The typical model for measurement noise in quantum error correction is to randomly flip the binary measurement outcome. In experiments, measurements yield much richer information - e.g., continuous current values, discrete photon counts - which is th
en mapped into binary outcomes by discarding some of this information. In this work, we consider methods to incorporate all of this richer information, typically called soft information, into the decoding of quantum error correction codes, and in particular the surface code. We describe how to modify both the Minimum Weight Perfect Matching and Union-Find decoders to leverage soft information, and demonstrate these soft decoders outperform the standard (hard) decoders that can only access the binary measurement outcomes. Moreover, we observe that the soft decoder achieves a threshold 25% higher than any hard decoder for phenomenological noise with Gaussian soft measurement outcomes. We also introduce a soft measurement error model with amplitude damping, in which measurement time leads to a trade-off between measurement resolution and additional disturbance of the qubits. Under this model we observe that the performance of the surface code is very sensitive to the choice of the measurement time - for a distance-19 surface code, a five-fold increase in measurement time can lead to a thousand-fold increase in logical error rate. Moreover, the measurement time that minimizes the physical error rate is distinct from the one that minimizes the logical performance, pointing to the benefits of jointly optimizing the physical and quantum error correction layers.
Estimating and reducing the overhead of fault tolerance (FT) schemes is a crucial step toward realizing scalable quantum computers. Of particular interest are schemes based on two-dimensional (2D) topological codes such as the surface and color codes
which have high thresholds but lack a natural implementation of a non-Clifford gate. In this work, we directly compare two leading FT implementations of the T gate in 2D color codes under circuit noise across a wide range of parameters in regimes of practical interest. We report that implementing the T gate via code switching to a 3D color code does not offer substantial savings over state distillation in terms of either space or space-time overhead. We find a circuit noise threshold of 0.07(1)% for the T gate via code switching, almost an order of magnitude below that achievable by state distillation in the same setting. To arrive at these results, we provide and simulate an optimized code switching procedure, and bound the effect of various conceivable improvements. Many intermediate results in our analysis may be of independent interest. For example, we optimize the 2D color code for circuit noise yielding its largest threshold to date 0.37(1)%, and adapt and optimize the restriction decoder finding a threshold of 0.80(5)% for the 3D color code with perfect measurements under Z noise. Our work provides a much-needed direct comparison of the overhead of state distillation and code switching, and sheds light on the choice of future FT schemes and hardware designs.
The surface code is a prominent topological error-correcting code exhibiting high fault-tolerance accuracy thresholds. Conventional schemes for error correction with the surface code place qubits on a planar grid and assume native CNOT gates between
the data qubits with nearest-neighbor ancilla qubits. Here, we present surface code error-correction schemes using $textit{only}$ Pauli measurements on single qubits and on pairs of nearest-neighbor qubits. In particular, we provide several qubit layouts that offer favorable trade-offs between qubit overhead, circuit depth and connectivity degree. We also develop minimized measurement sequences for syndrome extraction, enabling reduced logical error rates and improved fault-tolerance thresholds. Our work applies to topologically protected qubits realized with Majorana zero modes and to similar systems in which multi-qubit Pauli measurements rather than CNOT gates are the native operations.
In statistical mechanics, a small system exchanges conserved quantities---heat, particles, electric charge, etc.---with a bath. The small system thermalizes to the canonical ensemble, or the grand canonical ensemble, etc., depending on the conserved
quantities. The conserved quantities are represented by operators usually assumed to commute with each other. This assumption was removed within quantum-information-theoretic (QI-theoretic) thermodynamics recently. The small systems long-time state was dubbed ``the non-Abelian thermal state (NATS). We propose an experimental protocol for observing a system thermalize to the NATS. We illustrate with a chain of spins, a subset of which form the system of interest. The conserved quantities manifest as spin components. Heisenberg interactions push the conserved quantities between the system and the effective bath, the rest of the chain. We predict long-time expectation values, extending the NATS theory from abstract idealization to finite systems that thermalize with finite couplings for finite times. Numerical simulations support the analytics: The system thermalizes to the NATS, rather than to the canonical prediction. Our proposal can be implemented with ultracold atoms, nitrogen-vacancy centers, trapped ions, quantum dots, and perhaps nuclear magnetic resonance. This work introduces noncommuting conserved quantities from QI-theoretic thermodynamics into quantum many-body physics: atomic, molecular, and optical physics and condensed matter.
Three-dimensional (3D) color codes have advantages for fault-tolerant quantum computing, such as protected quantum gates with relatively low overhead and robustness against imperfect measurement of error syndromes. Here we investigate the storage thr
eshold error rates for bit-flip and phase-flip noise in the 3D color code on the body-centererd cubic lattice, assuming perfect syndrome measurements. In particular, by exploiting a connection between error correction and statistical mechanics, we estimate the threshold for 1D string-like and 2D sheet-like logical operators to be $p^{(1)}_mathrm{3DCC} simeq 1.9%$ and $p^{(2)}_mathrm{3DCC} simeq 27.6%$. We obtain these results by using parallel tempering Monte Carlo simulations to study the disorder-temperature phase diagrams of two new 3D statistical-mechanical models: the 4- and 6-body random coupling Ising models.
