PAEMS: Precise and Adaptive Error Model for Superconducting Quantum Processors

Abstract

Superconducting quantum processor units (QPUs) are incapable of producing massive datasets for quantum error correction (QEC) because of hardware limitations. Thus, QEC decoders heavily depend on synthetic data from qubit error models. Classic depolarizing error models with polynomial complexity present limited accuracy. Coherent density matrix methods suffer from exponential complexity $\propto O(4^n)$ where $n$ represents the number of qubits. This paper introduces PAEMS: a precise and adaptive qubit error model. Its qubit-wise separation framework, incorporating leakage propagation, captures error evolvements crossing spatial and temporal domains. Utilizing repetition-code experiment datasets, PAEMS effectively identifies the intrinsic qubit errors through an end-to-end optimization pipeline. Experiments on IBM's QPUs have demonstrated a 19.5$\times$, 9.3$\times$, and 5.2$\times$ reduction in timelike, spacelike, and spacetime error correlation, respectively, surpassing all of the previous works. It also outperforms the accuracy of Google's SI1000 error model by 58$\sim$73\% on multiple quantum platforms, including IBM's Brisbane, Sherbrooke, and Torino, as well as China Mobile's Wuyue and QuantumCTek's Tianyan.

Figures (5)

• Figure 1: Errors of a superconducting qubit array. a Two coupled transmon qubits. Each qubit employs XY RF pulses and Z flux pulses for the single-qubit gates ($e.g.$ Hadamard gate). A tunable coupler enables the two-qubit gates ($e.g.$ CNOT, CZ). The RF dispersive reflectometer measures the qubit state. b Error mechanisms of a single qubit: besides the excitation ($\Gamma_{0\rightarrow1}$), relaxation ($\Gamma_{1\rightarrow0}$), and dephasing ($\Gamma_{\phi}$), leakage beyond the computational subspace is critical. These errors come from unwanted transition and phase shift in a multi-level system. c A surface code topology with data and ancilla qubits. Z-stabilizers (yellow) and X-stabilizers (green) are implemented through coupled qubit pairs of a. d Error syndromes reflect the origin of qubit errors: Z-stabilizers detect Pauli-X errors, while X-stabilizers detect Pauli-Z errors. Different errors and locations lead to varying time, space, and spacetime error syndromes and the corresponding error pairs. In particular, a leakage event can persist across multiple rounds and propagate through entangling gates, generating multiple correlated error pairs spanning over space and time.
• Figure 2: Repetition code experiment with 21 qubits and 30 detection rounds. a$\sim$c Correlation matrices of experiment, SI1000 model with $p$=0.02 and PAEMS model. Different element sets of the symmetric correlation matrix correspond to different errors, where $p_{ij}$ denotes the two-point correlations between detection events. d Correlation strength differences between the experiment and the qubit error models. PAEMS, Circuit, Code-capacity (CC), Phenomenological (Phe), SD6, and SI1000 models are simulated in time, space, and spacetime differences. e Detection event fraction indicating the event intensity over 30 successive rounds of repetition code. The experiment results, PAEMS, Circuit ($p$=0.025), CC ($p$=0.15), Phe ($p$=0.075), SD6 ($p$=0.02), and SI1000 ($p$=0.015) models are plotted.
• Figure 3: Cross-platform adaptivity with single-round repetition code. a X basis and b Z basis repetition-codes with $N$ = 5, 9, 13, 17, 21 qubits. 40960 shots are measured for each case. c$\sim$e Measured probability distributions of X basis repetition code with $N=13$ on Brisbane, Wuyue, and Tianyan, respectively, compared with PAEMS model. f$\sim$h Total variation distance (TVD) between experiment and model prediction. PAEMS and SI1000 models are compared in both X and Z basis. SI1000 model is simulated with the optimal $p$.
• Figure 4: Framework of PAEMS qubit error model. a Circuit-level representation of error channels, which is parameterized per qubit and explicitly resolved in the time domain. Decoherence errors are modeled by asymmetric depolarizing channels (ADC). Gate errors are modeled by symmetric depolarizing channels (SDC). State preparation and measurement (SPAM) errors are modeled by stochastic Pauli-$X$ channels. Leakage and seepage processes evolve in parallel with Pauli error channels, and their error propagation is simulated by randomizing the outcomes of two-qubit gate operations and measurements to $\lvert 0\rangle$ or $\lvert 1\rangle$ with equal probability. b Parameter mapping of ADC channels, where decoherence errors of qubit $n$ are initialized by the relaxation time $T_1$, dephasing time $T_2$, and gate or idle duration time $t$ from qubit calibration. It leads to stochastic Pauli error channels with realistic non-uniformity. c Parameter mapping of SDC channels for single-qubit and two-qubit gates, parameterized by gate fidelities $f(F)_n$ and $f(F)_c$, respectively. It results in stochastic Pauli error channels with equal probabilities. $n$ is the qubit index, while $c$ labels the coupler index.
• Figure 5: Training of PAEMS model. a End-to-end workflow of model training. The experiment inputs (gray boxes) include the qubit calibration and the sampled repetition-code datasets. The training pipeline (red boxes) starts with the parameter mapping, repetition-code simulation, and error-syndrome generation. Model optimization is performed in three stages: coarse optimization of leakage parameters, parallel fine-grained optimization using time, space, and spacetime loss terms, and final global fine-tuning. All stages employ the CMA-ES algorithm. b The 21-qubit repetition code topology for dataset generation, with platform-calibrated parameter collection performed on each physical qubit and coupler. c Quantum circuit of multi-round 21-qubit repetition code, including 30 detection rounds with 4096 shots per run, repeated 25 times. Two different qubit sequences are tested on each quantum chip. d,e Comparison between platform-calibrated and optimized decoherence times for the Torino processor, including the relaxation time $T_1$ and dephasing time $T_2$ of the 21 tested qubits. f Comparison between platform-calibrated and optimized SPAM error rates of the ancilla qubits, and the error rate of couplers, from the Torino processor.