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.
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
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.
This study examines how noise affects the reliability of quantum circuits through two complementary approaches: a controlled simulation using Qiskit Aer and an execution study on real IBM Quantum hardware. In simulation, five noise models (bit-flip, phase-flip, depolarizing, readout, and thermal-relaxation) were applied to a parameterized benchmark circuit across varying qubit counts, circuit depths, and noise strengths, producing 6,000 experimental observations. Success probability declined substantially as qubit count and circuit depth increased, with bit-flip, phase-flip, and depolarizing noise degrading performance far more severely (mean success probabilities of 0.597, 0.580, and 0.607) than readout noise (0.917) or thermal-relaxation noise (0.982). The same benchmark circuit was then executed on the ibm_marrakesh processor across 150 hardware runs (150,000 shots). Hardware performance remained consistently high (mean success probability 0.992), declining with qubit count (0.998 at two qubits to 0.986 at six) but showing little sensitivity to circuit depth. These results indicate that controlled noise mechanisms and real hardware degrade circuit performance through different dynamics, with implications for interpreting simulated noise models relative to physical device behavior.
Quantum error correction (QEC) is essential for realizing scalable quantum computation. However, when evaluating its benefits, most analyses assume idealized components, overlooking the imperfections inherent in realistic fault-tolerant (FT) implementations. In this paper, we investigate the performance of QEC schemes taking into account that quantum gates and measurements are themselves error-prone. We derive bounds for the failure probability of Shor-style FT-QEC schemes using limited structural information, such as the number of flag qubits and quantum gates. Our analysis separates and quantifies two key contributors to the failure rate: decoding errors and residual errors arising from circuit-level faults. The derived bounds highlight fundamental limitations in Shor-style FT-QEC performance and quantify how circuit imperfections degrade error correction capabilities, under the assumption of depolarizing noise.
Quantum Error Correction (QEC) is essential for achieving fault-tolerant quantum computation, enabling the protection of fragile quantum information from decoherence and noise. While classical Forward Error Correction (FEC) codes have been highly effective in digital communication systems, their direct application to quantum systems is inadequate mainly due to the no cloning theorem and the continuous nature of quantum errors. Unlike classical FECs, QEC is designed to detect and correct both bit-flip and phase-flip errors without disturbing the encoded quantum state. This paper presents the code structure design and simulation of two quantum error correction codes—the Planar surface code and the 5-qubit Perfect code—using Qiskit and Stim on Google Colab. It is worth mentioning here that the performance of the Steane [ [7], [1], [3]] code is included in the current work as a benchmark. The codes are evaluated under statistical noise models, including amplitude damping and dephasing, which closely mirror the physical behavior of quantum hardware. The primary performance metric, for performance assessment, is the logical error rate, evaluated across a range of physical error rates and code distances under each noise model. The simulation results show that increasing the distance of the Planar code significantly reduces the logical error rates, demonstrating superior performance compared to other similar codes under this study.