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Bulletin of the American Physical Society

Numerical Analysis of Robust Phase Estimation

Kenneth Rudinger, Shelby Kimmel

Published Mar 15, 2016• 0 citations
OpenAlex

Abstract

SNL), Shelby Kimmel (QuICS) SAND2016-1639C Main question  You want to characterize single-qubit operations: G x (α), G y (ε), θ=arccos(XY)  Desire self-consistently, efficiently (Heisenberg-limited accuracy)). There are (at least!) two good options:  Robust Phase Estimation (RPE)  Gate Set Tomography (GST) When should you use RPE and when should you use GST?

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SNL), Shelby Kimmel (QuICS) SAND2016-1639C Main question  You want to characterize single-qubit operations: G x (α), G y (ε), θ=arccos(XY)  Desire self-consistently, efficiently (Heisenberg-limited accuracy)). There are (at least!) two good options:  Robust Phase Estimation (RPE)  Gate Set Tomography (GST) When should you use RPE and when should you use GST?
The robust phase estimation (RPE) protocol was designed to be an efficient and robust way to calibrate quantum operations. The robustness of RPE refers to its ability to estimate a single parameter, usually gate amplitude, even when other parameters are poorly calibrated or when the gate experiences significant errors. Here we demonstrate the robustness of RPE to errors that affect initialization, measurement, and gates. In each case, the error threshold at which RPE begins to fail matches quantitatively with theoretical bounds. We conclude that RPE is an effective and reliable tool for calibration of one-qubit rotations and that it is particularly useful for automated calibration routines and sensor tasks.
An important step in building a quantum computer is calibrating experimentally implemented quantum gates to produce operations that are close to ideal unitaries. The calibration step involves estimating the error in gates and then using controls to correct the implementation. Quantum process tomography is a standard technique for estimating these errors, but is both time consuming, (when one only wants to learn a few key parameters), and requires resources, like perfect state preparation and measurement, that might not be available. With the goal of efficiently estimating specific errors using minimal resources, we develop a parameter estimation technique, which can gauge two key parameters (amplitude and off-resonance errors) in a single-qubit gate with provable robustness and efficiency. In particular, our estimates achieve the optimal efficiency, Heisenberg scaling. Our main theorem making this possible is a robust version of the phase estimation procedure of Higgins et al. [B. L. Higgins, New J. Phys. 11, 073023 (2009)].
We demonstrate an experimental implementation of robust phase estimation (RPE) to learn the phase of a single-qubit rotation on a trapped Yb^{+} ion qubit. We show this phase can be estimated with an uncertainty below 4×10^{-4} rad using as few as 176 total experimental samples, and our estimates exhibit Heisenberg scaling. Unlike standard phase estimation protocols, RPE neither assumes perfect state preparation and measurement, nor requires access to ancillae. We crossvalidate the results of RPE with the more resource-intensive protocol of gate set tomography.
We introduce and demonstrate experimentally: (1) a framework called "gate set tomography" (GST) for self-consistently characterizing an entire set of quantum logic gates on a black-box quantum device; (2) an explicit closed-form protocol for linear-inversion gate set tomography (LGST), whose reliability is independent of pathologies such as local maxima of the likelihood; and (3) a simple protocol for objectively scoring the accuracy of a tomographic estimate without reference to target gates, based on how well it predicts a set of testing experiments. We use gate set tomography to characterize a set of Clifford-generating gates on a single trapped-ion qubit, and compare the performance of (i) standard process tomography; (ii) linear gate set tomography; and (iii) maximum likelihood gate set tomography.