Robust Extraction of Tomographic Information via Randomized Benchmarking

We describe how randomized benchmarking can be used to reconstruct the unital part of any trace-preserving quantum map, which in turn is sufficient for the full characterization of any unitary evolution or, more generally, any unital trace-preserving evolution. This approach inherits randomized benchmarking’s robustness to preparation, measurement, and gate imperfections, thereby avoiding systematic errors caused by these imperfections. We also extend these techniques to efficiently estimate the average fidelity of a quantum map to unitary maps outside of the Clifford group. The unitaries we consider correspond to large circuits commonly used as building blocks to achieve scalable, universal, and fault-tolerant quantum computation. Hence, we can efficiently verify all such subcomponents of a circuit-based universal quantum computer. In addition, we rigorously bound the time and sampling complexities of randomized benchmarking procedures, proving that the required nonlinear estimation problem can be solved efficiently.

Popular Summary

Quantum process tomography aims at reconstructing an unknown quantum dynamical process or operation by measuring its effect on known states of a quantum processing device. In practice, it often runs into a “chicken-or-the-egg” problem: To characterize the operation of interest, one must know the states of the device before the operation, but to characterize these states, one must be able to apply well-characterized operations (similar problems apply to the measurements used). Incomplete knowledge of the state preparations and measurements leads to systematic errors in quantum process tomography. Similarly, simultaneous characterization of state preparation, measurement, and the operation is a nonlinear estimation problem with local minima. In this theoretical paper, we present a new method, building on the existing concept of “randomized benchmarking,” that reliably reconstructs a quantum operation even in the absence of nearly any information about the initial state or the measurement observable.

Our approach relies on two insights about quantum operations. (i) Rather than characterizing an operation by carefully measuring how it transforms various input states, one can characterize an operation by comparing its effect to other known reference operations, even if these reference operations have small errors. This idea is at the heart of randomized benchmarking. While standard randomized benchmarking can only extract limited information about limited types of operations, we show that (ii) by comparing the operation of interest to a set of finite and easily implementable reference operations, it is possible to determine the vast majority of parameters describing any operation.

Looking ahead, we expect that extensions of this work will be useful not just for quantum process characterization but also for the characterization of state preparations and measurements in quantum computing experiments. These techniques may eventually provide necessary insights into errors that plague those experiments.

Figure 1

RB decays used to estimate the fidelity between an ideal Hadamard gate and ${\widehat{C}}_0$ (green circles, with $p=\frac{1}{3}$) and ${\widehat{C}}_1$ (orange squares, with $p=-\frac{1}{3}$). The decays corresponding to each of the remaining average fidelities coincide with one of these two representative decays. Note that these decays are much faster than decays previously estimated in RB, as they correspond to the average fidelities between very different maps. The data points are offset along the $x$ axis for clarity.

Figure 2

Bounds on ${\chi }_{0,0}^{\mathcal{A}}$ versus ${\chi }_{0,0}^{\mathcal{A}\circ \mathcal{B}}$, when ${\chi }_{0,0}^{\mathcal{B}}$ is fixed at 0.995. Our bounds are shown as a solid blue line, while the bounds of Ref. [12] are shown as a red dashed line. The bounds of Ref. [12] are valid in the green shaded region, while our bounds are valid for all values of ${\chi }_{0,0}^{\mathcal{A}\circ B}$.