### Error mitigation for near-term quantum computing

Digital quantum computing promises to eventually outperform the best classical algorithms for tasks like simulation of dynamics of many-body quantum systems or prime factoring. At the same time, the potential of current quantum devices is severely limited by decoherence and imperfect control of quantum gates called noise. Therefore, methods reversing the noise effects are crucial for obtaining a quantum advantage. Quantum computers with low enough qubit error rates and high enough qubit counts enable quantum error correction capable of on-fly detection and correction of errors that make possible error-free quantum computation in a quantum advantage regime. Nevertheless, to make that possible, one needs to increase the quantum computers' qubit counts by orders of magnitude and decrease their error rates. As such improvements in the quantum hardware are not expected in the near future, alternative resource-efficient techniques to decrease the noise impact are necessary to tackle challenging computational tasks. Such techniques are called quantum error mitigation.

Multiple approaches to quantum error mitigation have been proposed. Among the most promising ones is data-driven quantum error mitigation that partially removes the effects of the noise by classical post-processing of noisy quantum computation outcomes (see P. Czarnik et al., Quantum 5, 592). Our group develops such cutting-edge error mitigation methods, increasing their power (D. Bultrini et al., Quantum 7, 1034), improving their robustness (P. Czarnik et al. arXiv:2307.05302), and enhancing their resource efficiency (P. Czarnik et al., arXiv:2204.07109). This research is conducted in collaboration with Los Alamos National Laboratory.

Fig. 1. An example of data-driven error mitigation is a learning-based approach introduced by a group member in collaboration with Los Alamos National Laboratory. It uses classically simulable circuits like near-Clifford circuits similar to the circuit of interest to learn a model enabling correction of the noise effects on an observable of interest. This model is then applied to correct the observable of interest for the circuit of interest that is not simulable classically.

The figure is taken from P. Czarnik, A. Arrasmith, P. J. Coles, and L. Cincio, "Error mitigation with Clifford quantum-circuit data", Quantum 5, 592 (2021).

Fig. 2. The learning-based quantum error mitigation is applied to a 16-qubit simulation of the ground state of a transverse Quantum Ising model with IBM's Almaden quantum computer. The left plot shows the energies of four variational ground state search runs. The red curves are the results of the noisy computation, while the blue ones are obtained by error mitigating those results. The error mitigation decreases the error by an order of magnitude, as shown in the right plot.

The figure is taken from P. Czarnik, A. Arrasmith, P. J. Coles, and L. Cincio, "Error mitigation with Clifford quantum-circuit data", Quantum 5, 592 (2021).

Another promising approach to mitigating the effects of noise is exploiting recent advances in quantum error correction. While the power of current quantum error correction implementations is insufficient for quantum advantage with the error-corrected qubits only, their combination with standard noisy qubits enables reaching the qubit numbers large enough for classically intractable problems. At the same time, a quantum computer with error-corrected qubits makes possible partial suppression of the errors, improving upon an all-the-noisy device. The group, in collaboration with Los Alamos National Laboratory, develops quantum computing primitives and algorithms tailored for such an architecture (see D. Bultrini et al., Quantum 7, 1060, and N. Koukoulekidis et al., arXiv:2306.15531).

Fig. 3. The diagram shows an example of a logical CNOT gate between noisy and error-corrected (clean) logical qubits. The gate is implemented with native quantum computers' CNOT gates. Here, we illustrate the concept using an example of quantum error correction realized with a three-qubit repetition code.

The figure is taken from N. Koukoulekidis, S. Wang, T. O'Leary, D. Bultrini, L. Cincio, and P. Czarnik, "A framework of partial error correction for intermediate-scale quantum computers", arXiv:2306.15531, (2023).