Exact and Efficient Stabilizer Simulation of Thermal-Relaxation Noise for Quantum Error Correction

TL;DR

Thermal-relaxation noise is pervasive in quantum devices and challenging to simulate within stabilizer frameworks. This work presents an exact Clifford-compatible model by unifying amplitude damping and dephasing into a single relaxation channel, enabling scalable, accurate stabilizer-based simulations of QEC under realistic thermal noise characterized by $T_1$ and $T_2$. It analyzes Pauli-twirling approximation and quasi-probabilistic decomposition, showing that combining $T_1$ and $T_2$ reduces sampling overhead and that a reset-based positive approximation can outperform PTA in finite-temperature regimes. The approach is implemented in a GPU- and MPI-accelerated stabilizer simulator and applied to rotated surface codes and BB codes, revealing that PTA can misestimate logical error rates by factors of a few and highlighting the importance of noise-informed decoders for future fault-tolerant architectures.

Abstract

Stabilizer-based simulation of quantum error-correcting codes typically relies on the Pauli-twirling approximation (PTA) to render non-Clifford noise classically tractable, but PTA can distort the behavior of physically relevant channels such as thermal relaxation. Physically accurate noise simulation is needed to train decoders and understand the noise suppression capabilities of quantum error correction codes. In this work, we develop an exact and stabilizer-compatible model of qubit thermal relaxation noise and show that the combined amplitude damping and dephasing channel admits a fully positive probability decomposition into Clifford operations and reset whenever $T_2 \leqslant T_1$. For $T_2 > T_1$, the resulting decomposition is negative, but allows a smaller sampling overhead versus independent channels. We further introduce an approximated error channel with reset that removes the negativity of the decomposition while achieving higher channel fidelity to the true thermal relaxation than PTA, and extend our construction to finite temperature relaxation. We apply the exact combined model to investigate large surface codes and bivariate bicycle codes on superconducting platforms with realistic thermal relaxation error. The differing logical performances across code states further indicate that noise-model-informed decoders will be essential for accurately capturing thermal-noise structure in future fault-tolerant architectures.

Figures (10)

• Figure 1: The effect of the thermal relaxation error channel and the channel after PTA. (a) The Bloch sphere after applying the thermal relaxation channel and the channel after PTA on a single qubit. We notice that the PTA channel can nicely capture the distortion on the Pauli-X and Y directions, while the directional relaxation along the Pauli-Z axis is not. The gray-shaded sphere shows the unit Bloch sphere. (b) The trace distance difference ($\Delta D$) between the two channels. We focus on the states whose Bloch vectors lie along the red great circle in (a). Parameters if not specified: $\tau/T_1 = 1.0$, $T_2/T_1 = 1.5$, thermal temperature $T_b = 0$.
• Figure 2: (a) Sampling cost to offset estimator variance $\propto\Gamma^{2 n_c}$ if all gates of a code have a uniform time cost, i.e., $\Gamma$ is the same for every type of gate. Vertical lines denote approximately the current time cost of measurement and 2-qubit gates in relation to $T_1$ coherence on IBM devices. In reality, the time cost will fall in between measurement (M) and 2-qubit gates, depending on the relative amount of those gates for each code. Overhead assumes $d$ rounds of syndrome extraction with a distance $d$ code. (b) Approximate quasi-probability sampling cost per-memory experiment. As opposed to (a), $\Gamma$ here is calculated for each gate time, for the number of each gate type in each code, representing the actual simulation. Measurement is approximated to take $T_1/100$, while 2-qubit gates take $T_1/1000$. The uncombined quasi-probability variance overhead is independent of $T_2$ as the probability decomposition for $T_2$ is always positive.
• Figure 3: Sampling overhead for the quasi-probability relaxation channel on a single qubit, decomposed using Eq. \ref{['eq:th0_qpd']}. The overhead is measured by the negativity of the quasi-probabilistic distribution, which is captured by $\Gamma$. When $T_2/T_1 \leqslant 1$, the QP distribution is completely positive, while $T_2/T_1 = 2$, there is no additional dephasing, the overhead is the same as the QP distribution of an amplitude damping channel.
• Figure 4: We compare the channel fidelity of the reset error and the Pauli-twirled thermal channel to the exact thermal channel at zero temperature. We plot the difference of the channel fidelity $\Delta F = F_{\text{reset}} - F_{\text{pw}}$. In this regime, the reset channel can always give a better channel fidelity to the exact thermal relaxation error channel compared to the Pauli-twirled error channel.
• Figure 5: The channel fidelity gain of the reset-based Clifford channel to the PTA channel to approximate a finite temperature thermal relaxation channel. We set $T_2 = 1.5 T_1$. We sweep the excited state population up to $p_1 = 0.1$. To give a more intuitive understanding of experimental relevance, we compute the corresponding thermal temperature assuming the qubit frequency is $5$ GHz, which is labeled on the right.