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 relaxation, which is irreversible. Current studies focus on coherent gate errors caused by control misalignment, which accumulate with circuit depth but can, in principle, be corrected. 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 build up 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 coherent error propagation is also developed. The method tracks error distributions through 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 circuits and limits under error correction. Our results show that continuous coherent noise can degrade logical performance more strongly than Pauli noise. They also clarify when simplified propagation models succeed and when they break down.
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 relaxation, which is irreversible. Current studies focus on coherent gate errors caused by control misalignment, which accumulate with circuit depth but can, in principle, be corrected. 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 build up 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 coherent error propagation is also developed. The method tracks error distributions through 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 circuits and limits under error correction. Our results show that continuous coherent noise can degrade logical performance more strongly than Pauli noise. They also clarify when simplified propagation models succeed and when they break down.
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.
A quantum error correction code is assessed over its ability to correct errors in noisy quantum circuits. This task requires extensive simulations of faulty quantum circuits, which are often made tractable by considering stochastic Pauli noise models, as they are compatible with efficient classical simulation techniques. However, such noise models do not fully capture the variety of physical error mechanisms encountered in realistic quantum platforms. In this work, we extend circuit-level noise modeling beyond the Pauli framework by estimating the threshold of the hexagonal color code under more general noise models. Specifically, we consider two representative non-Pauli error channels: a systematic X-rotation model that introduces coherent over-rotations, and an amplitude damping channel that captures relaxation processes. These models are incorporated at the circuit level into color code circuits using a Tree Tensor Network ansatz. Our simulations demonstrate that tensor network simulations enable accurate threshold estimation under non-Pauli noise for color codes up to distance d=7 (73 qubits). Comparing our results with the Pauli twirling approximations of the noise models, we find that coherent over-rotations yield systematically higher error rates, deviating from the Pauli twirling approximation as the code distance increases.
Stabilizer simulation of Clifford quantum circuits - error-correction circuits, Clifford subroutines, etc. - on classical computers has played a central role in our understanding of circuit performance. The stabilizer description, however, restricts the accessible noise one can incorporate into the simulation to Pauli-type noise. More general noise, including coherent errors, may have more severe impact on circuit performance than Pauli noise; yet, such general noise have been difficult to access, much less investigate fully, in numerical simulations. Here, through the use of stratified importance sampling, we show how general noise can be simulated within the stabilizer formalism in reasonable time, with non-unitary noise being nearly as cheap as Pauli noise. Unitary (or coherent) noise can require an order of magnitude more time for the simulation, but nevertheless completes in very reasonable times, a drastic improvement over past approaches that typically fail to converge altogether. Our work thus enables detailed beyond-Pauli understanding of circuit performance in the presence of real device noise, which is rarely Pauli in nature. Among other examples, we present direct simulation results for the performance of the popular rotated planar surface codes under circuit-level general noise, previously available only in limited situations and/or through mappings to efficiently simulatable physical models.
Fault-tolerant quantum design relies on simulating noisy quantum error correction (QEC) circuits with hundreds to thousands of physical qubits and large circuit depths. High-throughput stabilizer simulators, such as Stim, are the de facto tools for this regime. However, these simulators typically support only Clifford gates and stochastic Pauli noise. As a result, more realistic device noise models-such as coherent control errors and thermal relaxation-are often approximated using Pauli Twirling. This replacement offers speed but can introduce inaccuracies.
To address this, we propose QuasiStim for scalable exact QEC simulation of realistic noise, with QuasiFold as a compiler frontend and DeltaFrame as a sampling backend. QuasiFold lowers sampling overhead by merging equivalent non-Pauli sites in the quantum circuit. DeltaFrame samples the resulting stabilizer trajectories by reusing precomputed predicted references instead of resimulating the full tableau for each shot.
QuasiFold can reduce the number of shots by over 100× on bivariate-bicycle (BB) codes. Compared to a per-shot tableau simulator, DeltaFrame is 166×--1331× faster on rotated surface-code memory under coherent RZ noise as d grows from 5 to 35. QuasiStim also makes the PTA bias measurable: the gap is non-negligible and varies with noise strength, decoder, and circuit schedule.
QuasiStim will be open-sourced upon publication at https://github.com/isaachyw/QuasiStim
Accurately estimating observables on noisy quantum devices remains a central challenge for near-term quantum algorithms. While quantum error mitigation techniques can reduce noise-induced bias, they often rely on unverifiable assumptions about the circuit noise, and cannot guarantee the magnitude of residual bias error. Here, rather than using classical resources to mitigate a noisy quantum circuit execution, we propose a hybrid algorithm that uses quantum resources to improve the accuracy of approximate classical Pauli-path simulation. Our protocol, Quantum Enhanced Pauli Propagation (QuEPP), uses Clifford perturbation theory (CPT) to construct a classically simulable ensemble of Clifford circuits from the low-order terms in CPT, which directly provide the approximate classical Pauli-path simulation of the target circuit. Noisy quantum expectation values of this ensemble are then used to infer a global rescaling factor that corrects quantum execution of the target circuit, providing higher-order contributions absent from the truncated low-order classical simulation. This approach requires no noise characterization, applies to arbitrary circuits, and provides a provable route to asymptotically unbiased estimates. Using IBM Heron hardware, we demonstrate QuEPP on 2D random mirror circuits of up to 49 qubits and circuit depth 80, as well as Trotterized Hamiltonian evolution, showing consistent improvements beyond classical CPT and unmitigated quantum results. QuEPP offers a simple, scalable, and model-free framework for enabling accurate quantum computation in the pre-fault-tolerant era.