Machine Learning for Practical Quantum Error Mitigation

TL;DR

Quantum errors constrain near-term quantum computing, motivating quantum error mitigation (QEM). This work introduces ML-QEM, learning to predict noise-free expectation values from noisy data using four models (OLS, RF, MLP, GNN) and demonstrates that RF often matches or surpasses digital ZNE while substantially reducing runtime overhead. The approach scales to hardware with up to 100 qubits, enables mitigation of unseen Pauli observables, and improves variational algorithms like VQE; it also shows that ML-QEM can mimic existing QEM methods to further reduce overhead. Collectively, the results suggest ML-QEM as a practical pathway to extend quantum utility on noisy devices, with a roadmap for enhanced encoding, drift adaptation, and cross-method benchmarking.

Abstract

Quantum computers progress toward outperforming classical supercomputers, but quantum errors remain their primary obstacle. The key to overcoming errors on near-term devices has emerged through the field of quantum error mitigation, enabling improved accuracy at the cost of additional run time. Here, through experiments on state-of-the-art quantum computers using up to 100 qubits, we demonstrate that without sacrificing accuracy machine learning for quantum error mitigation (ML-QEM) drastically reduces the cost of mitigation. We benchmark ML-QEM using a variety of machine learning models -- linear regression, random forests, multi-layer perceptrons, and graph neural networks -- on diverse classes of quantum circuits, over increasingly complex device-noise profiles, under interpolation and extrapolation, and in both numerics and experiments. These tests employ the popular digital zero-noise extrapolation method as an added reference. Finally, we propose a path toward scalable mitigation by using ML-QEM to mimic traditional mitigation methods with superior runtime efficiency. Our results show that classical machine learning can extend the reach and practicality of quantum error mitigation by reducing its overheads and highlight its broader potential for practical quantum computations.

Figures (11)

• Figure 1: Execution and training for tractable and intractable circuits with ML-QEM. A quantum circuit (left) is passed to an encoder (top) that creates a feature set for the ML model (right) based on the circuit and the quantum processor unit (QPU) targeted for execution. The model and features are readily replaceable. The executed noisy expectation values $\langle \hat{O} \rangle^\mathrm{noisy}$ (middle) serve as the input to the model whose aim is to predict their noise-free value $\langle \hat{O} \rangle^\mathrm{mit}$. To achieve this, the model is trained beforehand (bottom, blue highlighted path) against target values $\langle \hat{O} \rangle^\mathrm{target}$ of example circuits. These are obtained either using noiseless simulations in the case of small-scale, tractable circuits or using the noisy QPU in conjunction with a conventional error mitigation strategy in the case of large-scale, intractable circuits. The training minimizes the loss function, typically the mean square error. The trained model operates without the need for additional mitigation circuits, thus reducing runtime overheads.
• Figure 2: Accuracy of QEM and ML-QEM in random circuits. Top: Error distribution for unmitigated and mitigated Pauli-$Z$ expectation values. Mitigation is performed using either a reference QEM method, digital zero-noise extrapolation (ZNE), or one of four ML-QEM models (explained in text). Inset: Example random circuits. Noisy execution is numerically simulated using a noise model derived from IBM QPU Lima, FakeLima. The error is defined as the $L_2$ distance between the vector of all ideal and noisy single-qubit expectations $\langle \hat{Z}_i \rangle$; i.e., $\Vert \langle \hat{Z} \rangle - \langle \hat{Z} \rangle_{\mathrm{ideal}} \Vert_2$. Distribution is over $n=2{,}000$ four-qubit random circuits, with two-qubit-gate depths sampled up to 18 with a step size of 2. The horizontal lower boundary, center line, and upper boundary of a colored box indicate the first, second, and third quantile, respectively. The horizontal lines outside indicate the $1.5\times$(interquartile range) from the nearest hinge. Black dots are outliers. Bottom: Mean $L_2$ error over the $n=2{,}000$ circuits for each method (using data from the top). The value is provided above each column. The errorbar represents the $95\%$ confidence intervals derived from $1{,}000$ bootstrap re-sampling.
• Figure 3: Mitigation accuracy under i) complexity of quantum noise and ii) ML-QEM interpolation and extrapolation for Trotter circuits. Top row: Average error performance on Trotter circuits (top-left inset) representing the time evolution dynamics of a four-site, 1D, transverse-field Ising model in numerical simulations. A Trotter step comprises four layers of CNOT gates (inset). Vertical dashed line separates experiments in the ML-QEM interpolation regime (left) from the extrapolation regime (right). The 3 curves represent the performance of the highest-performing ML-QEM method, the ZNE method, and the unmitigated experiments. They are averaged over 300 circuits, each with a randomly chosen Pauli measurement bases. The data is for all four weight-one expectations $\langle \hat{P}_i \rangle$. The error is defined as $L_2$ distance from the ideal expectations, $\Vert \langle \hat{P} \rangle - \langle \hat{P} \rangle_{\mathrm{ideal}} \Vert_2$, as also defined for the remainder of figures. From the left to right, the complexity of the device noise model increases to include additional realistic noise types. Coherent errors are introduced on CNOT gates. Bottom row: Corresponding typical data of the error-mitigated expectation values of the $\langle \hat{Z}_0 \rangle$ Trotter evolution; here, for Ising parameter ratio $J/h=0.15$. Each error bar shows $\pm$standard error.
• Figure 4: Accuracy and overhead on QPU hardware for ML-QEM and QEM Average execution error of Trotter circuits for experiments on QPU device ibm_algiers without mitigation and with ZNE or ML-QEM (RF) mitigation. Error performance is averaged over $250$ Ising circuits per Trotter step, each with sampled Ising parameters $J<h$ and each measured for all weight-one observables in a randomly chosen Pauli basis. Training is performed over $50$ circuits per Trotter step, which results in both a $40\%$ lower overall and $50\%$ lower runtime quantum resource overhead of RF compared to the overhead of the digital ZNE (see inset). Shaded regions represent $\pm$standard error.
• Figure 5: Application of ML-QEM to a) unseen expectation values and b) the variational quantum eigensolver (VQE). a) Top: Schematic of a Trotter circuit, which prepares a many-body quantum state on $n=6$ qubits (in 5 Trotter steps). Top right: Circle depicts the pool of all possible $4^n$ Pauli observables. Shadings depicts the fraction of observables used in training the ML model; the remaining observables are unseen prior to deployment in mitigation. Bottom: Average error of mitigated unseen Pauli observables versus the total number of distinct observables seen in training. Shaded regions represent the standard error. b) Top: Schematic of the VQE ansatz circuit for 2 qubits parametrized by 8 angles $\vec{\theta}$. Below, a depiction of the VQE optimization workflow optimizing the set of angles $\vec{\theta}$ on a simulated QPU, yielding the noisy chemical energy $\langle \hat{H}\rangle_{\vec{\theta}}^\mathrm{noisy}$, which is first mitigated by the ML-QEM or QEM before being used in the optimizer as $\langle \hat{H}\rangle_{\vec{\theta}}^\mathrm{mit}$. Compared to the ZNE method, the ML-QEM with RF method obviates the need for additional mitigation circuits at every optimization iteration at runtime. Bottom: Ground energy estimate from VQE of the $\text{H}_2$ molecule as a function of the molecular bond lengths. The absolute error between the ZNE-mitigated and RF-mitigated estimates was calculated.