Simulating Quantum Error Correction beyond Pauli Stochastic Errors

Abstract

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.

Figures (8)

• Figure 1: Efficient prediction of quantum error correction circuits with non-Pauli errors. (a) We use an efficient method for propagating errors through Clifford circuits to determine how non-Pauli errors impact a QEC circuit. This allows us to (approximately) map errors onto events in a detector error model, which can be efficiently sampled from to predict logical performance. (b) Our method outperforms DEMs generated from Pauli-twirled error models in predicting detection history distributions by roughly 100x (quantified by TVD between the predicted and true distributions). (c) We simulated the injection of an $S$ state (as a proxy for a $T$ state) followed by a double logical $H$ check in a $d=3$ color code with error models with the same gate infidelities, but different relative contributions of stochastic and coherent errors. We include depolarizing error plus additional error on each gate, and the dominant source of non-depolarizing error in these simulations was $ZZ$ error on $CZ$ gates. As the contribution of coherent error to the gate error increases, the discard rate and logical error probability both increase relative to a Pauli stochastic model (dashed lines).
• Figure 2: Surface Code logical performance with coherent and Pauli stochastic error. (a) We simulated surface code syndrome extraction with 10 random sparse CPTP error models. Here, we show the results from the models with the lowest and highest thresholds. (b) We simulated models with strong intrinsic ($ZX$, $IX$, $ZI$) CNOT error, but with different amounts of stochastic ($S_P$) and coherent ($H_P$) error. Error models with larger contributions from coherent error exhibit a lower threshold CNOT infidelity than the purely stochastic error model. (c) We compared two decoder priors for MWPM in our simulation with coherent error plus depolarizing error---priors from a DEM derived using our method and priors from a DEM derived from the Pauli-twirled version of the error model. We see a significant decrease in logical error probability with the DEM derived from our method at CNOT infidelities above threshold, but this difference rapidly drops off below threshold.
• Figure 3: Scaling of gross code logical error probability with coherent error. We simulate the [[144,12,12]] gross code (decoding using the Beam Decoder ye2025beam), changing the strengths of two coherent errors. (a) The logical error rate per qubit is minimized at a nonzero $H_{XX}$ error rate for the CNOT gates, with the location of the minimum being dependent on the rate of the idle $H_X$ error. (b) In contrast, in a Pauli-twirled analogue of the model, the $H_{XX}$ error rate minimizing the logical error rate does not shift with the $H_X$ idle error rate. In the regime where physical-level errors cancel, the Pauli-twirled model overestimates the logical error rate.
• Figure 4: Magic State Cultivation with Coherent Error. (a) High-level diagram of the cultivation protocol we simulated. We simulated the injection of an $S$ state (as a proxy for a $T$ state) followed by a double logical $H$ check in a $d=3$ color code. We model each $CZ$ gate as having a fixed infidelity of 0.00165, and we vary the relative contributions of stochastic and coherent $ZI$, $IZ$, and $ZZ$ error on CZ gates and Z error on single-qubit gates gates. (b) The sensitivity of the rate of postselection to coherent error at leading order is quadratic in the $H$ error parameters described by its sensitivity matrix. (c) The end-of-circuit logical observable is is particularly sensitive to $H_{IY}$ and $H_{YI}$ errors. Furthermore, it is more sensitive to $H_{YI}$ and $H_{ZZ}$ error than their $S$ analogues---suggesting that these $H$ errors lead to a higher logical error probability.
• Figure 5: Efficient prediction of quantum error correction circuits with non-Pauli errors. (a) We model errors in a QEC circuit with sparse error generator models, where each error channel is represented as an exponential of a generator that is sparse in the elementary error generator basis. (b) Our method estimates the impact of all physical-level errors on a circuit by first propagating all errors to the end of the circuit, then approximately combining them with a BCH expansion. Then, our method decomposes that circuit error channel approximately into individual error channels that each cause one DEM event, from which the probability of each DEM event can be efficiently estimated.

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Definition 1
• Lemma 3
• Lemma 4
• Lemma 5
• Theorem 6
• Lemma 7
• Lemma 8
• Lemma 9