Modeling and mitigation of cross-talk effects in readout noise with applications to the Quantum Approximate Optimization Algorithm

Filip B. Maciejewski1, Flavio Baccari2, Zoltán Zimborás3,4,5, and Michał Oszmaniec1

1Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland
2Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
3Wigner Research Centre for Physics, H-1525 Budapest, P.O.Box 49, Hungary
4BME-MTA Lendület Quantum Information Theory Research Group, Budapest, Hungary
5Mathematical Institute, Budapest University of Technology and Economics, P.O.Box 91, H-1111, Budapest, Hungary

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Measurement noise is one of the main sources of errors in currently available quantum devices based on superconducting qubits. At the same time, the complexity of its characterization and mitigation often exhibits exponential scaling with the system size. In this work, we introduce a correlated measurement noise model that can be efficiently described and characterized, and which admits effective noise-mitigation on the level of marginal probability distributions. Noise mitigation can be performed up to some error for which we derive upper bounds. Characterization of the model is done efficiently using Diagonal Detector Overlapping Tomography – a generalization of the recently introduced Quantum Overlapping Tomography to the problem of reconstruction of readout noise with restricted locality. The procedure allows to characterize $k$-local measurement cross-talk on $N$-qubit device using $O(k2^klog(N))$ circuits containing random combinations of X and identity gates. We perform experiments on 15 (23) qubits using IBM's (Rigetti's) devices to test both the noise model and the error-mitigation scheme, and obtain an average reduction of errors by a factor $>22$ ($>5.5$) compared to no mitigation. Interestingly, we find that correlations in the measurement noise do not correspond to the physical layout of the device. Furthermore, we study numerically the effects of readout noise on the performance of the Quantum Approximate Optimization Algorithm (QAOA). We observe in simulations that for numerous objective Hamiltonians, including random MAX-2-SAT instances and the Sherrington-Kirkpatrick model, the noise-mitigation improves the quality of the optimization. Finally, we provide arguments why in the course of QAOA optimization the estimates of the local energy (or cost) terms often behave like uncorrelated variables, which greatly reduces sampling complexity of the energy estimation compared to the pessimistic error analysis. We also show that similar effects are expected for Haar-random quantum states and states generated by shallow-depth random circuits.

State of the art quantum devices are affected by a significant amount of noise, which often prohibits the implementation of useful quantum information processing tasks. It is, therefore, not surprising that recently the field of characterization and mitigation of various types of noise has been developing rapidly. In this work, we make a few contributions that focus specifically on the characterization and reduction of the noise affecting the quantum measurement process.

We propose a readout noise model that can capture multiqubit cross-talk while remaining efficiently describable under the assumption that the locality of correlations in noise is not too high. Inspired by recent advances of the so-called Quantum Overlapping Tomography, we show how to characterize k-local correlations in measurement noise using random combinations of elementary quantum gates (identity gate and NOT gate), and we prove that the number of circuits required to do so scales only logarithmically with the number of qubits.

The characterization of the noise can be used to perform error mitigation on marginal probability distributions in a manner that does not exhibit exponential sampling complexity – exactly because it is performed on the level of marginals (contrary to standard methods that operate on global probability distributions). While this correction is imperfect, we provide upper bounds on the errors that can be calculated directly from the noise-characterization results. We demonstrate the effectiveness of noise mitigation in experiments on 15- and 23-qubit systems and conclude great improvements.

While mitigating noise on marginals does not allow correcting results of arbitrary experiments, currently the most promising quantum algorithms belong to the class of hybrid quantum-classical variational algorithms, where one usually restricts the locality of the quantities that are to be estimated. Here we investigate an example of such algorithms, i.e., Quantum Approximate Optimization Algorithm (QAOA). We numerically demonstrate that the correlated readout noise affects both the convergence and the final estimation of QAOA, and find that error mitigation helps to reduce such effects.

Finally, we study the sample complexity of estimation of local observables, from the point of view of quantum correlations. Based on recent results on correlations spreading in QAOA circuits, we show that for shallow-depth QAOA, one should expect that the estimators of expected values of local observables will not be correlated. Consequently, such estimators effectively behave as uncorrelated variables, highly reducing the sampling complexity of such estimation. We show that similar effects should be expected also from random quantum states.

Our work is accompanied by an open-source GitHub repository QREM – Quantum Readout Error Mitigation (, where we develop a Python code that allows performing efficient measurement noise characterization and mitigation.

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[3] Kerstin Beer, Daniel List, Gabriel Müller, Tobias J. Osborne, and Christian Struckmann, "Training Quantum Neural Networks on NISQ Devices", arXiv:2104.06081.

[4] Benjamin Nachman and Michael R. Geller, "Categorizing Readout Error Correlations on Near Term Quantum Computers", arXiv:2104.04607.

[5] Tanmay Singal, Filip B. Maciejewski, and Michał Oszmaniec, "Implementation of quantum measurements using classical resources and only a single ancillary qubit", arXiv:2104.05612.

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