Can rare event simulation access low logical failure rates in QEC circuits?

Why Monte Carlo stalls at low physical error rates

Standard Monte Carlo estimates a logical failure rate by running many noisy circuit samples and counting failures. That works while failures are frequent enough to observe. The paper notes that many circuit-model simulations therefore go no lower than end-to-end logical failure rates around 10^-6, and that reaching the teraquop regime would require on the order of a hundred trillion independent runs.

The difficulty grows with the circuit, not just with the target number. As the physical failure rate of individual gates falls, or as the code's correcting power rises, failures become rarer and the sample count needed for an estimate becomes exorbitant. The paper also points to a subtler obstacle: for a strictly fault-tolerant circuit the logical failure rate scales as Omega(p^ceil(d/2)), so the dominant contributions come from fault sets of size near ceil(d/2), and estimating the conditional failure probability for those sets is itself hard. In one illustration given for 10,000 faulty gates and distance d = 13, there are potentially about 2x10^24 candidate sets of size 7, and roughly 2x10^20 samples would be needed on average to see a single failing event.

Splitting turns one hard estimate into many easier ratios

The method the authors adopt, following Bravyi and Vargo, does not estimate the tiny target failure rate directly. It starts at a higher physical failure rate p1 where Monte Carlo can produce an output logical failure rate in reasonable time, then walks down a sequence of physical rates p1, p2, ..., pt. At each step it estimates the ratio of consecutive output failure rates rather than the absolute value, and multiplies the ratios together.

The step size is set by a heuristic: p_{i+1} = p_i * 2^(-1/sqrt(w_i)), with w_i = max(d/2, p_i * G), where G is the number of faulty gates in the encoded circuit. The intuition given is to start near the expected number of gates that cause logical failure. The paper's contribution is not this ratio idea itself but its extension: earlier work applied the splitting technique to the code capacity and phenomenological noise models, which the authors describe as insufficient for the circuit-noise model. They state that, to their knowledge, this is the first full prescription of rare event simulation by splitting for circuit-based noise.

What circuit noise adds to the problem

The three noise models differ in where errors are allowed. In code capacity, errors occur on data qubits and syndrome extraction is treated as error-free. In the phenomenological model, syndrome extraction is a black box that can produce measurement errors. In the circuit model, syndrome extraction is itself a quantum circuit of one- and two-qubit gates, each modeled as an ideal gate followed by a Pauli noise channel, with gates failing independently and errors propagating between data and ancilla qubits through couplings such as CNOT, CPHASE, or ISWAP.

That structure is why the earlier splitting prescription does not transfer unchanged. The state space the sampler must explore is now made of gate faults inside a circuit, and the paper works out the Metropolis-Hastings acceptance rules for moving between fault sets in this setting. The authors also describe extensions for leakage and for post-selection, noting that single-element alterations can leave the Markov chain trapped in islands when post-selection requires a zero syndrome; those extensions are presented as prescriptions, with simulated results left to future work.

Agreement with Monte Carlo where both can run

The empirical check is a comparison, not an absolute validation. For a rotated surface code under symmetric circuit noise, where all gates in the syndrome extraction circuits share the same failure probability p, the paper compares the logical Z error rate from rare event simulation against Monte Carlo over p in [10^-4, 10^-3], and then extends the rare event estimates down to p in [10^-5, 10^-3]. The Monte Carlo points use at least 1,000 failures and an unbiased negative binomial estimator.

The rare event estimates agree with Monte Carlo in the accessible regime, and the paper reports that the method continues to produce valid results at lower p where standard Monte Carlo would be practically impossible. The authors also report 20 rare event runs for d = 7, summarized as mean plus or minus one standard deviation, alongside 10 Monte Carlo runs with at least 100 failures each. The comparison therefore supports the method where the two approaches overlap; below that overlap, the rare event curve is the only estimate available in this study.

Fault-set sizes shift as noise drops

A diagnostic in the paper shows what the sampler is actually visiting. Count histograms of fault-set sizes for a distance-7 surface code at p_phys = 10^-3 and p_phys = 10^-4 show that at the lower physical error rate, faults close to size ceil(d/2) account for a larger portion of events. This is consistent with the scaling argument: as p falls, the failures that matter are concentrated in the smallest fault sets capable of defeating the code.

That shift is also a caution about cost. The paper notes that at k = ceil(d/2), many of the size-k gate sets may not lead to failure, and that for large code distances without guarantees of strict fault tolerance the estimator can stall, effectively assuming a conditional failure probability of zero. The reported results were generated without leakage or multi-gate fault selection, so the histograms and comparisons describe that configuration.

The 10^-20 claim and what stands behind it

The headline result is that the approach can access logical failure rates below 10^-20 in the circuit-model prescription. That figure comes from the rare event simulation itself; it is not a Monte Carlo measurement, because Monte Carlo cannot reach that regime in this setup. The evidence supporting the extrapolation is the agreement with Monte Carlo at p in [10^-4, 10^-3] and the continued plausible behavior of the estimates at lower p, together with the structural argument that the splitting ratios are estimated at each step.

The authors are explicit about the open statistical question. Because the ratios are multiplied, errors accumulate across the sequence, and the paper lists as open both ensuring enough jumps for Markov chain convergence and quantifying confidence in the estimates. A convergence diagnostic such as the Gelman-Rubin statistic is mentioned as intuitive and computable but not a confidence measure, and it is described as sensitive to long-tailed iterative statistics or differing within-chain variance. The splitting heuristic itself, taken from the earlier work for its tradeoff between number of steps and statistical error, is another place where asymmetric noise could motivate a different tradeoff.

Where the method changes a practical comparison

The practical difference is a change in what can be compared. With standard Monte Carlo, a circuit-model study is bounded near 10^-6 end-to-end logical failure rates, so designs whose differences appear only at lower physical error rates cannot be separated by direct sampling. The splitting approach moves the comparison to a sequence of ratios anchored at a rate Monte Carlo can measure, and the paper's results show that this anchor can be carried down through p = 10^-5 and, per the authors, below 10^-20 in logical failure rate.

The priority the paper establishes is therefore not a claim that every low-rate estimate is trustworthy, but a specific ordering: use Monte Carlo to anchor and check the method in the accessible regime, then use the splitting sequence to extend the comparison downward, while treating convergence and confidence quantification as unresolved. The extensions to leakage and post-selection, and the asymmetric-noise tradeoff in choosing interval points, are the scenarios the authors flag as the next places this comparison would need its own validation.

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