What is being claimed
John Preskill coined "quantum supremacy" in 2012 to mean a quantum device performing a task that no classical computer could complete in any feasible time, regardless of whether the task is useful. "Quantum advantage" is now the more common phrase, and it is used loosely to mean anything from that strict definition down to "faster than the classical method we happened to compare against". Much of the confusion in this area comes from those two meanings being run together, so the first thing we do with any claim is ask which one is intended.
Sycamore and the first reply
Google's 2019 paper in Nature reported that its 53-qubit Sycamore processor sampled the output of random quantum circuits in about two hundred seconds, and estimated that the same task would take a state-of-the-art supercomputer around ten thousand years. IBM replied within days with an analysis arguing that, by using the disk storage of the Summit supercomputer to hold the full state vector, the task could be done classically in a matter of days. Neither side was wrong on its own terms; the disagreement was about which classical method to compare against.
Over the following two or three years, groups in China and elsewhere published tensor-network methods that reproduced Sycamore-scale sampling on large GPU clusters in hours, exploiting the fact that the experiment's fidelity was low and that a classical simulation only needs to match that fidelity, not perfection. The task remains hard, and the quantum machine remains impressive, but the ten-thousand-year gap did not survive. That pattern, a large claimed gap narrowing as classical algorithms catch up, recurs throughout this sheet.
Boson sampling: Jiuzhang and Borealis
The University of Science and Technology of China reported Gaussian boson sampling on its Jiuzhang photonic device in Science in 2020, with later versions following, and the same institution reported random circuit sampling on its superconducting Zuchongzhi processors from 2021. Xanadu published Gaussian boson sampling on its programmable Borealis machine in Nature in 2022. These are genuinely different experiments from Sycamore, using photons rather than qubits, and they are harder to attack with the same tensor-network tools.
They have nonetheless attracted classical replies, including algorithms that exploit photon loss and the specific structure of the sampled distribution. Our reading is that boson sampling claims stand on firmer ground than random circuit sampling in terms of the raw gap, but on weaker ground when it comes to verification, because there is no efficient way to check that the photonic device is sampling from the right distribution at all. A speed-up you cannot verify is a claim of a different kind.
The IBM utility paper
In 2023 IBM published in Nature an experiment on its 127-qubit Eagle processor simulating the dynamics of a kicked Ising model, arguing that error mitigation let the machine produce useful results beyond what brute-force classical methods could reach. Unlike the sampling experiments, this was a physics problem someone might want solved. Within weeks, several groups had reproduced the results using tensor networks, belief propagation and Pauli-path methods on ordinary hardware, in some cases on a laptop.
We think this episode is more informative than either side's framing. It showed that noisy processors can run circuits at a scale where the answer is not trivially known, which is real progress, and it also showed that the classical community responds fast when given a concrete target. The paper's title claimed evidence of utility, not proof, and on that narrow reading it held up.
Why the goalposts move
Three reasons. Classical algorithms improve, particularly when a specific quantum experiment gives them something to aim at. Quantum experiments run at low fidelity, and a classical method only needs to match that fidelity to count as a spoof. And the tasks chosen for advantage demonstrations are chosen because they are easy for the quantum machine, which often means they have structure a clever classical method can exploit. None of this means advantage is impossible. It means that a claim published this year should be read as provisional until it has survived a year or two of classical attention.
Benchmarks that help, and their limits
Randomised benchmarking measures average gate error and is well understood. Cross-entropy benchmarking, used in the sampling experiments, estimates fidelity by comparing outputs to an ideal simulation, which by construction only works at scales where that simulation is possible. Quantum volume, proposed by IBM in 2019, combines qubit count, connectivity and error rates into one number by asking how large a square random circuit a machine can run successfully; it is useful for tracking one machine over time and less useful for comparing platforms, since it favours particular circuit shapes. We would not treat any single benchmark as a ranking.
Weighing the next claim
Ask whether the task has a use, or whether usefulness is being inferred from difficulty. Ask what classical baseline was used and whether the authors invited attempts to beat it. Ask whether the result can be verified independently of the device that produced it. And note whether the machine used error correction or error mitigation, because the two scale in opposite directions. Useful, verified, error-corrected advantage has not been demonstrated to the field's satisfaction. Each of the results above moved the frontier; none of them, on our reading, crossed it.