Linear Cross-Entropy Benchmarking (LXEB)

• LXEB is a quantum benchmarking method that quantifies a sampler’s bias toward outputs with higher ideal probabilities, providing a proxy for circuit fidelity.
• It is computed by averaging ideal output probabilities from samples, with normalization by 2^n, where chaotic circuits yield scores near 1 and uniform samplers score 0.
• LXEB has been extended to various architectures including randomized and photonic circuits, though it faces limitations such as spoofing and challenges in faithfully tracking true fidelity.

Linear Cross-Entropy Benchmarking (LXEB), usually written Linear XEB, is a family of benchmark statistics that quantify how strongly samples produced by a device or simulator are biased toward outputs having larger ideal probabilities than average. In the standard random-circuit-sampling setting, if an ideal $n$-qubit circuit $U$ induces probabilities $p_U(x)$ and an experiment produces a distribution $q_U(x)$, the normalized score is

$\chi = 2^n\sum_x p_U(x)q_U(x)-1,$

or, from samples $x_i\sim q_U$,

$$F_{\mathrm{XEB}}=\frac{2^n}{T}\sum_{i=1}^T p_U(x_i)-1.$$

A uniform sampler gives baseline score $0$, while ideal chaotic sampling is approximately $1$ under Porter–Thomas statistics (Barak et al., 2020). Originally introduced as an experimentally convenient proxy for fidelity in random-circuit sampling, LXEB has since been reinterpreted through randomized benchmarking, generalized via ergodicity arguments, and extended to monitored circuits and photonic sampling, while also becoming a focal point in debates about spoofing, hardness, and the limits of benchmark-based claims (Cheng et al., 13 Feb 2025).

1. Standard formulation and normalization

In the random-circuit-sampling literature, an $n$-qubit circuit $U$0 defines ideal output probabilities

$U$1

For any candidate sampler $U$2 over $U$3, the linear cross-entropy fidelity is

$U$4

The uniform distribution satisfies $U$5, while ideal sampling $U$6 gives

$U$7

which is approximately $U$8 for chaotic random circuits with Porter–Thomas output statistics (Barak et al., 2020).

A closely related formulation appears as XHOG (“Linear Cross-Entropy Heavy Output Generation”), where one asks for $U$9 distinct outputs $p_U(x)$0 satisfying

$p_U(x)$1

This makes explicit that LXEB is linear in the ideal probabilities of the observed strings, rather than logarithmic in those probabilities (Aaronson et al., 2019).

The experimental appeal of LXEB is that it can be estimated from samples by computing the ideal probabilities of the observed outputs and averaging them. This is the benchmark that became prominent in Google’s 2019 random-circuit-sampling experiment; in the normalization above, Google’s noisy 53-qubit circuit family achieved a reported value of $p_U(x)$2 (Barak et al., 2020).

An important structural caveat is that LXEB is not a proper distance to the ideal distribution. The ideal distribution $p_U(x)$3 does not maximize $p_U(x)$4; concentrating all mass on the mode of $p_U(x)$5 can yield a much larger score. LXEB therefore rewards correlation with high-probability outputs, not faithful reproduction of the full output law (Barak et al., 2020).

2. Fidelity interpretation, ergodicity, and ideal-circuit guarantees

A major line of work treats LXEB as a special case of a broader benchmarking family indexed by a post-processing function $p_U(x)$6. In that framework, choosing

$p_U(x)$7

recovers the standard linear-XEB statistic

$p_U(x)$8

The same framework defines a deviation-of-ergodicity observable and shows that, under global depolarizing noise or weakly correlated noise satisfying the paper’s sufficient condition, one obtains

$p_U(x)$9

so linear XEB estimates circuit fidelity in the weak-noise regime (Cheng et al., 13 Feb 2025).

That identification is not unconditional. In a “benign” noisy-circuit setting, one analysis shows that XEB, fidelity, and the no-error probability agree only under additional assumptions, summarized by the condition

$q_U(x)$0

where $q_U(x)$1 is the local error rate and $q_U(x)$2 depends on architecture and scrambling properties. The same work shows that XEB and fidelity scale differently with system size: for weakly correlated composite systems, fidelity is multiplicative, while small XEB scores are approximately additive (Gao et al., 2021).

A sharper spectral analysis finds that the breakdown of XEB as a fidelity proxy occurs through a phase transition in the scaling variable $q_U(x)$3. For Haar-random two-qubit gates in an all-to-all architecture, the critical point is

$q_U(x)$4

In one-dimensional architectures the maximal threshold found is

$q_U(x)$5

while in all-to-all circuits the threshold can reach

$q_U(x)$6

for sufficiently favorable gate ensembles. Below threshold, XEB tracks the white-noise fidelity decay; above threshold, it is controlled by a different transfer-matrix eigenmode and ceases to be a reliable fidelity proxy (Ware et al., 2023).

A complementary result concerns the ideal, noiseless benchmark target itself. For 1D brickwork random circuits with Haar-random two-qubit gates, ideal circuits pass the LXEB test with high probability: depth $q_U(x)$7 yields success probability

$q_U(x)$8

while depth $q_U(x)$9 improves this to

$\chi = 2^n\sum_x p_U(x)q_U(x)-1,$0

The proof uses approximate unitary designs, strong concentration of the collision probability $\chi = 2^n\sum_x p_U(x)q_U(x)-1,$1, and tail bounds showing $\chi = 2^n\sum_x p_U(x)q_U(x)-1,$2 at near-quadratic depth (Hunter-Jones et al., 26 Feb 2026).

3. Hardness assumptions, spoofing, and verification limits

The earliest hardness analysis isolates spoofing LXEB as a narrower task than full random-circuit simulation. Under XQUATH—the conjecture that no polynomial-time classical algorithm can estimate

$\chi = 2^n\sum_x p_U(x)q_U(x)-1,$3

with mean-squared error improved by $\chi = 2^n\sum_x p_U(x)q_U(x)-1,$4 over the trivial estimator $\chi = 2^n\sum_x p_U(x)q_U(x)-1,$5—no polynomial-time classical algorithm can solve XHOG with success probability

$\chi = 2^n\sum_x p_U(x)q_U(x)-1,$6

and

$\chi = 2^n\sum_x p_U(x)q_U(x)-1,$7

This gives a conditional hardness result for spoofing nontrivial LXEB scores (Aaronson et al., 2019).

Subsequent work showed that positive LXEB can nevertheless be substantially easier than faithful simulation. For shallow random circuits with light-cone size $\chi = 2^n\sum_x p_U(x)q_U(x)-1,$8, one classical randomized algorithm chooses $\chi = 2^n\sum_x p_U(x)q_U(x)-1,$9 output qubits with disjoint light cones, computes their exact one-bit marginals, samples those bits from the true marginals, and fills the rest uniformly. Its output distribution is

$x_i\sim q_U$0

and it achieves

$x_i\sim q_U$1

For 2D circuits of depth $x_i\sim q_U$2, this becomes a polynomial-time spoofing algorithm with superconstant LXEB fidelity $x_i\sim q_U$3 (Barak et al., 2020).

A broader critique distinguishes the benign setting from the adversarial one. In the adversarial setting, an efficient classical algorithm can omit carefully chosen entangling gates, factor the circuit into smaller subcircuits, and use post-processing to amplify the resulting linear cross-entropy score. Reported implementations obtained $x_i\sim q_U$4–$x_i\sim q_U$5 of the XEB values achieved in contemporary quantum experiments, using one GPU within $x_i\sim q_U$6 seconds, while remaining far from faithful many-body simulation (Gao et al., 2021).

A related caution comes from the system linear cross-entropy score (sXES) proposed for mQSVT-based Hamiltonian simulation. That work proves that sXQUATH fails for sublinear-depth single-block mQSVT circuits and gives an efficient classical spoofing algorithm whenever the noisy experiment’s sXES falls below a baseline-plus-$x_i\sim q_U$7 threshold. The result is comparative: moving from standard LXEB to a closely related linear cross-correlation benchmark does not, by itself, restore soundness (Tanggara et al., 2024).

4. Randomized-benchmarking reinterpretations and scalable variants

LXEB has also been absorbed into more general randomized-benchmarking frameworks. Universal randomized benchmarking replaces the recovery-gate paradigm by a sequence-dependent post-processing POVM and explicitly identifies linear XEB as a URB protocol with an effectively factorizable post-processing measurement. Under the theorem’s assumptions—approximate factorization, near-ideal implementation, and a twirling map within unit distance of the Haar twirl—one gets an approximate single-exponential decay

$x_i\sim q_U$8

for the expected benchmarking signal (Chen et al., 2022).

A more specialized filtered randomized benchmarking formalism then shows that LXEB is the adjoint-irrep case: $x_i\sim q_U$9 Under spectral-gap and perturbative-noise conditions, the expected signal has a dominant exponential term plus a decaying mixing transient, and for brickwork random circuits the sufficient circuit depth can be linear in qubit number (Heinrich et al., 2022).

To address the classical intractability of simulating generic random circuits, Clifford XEB restricts the circuit ensemble to $$F_{\mathrm{XEB}}=\frac{2^n}{T}\sum_{i=1}^T p_U(x_i)-1.$$0 while keeping the same normalized estimator

$$F_{\mathrm{XEB}}=\frac{2^n}{T}\sum_{i=1}^T p_U(x_i)-1.$$1

Because Clifford circuits are classically simulable, this variant scales to much larger systems; numerical simulations reached $$F_{\mathrm{XEB}}=\frac{2^n}{T}\sum_{i=1}^T p_U(x_i)-1.$$2 qubits on a workstation, and in low-noise regimes the observed exponential decay rates tracked per-cycle noise under a digital error model (Chen et al., 2022).

A different scalability route imposes a symmetry constraint. For particle-number-conserving random circuits, the dynamics is confined to a sector of dimension

$$F_{\mathrm{XEB}}=\frac{2^n}{T}\sum_{i=1}^T p_U(x_i)-1.$$3

and the modified score

$$F_{\mathrm{XEB}}=\frac{2^n}{T}\sum_{i=1}^T p_U(x_i)-1.$$4

accounts both for in-sector ideal probabilities and for leakage out of the target sector through the prefactor $$F_{\mathrm{XEB}}=\frac{2^n}{T}\sum_{i=1}^T p_U(x_i)-1.$$5. When $$F_{\mathrm{XEB}}=\frac{2^n}{T}\sum_{i=1}^T p_U(x_i)-1.$$6, this makes ideal simulation feasible for $$F_{\mathrm{XEB}}=\frac{2^n}{T}\sum_{i=1}^T p_U(x_i)-1.$$7, and in the reported implementation even up to $$F_{\mathrm{XEB}}=\frac{2^n}{T}\sum_{i=1}^T p_U(x_i)-1.$$8 qubits in noiseless simulation (Kaneda et al., 16 May 2025).

5. Domain-specific variants in monitored circuits and photonic sampling

In monitored random circuits used to study measurement-induced phase transitions, the relevant object is not the usual output-bitstring XEB fidelity but a normalized linear cross-entropy correlator between measurement-record distributions. For a monitored circuit $$F_{\mathrm{XEB}}=\frac{2^n}{T}\sum_{i=1}^T p_U(x_i)-1.$$9, the benchmark is

$0$0

where $0$1 is the device-generated record distribution from a hard nonstabilizer input $0$2, and $0$3 is a classically simulated stabilizer reference. This protocol was demonstrated on IBM hardware up to $0$4 physical qubits, or $0$5 effective qubits after compression. Finite-size scaling of the disorder-averaged $0$6 yielded

$0$7

for 1D connectivity, and

$0$8

for all-to-all connectivity (Kamakari et al., 2024).

For Gaussian Boson Sampling (GBS), a sector-conditioned linear cross-entropy score benchmarks samples directly against the ideal, lossless distribution rather than against a noisy ground-truth model. In photon-number sector $0$9,

$1$0

The associated normalized score uses the sector-uniform baseline, which equals $1$1, and for the ideal no-vacuum asymptotic case the score becomes

$1$2

This construction is motivated by the desire to validate directly against the ideal hard distribution rather than against a possibly classically tractable noisy model (Martínez-Cifuentes et al., 2024).

A more general photonic framework gives exact Haar-averaged LXEB reference values for bosonic sampling schemes in arbitrary regimes, including the saturated regime $1$3. For $1$4-boson inputs $1$5,

$1$6

and the two-copy Haar averages reduce to bosonic-swap expectation values, hence to purities of particle-reduced density matrices. The same second-moment machinery proves average-case anticoncentration for standard Fock-state Boson Sampling, with

$1$7

but shows that for GBS second moments are not sufficient to obtain a comparably strong anticoncentration statement (Kolarovszki et al., 16 Apr 2026).

6. Scope, ambiguities, and open problems

Across the literature, the common conclusion is that LXEB is informative but limited. It is useful as a coarse indicator of correlation with the ideal distribution, and in carefully delimited weak-noise regimes it can estimate fidelity. Yet it is not a proper distance to the ideal output law, it can be substantially easier to spoof than full simulation, and benchmark variants built on similar linear cross-correlation ideas can inherit the same weakness. This has led to repeated calls for stronger verification tasks with more secure complexity-theoretic guarantees (Tanggara et al., 2024).

Several open problems remain explicit. One is to characterize precisely when noisy LXEB still tracks fidelity outside weakly correlated or effectively depolarizing regimes. Another is to prove concentration and finite-sample guarantees for photonic and monitored-circuit LXEB variants, where current evidence is partly numerical (Kolarovszki et al., 16 Apr 2026). A third is to understand spoofing hardness for generalized or architecture-specific linear cross-entropy benchmarks, rather than assuming that positive correlation scores automatically imply classical intractability (Gao et al., 2021).

A separate, unrelated usage of “linear cross entropy” appears in classical machine learning. One example proposes the Linearly Adaptive Cross Entropy Loss

$1$8

equivalently

$1$9

for one-hot classification. In the reported CIFAR-100/ResNet-18 experiment, the paper gives top-5 error $n$0 for the adaptive loss versus $n$1 for standard cross entropy (Shim, 10 Jul 2025). This suggests a terminological overlap around “linear cross-entropy,” rather than identity with the quantum-sampling benchmark normally denoted LXEB.