As quantum computers approach the fault-tolerance threshold, diagnosing and characterizing the noise on large-scale quantum devices is increasingly important.One of the most important classes of noise channels is the class of Pauli channels, for reasons of both theoretical tractability and experimental relevance.Here we present a practical algorithm for estimating the s nonzero Pauli error rates in an s-sparse, n-qubit Pauli noise channel, or more generally the s largest Pauli error rates.The algorithm comes with rigorous recovery guarantees and uses only O(n 2 ) measurements, O(sn 2 ) classical processing time, and Clifford quantum circuits.We experimentally validate a heuristic version of the algorithm that uses simplified Clifford circuits on data from an IBM 14-qubit superconducting device and our open-source implementation.These data show that accurate and precise estimation of the probability of arbitrary-weight Pauli errors is possible even when the signal is 2 orders of magnitude below the measurement noise floor.
As quantum computers approach the fault-tolerance threshold, diagnosing and characterizing the noise on large-scale quantum devices is increasingly important.One of the most important classes of noise channels is the class of Pauli channels, for reasons of both theoretical tractability and experimental relevance.Here we present a practical algorithm for estimating the s nonzero Pauli error rates in an s-sparse, n-qubit Pauli noise channel, or more generally the s largest Pauli error rates.The algorithm comes with rigorous recovery guarantees and uses only O(n 2 ) measurements, O(sn 2 ) classical processing time, and Clifford quantum circuits.We experimentally validate a heuristic version of the algorithm that uses simplified Clifford circuits on data from an IBM 14-qubit superconducting device and our open-source implementation.These data show that accurate and precise estimation of the probability of arbitrary-weight Pauli errors is possible even when the signal is 2 orders of magnitude below the measurement noise floor.
Pauli channels are ubiquitous in quantum information, both as a dominant noise source in many computing architectures and as a practical model for analyzing error correction and fault tolerance. Here, we prove several results on efficiently learning Pauli channels and more generally the Pauli projection of a quantum channel. We first derive a procedure for learning a Pauli channel on n qubits with high probability to a relative precision ϵ using O (ϵ -2 n2 n ) measurements, which is efficient in the Hilbert space dimension. The estimate is robust to state preparation and measurement errors, which, together with the relative precision, makes it especially appropriate for applications involving characterization of high-accuracy quantum gates. Next, we show that the error rates for an arbitrary set of s Pauli errors can be estimated to a relative precision ϵ using O (ϵ -4 log s log s/ϵ) measurements. Finally, we show that when the Pauli channel is given by a Markov field with at most k -local correlations, we can learn an entire n -qubit Pauli channel to relative precision ϵ with only O k (ϵ -2 n 2 log n ) measurements, which is efficient in the number of qubits. These results enable a host of applications beyond just characterizing noise in a large-scale quantum system: they pave the way to tailoring quantum codes, optimizing decoders, and customizing fault tolerance procedures to suit a particular device.
Motivated by estimation of quantum noise models, we study the problem of learning a Pauli channel, or more generally the Pauli error rates of an arbitrary channel. By employing a novel reduction to the "Population Recovery" problem, we give an extremely simple algorithm that learns the Pauli error rates of an n -qubit channel to precision ϵ in ℓ∞ using just O(1/ϵ2)log(n/ϵ) applications of the channel. This is optimal up to the logarithmic factors. Our algorithm uses only unentangled state preparation and measurements, and the post-measurement classical runtime is just an O(1/ϵ) factor larger than the measurement data size. It is also impervious to a limited model of measurement noise where heralded measurement failures occur independently with probability ≤1/4 .We then consider the case where the noise channel is close to the identity, meaning that the no-error outcome occurs with probability 1−η . In the regime of small η we extend our algorithm to achieve multiplicative precision 1±ϵ (i.e., additive precision ϵη ) using just O(1ϵ2η)log(n/ϵ) applications of the channel.
Benchmarking physical devices and verifying logical algorithms are important tasks for scalable fault-tolerant quantum computing. Numerous protocols exist for benchmarking devices before running actual algorithms. In this work, we show that both physical and logical errors of fault-tolerant circuits can even be characterized in-situ using syndrome data. To achieve this, we map general fault-tolerant Clifford circuits to subsystem codes using the spacetime code formalism and develop a scheme for estimating Pauli noise in Clifford circuits using syndrome data. We give necessary and sufficient conditions for the learnability of physical and logical noise from given syndrome data, and show that we can accurately predict logical fidelities from the same data. Importantly, our approach requires only a polynomial sample size, even when the logical error rate is exponentially suppressed by the code distance, and thus gives an exponential advantage against methods that use only logical data such as direct fidelity estimation. We demonstrate the practical applicability of our methods in various scenarios using synthetic data as well as the experimental data from a recent demonstration of fault-tolerant circuits by Bluvstein et al. [Nature 626, 7997 (2024)]. Our methods provide an efficient, in-situ way of characterizing a fault-tolerant quantum computer to help gate calibration, improve decoding accuracy, and verify logical circuits.
Noise is the fundamental challenge to quantum computation. Large-scale quantum computers will overcome this challenge with fault-tolerant architectures that leverage quantum error correction. This thesis introduces and investigates a scalable noise characterisation protocol suited to this key context. First, we introduce this protocol by building upon averaged circuit eigenvalue sampling (ACES), a framework for noise characterisation experiments that simultaneously estimates the Pauli error probabilities of all gates in a Clifford circuit and captures averaged spatial correlations between gates implemented simultaneously in the layers of the circuit. We demonstrate the scalability and performance of our protocol through circuit-level numerical simulations of the entire noise characterisation procedure. Then we demonstrate in circuit-level numerical simulation that this protocol is practically capable of calibrating a fast correlated matching decoder, enabling noise-aware decoding. We find that noise-aware decoding increases the error suppression factor of the code, leading to reductions in the logical error rate that increase exponentially with the code distance. Finally, we present several results obtained from experimentally implementing this protocol. We use the noise characterisation results to design an improved syndrome extraction circuit for a heavy hexagon memory that is adapted to the noise characteristics of the quantum device. The model error of our noise characterisation results suggests that the circuit-level Pauli noise model estimated by ACES can describe the essential features of quantum noise. We also operate a heavy hexagon memory and use noise estimates to predict performance and inform decoding. These results demonstrate that Pauli noise estimates can calibrate decoders to enable noise-aware decoding and inform the co-design of quantum error correcting codes, decoders, fault-tolerant circuits, and quantum devices.
Characterizing errors in quantum circuits is essential for device calibration, yet detecting rare error events requires a large number of samples. This challenge is particularly severe in calibrating fault-tolerant, error-corrected circuits, where logical error probabilities are suppressed to higher order relative to physical noise and are therefore difficult to calibrate through direct logical measurements. Recently, Wagner et al. [PRL 130, 200601 (2023)] showed that, for phenomenological Pauli noise models, the logical channel can instead be inferred from syndrome measurement data generated during error correction. Here, we extend this framework to realistic circuit-level noise models. From a unified code-theoretic perspective and spacetime code formalism, we derive necessary and sufficient conditions for learning the logical channel from syndrome data alone and explicitly characterize the learnable degrees of freedom of circuit-level Pauli faults. Using Fourier analysis and compressed sensing, we develop efficient estimators with provable guarantees on sample complexity and computational cost. We further present an end-to-end protocol and demonstrate its performance on several syndrome-extraction circuits, achieving orders-of-magnitude sample-complexity savings over direct logical benchmarking. Our results establish syndrome-based learning as a practical approach to characterizing the logical channel in fault-tolerant quantum devices.
Quantum computers have enabled solving problems beyond the current machines' capabilities. However, this requires handling noise arising from unwanted interactions in these systems. Several protocols have been proposed to address efficient and accurate quantum noise profiling and mitigation. In this work, we propose a novel protocol that efficiently estimates the average output of a noisy quantum device to be used for quantum noise mitigation. The multi-qubit system average behavior is approximated as a special form of a Pauli Channel where Clifford gates are used to estimate the average output for circuits of different depths. The characterized Pauli channel error rates, and state preparation and measurement errors are then used to construct the outputs for different depths thereby eliminating the need for large simulations and enabling efficient mitigation. We demonstrate the efficiency of the proposed protocol on four IBM Q 5-qubit quantum devices. Our method demonstrates improved accuracy with efficient noise characterization. We report up to 88% and 69% improvement for the proposed approach compared to the unmitigated, and pure measurement error mitigation approaches, respectively.