Qubit readout is commonly performed by thresholding a collection of analog detector signals to obtain a sequence of single-shot bit values. The intrinsic irreversibility of the mapping from analog to digital signals discards soft information associated with an a posteriori confidence that can be assigned to each bit value when a detector is well characterized. Accounting for soft information, we show significant improvements in enhanced state detection with the quantum repetition code as well as quantum state or parameter estimation. These advantages persist in spite of non-Gaussian features of realistic readout models, experimentally relevant small numbers of qubits, and finite encoding errors. These results show useful and achievable advantages for a wide range of current experiments on quantum state tomography, parameter estimation, and qubit readout.
Qubit readout is commonly performed by thresholding a collection of analog detector signals to obtain a sequence of single-shot bit values. The intrinsic irreversibility of the mapping from analog to digital signals discards soft information associated with an a posteriori confidence that can be assigned to each bit value when a detector is well characterized. Accounting for soft information, we show significant improvements in enhanced state detection with the quantum repetition code as well as quantum state or parameter estimation. These advantages persist in spite of non-Gaussian features of realistic readout models, experimentally relevant small numbers of qubits, and finite encoding errors. These results show useful and achievable advantages for a wide range of current experiments on quantum state tomography, parameter estimation, and qubit readout.
Quantum error correction enables the preservation of logical qubits with a lower logical error rate than the physical error rate, with performance depending on the decoding method. Traditional decoding approaches rely on the binarization (“hardening”) of readout data, thereby ignoring valuable information embedded in the analog (“soft”) readout signal. We present experimental results showcasing the advantages of incorporating soft information into the decoding process of a distance-3 ( d = 3 ) bit-flip surface code with flux-tunable transmons. We encode each of the 16 computational states that make up the logical state | 0 L ⟩ , and protect them against bit-flip errors by performing repeated Z -basis stabilizer measurements. To infer the logical fidelity for the | 0 L ⟩ state, we average across the 16 computational states and employ two decoding strategies: minimum-weight perfect matching and a recurrent neural network. Our results show a reduction of up to 6.8% in the extracted logical error rate with the use of soft information. Decoding with soft information is widely applicable, independent of the physical qubit platform, and could allow for shorter readout durations, further minimizing logical error rates. Published by the American Physical Society 2024
Quantum error correction is of crucial importance for fault-tolerant quantum computers.As an essential step toward the implementation of quantum error-correcting codes, quantum nondemolition measurements are needed to efficiently detect the state of a logical qubit without destroying it.Here we implement quantum nondemolition measurements in a Si/SiGe two-qubit system, with one qubit serving as the logical qubit and the other serving as the ancilla.Making use of a two-qubit controlled-rotation gate, the state of the logical qubit is mapped onto the ancilla, followed by a destructive readout of the ancilla.Repeating this procedure enhances the logical readout fidelity from 75.5 Æ 0.3% to 94.5 Æ 0.2% after 15 ancilla readouts.In addition, we compare the conventional thresholding method with an improved signal processing method called soft decoding that makes use of analog information in the readout signal to better estimate the state of the logical qubit.We demonstrate that soft decoding leads to a significant reduction in the required number of repetitions when the readout errors become limited by Gaussian noise, for instance, in the case of readouts with a low signal-to-noise ratio.These results pave the way for the implementation of quantum error correction with spin qubits in silicon.
Quantum error correction allows for quantum information to be preserved using logical qubits, which are subject to lower error rates than their constituent physical qubits. The degree of error suppression depends on the choice of error correcting code and distance, the underlying physical error rate, and the accuracy of the decoder. While traditional decoders utilise a binary (hard) syndrome, recent work shows that additional (soft) information captured during qubit readout can be effectively utilised to improve decoding accuracy. In this work, we present experimental results from a distance-three surface code implemented on transmon qubits, where we perform Z-stabiliser measurements to protect the state of the logical qubit against bit-flip errors. We initialise the logical qubit in one of 16 possible computational states representing the logical zero state, and perform repeated stabiliser checks over a variable number of rounds to preserve the state over time. We compare the decoding performance for a hard minimum-weight perfect matching decoder against a soft variant where rich measurement information is incorporated, and demonstrate an improved logical fidelity. Additionally, we employ a recurrent neural network decoder with both soft and hard variants and observe improved performance when soft information is used. The general nature of soft information makes it widely applicable to different physical qubit platforms, where it can be leveraged to shorten measurement times and improve the logical fidelity in quantum error correction experiments. Pre-print available at arXiv:2403.00706.
Quantum error correction enables the preservation of logical qubits with a lower logical error rate than the physical error rate, with performance depending on the decoding method. Traditional error decoding approaches, relying on the binarization (`hardening') of readout data, often ignore valuable information embedded in the analog (`soft') readout signal. We present experimental results showcasing the advantages of incorporating soft information into the decoding process of a distance-three (d=3) bit-flip surface code with transmons. To this end, we use the 3×3 data-qubit array to encode each of the 16 computational states that make up the logical state ∣0L⟩, and protect them against bit-flip errors by performing repeated Z-basis stabilizer measurements. To infer the logical fidelity for the ∣0L⟩ state, we average across the 16 computational states and employ two decoding strategies: minimum weight perfect matching and a recurrent neural network. Our results show a reduction of up to 6.8% in the extracted logical error rate with the use of soft information. Decoding with soft information is widely applicable, independent of the physical qubit platform, and could reduce the readout duration, further minimizing logical error rates.
Readout errors are one of the dominant sources of noise in current quantum processors, limiting both expectation-value estimation and sampling-based applications. Since they affect only the classical measurement outcomes, they can be addressed using classical coding techniques: immediately before measurement, each data qubit is redundantly encoded with ancilla qubits, and the resulting bit string is decoded either by post-selection or by majority voting. Unlike conventional readout error mitigation, which corrects only aggregate quantities such as expectation values, this approach operates on individual measurement shots and can therefore produce approximately corrected samples. We present a systematic cross-platform and cross-generation experimental evaluation of repetition-code readout error detection and correction. We benchmark the same protocol on IBM Heron r1-r3 superconducting processors and Quantinuum H1 and H2 trapped-ion processors while independently varying the code distance, hardware generation, and encoding layout. We find that both error detection and correction improve readout fidelity on every device and generation tested, even as the unencoded baseline improves substantially across successive hardware releases. At the same time, the value of additional redundancy depends strongly on the underlying hardware. On superconducting processors, the extra gate errors introduced by the encoding rapidly offset its benefits, whereas on trapped-ion processors the much lower gate error rates allow larger code distances to remain advantageous.
The typical model for measurement noise in quantum error correction is to randomly flip the binary measurement outcome. In experiments, measurements yield much richer information - e.g., continuous current values, discrete photon counts - which is then mapped into binary outcomes by discarding some of this information. In this work, we consider methods to incorporate all of this richer information, typically called soft information, into the decoding of quantum error correction codes, and in particular the surface code. We describe how to modify both the Minimum Weight Perfect Matching and Union-Find decoders to leverage soft information, and demonstrate these soft decoders outperform the standard (hard) decoders that can only access the binary measurement outcomes. Moreover, we observe that the soft decoder achieves a threshold 25\% higher than any hard decoder for phenomenological noise with Gaussian soft measurement outcomes. We also introduce a soft measurement error model with amplitude damping, in which measurement time leads to a trade-off between measurement resolution and additional disturbance of the qubits. Under this model we observe that the performance of the surface code is very sensitive to the choice of the measurement time - for a distance-19 surface code, a five-fold increase in measurement time can lead to a thousand-fold increase in logical error rate. Moreover, the measurement time that minimizes the physical error rate is distinct from the one that minimizes the logical performance, pointing to the benefits of jointly optimizing the physical and quantum error correction layers.