Quantum error correction is a key challenge for the development of practical quantum computers, a direction in which significant experimental progress has been made in recent years. In solid-state qubits, one of the leading information loss mechanisms is dephasing, usually modeled by phase-flip errors. Here, we introduce quasistatic phase damping, a more subtle error model which describes the effect of Larmor frequency fluctuations due to
1/f noise. We show how this model is different from a simple phase-flip error model, in terms of multicycle error correction. Considering the surface code, we provide numerical evidence for an error threshold, in the presence of quasistatic phase damping and readout errors. We discuss the implications of our results for spin qubits and superconducting qubits.
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.
The self-energy, spectral functions, and susceptibilities of two-dimensional systems with strong ferromagnetic fluctuations are considered within the quasistatic approach. The self-energy at low temperatures
T has a non-Fermi-liquid form in the energy window $\ensuremath{\mid}\ensuremath{\omega}\ensuremath{\mid}\ensuremath{\lesssim}{\ensuremath{\Delta}}_{0}$ near the Fermi level, where ${\ensuremath{\Delta}}_{0}$ is the ground-state spin splitting for magnetically ordered ground state, and ${\ensuremath{\Delta}}_{0}\ensuremath{\propto}{T}^{1∕2}\phantom{\rule{0.2em}{0ex}}{\mathrm{ln}}^{1∕2}({v}_{F}∕T)$ in the quantum critical regime (
vF is the Fermi velocity). Spectral functions have a two-peak structure at finite
T above the magnetically ordered ground state, which implies quasisplitting of the Fermi surface in the paramagnetic phase in the presence of strong ferromagnetic fluctuations. The triplet pairing amplitude in the quasistatic approximation increases with increasing correlation length; at low temperatures $T⪡{\ensuremath{\Delta}}_{0}$ the vertex corrections become important and the Eliashberg approach is not justified. The results for the spectral properties and susceptibilities in the quantum critical regime near charge (spin) instabilities with large enough correlation length $\ensuremath{\xi}⪢{(T∕{v}_{F})}^{\ensuremath{-}1∕3}$ are obtained.