# Error mitigation extends the computational reach of a noisy quantum processor

## Abstract

Quantum computation, a paradigm of computing that is completely different from classical methods, benefits from theoretically proved speed-ups for certain problems and can be used to study the properties of quantum systems1. Yet, because of the inherently fragile nature of the physical computing elements (qubits), achieving quantum advantages over classical computation requires extremely low error rates for qubit operations, as well as substantial physical qubits, to realize fault tolerance via quantum error correction2,3. However, recent theoretical work4,5 has shown that the accuracy of computation (based on expectation values of quantum observables) can be enhanced through an extrapolation of results from a collection of experiments of varying noise. Here we demonstrate this error mitigation protocol on a superconducting quantum processor, enhancing its computational capability, with no additional hardware modifications. We apply the protocol to mitigate errors in canonical single- and two-qubit experiments and then extend its application to the variational optimization6,7,8 of Hamiltonians for quantum chemistry and magnetism9. We effectively demonstrate that the suppression of incoherent errors helps to achieve an otherwise inaccessible level of accuracy in the variational solutions using our noisy processor. These results demonstrate that error mitigation techniques will enable substantial improvements in the capabilities of near-term quantum computing hardware.

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## Data availability

The data that support the findings of this study are available from the corresponding author on reasonable request.

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## Acknowledgements

We thank J. Rozen, M. Takita, and O. Jinka for experimental contributions, B. Abdo for design and characterization of the Josephson parametric converters, and M. Brink for device fabrication. We acknowledge discussions with S. Brayvi, E. Magesan, S. Sheldon, and M. Takita. We acknowledge support from the IBM Research Frontiers Institute. The research is based on work supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via the Army Research Office contract W911NF-10-1-0324.

## Author information

Authors

### Contributions

A.K., K.T. and J.M.G. developed the experimental protocol, planned the experiments and analysed the experimental data. A.K. performed the experiments. A.D.C. contributed to the experimental set-up. A.M. contributed to the data analysis. A.K., K.T., J.M.C., and J.M.G. wrote the manuscript, with contributions from all authors.

### Corresponding author

Correspondence to Abhinav Kandala.

## Ethics declarations

### Competing interests

The authors declare that elements of this work are included in a patent filed by the International Business Machines Corporation with the US Patent and Trademark office.

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Gate characterization.

a, b, Single-qubit (1Q, a) and two-qubit (2Q, b) error per gate (EPG), for a range of stretch factors c, obtained by randomized benchmarking (RB). The error bars represent the standard deviation of the exponential fits to the  RB decay, for each stretch factor. The considered 1Q gates in a are for qubits Q0 (red), Q1 (yellow), Q2 (green), Q3 (blue) and the 2Q gates in b are CR0-1 (red), CR1-2 (green), CR3-2 (yellow).

### Extended Data Fig. 2 Coherence and relaxation time fluctuations in superconducting qubits.

a, Repeated measurements of T1 and echo T2 show fluctuations that are larger than the error bars to the fits. The error bars represent the standard deviation of the exponential fits to each instance of the T1 and echo T2 decay curves. b, c, Decay of excited-state projector $$\langle {\Pi }_{1}\rangle$$ for a standard T1 sequence (b) and T2 echo sequence (c), for stretch factors c = 1, 2 obtained using 105 samples. In both cases, the average decay over the total sampling duration for the different stretch factors is the same only when their measurements are grouped together (run 1), in contrast to when sampled separately (run 2 and run 3).

### Extended Data Fig. 3 Hamiltonian tomography of the CR drive.

Amplitude dependence of the interaction terms in the cross-resonance drive CR3−2. The error bars represent the standard deviation of the interaction strengths, estimated from fits to Hamiltonian tomography20 for each CR drive amplitude. The dominant interactions in the drive are ZX and IX, with discernible nonlinearities in the strong drive limit. (a.u., arbitrary units.).

### Extended Data Fig. 4 Error mitigation with a nonlinear CR drive.

Time evolution of $$\left\langle IZ\right\rangle$$ under a simplistic CR drive JZX = −0.0159 + 1.0541 × 10−63, and fixed amplitude damping and dephasing channels, for drive amplitudes calibrated to ZXπ/2 gate times Tgate values 2J−1 (a), 3J−1 (b) and 6J−1 (c) and with stretch factors c = 1 (blue) and c = 2 (green). The mitigated time evolution (red) goes out of bounds for fast gates (large drive amplitudes), owing to the nonlinearity in the amplitude dependence.

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Kandala, A., Temme, K., Córcoles, A.D. et al. Error mitigation extends the computational reach of a noisy quantum processor. Nature 567, 491–495 (2019). https://doi.org/10.1038/s41586-019-1040-7

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