Quantum error-correcting (QEC) stabilizer codes enable protection of quantum information against errors during storage and processing. Simulation of noisy QEC codes is used to identify the noise parameters necessary for advantageous operation of logical qubits in realistic quantum computing architectures. Typical quantum error-correction techniques contain intermediate measurements and classical feedback that determine the actual noisy circuit sequence in an instance of performing the protocol. Dynamical subset sampling enables efficient simulation of such nondeterministic quantum error-correcting protocols for any type of quantum circuit and incoherent noise of low strength. As an importance sampling technique, dynamical subset sampling allows one to effectively make use of computational resources to only sample the most relevant sequences of quantum circuits in order to estimate a protocol's logical failure rate with well-defined error bars. We demonstrate the capabilities of dynamical subset sampling with examples from fault-tolerant (FT) QEC. We show that, in a typical stabilizer simulation with incoherent Pauli noise of strength p=10−3 , our method can reach a required sampling accuracy on the logical failure rate with two orders of magnitude fewer samples than direct Monte Carlo simulation. Furthermore, dynamical subset sampling naturally allows for efficient simulation of realistic multi-parameter noise models describing faulty quantum processors. It can be applied not only for QEC in the circuit model but any noisy quantum computing framework with incoherent fault operators including measurement-based quantum computation and quantum networks. Published by the American Physical Society 2024
Quantum error-correcting (QEC) stabilizer codes enable protection of quantum information against errors during storage and processing. Simulation of noisy QEC codes is used to identify the noise parameters necessary for advantageous operation of logical qubits in realistic quantum computing architectures. Typical quantum error-correction techniques contain intermediate measurements and classical feedback that determine the actual noisy circuit sequence in an instance of performing the protocol. Dynamical subset sampling enables efficient simulation of such nondeterministic quantum error-correcting protocols for any type of quantum circuit and incoherent noise of low strength. As an importance sampling technique, dynamical subset sampling allows one to effectively make use of computational resources to only sample the most relevant sequences of quantum circuits in order to estimate a protocol's logical failure rate with well-defined error bars. We demonstrate the capabilities of dynamical subset sampling with examples from fault-tolerant (FT) QEC. We show that, in a typical stabilizer simulation with incoherent Pauli noise of strength p=10−3 , our method can reach a required sampling accuracy on the logical failure rate with two orders of magnitude fewer samples than direct Monte Carlo simulation. Furthermore, dynamical subset sampling naturally allows for efficient simulation of realistic multi-parameter noise models describing faulty quantum processors. It can be applied not only for QEC in the circuit model but any noisy quantum computing framework with incoherent fault operators including measurement-based quantum computation and quantum networks. Published by the American Physical Society 2024
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.
Logical qubits can be protected from decoherence by performing quantum error-correction (QEC) cycles repeatedly. Algorithms for fault-tolerant QEC must be compiled to the specific hardware platform under consideration in order to practically realize a quantum memory that operates for in principle arbitrary long times. All circuit components must be assumed as noisy unless specific assumptions about the form of the noise are made. Modern QEC schemes are challenging to implement experimentally in physical architectures where in-sequence measurements and feed forward of classical information cannot be reliably executed fast enough or even at all. Here we provide a novel scheme to perform QEC cycles without the need of measuring qubits that is fully fault-tolerant with respect to all components used in the circuit. Our scheme can be used for any low-distance CSS code since its only requirement towards the underlying code is a transversal gate. Similarly to Steane-type EC, we coherently copy errors to a logical auxiliary qubit but then apply a coherent feedback operation from the auxiliary system to the logical data qubit. The logical auxiliary qubit is prepared fault tolerantly without measurements, too. We benchmark logical failure rates of the scheme in comparison to a flag-qubit-based EC cycle. We map out a parameter region where our scheme is feasible and estimate physical error rates necessary to achieve the break-even point of beneficial QEC with our scheme. We outline how our scheme could be implemented in ion traps and with neutral atoms in a tweezer array. For recently demonstrated capabilities of atom shuttling and native multiatom Rydberg gates, we achieve moderate circuit depths and beneficial performance of our scheme while not breaking fault tolerance. These results thereby enable practical fault-tolerant QEC in hardware architectures that do not support midcircuit measurements. Published by the American Physical Society 2024
We demonstrate that Pauli frames, a mathematical object used to efficiently sample qubit stabilizer circuits, can be used to efficiently sample qudit circuits of any prime dimension. Quantum error correcting codes are a central object of study in fault-tolerance quantum computing, which is necessary to realize useful universal quantum computing. Stabilizer codes belong to a large class of well-understood quantum codes that are crucially efficiently simulable on classical hardware using the so-called tableau representation. Popular simulation software like Stim exclusively studies and characterizes qubit codes, which entails repeatedly sampling a noisy quantum circuit with Pauli frames. We highlight the key differences (and fundamental similarities) between qubit and qudit stabilizer tableaus. It then teaches how Pauli frames generate batches of samples given a qudit tableau via two guided examples: one with a random measurement outcome but no noise, and one with a single noise gate.
Quantum error correction (QEC) enables reliable computation on noisy hardware by encoding logical information across many physical qubits and periodically measuring parities to detect errors. A decoder is the classical algorithm that uses these measurements to infer which error most likely occurred, so that the system can correct it. The decoder's accuracy-how rarely it makes the wrong guess-directly determines the scale of quantum computation that can be reliably executed. With a wealth of competing decoding algorithms, a QEC system designer needs reliable methods to evaluate them. Today, the dominant approach is to evaluate decoders using Monte Carlo simulation. However, simulation has several drawbacks such as requiring many samples to produce low variance estimates.
In this work, we develop a new systematic analysis for evaluating decoders. We introduce a novel formal semantics of a core language for QEC programs that captures the de facto standard Stim circuit format, providing a principled theoretical foundation for the emerging space of fault-tolerant quantum systems design. Given a QEC program and a decoder, our verifier can quantify both the decoder accuracy and the decoder robustness to drift in physical error rate. Our approach has two key components: (i) a structured search over the space of possible errors; and (ii) a constrained polynomial optimization kernel. A thorough empirical evaluation of our approach suggests that it can outperform simulation, especially in low error rate regimes, and that it can be deployed to quantify decoder robustness over an interval of physical error rates.
A universal, scalable quantum computer will require the use of quantum error correction in order to achieve fault tolerance. The assessment and comparison of error-correcting strategies is performed by classical simulation. However, due to the prohibitive exponential scaling of general quantum circuits, simulations are restrained to specific subsets of quantum operations. This creates a gap between accuracy and efficiency which is particularly problematic when modeling noise, because most realistic noise models are not efficiently simulable on a classical computer. We have introduced extensions to the Pauli channel, the traditional error channel employed to model noise in simulations of quantum circuits. These expanded error channels are still computationally tractable to simulate, but result in more accurate approximations to realistic error channels at the single qubit level. Using the Steane [[7,1,3]] code, we have also investigated the behavior of these expanded channels at the logical error-corrected level. We have found that it depends strongly on whether the error is incoherent or coherent. In general, the Pauli channel will be an excellent approximation to incoherent channels, but an unsatisfactory one for coherent channels, especially because it severely underestimates the magnitude of the error. Finally, we also studied the honesty and accuracy of the expanded channels at the logical level. Our results suggest that these measures can be employed to generate lower and upper bounds to a quantum code's threshold under the influence of a specific error channel.
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