Abstract Quantum computing systems remain fragile due to decoherence and operational noise. Although large-scale quantum error correction offers robustness against general error models, its qubit and circuit overhead is often prohibitive for near-term devices. In this work, we present a scalable bit-flip error correction methodology that protects an arbitrary number of data qubits using only a single ancilla qubit, independent of register size. The method extends a previously introduced 4-qubit correction circuit for a 3-qubit data block and introduces a novel similarity circuit that enforces intra-triplet uniformity (always producing outputs of the form 000 or 111). Dedicated tail subcircuits handle the cases N = 3n + 1 and N = 3n + 2.
Abstract Quantum computing systems remain fragile due to decoherence and operational noise. Although large-scale quantum error correction offers robustness against general error models, its qubit and circuit overhead is often prohibitive for near-term devices. In this work, we present a scalable bit-flip error correction methodology that protects an arbitrary number of data qubits using only a single ancilla qubit, independent of register size. The method extends a previously introduced 4-qubit correction circuit for a 3-qubit data block and introduces a novel similarity circuit that enforces intra-triplet uniformity (always producing outputs of the form 000 or 111). Dedicated tail subcircuits handle the cases N = 3n + 1 and N = 3n + 2.
Abstract Quantum systems are inherently prone to errors caused by decoherence, noise, and imperfect gate operations. Effective error correction is essential to preserving quantum information, especially under hardware constraints. In this paper, we present a resource-efficient 4-qubit quantum circuit designed to detect and correct single-qubit bit-flip errors within a 3-qubit data block. The circuit uses Feynman (CNOT) and NOT gates alongside an ancillary qubit to implement full correction logic with minimal overhead. Compared to well-known approaches like the 9-qubit Shor code, our design achieves equivalent fidelity (100%) using fewer qubits, making it highly suitable for near-term quantum devices. Simulation results, obtained using IBM Quantum Composer and Qiskit, confirm the circuit’s correctness and reliability across all bit-flip scenarios. The design also introduces a reusable unitary block for efficient implementation. This work offers a scalable and practical solution to quantum error correction using minimal hardware resources.
To solve problems of practical importance1,2, quantum computers probably need to incorporate quantum error correction, in which a logical qubit is redundantly encoded in many noisy physical qubits3–5. The large physical-qubit overhead associated with error correction motivates the search for more hardware-efficient approaches6–18. Here, using a superconducting quantum circuit19, we realize a logical qubit memory formed from the concatenation of encoded bosonic cat qubits with an outer repetition code of distance d = 5 (ref. 10). A stabilizing circuit passively protects cat qubits against bit flips20–24. The repetition code, using ancilla transmons for syndrome measurement, corrects cat qubit phase flips. We study the performance and scaling of the logical qubit memory, finding that the phase-flip correcting repetition code operates below the threshold. The logical bit-flip error is suppressed with increasing cat qubit mean photon number, enabled by our realization of a cat-transmon noise-biased CX gate. The minimum measured logical error per cycle is on average 1.75(2)% for the distance-3 code sections, and 1.65(3)% for the distance-5 code. Despite the increased number of fault locations of the distance-5 code, the high degree of noise bias preserved during error correction enables comparable performance. These results, where the intrinsic error suppression of the bosonic encodings enables us to use a hardware-efficient outer error-correcting code, indicate that concatenated bosonic codes can be a compelling model for reaching fault-tolerant quantum computation. Bosonic qubits can be engineered to feature intrinsic protection against certain kinds of errors, which makes quantum error correction across many bosonic qubits possible with less overhead.
Abstract Quantum computers are powerful tools for certain computational problems, but their fragile quantum states are easily disrupted by environmental noise, causing errors that degrade computational accuracy. Quantum error correction (QEC) addresses this challenge by encoding one logical qubit across multiple physical qubits, so that individual errors can be identified and reversed through redundancy. We used Qiskit Aer, a free open-source quantum circuit simulator, to test the repetition code, one of the simplest QEC methods, under three types of noise: bit-flip, phase-flip, and depolarizing. We compared an unprotected single-qubit baseline (n = 1) to 3-qubit (n = 3) and 5-qubit (n = 5) repetition codes, sweeping the physical error rate p from 0 to 0.15 with 4,000 simulation shots per data point. We hypothesized that larger codes would achieve higher logical success probabilities at all tested noise levels, with n = 5 outperforming n = 3, which would outperform the unprotected baseline. Our results confirmed this scaling trend across all three noise types. At p = 0.10 under bit-flip noise, the baseline achieved 0.713 logical success, while n = 3 reached 0.791 and n = 5 reached 0.843. At low noise (p = 0.02), the n = 5 code approached 0.994 logical success. Simulation results closely matched exact theoretical predictions derived from the binomial failure probability formula, validating our experimental implementation. These findings confirm that even the simplest QEC methods provide meaningful error reduction and that larger codes offer greater protection, demonstrating a foundational principle of quantum error correction using only free simulation tools.
Abstract Quantum computers are powerful tools for certain computational problems, but their fragile quantum states are easily disrupted by environmental noise, causing errors that degrade computational accuracy. Quantum error correction (QEC) addresses this challenge by encoding one logical qubit across multiple physical qubits, so that individual errors can be identified and reversed through redundancy. We used Qiskit Aer, a free open-source quantum circuit simulator, to test the repetition code, one of the simplest QEC methods, under three types of noise: bit-flip, phase-flip, and depolarizing. We compared an unprotected single-qubit baseline (n = 1) to 3-qubit (n = 3) and 5-qubit (n = 5) repetition codes, sweeping the physical error rate p from 0 to 0.15 with 4,000 simulation shots per data point. We hypothesized that larger codes would achieve higher logical success probabilities at all tested noise levels, with n = 5 outperforming n = 3, which would outperform the unprotected baseline. Our results confirmed this scaling trend across all three noise types. At p = 0.10 under bit-flip noise, the baseline achieved 0.713 logical success, while n = 3 reached 0.791 and n = 5 reached 0.843. At low noise (p = 0.02), the n = 5 code approached 0.994 logical success. Simulation results closely matched exact theoretical predictions derived from the binomial failure probability formula, validating our experimental implementation. These findings confirm that even the simplest QEC methods provide meaningful error reduction and that larger codes offer greater protection, demonstrating a foundational principle of quantum error correction using only free simulation tools.
We propose a scheme for quantum error correction using robust continuous variable probe modes, rather than fragile ancilla qubits, to detect errors without destroying data qubits. The use of such probe modes reduces the required number of expensive qubits in error correction and allows efficient encoding, error detection, and error correction. Moreover, the elimination of the need for direct qubit interactions significantly simplifies the construction of quantum circuits. We will illustrate how the approach implements three existing quantum error correcting codes: the three-qubit bit-flip (phase-flip) code, the Shor code, and an erasure code.
Abstract Future large-scale quantum computers will rely on quantum error correction (QEC) to protect the fragile quantum information during computation1,2. Among the possible candidate platforms for realizing quantum computing devices, the compatibility with mature nanofabrication technologies of silicon-based spin qubits offers promise to overcome the challenges in scaling up device sizes from the prototypes of today to large-scale computers3–5. Recent advances in silicon-based qubits have enabled the implementations of high-quality one-qubit and two-qubit systems6–8. However, the demonstration of QEC, which requires three or more coupled qubits1, and involves a three-qubit gate9–11 or measurement-based feedback, remains an open challenge. Here we demonstrate a three-qubit phase-correcting code in silicon, in which an encoded three-qubit state is protected against any phase-flip error on one of the three qubits. The correction to this encoded state is performed by a three-qubit conditional rotation, which we implement by an efficient single-step resonantly driven iToffoli gate. As expected, the error correction mitigates the errors owing to one-qubit phase-flip, as well as the intrinsic dephasing mainly owing to quasi-static phase noise. These results show successful implementation of QEC and the potential of a silicon-based platform for large-scale quantum computing.