A volumetric framework for quantum computer benchmarks

We propose a very large family of benchmarks for probing the performance of quantum computers. We call them \emph{volumetric benchmarks} (VBs) because they generalize IBM's benchmark for measuring quantum volume \cite{Cross18}. The quantum volume benchmark defines a family of \emph{square} circuits whose depth $d$ and width $w$ are the same. A volumetric benchmark defines a family of \emph{rectangular} quantum circuits, for which $d$ and $w$ are uncoupled to allow the study of time/space performance tradeoffs. Each VB defines a mapping from circuit shapes -- $(w,d)$ pairs -- to test suites $\mathcal{C}(w,d)$. A test suite is an ensemble of test circuits that share a common structure. The test suite $\mathcal{C}$ for a given circuit shape may be a single circuit $C$, a specific list of circuits $\{C_1\ldots C_N\}$ that must all be run, or a large set of possible circuits equipped with a distribution $Pr(C)$. The circuits in a given VB share a structure, which is limited only by designers' creativity. We list some known benchmarks, and other circuit families, that fit into the VB framework: several families of random circuits, periodic circuits, and algorithm-inspired circuits. The last ingredient defining a benchmark is a success criterion that defines when a processor is judged to have "passed" a given test circuit. We discuss several options. Benchmark data can be analyzed in many ways to extract many properties, but we propose a simple, universal graphical summary of results that illustrates the Pareto frontier of the $d$ vs $w$ tradeoff for the processor being benchmarked. [1] A. Cross, L. Bishop, S. Sheldon, P. Nation, and J. Gambetta, arXiv:1811.12926 (2018)