2D Compass Codes

The compass model on a square lattice provides a natural template for building subsystem stabilizer codes. The surface code and the Bacon-Shor code represent two extremes of possible codes depending on how many gauge qubits are fixed. We explore threshold behavior in this broad class of local codes by trading locality for asymmetry and gauge degrees of freedom for stabilizer syndrome information. We analyze these codes with asymmetric and spatially inhomogeneous Pauli noise in the code capacity and phenomenological models. In these idealized settings, we observe considerably higher thresholds against asymmetric noise. At the circuit level, these codes inherit the bare-ancilla fault tolerance of the Bacon-Shor code.

Popular Summary

Quantum computing promises to be a powerful tool in computer science, physics, and quantum chemistry. However, quantum information is also extremely delicate and is easily perturbed by its environment. If we want to realize the power of quantum computing, we must protect and preserve quantum information against these confounding factors. Quantum error-correcting codes can produce robust quantum bits by encoding them into many more nonrobust quantum bits. We explore a large class of such codes that interpolate between two of the most popular quantum codes, the surface code and the Bacon-Shor code.

We call this family “compass codes,” because of their similarity to the quantum compass model, a theory of matter in which couplings between spins depend on orientation. We establish many fundamental properties of these codes, including thresholds below which these encodings become much more robust. Furthermore, as many popular architectures have quantum bits arrayed on a physical chip, each physical quantum bit is likely to experience a different type of noise, depending on its immediate surroundings. We demonstrate that, when taking certain noisy factors into account, the malleability of this family of codes performs extremely well in idealized settings.

In the future, we hope that such codes can be developed around noise sources that are asymmetric (which depend on qubit placement within a chip) or biased (which favor one error type over another)—both of which are relevant to many proposals for quantum computing.

Figure 1

An example of a compass code on a $9\times 9$ lattice with qubits placed on vertices. Red and blue plaquettes represent cuts in the $Z$-type and $X$-type stabilizers, respectively. Equivalently, $Z$-type stabilizers are formed by blue plaquettes on the left, and $X$-type stabilizers are formed by red plaquettes on the right. Bare horizontal lines on the left represent horizontal $ZZ$ stabilizers or “0”-area blue plaquettes; the analogous vertical lines on the right form $XX$ stabilizers. As there are no blank plaquettes in the combined picture, all of the gauge d.o.f. are fixed.

Figure 2

The square represents the decoder graph (unseen) with north-south open boundary conditions. First, the clusters (dashed enclosures) are grown to an erasure consistent with the (red) marked syndromes. We then find maximum-weight spanning trees, with unmarked syndromes in black. After peeling the leaf nodes, we decode the trees using dynamic programming.

Figure 3

A comparison of logical error rates for the asymmetric decoder (solid lines) versus the symmetric decoder (dashed lines). Each data point was generated with $10^6$ independent Monte Carlo trials.

Figure 4

The left-hand side represents the graph describing the two-body Ising model. The right-hand side represents its dual, the decoder graph. The blue squares represent cuts in the $X$-type stabilizers on a $9\times 9$ lattice. The connectivity on the left-hand graph determines the sparsity on the right.

Figure 5

Scaling of the critical disorder $p_c$ with respect to the surface density $q_{\text{surf}}$. Autocorrelation is checked using a binning analysis; the fit is linear through the origin. The widest error bars are of total width of approximately 1%. At $q_{\text{surf}}=1$, the results closely match the established critical point at $p_c=0.1094\pm 0.0002$ [51].

Figure 6

Scaling of the estimated threshold $p$ with respect to the Shor density $q_{\text{shor}}$. The fit is quadratic through the origin; the finite-size effects are apparent. All points were obtained on $81\times 81$-size lattices, except for $q_{\text{shor}}=0.9$. To emphasize finite-size effects, this was performed on a $631\times 631$ lattice, which is greater than necessary for fault-tolerant computation [10].

Figure 7

Error rates for gradually ($w=0.25$, top panel) and steeply ($w=0.10$, bottom panel) inclined linear noise, computed on a lattice of size $33\times 33$ using union-find decoding in the high-noise regime. Here, $p_{\text{fail}}$ represents the total probability of a failure in either the $X$- or $Z$-type decoders.