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.
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.
Fault-tolerant quantum computation requires quantum error correction (QEC), and while the performance of various QEC codes has been well characterized under idealized noise conditions, it has not been sufficiently characterized by simulation under realistic noise conditions.In this paper, the performance of three basic QEC codes (3-qubit Bit-Flip, 3-qubit Phase-Flip and Shor's 9-qubit [[9,1,3]]) under simulated depolarizing noise, by using Qiskit Aer, is analyzed systematically in three stages.Stage 1 is a validation of all three implementations under ideal circumstances.Stage 2 tests for 6 different error rates of 0% to 5%, and runs 2048 shots per error rate.Stage 3 brings the analysis from stage 2 to 0% to 10%, but with an uncorrected baseline for comparison.Results verify that Shor's > Bit-Flip > Phase-Flip is consistent over practically relevant error rates.At 1% noise, within current IBM Quantum hardware range, all three codes achieve a logical fidelity of over 98.7%.An error rate of 7-8% is used to identify a cross-over threshold above which the simpler Bit-Flip code can match Shor's code in fidelity with only one-third the number of qubits.
Quantum error correction (QEC) is a prerequisite for fault-tolerant quantum computation, yet the overhead cost—the number of physical qubits required to encode and protect a single logical qubit—varies substantially across QEC code families. This paper presents a systematic simulation-based comparison of two paradigmatic QEC codes: the three-qubit bit-flip repetition code and the distance-3 surface code. The bit-flip code is simulated by literal Qiskit Aer circuit execution (encoding, injected depolarizing noise, and dynamic classical feed-forward correction). The distance-3 surface code is instead simulated with a custom, vectorized stabilizer-formalism engine, because its 17-qubit noisy circuit would be far more expensive to run shot-by-shot in Aer at the scale needed here. This engine implements the same Aaronson-Gottesman algorithm underlying Aer's stabilizer backend and is cross-validated against genuine Aer circuit execution at a representative operating point; no surface-code result in this paper comes from literal Aer execution. We apply an independent single-qubit depolarizing channel at physical error rates p ∈ [0.001, 0.30] and measure the resulting logical error rate P_L for each code across 200,000 independent trials per data point (95% Wilson confidence intervals reported throughout). We identify the pseudo-threshold p*—the physical error rate at which a code's logical error rate equals the uncorrected physical error rate, and compare simulated crossover points against closed-form analytical predictions. Our results reveal a more nuanced picture than the naive threshold-theorem intuition suggests. The three-qubit bit-flip code cannot correct phase-type errors under any circumstance: under a full quantum-state-protection criterion its logical error rate exceeds the physical error rate for every physically meaningful p, with the two curves crossing only at p ≈ 0.75 deep in the maximally-mixed regime and of no practical relevance. Under the narrower, but operationally standard, "bit-observable" criterion (protection of a definite computational-basis value only), the code is unconditionally beneficial (P_L < p) across the entire tested range, because the 2/3 fraction of depolarizing weight that manifests as a detectable bit-flip is compressed further by majority-vote correction. These two results are not contradictory: they evaluate different protection goals—preservation of a classical bit value versus preservation of an arbitrary quantum state—and every comparison in this paper that mixes the two is labeled accordingly. Under this study's idealized, single-round setup, the distance-3 surface code achieves a simulated finite-size pseudo-threshold of p*_SC ≈ 0.083 (8.3%)—a result specific to noiseless syndrome extraction, a single correction round, and an exact minimum weight decoder, and not a hardware representative or asymptotic threshold. That figure is roughly three orders of magnitude higher than an order-of-magnitude literature extrapolation of ≈ 5.6 × 10⁻⁵ (substituting the asymptotic literature threshold p_th ≈ 0.75% into a power law fit for a different, many-round, measurement noisy regime, not a dedicated prediction for our exact setup); the gap is attributable to precisely those idealizations. The surface code requires 17 physical qubits per logical qubit in our implementation (9 data + 8 ancilla) versus 3 for the bit-flip code, a ≈ 5.7× overhead, quantitatively illustrating the overhead-versus-protection tradeoff fundamental to near-term fault-tolerant architectures. All simulation code is available at https://github.com/shanjeev-b-u/qec-code-comparison-depolarizing-noise
Quantum systems operate using quantum bits (qubits), which are highly sensitive to external noises and disturbances and they are easily prone to errors. In the Noisy Intermediate-Scale Quantum (NISQ) era, these errors make difficult to build reliable, fault tolerant quantum circuits for real world deployment. This work provides an overview of need for Quantum Error Correction (QEC) in NISQ devices, as qubits commonly get affected by bit-flip, phase-flip, and combined errors that degrades system performance and examines Quantum Error Correction Code (QECC) techniques usage in communication, computation, and quantum multimedia applications such as image and video transmission. In addition, a simple three-qubit repetition circuit is introduced to demonstrate an error-mitigation approach. These findings, also show how QEC improves fidelity, reduces logical errors, and improves noise tolerance. The analysis also identifies challenges and outlines the future research direction aimed at achieving efficient and scalable fault tolerant quantum circuits.
Quantum computers have the possibility of a much reduced calculation load compared with classical computers in specific problems. Quantum error correction (QEC) is vital for handling qubits, which are vulnerable to external noise. In QEC, actual errors are predicted from the results of syndrome measurements by stabilizer generators, in place of making direct measurements of the data qubits. Here, we propose Quantum Golay code Error Correction (QGEC), a QEC method using Golay code, which is an efficient coding method in classical information theory. We investigated our method's ability in decoding calculations with the Transformer. We evaluated the accuracy of the decoder in a code space defined by the generative polynomials with three different weights sets and three noise models with different correlations of bit-flip error and phase-flip error. Furthermore, under a noise model following a discrete uniform distribution, we compared the decoding performance of Transformer decoders with identical architectures trained respectively on Golay and toric codes. The results showed that the noise model with the smaller correlation gave better accuracy, while the weights of the generative polynomials had little effect on the accuracy of the decoder. In addition, they showed that Golay code requiring 23 data qubits and having a code distance of 7 achieved higher decoding accuracy than toric code which requiring 50 data qubits and having a code distance of 5. This suggests that implementing quantum error correction using a Transformer may enable the Golay code to realize fault-tolerant quantum computation more efficiently.
Abstract Practical quantum computing will require error rates well below those achievable with physical qubits. Quantum error correction 1,2 offers a path to algorithmically relevant error rates by encoding logical qubits within many physical qubits, for which increasing the number of physical qubits enhances protection against physical errors. However, introducing more qubits also increases the number of error sources, so the density of errors must be sufficiently low for logical performance to improve with increasing code size. Here we report the measurement of logical qubit performance scaling across several code sizes, and demonstrate that our system of superconducting qubits has sufficient performance to overcome the additional errors from increasing qubit number. We find that our distance-5 surface code logical qubit modestly outperforms an ensemble of distance-3 logical qubits on average, in terms of both logical error probability over 25 cycles and logical error per cycle ((2.914 ± 0.016)% compared to (3.028 ± 0.023)%). To investigate damaging, low-probability error sources, we run a distance-25 repetition code and observe a 1.7 × 10 −6 logical error per cycle floor set by a single high-energy event (1.6 × 10 −7 excluding this event). We accurately model our experiment, extracting error budgets that highlight the biggest challenges for future systems. These results mark an experimental demonstration in which quantum error correction begins to improve performance with increasing qubit number, illuminating the path to reaching the logical error rates required for computation.