Error Bounds for Expectation Values

A. Weinstein

Michael Fielding Barnsley, Peter D. Robinson

Touheed Anwar Atif, Reuben Tate, Stephan Eidenbenz

Precise quantum expectation values are crucial for quantum algorithm development, but noise in real-world systems can degrade these estimations. While quantum error correction is resource-intensive, error mitigation strategies offer a practical alternative. This paper presents a framework that combines Virtual Channel Purification (VCP) technique with Conditional Value-at-Risk (CVaR) optimization to improve expectation value estimations in noisy quantum circuits. Our contributions are twofold: first, we derive conditions to compare CVaR values from different probability distributions, offering insights into the reliability of quantum estimations under noise. Second, we apply this framework to VCP, providing analytical bounds that establish its effectiveness in improving expectation values, both when the overhead VCP circuit is ideal (error-free) and when it adds additional noise. By introducing CVaR into the analysis of VCP, we offer a general noise-characterization method that guarantees improved expectation values for any quantum observable. We demonstrate the practical utility of our approach with numerical examples, highlighting how our bounds guide VCP implementation in noisy quantum systems.

T. S. Shankara

E. van den Berg, Zlatko Minev, Kristan Temme

Measurements on current quantum processors are subject to hardware imperfections that lead to readout errors. These errors manifest themselves as a bias in quantum expectation values. Here, we consider a very simple method that forces the bias in the expectation value to appear as a multiplicative factor that can be measured directly and removed at the cost of an increase in the sampling complexity for the observable. This method was previously discussed by Karalekas et al. [Quantum Sci. Technol. 5, 024003 (2020)] for the product of single-qubit readout-error models. Here, we formally analyze the method and show that it can be applied to general noise models. In particular, we show the method does not need to assume any specific form of the noise model and requires only that the noise is ``weak'' to avoid excessive sampling overhead. We prove bounds relating the error in the expectation value to the sample complexity.

Andrei Olegovich Barvinsky, G. A. Vilkovisky