Researchers have developed a way to verify quantum computers are doing quantum work - even when classical computers can easily fake the same results.
The problem they're solving is subtle but critical. When you run a quantum circuit, you get a distribution of measurement outcomes. A noisy quantum computer produces one distribution. A classical computer pretending to be quantum produces another. At shallow circuit depths - where quantum computers still make mistakes - those distributions can look very similar. How do you tell them apart without running an exponentially expensive classical simulation to check?
The new approach uses nonlinear cross-entropy. Instead of comparing raw output probabilities (which classical computers can spoof), it looks at how those probabilities cluster. Real quantum computers produce correlations that classical spoofing methods miss. The method detects these correlations even at circuit depths where sampling is hard for classical machines to fake convincingly.
Why Current Benchmarks Miss the Mark
Most quantum benchmarks today use linear cross-entropy - a measure that compares the quantum output distribution to the ideal, noise-free distribution. The problem: that comparison assumes you trust the quantum computer's output in the first place. If a classical adversary wanted to fake quantum results, linear cross-entropy wouldn't catch them unless you also ran a full classical simulation as ground truth.
At shallow depths, that's still feasible. You can simulate a 30-qubit circuit classically, compare the results, and declare the quantum computer valid. But as circuits get deeper, classical simulation becomes impossible. You lose your ground truth. Nonlinear cross-entropy sidesteps this by testing for quantum correlations directly - patterns that emerge from entanglement and interference, not from classical probability distributions.
The method works by sampling the quantum output and computing higher-order moments of the probability distribution. These moments reveal structure that classical spoofing methods - even sophisticated ones - can't reproduce without quantum resources. In practice, this means you can verify a quantum computer is doing quantum work without needing a classical supercomputer to check its homework.
What This Means for NISQ Machines
The immediate application is benchmarking noisy intermediate-scale quantum (NISQ) devices. These are the quantum computers we have right now - machines with 50 to 1,000 qubits, high error rates, and no error correction. They're powerful enough to do things classical computers struggle with, but not reliable enough to trust blindly.
Nonlinear cross-entropy gives researchers a way to assess these machines sample-efficiently. You don't need millions of measurement shots to detect quantum behaviour - a few thousand samples are enough to cleanly separate quantum from classical. That matters because every additional measurement costs time and money on real quantum hardware.
The paper demonstrates this on all-to-all random circuits - a standard benchmark where every qubit can interact with every other qubit. These circuits are hard to simulate classically, but they're also noisy in practice. The new method shows you can still verify quantum advantage even when the circuit is shallow enough that a classical computer could simulate it if it wanted to - the quantum correlations are the giveaway.
The Bigger Picture
This work sits at the intersection of quantum verification and classical skepticism. As quantum computers improve, the gap between "what they claim to do" and "what we can verify they're doing" widens. Nonlinear cross-entropy is one tool to close that gap without requiring exponential classical resources.
It also has implications for quantum supremacy claims. Google's 2019 demonstration used linear cross-entropy to argue their quantum processor outperformed classical computers. Skeptics pointed out that classical spoofing methods could approximate the output distribution closely enough to cast doubt on the claim. Nonlinear cross-entropy would make that argument cleaner - either the machine is producing quantum correlations, or it's not. No ambiguity.
For developers and researchers working with NISQ hardware, the practical takeaway is simple: you now have a way to test whether your quantum circuits are genuinely quantum, without needing a classical reference machine that can barely keep up. That's a small but significant step toward trusting these machines to do useful work.
Read the full paper on arXiv for the mathematical details and experimental results.