Analysis of quantum error correcting codes is typically done using a stochastic, Pauli channel error model for describing the noise on physical qubits. However, it was recently found that coherent errors (systematic rotations) on physical data qubits result in both physical and logical error rates that differ significantly from those predicted by a Pauli model. Here we examine the accuracy of the Pauli approximation for noise containing coherent errors (characterized by a rotation angle ) under the repetition code. We derive an analytic expression for the logical error channel as a function of arbitrary code distance d and concatenation level n , in the small error limit. We find that coherent physical errors result in logical errors that are partially coherent and therefore non-Pauli. However, the coherent part of the logical error is negligible at fewer than error correction cycles when the decoder is optimized for independent Pauli errors, thus providing a regime of validity for the Pauli approximation. Above this number of correction cycles, the persistent coherent logical error will cause logical failure more quickly than the Pauli model would predict, and this may need to be combated with coherent suppression methods at the physical level or larger codes.
Analysis of quantum error correcting codes is typically done using a stochastic, Pauli channel error model for describing the noise on physical qubits. However, it was recently found that coherent errors (systematic rotations) on physical data qubits result in both physical and logical error rates that differ significantly from those predicted by a Pauli model. Here we examine the accuracy of the Pauli approximation for noise containing coherent errors (characterized by a rotation angle ) under the repetition code. We derive an analytic expression for the logical error channel as a function of arbitrary code distance d and concatenation level n , in the small error limit. We find that coherent physical errors result in logical errors that are partially coherent and therefore non-Pauli. However, the coherent part of the logical error is negligible at fewer than error correction cycles when the decoder is optimized for independent Pauli errors, thus providing a regime of validity for the Pauli approximation. Above this number of correction cycles, the persistent coherent logical error will cause logical failure more quickly than the Pauli model would predict, and this may need to be combated with coherent suppression methods at the physical level or larger codes.
We compare the performance of quantum error correcting codes when memory errors are unitary with the more familiar case of dephasing noise. For a wide range of codes, we analytically compute the effective logical channel that results when the error correction steps are performed noiselessly. Our examples include the entire family of repetition codes, the five-qubit, Steane, Shor, and surface codes. When errors are measured in terms of the diamond norm, we find that the error correction is typically much more effective for unitary errors than for dephasing. We observe this behavior for a wide range of codes after a single level of encoding, and in the thresholds of concatenated codes using hard decoders. We show that this holds with great generality by proving a bound on the performance of any stabilizer code when the noise at the physical level is unitary. By comparing the diamond norm error ${D}_{\ensuremath{\diamond}}^{\ensuremath{'}}$ of the logical qubit with the same quantity at the physical level ${D}_{\ensuremath{\diamond}}$, we show that ${D}_{\ensuremath{\diamond}}^{\ensuremath{'}}\ensuremath{\le}c{D}_{\ensuremath{\diamond}}^{d}$ where d is the distance of the code and c is a constant that depends on the code but not on the error. This bound compares very favorably to the performance of error correction for dephasing noise and other Pauli channels, where an error correcting code of odd distance d will exhibit a scaling ${D}_{\ensuremath{\diamond}}^{\ensuremath{'}}\ensuremath{\sim}{D}_{\ensuremath{\diamond}}^{(d+1)/2}$.
Quantum noise is a central challenge in quantum computing across many applications. Extensive work has examined how qubits couple to their environment, leading to decoherence and irreversible relaxation. This work studies a continuous coherent noise model for quantum circuits and compares it with a discrete Pauli model. The focus is on small, coherent gate errors that accumulate across circuit depth. These errors are modeled as random rotations on the Bloch sphere using a von Mises-Fisher distribution. In the small-angle limit, the model reduces to an isotropic Gaussian distribution. We test the model on quantum error-correction circuits based on the [[5,1,3]] and [[7,1,3]] codes. A variant of Grover's search circuit with different qubit counts is also examined. To enable fair comparison, we introduce a model-independent matching scheme. Pauli and continuous noise channels are aligned using the binary entropy at readout. This isolates the effect of noise structure at fixed uncertainty. An approximate analytical method for the propagation of coherent errors is also developed. The method tracks error distributions both on Clifford and non-Clifford circuits without full Monte Carlo sampling. It reduces simulation cost while preserving accuracy for circuit-level error estimates. The approximation is validated against brute-force simulations, identifying its regime of validity with Clifford and non-Clifford circuits under error correction. Our results show that continuous coherent noise and Pauli noise lead to comparable logical-error trends in the stabilizer-code circuits studied here. This behavior is consistent with syndrome extraction projecting a general single-qubit error onto Pauli-error subspaces before recovery. They also show where simplified propagation models work well and where their accuracy is reduced.
Full accuracy simulations of quantum systems are very costly, and as a result most studies \nof quantum error correction assume a probabilistic Pauli error model, largely because such \nerrors can be e ciently simulated. Therefore, the behaviour of more general noise in a \nquantum error correcting code is poorly characterized. In this thesis, we present results which \ndemonstrate the scaling of the logical noise with respect to the physical delity, and argue \nthat the e ective logical noise approaches a Pauli channel as the code distance increases, \neven when no recovery operations are applied. As a result, we argue that the average logical \n delity can be used to accurately quantify the e ective logical noise, and to select recovery \noperations appropriate to the system. We further demonstrate that when physical noise \nacts on fewer than d qubits in an [[n; k; d]] code, the resultant noise is Pauli, and develop \na method for approximating the dominant contributions to the e ective logical noise up to \na speci ed precision in terms of the physical in delity. We derive conditions under which \nsets of recovery operations will produce equivalent logical noise channels, with examples of \nequivalencies in the 3 qubit repetition code, the 5 qubit code, the Steane code, and the Shor \ncode. We also provide a general expression for the e ective logical noise when the physical \nqubits undergo depolarizing or Pauli noise in a quantum error correcting code, examine the \nbehaviour of depolarizing noise under concatenation of the 5 qubit and Steane codes, and \npresent an algorithm for soft decoding which is not subject to statistical sampling, with an \nemphasis on the e ective behaviour of a concatenated 5 qubit code undergoing depolarizing \nnoise after applying a specialized version our soft decoding algorithm.
Quantum error correction (QEC), the lynchpin of fault-tolerant quantum computing (FTQC), is designed and validated against well-behaved Pauli stochastic error models. But in real-world deployment, QEC protocols encounter a vast array of other errors -- coherent and non-Pauli errors -- whose impacts on quantum circuits are vastly different than those of stochastic Pauli errors. The impacts of these errors on QEC and FTQC protocols have been largely unpredictable to date due to exponential classical simulation cost. Here, we show how to accurately and efficiently model the effects of coherent and non-Pauli errors on FTQC, and we study the effects of such errors on syndrome extraction for surface and bivariate bicycle codes, and on magic state cultivation. Our analysis suggests that coherent error can shift fault-tolerance thresholds, increase the space-time cost of magic state cultivation, and can increase logical error rates by an order of magnitude compared to equivalent stochastic errors. These analyses are enabled by a new technique for mapping any Markovian circuit-level error model with sufficiently small error rates onto a detector error model (DEM) for an FTQC circuit. The resulting DEM enables Monte Carlo estimation of logical error rates and noise-adapted decoding, and its parameters can be analytically related to the underlying physical noise parameters to enable approximate strong simulation.
The performance of a given quantum error correction (QEC) code depends upon the noise model that is assumed. Independent Pauli noise, applied after each quantum operation, is a simplistic noise model that is easy to simulate and understand in the context of stabilizer codes. Although such a noise model is artificial, it is equivalent to independent, random, unbiased qubit rotations. What about spatially or temporally correlated qubit rotations? Such a noise model is applicable to global operations (e.g., NMR or ESR), common control sources (e.g., lasers), or slow drift (e.g., charge or magnetic noise) in various qubit technologies. In the worst case, such errors can combine constructively and result in a post-correction failure rate that increases with the number of error correction cycles. However, we show that this worst case does not generally arise unless taking active corrective actions while performing QEC. That is, by employing virtual Pauli frame updates ("passive" error correction) rather than physical corrections ("active" error correction), coherent errors do not compound appreciably. Starting in a random Pauli frame is also advantageous. In fact, through perturbation theory arguments and supporting numerical simulations, we show that the logical qubit performance beyond distance 3 for correlated single-qubit Hamiltonian noise models (i.e., global errant qubit rotations), when employing these "lazy" strategies, essentially matches the performance of Pauli noise model with the same process fidelity (fidelity after one application). In a more general circuit model of noise, correlations may add constructively within syndrome extraction rounds but Pauli frame randomization from passive error correction mitigates this effect across multiple rounds.
Abstract Surface codes are building blocks of quantum computing platforms based on 2D arrays of qubits responsible for detecting and correcting errors. The error suppression achieved by the surface code is usually estimated by simulating toy noise models describing random Pauli errors. However, Pauli noise models fail to capture coherent processes such as systematic unitary errors caused by imperfect control pulses. Here we report the first large-scale simulation of quantum error correction protocols based on the surface code in the presence of coherent noise. We observe that the standard Pauli approximation provides an accurate estimate of the error threshold but underestimates the logical error rate in the sub-threshold regime. We find that for large code size the logical-level noise is well approximated by random Pauli errors even though the physical-level noise is coherent. Our work demonstrates that coherent effects do not significantly change the error correcting threshold of surface codes. This gives more confidence in the viability of the fault-tolerance architecture pursued by several experimental groups.