Cross-entropy benchmarking

Cross-entropy benchmarking (also referred to as XEB) is a quantum benchmarking protocol which can be used to demonstrate quantum supremacy.^[1] In XEB, a random quantum circuit is executed on a quantum computer multiple times in order to collect a set of ${\displaystyle k}$ samples in the form of bitstrings ${\displaystyle \{x_{1},\dots ,x_{k}\}}$. The bitstrings are then used to calculate the cross-entropy benchmark fidelity (${\displaystyle F_{\rm {XEB}}}$) via a classical computer, given by

${\displaystyle F_{\rm {XEB}}=2^{n}\langle P(x_{i})\rangle _{k}-1={\frac {2^{n}}{k}}\left(\sum _{i=1}^{k}|\langle 0^{n}|C|x_{i}\rangle |^{2}\right)-1}$,

where ${\displaystyle n}$ is the number of qubits in the circuit and ${\displaystyle P(x_{i})}$ is the probability of a bitstring ${\displaystyle {x_{i}}}$ for an ideal quantum circuit ${\displaystyle C}$. If ${\displaystyle F_{XEB}=1}$, the samples were collected from a noiseless quantum computer. If ${\displaystyle F_{\rm {XEB}}=0}$, then the samples could have been obtained via random guessing.^[2] This means that if a quantum computer did generate those samples, then the quantum computer is too noisy and thus has no chance of performing beyond-classical computations. Since it takes an exponential amount of resources to classically simulate a quantum circuit, there comes a point when the biggest supercomputer that runs the best classical algorithm for simulating quantum circuits can't compute the XEB. Crossing this point is known as achieving quantum supremacy; and after entering the quantum supremacy regime, XEB can only be estimated.^[3]

The Sycamore processor was the first to demonstrate quantum supremacy via XEB. Instances of random circuits with ${\displaystyle n=53}$ and 20 cycles were run to obtain an XEB of ${\displaystyle 0.0024}$.^[3] Generating samples took 200 seconds on the quantum processor when it would have taken 10,000 years on Summit at the time of the experiment. Improvements in classical algorithms have shortened the runtime to about a week on Sunway TaihuLight thus collapsing Sycamore's claim to quantum supremacy.^[4] As of 2021, the latest demonstration of quantum supremacy by Zuchongzhi 2.1 with ${\displaystyle n=60}$, 24 cycles and an XEB of ${\displaystyle 0.000366}$ holds. It takes around 4 hours to generate samples on Zuchonzhi 2.1 when it would take 10,000 years on Sunway.^[4]