the stabilizers can be measured directly and fault tolerantly with bare ancillary qubits by constructing circuits that follow the pattern of gauge operators. We then examine the implications of this method for small quantum error-correcting codes by comparing distance-3 versions of the rotated surface code and the Bacon-Shor code with the standard depolarizing model and in the context of a trapped-ion quantum computer. We find that for a simple circuit of prepare, error correct, and measure, the Bacon-Shor code outperforms the surface code by requiring fewer qubits, taking less time, and having a lower error rate.

Figure 1

The compass model with $ZZ$ bonds along the vertical axis and $XX$ bonds along the horizontal axis. Choices of gauge on a $3\times 3$ lattice lead to two well-known stabilizer codes: [[9,1,3]] surface code and [[9,1,3]] Bacon-Shor code. The underlying bonds of the compass model are a guide for how to fault tolerantly measure surface code and Bacon-Shor code stabilizers with bare ancillary qubits. Measuring stabilizers in order of gauge operators can help suppress hook errors on two-qubit gates in the stabilizer measurement circuit. The blue arrows show the circuit order for measuring an $X$-type stabilizer for both codes. (a) Compass model. (b) Surface. (c) Bacon-Shor.

Figure 2

Comparison of Bacon-Shor-13 and Surface-17 in a simple circuit simulation. (a) With one round of error correction, Bacon-Shor-13 shows a pseudothreshold of 0.9% and Surface-17 shows a pseudothreshold of 0.1%. The difference is mainly due to the difference in logical state preparation. (b) At a physical error rate of ${10}^{-3}$, Surface-17 starts to outperform Bacon-Shor-13 in more than nine rounds of error correction. (a) One round of error correction. (b) Multiple rounds of error correction.

Figure 3

Comparison of Bacon-Shor-13 and Surface-17 with a simple circuit under the influence of an ion-trap error model (see text). Logical error is plotted as a function of a control error in the Mølmer-Sørensen gate. Each plot includes an additional error due to ion heating ($\dot{\overline{n}}$) or gate-induced dephasing ($T_2$) that depends on gate time. (a) Ion heating error. (b) Spin dephasing error.

Figure 4

Qubit labeling for the [[9,1,3]] surface code and Bacon-Shor code. Data qubits, ancillary qubits for $X$-type stabilizers, and ancillary qubits for $Z$-type stabilizers are shown in black, blue, and orange circles, respectively. (a) Surface-17. (b) Bacon-Shor-13.