Practical quantum computers will require resource-efficient error-correcting codes. The rotated surface code uses approximately half the number of qubits as the unrotated surface code to create a logical qubit with the same error-correcting distance. However, instead of distance, a more useful qubit-saving metric would be based on logical error rates. In this work we find the well-below-threshold scaling of logical to physical error rates under circuit-level noise for both codes at high odd and even distances and then compare the number of qubits used by each code to achieve equal logical error rates. We perform Monte Carlo sampling of memory experiment circuits with all valid CNOT orders using the stabilizer simulator Stim and the uncorrelated minimum-weight perfect matching decoder PyMatching 2. We find that the rotated code uses about 74 % the number of qubits used by the unrotated code to achieve a logical error rate of p L = 10 − 12 at the operational physical error rate of p = 10 − 3 . The ratio remains ≈ 75 % for p values within a factor of two of p = 10 − 3 for all useful p L . Our work finds the low- p L scaling of the surface code and clarifies the qubit savings provided by the rotated surface code, providing numerical justification for its use in future implementations of the surface code.
Practical quantum computers will require resource-efficient error-correcting codes. The rotated surface code uses approximately half the number of qubits as the unrotated surface code to create a logical qubit with the same error-correcting distance. However, instead of distance, a more useful qubit-saving metric would be based on logical error rates. In this work we find the well-below-threshold scaling of logical to physical error rates under circuit-level noise for both codes at high odd and even distances and then compare the number of qubits used by each code to achieve equal logical error rates. We perform Monte Carlo sampling of memory experiment circuits with all valid CNOT orders using the stabilizer simulator Stim and the uncorrelated minimum-weight perfect matching decoder PyMatching 2. We find that the rotated code uses about 74 % the number of qubits used by the unrotated code to achieve a logical error rate of p L = 10 − 12 at the operational physical error rate of p = 10 − 3 . The ratio remains ≈ 75 % for p values within a factor of two of p = 10 − 3 for all useful p L . Our work finds the low- p L scaling of the surface code and clarifies the qubit savings provided by the rotated surface code, providing numerical justification for its use in future implementations of the surface code.
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
PyMatching provides a method for decoding quantum errors using the minimum-weight perfect matching algorithm. However, although the PyMatching supports various quantum error correction codes, no deep study has been conducted on the rotated planar surface codes that are receiving the latest attention. In this study, we define a procedure for detecting errors using PyMatching for rotated planar surface codes and analyze the correctness of error correction results. To compare error correction behavior by logical qubit size, we used logical qubit with distances of 3, 5, and 7, respectively. Furthermore, we have conducted comparative empirical studies by comparing the theoretical predictions of PyMatching with the actual execution results of classical state vector simulators.
Abstract To date, the surface code has become a promising candidate for quantum error correcting codes because it achieves a high threshold and is composed of only the nearest gate operations and low-weight stabilizers. Here, we have exhibited that the logical failure rate can be enhanced by manipulating the lattice size of surface codes that they can show an enormous improvement in the number of physical qubits for a noise model where dephasing errors dominate over relaxation errors. We estimated the logical error rate in terms of the lattice size and physical error rate. When the physical error rate was high, the parameter estimation method was applied, and when it was low, the most frequently occurring logical error cases were considered. By using the minimum weight perfect matching decoding algorithm, we obtained the optimal lattice size by minimizing the number of qubits to achieve the required failure rates when physical error rates and bias are provided .
Quantum computing offers unprecedented speed and power for solving complex problems, but its susceptibility to errors makes error-correction codes crucial. An important metric for evaluating the performance of error-correction codes is the code distance, which measures the number of errors a code can correct. Current qubit-plane architectures are implemented using the same code distance. However, a larger code distance reduces the logical error rate but increases resources. In fact, distance-5 code uses more resources but offers higher resilience to logical errors compared to distance-3 code. We propose a qubit-plane architecture for rotated and unrotated surface codes called Qentaurus that combines various code distances of logical qubits for robust error correction while maintaining qubit efficiency. Qentaurus is designed to be highly scalable regarding computation, layout, and code distance. We performed a case study using Qentaurus with distance-3 and -7 codes to evaluate the performance. Moreover, we additionally developed a qubit placement scheme that considers the critical path of the quantum circuit to reduce the logical error rate and gate count. Our main experimental results demonstrated that Qentaurus considerably reduces the overall logical error rate (by 15.76% on average and up to 25.25%) than that of the Distance-3 Checkerboard architecture. After additionally implementing the qubit placement scheme, the average reduction increases to 13.44% (up to 39.07%). Notably, the Qentaurus-37 maintains logical error rates comparable to those of the Distance-5 Checkerboard, achieving considerable qubit efficiency. The Distance-5 Checkerboard demands 227.7% more physical qubits than that required by the Distance-3 Checkerboard, while Qentaurus-37 utilizes just 131.2% of the physical qubits required by Distance-3 the Checkerboard, marking a substantial reduction of 96.5% in the resource gap. The overall gate count is reduced by an average of 1.74% with Qentaurus alone and by 3.33% on average with the qubit placement scheme. We believe that the proposed Qentaurus and qubit placement scheme have the potential to significantly improve the error-correction performance of large-scale quantum computers in practical applications.
Quantum error correction codes (QECCs) are essential for reliable quantum computing as they protect quantum states against noise and errors. Limited research has explored the resilience of QECCs to biased noise, critical for selecting optimal codes. We examine how different noise types impact QECCs, considering the varying susceptibility of quantum systems to specific errors. Our goal is to identify opportunities to minimize the resources—or overhead—needed for effective error correction. We conduct a detailed study on two QECCs—rotated and unrotated surface codes—under various noise models using simulations. Rotated surface codes generally perform better due to their simplicity and lower qubit overhead. They exceed the noise threshold of current quantum processors, making them more effective at lower error rates. This study highlights a hierarchy in surface code implementation based on resource demand, consistently observed across both code types. Our analysis ranks the code-capacity model as the most pessimistic and the circuit-level model as the most realistic, mapping error thresholds that show surface code advantages. Additionally, higher code distances improve performance without excessively increasing qubit overhead. Tailoring surface codes to align with the target logical error rate and the biased physical error profile is crucial for optimizing reliability and resource use.
Quantum error correction can suppress logical errors exponentially with code distance when physical noise is below threshold. Realizing this on hardware requires tight coordination among quantum gates, mid-circuit measurements, and qubit coherence. Device variation, noise asymmetry, and correlations across space and time complicate the translation of these capabilities into logical error suppression. Here, we study rotated surface-code memory on an IBM Nighthawk r2 processor. Performance varies across tested processor placements, which we attribute to differences in qubit quality. We optimize the code by combining short readout, fused measurement and reset, and an XX echo on data qubits during ancilla readout. Compared with standard readout without reset or echo, the optimized syndrome extraction circuit reduces the median per-round logical error rate by factors of 3.8 in the X basis and 1.4 in the Z basis. In memory experiments of up to 150 rounds, the distance advantage weakens, alongside long-lived correlations that may reflect a persistent error component. Under our optimization recipe, increasing the protected distance from 3 to 6 while holding the other distance fixed reduces the per-round logical error rate from 0.53% to 0.24% in Z memory and from 0.52% to 0.22% in X memory. The corresponding error-suppression factors under maximum-likelihood decoding are 1.68 and 1.85. These results connect circuit compilation and optimization to surface-code memory performance and limitations, and motivate QPU and QEC circuit codesign for fault-tolerant quantum computing primitives on current cloud hardware.