Few phrases in computing have been as consistently overused, misunderstood, and — in fairness to the skeptics — as consistently walked back as “quantum supremacy.” It’s a term with a precise technical meaning that gets flattened in press coverage into something closer to “quantum computers now beat regular computers,” which isn’t accurate and isn’t what the researchers making these claims actually mean. This article covers what quantum supremacy and quantum advantage actually mean as technical terms, walks through the major claimed milestones from 2019 through the most recent 2025 results, and is honest about the genuine controversies that have followed nearly every one of them.
Defining the Terms Precisely
Quantum supremacy, a term coined by Caltech physicist John Preskill in 2012, refers to a specific, narrow claim: a programmable quantum computer solving some computational task — any task, regardless of practical usefulness — faster than any classical computer could solve it in a feasible amount of time. Preskill deliberately chose a term meant to mark a scientific threshold, not a claim of general computational dominance, and he’s been explicit in subsequent writing that the term was never meant to imply quantum computers are now broadly useful or superior to classical computers for real-world problems.
Quantum advantage (sometimes “quantum utility” or “verifiable quantum advantage” in more recent framing) is a related but distinct and, in practice, more demanding claim: that a quantum computer solves a problem faster than classical alternatives and that the result is independently verifiable, ideally on a task with some connection to genuine scientific or practical value, rather than purely on a synthetic benchmark engineered specifically to be hard for classical computers and easy for quantum ones. Increasingly, researchers in the field — including Google’s own team — have shifted toward preferring “quantum advantage” or “beyond-classical” over “supremacy,” partly because of the connotations of the word “supremacy” and partly because it better reflects the incremental, contestable nature of these milestones.
It’s worth being direct about a distinction that gets lost constantly in popular coverage: neither term means “useful.” A quantum supremacy or advantage claim is about outperforming classical computers on a specific computational task, which is very often a deliberately constructed, practically useless benchmark chosen precisely because it’s hard to simulate classically — not because anyone needs the answer for any real-world purpose.
Google Sycamore, 2019: The First Claim
Google’s original quantum supremacy claim, published in Nature in October 2019, used a 53-qubit superconducting processor called Sycamore to perform random circuit sampling — running a specific, randomly generated quantum circuit many times and sampling from its output distribution. The task was deliberately chosen because sampling from the output distribution of a sufficiently complex random quantum circuit is believed to be exponentially hard for classical computers to replicate directly (a belief grounded in complexity-theoretic arguments, though not a mathematical proof), while a quantum computer can simply run the circuit and measure the outputs directly, with no such exponential difficulty.
Google’s claim was that Sycamore completed a specific instance of this sampling task in about 200 seconds, and that the same task would take the world’s most powerful classical supercomputer at the time (IBM’s Summit) approximately 10,000 years to replicate.
This claim was disputed almost immediately, and the dispute is genuinely instructive for understanding how these milestones tend to play out. IBM published a rebuttal arguing that, with better use of the classical supercomputer’s available disk storage rather than relying purely on memory, the same sampling task could be completed classically in about two and a half days rather than 10,000 years — a dramatic reduction, though still slower than Sycamore’s 200 seconds. This didn’t erase Google’s result, but it established an important and recurring lesson: classical simulation techniques keep improving after a quantum advantage claim is published, so any specific speedup figure is a snapshot in time, not a permanent, fixed boundary. Subsequent research over the following years further narrowed the gap for the specific Sycamore benchmark, with improved classical simulation algorithms eventually matching or approaching Sycamore’s original result on standard classical hardware, effectively “un-doing” much of the original claimed advantage for that specific circuit — though this took considerable classical algorithmic ingenuity to achieve, and doesn’t retroactively invalidate the genuine 2019 hardware achievement, only the durability of the specific speedup figure originally quoted.
Google Willow and the December 2024 “Below-Threshold” Claim
Google’s next major hardware milestone came with the Willow processor, a 105-qubit superconducting chip unveiled in December 2024. Willow’s headline result had two separate components worth distinguishing clearly, because they’ve sometimes been conflated in coverage.
The first was a below-threshold quantum error correction demonstration — a genuinely significant scientific result showing that, for the first time, adding more physical qubits to a logical error-correcting code actually reduced the logical error rate exponentially, rather than the error correction overhead itself introducing more noise than it removed. This is an important and widely respected milestone in the error-correction research community specifically, distinct from any supremacy or advantage claim, and represents genuine progress toward the kind of fault-tolerant hardware that large-scale useful quantum algorithms will eventually require.
The second was another random circuit sampling supremacy-style claim, with Google estimating the task would take a classical supercomputer on the order of $10^{25}$ years — a considerably larger claimed gap than the original 2019 Sycamore result, reflecting Willow’s larger qubit count and more complex circuit. This specific claim, however, became the subject of renewed controversy through late 2025, when researchers including groups affiliated with the Flatiron Institute and EPFL published classical simulation results that reproduced aspects of the claimed benchmark using classical methods, and other researchers demonstrated that similar tasks could be tractable on comparatively modest classical hardware when using more carefully optimized classical algorithms rather than the less-optimized classical baselines the original comparison had been made against. As with the 2019 Sycamore dispute, this doesn’t mean the Willow hardware itself is uninteresting or the below-threshold error-correction result is undermined — but it’s a clear, recent, concrete illustration of the same pattern recurring: the classical side of these comparisons is a moving target, and specific numerical speedup claims should be treated as provisional rather than settled.
Quantum Echoes, October 2025: A Different Kind of Claim
Google’s most recent major result, published in Nature in October 2025, used a 65-qubit subset of the Willow processor to run an algorithm called Quantum Echoes, based on measuring what’s called an out-of-time-order correlator (OTOC) — a quantity from many-body physics that characterizes how quantum information scrambles across a system over time. Google reported this computation running roughly 13,000 times faster than the best available classical estimate for the same task.
What makes this result methodologically distinct from the earlier random-circuit-sampling claims — and worth understanding as a genuine step forward in how these milestones are being validated — is that Google specifically designed this result to be independently verifiable through physical replication, including demonstrating the technique on separate quantum hardware, rather than resting the claim solely on a theoretical classical-cost estimate that later classical algorithmic improvements might undercut. The team also demonstrated an early proof-of-principle application connecting the technique to NMR (nuclear magnetic resonance) molecular structure analysis, though it’s important to be precise here too: this specific NMR application has not yet been shown to exceed classical performance in a practically useful sense — it’s a promising early bridge toward application relevance, not itself an established quantum advantage on a practical NMR task.
China’s Zuchongzhi Line and the Broader International Picture
Google is not the only group making these claims, and treating quantum advantage research as a purely American or purely corporate story would be inaccurate. Chinese research groups have pursued a parallel superconducting quantum processor program under the Zuchongzhi name. The Zuchongzhi 3.0 processor, a 105-qubit device, was the subject of a Physical Review Letters cover article reporting single-qubit, two-qubit, and readout fidelities of roughly 99.90%, 99.62%, and 99.13% respectively, alongside an 83-qubit random circuit sampling experiment that the authors estimated would take a leading classical supercomputer on the order of $10^{15}$ times longer to replicate classically — a smaller claimed gap than Google’s Willow figure, but still, if it holds up under continued scrutiny, a serious result from a research program that receives considerably less Western media attention than Google’s but has been methodologically serious and internationally published.
The existence of parallel, independent programs reaching broadly comparable milestones (high-fidelity superconducting processors in the 100–150 qubit range, random circuit sampling demonstrations with large claimed classical-cost gaps) is itself a useful data point: it suggests these results, while individually contestable in their specific numerical claims, reflect real and broadly reproducible hardware progress across multiple independent research groups, rather than an artifact specific to one company’s measurement methodology.
Why These Disputes Keep Happening: A Structural Explanation
It’s worth explaining why this pattern — bold claim, followed by classical rebuttal, followed by partial or full narrowing of the gap — recurs so reliably across nearly every major quantum supremacy announcement, because it’s not simply a story of researchers overclaiming.
The classical baseline is genuinely hard to establish rigorously. Proving that no classical algorithm can efficiently simulate a given quantum circuit is not something anyone currently knows how to do with mathematical certainty — the relevant complexity-theoretic conjectures underlying random circuit sampling’s presumed classical hardness are widely believed but not proven, in the same sense that P ≠ NP is widely believed but not proven. Every “classical cost” figure attached to a supremacy claim is therefore an estimate, based on the best currently known classical algorithms and available hardware, rather than a proven lower bound — which leaves genuine room for that estimate to later be beaten by cleverer classical algorithms or by using different classical hardware resources (memory versus disk, GPU versus CPU) more efficiently than the original comparison assumed.
The benchmarks are deliberately adversarial to classical computers, which invites classical counter-optimization. Random circuit sampling tasks are specifically chosen because they’re believed to be hard for generic, unoptimized classical simulation approaches. But once a specific claim is public, there’s a strong research incentive (and, frankly, an interesting research problem) for classical algorithms researchers to find structure or approximations specific to that particular benchmark that a generic simulator would miss — exactly what happened with Sycamore, and again more recently with aspects of the Willow random-circuit-sampling claim.
Practical usefulness and computational hardness are different axes entirely. A task can be extremely hard to simulate classically (satisfying the technical definition of quantum supremacy) while being of zero practical interest to anyone outside the narrow research question of whether quantum computers can outperform classical ones on that specific task. This is precisely why the field’s own vocabulary has shifted toward “quantum advantage” and “quantum utility” — terms meant to signal a higher, more demanding bar involving genuine verifiability and, ideally, some connection to a problem someone actually cares about solving, rather than “supremacy” on an arbitrary synthetic benchmark.
What These Milestones Do and Don’t Tell Us About Cryptographically Relevant Quantum Computers
This is worth addressing directly, because it’s one of the most common points of confusion connecting this topic back to the post-quantum cryptography migration discussed elsewhere in this series. None of the quantum supremacy or advantage results described above have any direct bearing on the cryptographic threat posed by Shor’s algorithm. Random circuit sampling and out-of-time-order correlator measurements are specifically chosen tasks that map well onto what current, noisy, non-error-corrected quantum hardware can actually do reliably. Breaking RSA or ECC via Shor’s algorithm at cryptographically relevant key sizes requires a fault-tolerant quantum computer with a very large number of high-quality logical qubits — a fundamentally different, and by essentially all serious estimates, still fundamentally more distant engineering milestone than anything demonstrated by the supremacy and advantage results covered here. Google’s own research team has been explicit on this point, noting that useful, cryptographically relevant quantum computing will require orders-of-magnitude improvements in system scale and performance beyond current below-threshold error correction demonstrations. Treat any claim connecting a specific quantum supremacy headline directly to “encryption is now breakable” with real skepticism — it’s a common and understandable but factually incorrect leap.
Trapped Ion and Other Non-Superconducting Advantage Claims
It’s worth noting that quantum supremacy and advantage discussions have historically centered heavily on superconducting hardware from Google and Chinese research groups, but other hardware modalities have made their own, generally more modest and more narrowly scoped claims worth being aware of. Trapped-ion companies, including Quantinuum, have published results demonstrating advantage on specific structured problems — including certain quantum simulation and random circuit sampling variants adapted to the all-to-all connectivity trapped ion systems naturally provide — though generally framed with less of the “beats every classical computer” rhetoric that characterized the original Sycamore announcement, and more explicitly scoped to particular benchmark tasks. Xanadu, working with photonic quantum computing rather than either superconducting or trapped-ion qubits, published a Gaussian boson sampling supremacy claim (Borealis, 2022) using squeezed light states, representing a third genuinely distinct hardware modality that has produced its own supremacy-style results, again subject to the same general pattern of subsequent classical algorithmic scrutiny that superconducting claims have experienced. The broader lesson is that supremacy and advantage claims are not a single-hardware-platform story, and comparing claims across genuinely different physical implementations (superconducting random circuit sampling versus photonic Gaussian boson sampling versus trapped-ion structured problems) requires care, since the specific computational task, the classical comparison baseline, and the verification methodology all differ meaningfully across these different lines of research.
What Would Actually Change the Picture
It’s worth being concrete about what kind of future result would represent a genuinely different order of milestone from what’s been achieved so far, since that helps calibrate expectations for what to watch for going forward. The results covered in this article all share a common feature: they demonstrate speed advantage on a task, but not on a task of independent practical importance that anyone outside the quantum computing research community has a direct stake in the answer to. A qualitatively more significant milestone — one that essentially the entire field agrees would represent a genuine turning point — would be a demonstrated quantum advantage on a problem with recognized real-world value: a quantum chemistry calculation producing a genuinely new, experimentally verified result for an industrially relevant molecule beyond the reach of classical methods (the subject of the companion article on quantum simulation), for instance, or an optimization result with measurable value in a real logistics or financial application, verified against the best available classical solvers rather than a weak baseline. No such result has been demonstrated as of the most recent public research, and closing that specific gap — from “beats classical computers on a synthetic benchmark” to “beats classical computers on a problem someone actually needed solved” — is generally regarded within the field as the more meaningful milestone still ahead, considerably more significant than any further improvement to the classical-cost estimate attached to a random circuit sampling task.
The Balanced Takeaway
The honest picture, after walking through the actual history rather than the headlines, looks like this: real, substantive, independently reproduced hardware progress has occurred across multiple research groups and hardware platforms since 2019 — coherence times, gate fidelities, and qubit counts have genuinely and measurably improved, and the below-threshold error correction result on Willow in particular represents authentic progress toward the error-correction milestones that matter most for eventually useful quantum computing. At the same time, essentially every specific “X times faster than the best classical computer” headline number attached to these results has proven to be a moving target rather than a settled, permanent fact, repeatedly narrowed or matched by subsequent classical algorithmic improvements. Neither of these facts cancels the other out. The right way to read quantum supremacy and advantage coverage, as a technically literate reader, is to separate the underlying hardware and error-correction progress (generally real and durable) from the specific comparative speedup figures attached to any given announcement (generally provisional, contested, and likely to shrink over time as classical methods catch up) — and to remain appropriately skeptical of any claim, however impressive, that treats a narrow synthetic benchmark result as evidence of broad, practically useful quantum computational superiority.
