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arXiv:2405.09287

Logical coherence in 2D compass codes

Bálint Pató, J. Wilson Staples, Kenneth R. Brown

Published May 15, 2024• 1 citation
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Abstract

2D compass codes are a family of quantum error-correcting codes that contain the Bacon-Shor codes, the $X$-Shor and $Z$-Shor codes, and the rotated surface codes. Previous numerical results suggest that the surface code has a constant accuracy and coherence threshold under uniform coherent rotation. However, having analytical proof supporting a constant threshold is still an open problem. It is analytically proven that the toric code can exponentially suppress logical coherence in the code distance $L$. However, the current analytical lower bound on the threshold for the rotation angle $θ$ is $|\sin(θ)| < 1/L$, which linearly vanishes in $L$ instead of being constant. We show that this lower bound is achievable by the $Z$-Shor code which does not have a threshold under stochastic noise. Compass codes provide a promising direction to improve on the previous bounds. We analytically determine thresholds for two new compass code families that provide upper and lower bounds to the rotated surface code's numerically established infidelity threshold. Furthermore, using a Majorana mode-based simulator, we use random families of compass codes to smoothly interpolate between the $Z$-Shor codes and the $X$-Shor codes.

DOI
10.48550/arxiv.2405.09287

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Logical coherence in two-dimensional compass codes

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Two-dimensional compass codes are a family of quantum error-correcting codes that contain the Bacon-Shor codes, the $X$-Shor and $Z$-Shor codes, and the rotated surface codes. Previous numerical results suggest that the surface code has a constant accuracy and coherence threshold under uniform coherent rotation. However, having analytical proof supporting a constant threshold is still an open problem. It is analytically proven that the toric code can exponentially suppress logical coherence in the code distance $L$. However, the current analytical lower bound on the threshold for the rotation angle $\ensuremath{\theta}$ is $|sin(\ensuremath{\theta})|<1/L$, which linearly vanishes in $L$ instead of being constant. We show that this lower bound is achievable by the $Z$-Shor code, which does not have a threshold under stochastic noise. Compass codes provide a promising direction to improve on the previous bounds. We analytically determine thresholds for two new compass code families that provide upper and lower bounds to the rotated surface code's numerically established infidelity threshold. Furthermore, using a Majorana mode-based simulator, we use random families of compass codes to smoothly interpolate between the $Z$-Shor codes and the $X$-Shor codes.

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2D compass codes are a family of quantum error-correcting codes that contain the Bacon-Shor codes, the XXX-Shor and ZZZ-Shor codes, and the rotated surface codes. Previous numerical results suggest that the surface code has a constant accuracy and coherence threshold under uniform coherent rotation. However, having analytical proof supporting a constant threshold is still an open problem. It is analytically proven that the toric code can exponentially suppress logical coherence in the code distance LLL. However, the current analytical lower bound on the threshold for the rotation angle θθθ is ∣sin⁡(θ)∣<1/L|\sin(θ)| < 1/L∣sin(θ)∣<1/L, which linearly vanishes in LLL instead of being constant. We show that this lower bound is achievable by the ZZZ-Shor code which does not have a threshold under stochastic noise. Compass codes provide a promising direction to improve on the previous bounds. We analytically determine thresholds for two new compass code families that provide upper and lower bounds to the rotated surface code's numerically established infidelity threshold. Furthermore, using a Majorana mode-based simulator, we use random families of compass codes to smoothly interpolate between the ZZZ-Shor codes and the XXX-Shor codes.
Two-dimensional compass codes are a family of quantum error-correcting codes that contain the Bacon-Shor codes, the XXX-Shor and ZZZ-Shor codes, and the rotated surface codes. Previous numerical results suggest that the surface code has a constant accuracy and coherence threshold under uniform coherent rotation. However, having analytical proof supporting a constant threshold is still an open problem. It is analytically proven that the toric code can exponentially suppress logical coherence in the code distance LLL. However, the current analytical lower bound on the threshold for the rotation angle $\ensuremath{\theta}$ is $|sin(\ensuremath{\theta})|<1/L$, which linearly vanishes in LLL instead of being constant. We show that this lower bound is achievable by the ZZZ-Shor code, which does not have a threshold under stochastic noise. Compass codes provide a promising direction to improve on the previous bounds. We analytically determine thresholds for two new compass code families that provide upper and lower bounds to the rotated surface code's numerically established infidelity threshold. Furthermore, using a Majorana mode-based simulator, we use random families of compass codes to smoothly interpolate between the ZZZ-Shor codes and the XXX-Shor codes.
Threshold analyses of quantum error-correcting codes are well established for stochastic error models, in which errors occur randomly with given probabilities. However, errors in actual devices can also be coherent, such as unwanted ZZZ rotations due to imperfect control, which are not captured by stochastic error models. For the surface code, numerical studies have indicated threshold behavior even under coherent errors, but a rigorous proof of threshold existence is lacking. Here we prove that a positive threshold for coherent ZZZ-rotation errors exists for quantum low-density parity-check codes with a bounded number of logical qubits, including the surface code and other topological codes. Specifically, we show that the maximum-likelihood Pauli recovery suppresses the entanglement infidelity exponentially in the code distance up to a prefactor linear in the number of physical qubits whenever the rotation angles lie below a constant that is independent of the code size. The proof combines Fourier analysis to retain the interference among the amplitudes of coherent errors with the cluster expansion of abstract polymer models. Our results expand the theoretical foundation of quantum error correction and offer a statistical-mechanical description of quantum error correction beyond stochastic errors.
Coherent noise on the physical qubits can lead to coherent error on the logical qubits in a quantum error-corrected system, even if the best recovery operator is found. An optimal decoder under coherent noise finds the minimal angle of the logical, coherent rotation for a given physical rotation angle under the uniform, coherent noise model. We show that one can optimally decode2D2\mathrm{D}2Dcompass codes without an explicit optimal decoder in polynomial complexity by simply flipping the sampled logical angle using a Majorana simulator and a sub-optimal decoder. Estimating the threshold requires tools different from MWPM decoding. As the infidelity curves don't intersect, we use finite-size scaling to evaluate optimal decoding performance by estimating the logical infidelity at the infinite distance limit. We motivate our finite size scaling ansatz based on repetition code-like compass codes for which we can exactly calculate the logical channel up to a distance of up to 1021. Under optimal decoding, our finite-size scaling methods predict a threshold of(0.274±0.010)π(0.274\pm 0.010)\pi(0.274±0.010)πfor the surface code and a threshold of(0.252±0.001)π(0.252\pm 0.001)\pi(0.252±0.001)πfor the m=2 Z-stacked Shor code. We compare our results to previous predictions and find a slight deviation in the case of the Z-stacked Shor code and a slight decrease in the case of the surface code.
We study the structure and transversal logical capabilities of stabilizer quantum error correcting codes. Among our results, we identify universal lower bounds on circuit depth to generate a full logical Clifford algebra, and develop novel constructions of logical transversal gates including a new depth-one transversal phase S‾\mathrm{\overline{S}}S gate in the rotated surface code and a depth-one intra-block CZ‾\mathrm{\overline{CZ}}CZ gate in the 2D-toric code that generalizes to all odd distances and all lengths L≥3L\ge3L≥3, respectively. Finally, we construct a high-rate non-LDPC CSS code family with parameters [[n,n,Θ(nβ)]][[n,\sqrt{n},Θ({n^β})]][[n,n​,Θ(nβ)]] where β≈0.2823β\approx 0.2823β≈0.2823 in one demonstrated case, that provably possesses a constant-depth complete 2-local transversal logical Clifford basis instruction set architecture (ISA) composed of all individually targeted S‾\mathrm{\overline{S}}S, SHS‾=X\mathrm{\overline{SHS}} = \sqrt{X}SHS=X​, and CZ‾\mathrm{\overline{CZ}}CZ gates. This ISA is depth-one for certain subfamilies that we design and generally constant-depth under certain conditions. The code family is built from a small code with parameters [[n0,2,d0]][[n_0, 2, d_0]][[n0​,2,d0​]], and is tunable in the standard way: it tiles out to form utility-scale logical qubit counts, and it scales up through concatenation to achieve higher distances and error suppression. We show that this construction preserves the depth-one complete transversal logical Clifford basis ISA when composed with these commuting construction actions, inheriting structure from the core codes so that at scale the complete logical Clifford basis ISA remains depth-one up to depth-two addressable operations between tiled cores. We call these Quantum Logic 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.
Statistical mechanics mappings provide key insights on quantum error correction. However, existing mappings assume incoherent noise, thus ignoring coherent errors due to, e.g., spurious gate rotations. We map the surface code with coherent errors, taken as XXX or ZZZ rotations (replacing bit or phase flips), to a two-dimensional (2D) Ising model with complex couplings, and further to a 2D Majorana scattering network. Our mappings reveal both commonalities and qualitative differences in correcting coherent and incoherent errors. For both, the error-correcting phase maps, as we explicitly show by linking 2D networks to 1D fermions, to a Z2{\mathbb{Z}}_{2}Z2​-nontrivial 2D insulator. However, beyond a rotation angle ${\ensuremath{\phi}}_{\mathrm{th}}$, instead of a Z2{\mathbb{Z}}_{2}Z2​-trivial insulator as for incoherent errors, coherent errors map to a Majorana metal. This ${\ensuremath{\phi}}_{\mathrm{th}}$ is the theoretically achievable storage threshold. We numerically find ${\ensuremath{\phi}}_{\mathrm{th}}\ensuremath{\approx}0.14\ensuremath{\pi}$. The corresponding bit-flip rate ${\mathrm{sin}}^{2}({\ensuremath{\phi}}_{\mathrm{th}})\ensuremath{\approx}0.18$ exceeds the known incoherent threshold ${p}_{\mathrm{th}}\ensuremath{\approx}0.11$.