Quantum information processors promise fast algorithms for problems inaccessible to classical computers. But since qubits are noisy and error-prone, they will depend on fault-tolerant quantum correction (FTQEC) to compute reliably. Quantum correction can protect against general noise if -- and only if -- the in each physical qubit operation is smaller than a certain threshold. The threshold for general errors is quantified by their diamond norm. Until now, qubits have been assessed primarily by randomized benchmarking (RB), which reports a different error that is not sensitive to all errors, cannot be compared directly to diamond norm thresholds, and cannot efficiently certify a qubit for FTQEC. We use gate set tomography (GST) to completely characterize the performance of a trapped-Yb$^+$-ion qubit and certify it rigorously as suitable for FTQEC by establishing that its diamond norm rate is less than $6.7\times10^{-4}$ with $95\%$ confidence.
Quantum information processors promise fast algorithms for problems inaccessible to classical computers. But since qubits are noisy and error-prone, they will depend on fault-tolerant quantum correction (FTQEC) to compute reliably. Quantum correction can protect against general noise if -- and only if -- the in each physical qubit operation is smaller than a certain threshold. The threshold for general errors is quantified by their diamond norm. Until now, qubits have been assessed primarily by randomized benchmarking (RB), which reports a different error that is not sensitive to all errors, cannot be compared directly to diamond norm thresholds, and cannot efficiently certify a qubit for FTQEC. We use gate set tomography (GST) to completely characterize the performance of a trapped-Yb+-ion qubit and certify it rigorously as suitable for FTQEC by establishing that its diamond norm rate is less than 6.7×10−4 with 95% confidence.
Abstract Quantum information processors promise fast algorithms for problems inaccessible to classical computers. But since qubits are noisy and error-prone, they will depend on fault-tolerant quantum error correction (FTQEC) to compute reliably. Quantum error correction can protect against general noise if—and only if—the error in each physical qubit operation is smaller than a certain threshold. The threshold for general errors is quantified by their diamond norm. Until now, qubits have been assessed primarily by randomized benchmarking, which reports a different error rate that is not sensitive to all errors, and cannot be compared directly to diamond norm thresholds. Here we use gate set tomography to completely characterize operations on a trapped-Yb + -ion qubit and demonstrate with greater than 95% confidence that they satisfy a rigorous threshold for FTQEC (diamond norm ≤6.7 × 10 −4 ).
Abstract Contemporary methods for benchmarking noisy quantum processors typically measure average error rates or process infidelities. However, thresholds for fault-tolerant quantum error correction are given in terms of worst-case error rates—defined via the diamond norm—which can differ from average error rates by orders of magnitude. One method for resolving this discrepancy is to randomize the physical implementation of quantum gates, using techniques like randomized compiling (RC). In this work, we use gate set tomography to perform precision characterization of a set of two-qubit logic gates to study RC on a superconducting quantum processor. We find that, under RC, gate errors are accurately described by a stochastic Pauli noise model without coherent errors, and that spatially correlated coherent errors and non-Markovian errors are strongly suppressed. We further show that the average and worst-case error rates are equal for randomly compiled gates, and measure a maximum worst-case error of 0.0197(3) for our gate set. Our results show that randomized benchmarks are a viable route to both verifying that a quantum processor’s error rates are below a fault-tolerance threshold, and to bounding the failure rates of near-term algorithms, if—and only if—gates are implemented via randomization methods which tailor noise.
The promise of quantum computing with imperfect qubits relies on the ability of a quantum computing system to scale cheaply through error correction and fault tolerance. While fault tolerance requires relatively mild assumptions about the nature of qubit errors, the overhead associated with coherent and non-Markovian errors can be orders of magnitude larger than the overhead associated with purely stochastic Markovian errors. One proposal to address this challenge is to randomize the circuits of interest, shaping the errors to be stochastic Pauli errors but leaving the aggregate computation unaffected. The randomization technique can also suppress couplings to slow degrees of freedom associated with non-Markovian evolution. Here, we demonstrate the implementation of Pauli-frame randomization in a superconducting circuit system, exploiting a flexible programming and control infrastructure to achieve this with low effort. We use high-accuracy gate-set tomography to characterize in detail the properties of the circuit error, with and without the randomization procedure, which allows us to make rigorous statements about Markovianity as well as the nature of the observed errors. We demonstrate that randomization suppresses signatures of non-Markovian evolution to statistically insignificant levels, from a Markovian model violation ranging from $43\ensuremath{\sigma}$ to $1987\ensuremath{\sigma}$, down to violations between $0.3\ensuremath{\sigma}$ and $2.7\ensuremath{\sigma}$ under randomization. Moreover, we demonstrate that, under randomization, the experimental errors are well described by a Pauli error model, with model violations that are similarly insignificant (between $0.8\ensuremath{\sigma}$ and $2.7\ensuremath{\sigma}$). Importantly, all these improvements in the model accuracy were obtained without degradation to fidelity, and with some improvements to error rates as quantified by the diamond norm. This demonstrates the ability of Pauli-frame randomization to shape noise into forms that are more benign for quantum error correction and fault tolerance.
Fault-tolerant quantum error correction (FTQEC) is pivotal for ensuring the reliability and scalability of quantum computers, which harness quantum mechanics principles. This paper presents a comprehensive analysis of FTQEC methodologies, integrating theoretical investigations, experimental validations, error correction capabilities, and simulation comparisons. Theoretical frameworks reveal high error correction capabilities, with success rates ranging from 0.88 to 0.95 and significant error rate reductions of up to 98%. Experimental validations across diverse quantum hardware platforms demonstrate promising results, with high success rates and notable error correction efficiency, albeit challenges in scalability. Analysis of error correction capabilities highlights robustness against errors and preservation of qubits integrity during error correction procedures. Simulation comparisons further validate the efficacy of FTQEC protocols, showing significant improvements in error rate reduction and gate fidelity. However, scalability and resource overhead remain concerns, necessitating further research for efficient and scalable FTQEC protocols. Overall, this study contributes to the understanding and advancement of fault-tolerant quantum error correction, crucial for realizing the transformative potential of quantum computing technology.
Abstract The ubiquitous noise in quantum system hinders the advancement of quantum information processing and has driven the emergence of different quantum error correction protocols. Among them, quantum error correction codes tailored for noise-biased qubits exhibit comparatively high error thresholds, making them a promising platform to achieve fault-tolerance. Nevertheless, their quantum operations are challenging and the demonstration of their performance beyond the fault-tolerant threshold remains incomplete. Here, we leverage Schrödinger cat states in scalable planar superconducting circuits to thoroughly characterize the high-fidelity single-qubit quantum operations on noise-biased qubits with systematic quantum tomography and benchmarking tools, demonstrating their state-of-the-art performance beyond the fault-tolerant threshold of the surface code. These results thus embody a transformative milestone in the exploration of quantum systems with structured noises. Notably, our framework is extensible to other structured-noise systems, paving the way for systematic characterization and validation of novel quantum platforms with structured noise.
Quantum error correction (QEC) can strongly suppress noise, but predicting whether a computation succeeds requires understanding not only how much effective noise persists after QEC, but what form that noise takes. To thoroughly characterize this noise, we introduce logical gate set tomography (``logical GST''). Logical GST produces detailed representations of noisy logical operations that can be used to derive higher-level metrics, such as gate fidelity and diamond distance, and to test for non-Markovian and coherent logical errors. We use logical GST to obtain the first full, self-consistent characterization of a universal single-qubit logical gate set, implemented in the seven-qubit Steane code using Quantinuum System Model H1, a 20-qubit trapped-ion quantum processor. In the experimental data from H1, logical GST identified predominantly logical Pauli-stochastic errors, found no evidence of logical non-Markovianity, and detected weak potential signatures of coherent logical error. By contrast, higher-shot datasets generated with Quantinuum's detailed system emulator, identify mechanisms for weak but statistically significant logical non-Markovianity and strong evidence of coherent logical error. Finally, we use logical GST to perform error attribution with the system emulator, revealing how specific physical error sources shape logical error mechanisms.