Cross-Entropy Benchmarking

Cross-entropy benchmarking (XEB) scores a whole device by how faithfully it can sample from a random quantum circuit. It became famous as the protocol behind the 2019 quantum-advantage experiment, because it gives a single fidelity number for circuits far too large for tomography or RB to touch.

Random circuits and the Porter–Thomas distribution

Run a random circuit of sufficient depth on $n$ qubits and read out a bitstring. Ideally the output-probability spectrum follows the Porter–Thomas distribution: the probabilities $p(x) = |\langle x|\psi\rangle|^2$ are exponentially distributed, so a few bitstrings are much more likely than the uniform value $2^{-n}$ and most are much less likely. The ideal circuit thus has "speckle" — sharp, structured peaks in an otherwise flat-looking landscape.

A noisy device washes out this speckle. In the fully-decohered limit it samples uniformly, with every bitstring equally likely. XEB measures how much speckle survives, and reads that as a fidelity.

The linear cross-entropy fidelity

The standard estimator is the linear cross-entropy benchmarking fidelity. Let the device produce samples $x_1, \dots, x_S$ , and let $p_{\text{ideal}}(x)$ be the probability the ideal circuit assigns to $x$ (computed classically by simulation). Then

F_{\text{XEB}} = 2^{\,n}\,\big\langle p_{\text{ideal}}(x) \big\rangle_{\text{samples}} - 1,

the average ideal probability of the observed bitstrings, rescaled. The intuition:

A perfect device preferentially returns the high-probability bitstrings, so the average ideal probability of its samples is large, and $F_{\text{XEB}} \to 1$ .
A fully noisy device samples uniformly; the average ideal probability is just $2^{-n}$ , so the rescaling gives $F_{\text{XEB}} \to 0$ .

The $2^n$ factor and the $-1$ are exactly the normalization that maps "samples uniform" to $0$ and "samples ideal" to $1$ , using the known statistics of the Porter–Thomas distribution.

XEB versus RB

XEB and randomized benchmarking are complementary:

RB uses Clifford circuits that return to a known state; it is SPAM-robust and isolates a gate error rate, but Clifford circuits are classically easy and so cannot probe the hard regime.
XEB uses generic random circuits that explore the Porter–Thomas regime; it captures the holistic fidelity of running a deep, entangling circuit — including crosstalk and correlated errors that per-gate RB can miss — at the price of an expensive classical reference.

In the Google experiment the per-cycle and per-gate fidelities measured by XEB on small circuits were multiplied together to predict the fidelity of the full, classically-intractable circuit; the agreement between prediction and the directly-measured small-circuit fidelity was the evidence that the big circuit was working as intended.

The big picture

XEB shifts benchmarking from "how good is one gate?" to "how well does this device execute a deep random computation as a whole?" That whole-device perspective is shared by the next lesson's metric, quantum volume, which folds width and depth together into a single benchmark number.