Optimization of Quantum Error Correcting Code under Temporal Variation of Qubit Quality

TL;DR

Problem: Fixed-distance QEC is inefficient on devices with per-qubit and temporal error variation. Approach: adapt code distance per qubit using daily calibration data and a target logical error rate $p_L = 10^{-6}$, with rotated surface codes and Stim-based simulations. Contributions: quantify Pauli-X and CNOT variability on ibm_kyiv, implement per-qubit distance assignment with practical caps, and demonstrate overhead reductions up to about 52% on ibm_kyiv (and up to about 71% on two additional devices) while preserving usability above 80%. Significance: provides a practical hardware-aware path to QEC for the NISQ era, enabling more efficient resource usage and deployment of fault-tolerance concepts on real devices.

Abstract

Error rates in current noisy quantum hardware are not static; they vary over time and across qubits. This temporal and spatial variation challenges the effectiveness of fixed-distance quantum error correction (QEC) codes. In this paper, we analyze 12 days of calibration data from IBM's 127-qubit device (ibm_kyiv), showing the fluctuation of Pauli-X and CNOT gate error rates. We demonstrate that fixed-distance QEC can either underperform or lead to excessive overhead, depending on the selected qubit and the error rate of the day. We then propose a simple adaptive QEC approach that selects an appropriate code distance per qubit, based on daily error rates. Using logical error rate modeling, we identify qubits that cannot be used and qubits that can be recovered with minimal resources. Our method avoids unnecessary resource overhead by excluding outlier qubits and tailoring code distances. Across 12 calibration days on ibm_kyiv, our adaptive strategy reduces physical qubit overhead by over 50% per logical qubit while maintaining access to 85-100% of usable qubits. To further validate the method, we repeat the experiment on two additional 127-qubit devices, ibm_brisbane and ibm_sherbrooke, where the overhead savings reach up to 71% while still preserving over 80% qubit usability. This approach offers a practical and efficient path forward for Noisy Intermediate-Scale Quantum (NISQ)-era QEC strategies.

Figures (6)

• Figure 1: Qubit requirement with increasing code distance. As the distance of the code increases to improve error tolerance, the number of qubits required for both data and ancilla qubits rises, with ancilla qubits contributing significantly to the total overhead chatterjee2024quantum.
• Figure 2: Representation of distance-3 surface codes. Unrotated surface code (left) and rotated surface code (right). The gray blobs depicting qubits are acted upon by the pink surfaces representing Z-stabilizers and green lines/surfaces indicating X-stabilizers. chatterjee2025q.
• Figure 3: (a) Pauli-X error rate across 12 calibration days for all 127 qubits in the ibm_kyiv processor. Each dot represents the Pauli-X error of one qubit per day. Qubit 8 (red dashed line) shows large temporal variation, while Qubit 80 (blue dashed line) maintains consistently high error. A dotted horizontal line marks the reference threshold at $10^{-3}$. (b) CNOT error rate across 12 calibration days for all 144 CNOT links. Each dot represents the CNOT error of one link per day. Link indices are assigned arbitrarily based on their order in the calibration data.
• Figure 4: Logical error rate as a function of physical Pauli-X error rate for surface code distances ranging from 3 to 21. The horizontal dashed line shows a target logical error rate of $10^{-6}$. The vertical dashed line marks the cutoff physical error rate above which QEC is ineffective.
• Figure 5: Percentage of usable qubits on each calibration day across four code distances. A qubit is considered usable if its Pauli-X physical error rate is below the threshold required to achieve a target logical error rate of $10^{-6}$ at that code distance. These threshold values, obtained from simulation, are: $7\times10^{-4}$ for distance-7, $10^{-3}$ for distance-9, $2\times10^{-3}$ for distance-11, and $7\times10^{-3}$ for distance-13. Higher distances tolerate more physical error, allowing more qubits to be used, but increase resource cost.