Quantum Computing RF

Cross-Entropy Benchmarking

Q: How does linear XEB fidelity differ from randomized benchmarking?

Randomized benchmarking (RB) applies sequences of Clifford gates that compose to the identity, then fits the survival probability decay to extract an average error per Clifford. RB is largely insensitive to state-preparation and measurement (SPAM) error and works best for one or two qubits because the Clifford group grows rapidly. Cross-entropy benchmarking instead runs deeply random, non-Clifford circuits that scramble the state toward a Porter-Thomas output distribution, then scores the measured bitstrings against simulated ideal probabilities. Linear XEB fidelity is the slope of measured probability versus ideal probability and decays exponentially with circuit depth and qubit count, F = (e1)^(g1) for g gates of error e. XEB scales to dozens of qubits, captures correlated and crosstalk errors that RB can miss, and is the protocol Google used to characterize the 53-qubit Sycamore processor, but it requires classical simulation of the ideal circuit, which becomes intractable past roughly 50 qubits.

Q: What microwave control specifications limit measured XEB fidelity?

XEB fidelity is set by the accumulated single-qubit and two-qubit gate error, which is dominated by the quality of the microwave control chain. Single-qubit XZ gates on a transmon are driven by shaped microwave pulses near 4 to 6 GHz delivered from an arbitrary waveform generator and IQ mixer; pulse amplitude and phase errors below 0.1 percent and timing jitter under 10 ps are needed to reach single-qubit errors near 0.1 percent. Two-qubit gates are limited by flux pulse settling and control-line crosstalk. Local oscillator phase noise integrated over the gate window must stay below about minus 100 dBc per Hz at 1 MHz offset, and DAC vertical resolution of 14 bits or more keeps quantization error from degrading pulse fidelity. Sycamore reported per-cycle XEB errors near 0.3 percent for single-qubit and 0.6 percent for two-qubit gates, a budget only reachable with carefully calibrated cryogenic RF wiring and attenuation.

Q: Why does the XEB score follow a Porter-Thomas distribution?

A sufficiently deep random circuit produces an output state whose probabilities behave like a random pure state in the 2^n dimensional Hilbert space. The probabilities of the 2^n bitstrings then follow the Porter-Thomas distribution, an exponential distribution with mean 1 over 2^n. Sampling from this distribution preferentially returns high-probability bitstrings, so a faithful quantum processor produces a positive cross-entropy signal. A noisy processor that outputs uniformly random bitstrings yields a linear XEB fidelity of zero. The expected linear XEB of an ideal sampler equals 1, and the ratio of the measured value to 1 gives the circuit fidelity, which is why XEB is also called the fidelity estimator for random circuit sampling.

/kraws EN-truh-pee BENCH-mar-king/ (XEB)

Used to certify the fidelity of a quantum processor, this protocol runs an ensemble of deeply random gate circuits, measures many output bitstrings, and scores them against the ideal probabilities computed by classical simulation. The resulting linear XEB fidelity decays exponentially with circuit depth and qubit count, so it directly exposes accumulated single-qubit and two-qubit gate error. Because every gate on a superconducting transmon qubit is a shaped microwave pulse, the measured score is ultimately limited by the control chain: arbitrary waveform generators, IQ mixers, local-oscillator phase noise, and cryogenic readout. XEB scaled random circuit sampling to 53 qubits in Google's 2019 Sycamore demonstration, where ideal classical simulation begins to break down past roughly 50 qubits.

Category: Quantum Computing RF

2-qubit per-cycle error: ≈ 0.6%

Ideal linear XEB: F_XEB = 1

How XEB Estimates Quantum Gate Fidelity

Cross-entropy benchmarking grew out of the random circuit sampling problem, the task chosen to demonstrate that a quantum processor can perform a computation no classical supercomputer can match in reasonable time. The protocol selects a circuit at random from a fixed gate set (typically single-qubit rotations drawn from {√X, √Y, √W} interleaved with a fixed entangling two-qubit gate), executes it tens of thousands of times, and records the resulting bitstrings. Because a deep random circuit drives the output state toward a Haar-random pure state, the bitstring probabilities follow the Porter-Thomas distribution, an exponential law with mean 1/2ⁿ across the 2ⁿ possible outcomes. A working processor preferentially samples the high-probability strings; pure noise samples uniformly and scores zero.

The linear cross-entropy fidelity F_XEB is computed by averaging the simulated ideal probability of each measured bitstring, scaling by 2ⁿ, and subtracting one. An ideal noiseless sampler returns F_XEB = 1, while a fully depolarized output returns 0. Critically, the fidelity multiplies across the circuit, so for a circuit containing g gates each with average error e, the expected score is approximately (1 e)^g. This exponential decay is what lets an experimenter extract a single per-gate or per-cycle error figure by fitting F_XEB against circuit depth, exactly as one fits an exponential to extract a decay constant.

What makes XEB an RF problem rather than a purely algorithmic one is that every operation in the circuit is a physical microwave event. A single-qubit gate is a 15 to 25 ns shaped pulse near the 4 to 6 GHz transmon transition, synthesized by an arbitrary waveform generator and upconverted through an IQ mixer. Errors in pulse amplitude, phase, or timing, along with control-line crosstalk and local-oscillator phase noise, accumulate directly into the XEB error budget. Reaching the sub-percent per-cycle errors reported for state-of-the-art devices demands a control and readout chain calibrated to the same tolerances RF Essentials engineers apply to precision millimeter-wave subsystems.

Linear XEB Fidelity and Per-Cycle Error

Porter-Thomas output distribution:
P(p) = N × e^−Np, N = 2ⁿ, mean probability = 1/2ⁿ

Linear cross-entropy fidelity:
F_XEB = 2ⁿ × ⟨ P_ideal(x_i) ⟩_measured − 1

Fidelity vs. circuit depth (g gates, error e per gate):
F_XEB ≈ (1 − e)^g ≈ e^−g×e

Where n = number of qubits, P_ideal(x_i) = simulated probability of measured bitstring x_i, e = average per-gate error, g = gate count. Ideal sampler: F_XEB = 1. Uniform noise: F_XEB = 0. Example: 53 qubits, 20 cycles, per-cycle error ≈ 0.6% → whole-circuit F_XEB ≈ 0.2%.

XEB Versus Other Benchmarking Protocols

Protocol	Circuit type	Qubit scaling	SPAM sensitivity	Captures crosstalk	Best use
Cross-entropy (XEB)	Deep random, non-Clifford	10s of qubits	Moderate	Yes	System-level fidelity, supremacy
Randomized benchmarking	Random Clifford, identity	1 to 2 qubits	Low	No	Average gate error
Gate set tomography	Designed informationally complete	1 to 2 qubits	Low	Partial	Full gate characterization
Process tomography	Full input/output basis	1 to 2 qubits	High	Yes	Detailed single-gate maps
Mirror benchmarking	Random then inverse	Many qubits	Low	Yes	Scalable holistic check

Common Questions

Frequently Asked Questions

How does linear XEB fidelity differ from randomized benchmarking?

Randomized benchmarking runs Clifford sequences that compose to the identity and fits the survival decay to an average error per Clifford; it is SPAM-insensitive but practical only for one or two qubits. XEB instead runs deep non-Clifford circuits that scramble toward a Porter-Thomas distribution, then scores bitstrings against simulation, with F_XEB ≈ (1 − e)^g. XEB scales to dozens of qubits and captures correlated crosstalk errors, but it needs classical simulation of the ideal circuit, which becomes intractable past roughly 50 qubits.

What microwave control specifications limit measured XEB fidelity?

The score is set by accumulated gate error, dominated by the microwave control chain. Single-qubit gates are shaped 4 to 6 GHz pulses from an arbitrary waveform generator and IQ mixer; amplitude and phase errors below 0.1% and timing jitter under 10 ps are needed for single-qubit errors near 0.1%. Local-oscillator phase noise must stay below roughly −100 dBc/Hz at 1 MHz offset, and 14-bit or finer DAC resolution keeps quantization from degrading the pulse. Sycamore reported per-cycle errors near 0.3% single-qubit and 0.6% two-qubit.

Why does the XEB score follow a Porter-Thomas distribution?

A deep random circuit produces an output state that behaves like a random pure state in the 2ⁿ-dimensional Hilbert space, so the bitstring probabilities follow the Porter-Thomas law, an exponential distribution with mean 1/2ⁿ. A faithful processor preferentially samples high-probability strings and yields a positive cross-entropy signal, whereas a processor outputting uniform noise gives F_XEB = 0. The ideal sampler scores F_XEB = 1, so the measured value is itself the circuit fidelity.

Related Terms

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