The Quantum Computer: Computing With the Impossible

A classical bit is 0 or 1. A qubit exists as a superposition, written in the standard notation as |psi> = alpha|0> + beta|1>: a linear combination of the two basis states, weighted by complex-number amplitudes. Because those weights are complex numbers rather than plain probabilities, they can interfere, adding constructively or cancelling destructively. That is the entire difference, and it is fundamental rather than practical. A qubit is not a classical bit whose single definite value merely happens to be unknown to us. It is a different mathematical object, and the interference is the part that does the work.

There is a hard limit built in from the beginning, and it has no classical counterpart at all. The no-cloning theorem states that an unknown quantum state cannot be copied. A classical bit can always be copied, freely and endlessly, which is why every file anybody owns can have a backup. An unknown qubit cannot be copied at all. This is a fundamental prohibition in the physics, not an engineering shortfall waiting on better hardware.

Two or more qubits can share a joint state that cannot be described as separate individual states at all. The textbook case is the Bell state |Phi+> = (1/sqrt2)(|00> + |11>), in which neither qubit has a state of its own and only the pair does. That this is a real feature of the world rather than a bookkeeping convenience is settled experimentally. Bell's 1964 theorem made the difference testable, and the experiments delivered: Clauser in 1972, Aspect in 1982, and Zeilinger and colleagues in 2015 all found violations of Bell inequalities. Local hidden variables, the idea that the particles were carrying ordinary predetermined values all along, does not survive those results.

That work was recognized formally in 2022, when Alain Aspect, John F. Clauser, and Anton Zeilinger shared the Nobel Prize in Physics "for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science." Zeilinger's group also produced the first demonstration of quantum teleportation, in 1997. That prize is worth carrying through the rest of this article. Whatever remains unproven about quantum computers, and a great deal does, the physics they are built on is not in the speculative column.

One thing entanglement does not do, and the correction belongs here rather than buried later. Entanglement cannot transmit information faster than light. The correlations between entangled particles only become apparent when the two sets of measurement results are brought together and compared, and that comparison travels down an ordinary classical channel at ordinary light-limited speed. Nothing is signalled by the measurement itself. The strangeness is real. The loophole is not.

The fourth property is the one that turns the other three into a computer. A quantum algorithm is an arrangement of operations designed so that the paths leading to wrong answers interfere destructively and cancel, while the paths leading to the right answer interfere constructively and amplify. The machine does not try every possibility at once and pick the best one. It arranges for the useless possibilities to erase each other. That distinction is lost in almost every popular account of the subject, and losing it is where most of the exaggeration in this territory begins.

Decoherence is quantum information leaking out of the machine into everything around it: thermal noise, electromagnetic fluctuations, even cosmic rays. It is the central practical obstacle to quantum computing, and it sets the clock every quantum processor runs against. As order-of-magnitude physics rather than any single laboratory's current benchmark, superconducting qubits hold coherence for roughly 100 microseconds, trapped ions for seconds to minutes, and some nuclear-spin systems for hours. Whatever a machine is going to do, it has to finish inside that window.

The answer the field settled on is quantum error correction, and its price is the most under-reported number in the subject. The scheme spreads the information of one logical qubit across many physical qubits; for the surface code, the leading approach, the estimate runs from roughly 1,000 to 10,000 physical qubits for a single logical one. The threshold theorem sets the condition that makes it worth doing: if the error rate per operation can be pushed below roughly 0.1 to 1 percent, error correction can in principle succeed rather than adding more noise than it removes. That was demonstrated in principle long before anyone demonstrated it at scale, and at scale is exactly where this article's argument lives.

Superconducting qubits are circuits rather than particles: loops of superconducting metal interrupted by Josephson junctions, held at about 15 millikelvin, roughly a hundredth of a degree above absolute zero, and controlled with microwave pulses. This is the most advanced platform by qubit count, having passed 1,000, and the fastest by gate time, at around 20 nanoseconds per operation. IBM, Google, and Rigetti are the leading builders. Almost every dated milestone later in this file was set on this platform.

Trapped-ion qubits are individual electrically charged atoms held in electromagnetic traps and manipulated with laser pulses. The platform's historic advantage is accuracy: its gate operations have been the most faithful of any approach, at around 99.9 percent, bought at the cost of speed, with gates measured in microseconds rather than nanoseconds. IonQ and Quantinuum, formerly Honeywell, are the leading builders.

Our own research document's trapped-ion figures have aged, and this article states the newer ones because they were independently checked, not because the file says so. In October 2025 IonQ, working with the Oxford Ionics team it had acquired, demonstrated better than 99.99 percent two-qubit gate fidelity, and did it without needing ground-state cooling. In November 2025 Quantinuum launched Helios, an all-to-all-connected ytterbium-ion processor reporting 99.921 percent two-qubit fidelity across all 98 qubit pairs, with coherence times measured in seconds to minutes. Those are the kind of numbers the threshold theorem cares about.

Photonic qubits encode information in the polarization or the path of individual photons. The platform runs at room temperature, which removes the cryogenics problem entirely, is naturally suited to carrying quantum information over a distance, and computes through a measurement-based approach rather than a sequence of gates on a static register. PsiQuantum and Xanadu are the leading builders.

Neutral-atom qubits are individual uncharged atoms held in optical tweezers or optical lattices, made to interact by exciting them into high-energy Rydberg states. The platform is fast-scaling and highly reconfigurable, and it has passed 1,000 qubits. QuEra, Pasqal, and Atom Computing are the leading builders. Two named systems make the scale checkable: QuEra's Aquila operates up to 256 qubits in analog mode using Rydberg interactions and optical-tweezer arrays, and Pasqal's platform has demonstrated more than 100 qubits in analog mode.

The fifth approach would be the most elegant of all if it worked. Topological qubits would store information not in the state of any single particle but in the braiding pattern of exotic quasiparticles, non-abelian anyons, of which Majorana zero modes are the sought-after example. Because the information would live in the topology rather than in a local property, it is theoretically the most error-resistant qubit encoding possible. On February 19, 2025, Microsoft announced Majorana 1, described as the first quantum processor built on eight such topological qubits, with a companion paper in Nature. That claim is under serious, named, published dispute. The physicist Henry Legg and others argue that Microsoft's own topological gap protocol can be fooled by non-topological doppelganger signatures that look the same in the data. Winfried Hensinger of the University of Sussex states that the peer-reviewed paper itself contains no proof for topological qubits. And Microsoft made, and later retracted, a similar claim in 2018. This file records the announcement, records the dispute, and crowns neither. What it will not do is list Majorana 1 as a fifth working platform beside the four above, because the field has not agreed that it is one.

In 2019 Google's 53-qubit Sycamore processor performed a specific sampling task in about 200 seconds. Google's team, publishing in Nature, claimed that a classical supercomputer would need roughly 10,000 years to reproduce it, and called the result quantum supremacy. That estimate did not survive contact with IBM, which publicly disputed it and argued that the same task could be simulated classically in about 2.5 days with a better algorithm and more storage. A group led by Pan Jianwei at USTC later demonstrated further improvements in classical simulation, narrowing the gap again. What is verified here is the run: 53 qubits, about 200 seconds, published as Arute and colleagues in Nature, volume 574, pages 505 to 510, in 2019. The 10,000-year figure is a contested estimate, not a measurement, and this file does not repeat it as fact. One footnote on that citation, because the point of this file is that sourcing is checkable: our own research document lists a DOI for that paper which resolves to Nature's news story about the result rather than to the research paper itself. The link in the sources below points at the paper.

In December 2020 a team at USTC led by Pan Jianwei answered with entirely different hardware. Jiuzhang, a 76-photon photonic quantum computer, performed Gaussian boson sampling in about 200 seconds, a task estimated at the time to take the Sunway TaihuLight supercomputer about 2.5 billion years. Photons rather than superconducting circuits, and no refrigerator. Two things belong in the same breath as that number. The first is that this was published in Science and stands as an independent demonstration on a second physical platform, which matters more than the size of the speedup. The second is that Gaussian boson sampling has no known practical application. It is a problem chosen because it is hard for classical computers, not because anybody needed the answer.

On December 4, 2023, at the IBM Quantum Summit, IBM unveiled Condor: 1,121 superconducting qubits on a single chip, built on the company's heavy-hexagonal architecture, and the first quantum chip to exceed 1,000 qubits. IBM said in the same announcement that it would now prioritize error resistance over further growth in raw qubit count, which is the most informative sentence in the whole release. The company that had just won the qubit-count race said the qubit-count race was not the one that mattered.

On December 9, 2024, Google announced Willow, a 105-qubit superconducting processor, and the part of the announcement that matters is not the part that got quoted. Willow demonstrated below-threshold quantum error correction for the first time: as the array scaled from a 3x3 to a 5x5 to a 7x7 grid, the logical error rate fell exponentially rather than rising. Adding more physical qubits made the encoded qubit better instead of worse, which is the behaviour the threshold theorem predicts and which nobody had shown across scaling before. Google also reported that Willow completed a standard benchmark computation in under five minutes that the company estimated a classical supercomputer would take roughly 10²⁵ years to match. That second figure is Google's own estimate, of exactly the kind that was contested in 2019, and this file states it as an estimate and leaves it there. The error-correction result is the one that changes anything.

One sentence carries more weight than the entire milestone list above it, and our own research file states it without softening. Quantum supremacy, or quantum advantage, has so far been demonstrated only for artificial, specially designed sampling problems with no known practical use. No quantum computer has outperformed a classical computer on a commercially relevant problem. Not one, on any platform, anywhere. That is not a hedge and it is not pessimism. It is the mainstream position of the field, and every genuine achievement above sits underneath it.

The field has a precise name for where it is, and it was not coined by a marketing department. In 2018 the physicist John Preskill published "Quantum Computing in the NISQ Era and Beyond" in the journal Quantum, introducing NISQ: Noisy Intermediate-Scale Quantum. It describes processors in the range of fifty to a few hundred qubits, large enough to show genuine quantum behaviour and noisy enough that imperfect qubit control limits what they can reliably compute. Raw qubit counts have since passed the top of that band, as Condor shows. The constraint the term actually names, noise limiting what a machine can be trusted to compute, is the one still standing. The physicist who coined it was not lowering expectations. He was naming the era accurately, so the next one could be recognized when it arrived.

The founding idea came from Richard Feynman in 1982, in a paper called "Simulating Physics with Computers" in the International Journal of Theoretical Physics. His observation was not about speed. It was that a classical computer cannot efficiently represent a quantum system at all once that system passes roughly fifty particles, and that the natural instrument for simulating a quantum system is therefore another quantum system. Molecules and materials are quantum systems. Quantum simulation, rather than anything to do with codes, is the application that follows most directly from the physics.

The result that turned quantum computing from a curiosity into a funded field came from Peter Shor in 1994. Shor's algorithm factors large integers into their primes with an exponential speedup over the best classical methods known. It was published as "Algorithms for Quantum Computation: Discrete Logarithms and Factoring" in the proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science, pages 124 to 134, and expanded into a full paper in the SIAM Journal on Computing in 1997. Factoring became the textbook case for quantum advantage for a straightforward reason: it is a problem where no classical method is known that does the job efficiently, and the quantum route is exponentially shorter.

Factoring is not an academic exercise, which is why this is the algorithm everybody has heard of. RSA public-key encryption, which a large enough fault-tolerant quantum computer running Shor's algorithm would break, underlies a very large share of ordinary internet security: banking, e-commerce, and the TLS connections behind everyday web traffic. Our own research file puts a figure on that, roughly three trillion dollars of daily commerce, and this article states it as the file's own order-of-magnitude estimate rather than a sourced number, because independent checking confirmed RSA's foundational role and did not turn up a primary source for that precise figure. Trillions of dollars a day is the honest form of the claim. The response is already in motion: NIST finalized its first three post-quantum cryptography standards on August 13, 2024, after an eight-year evaluation, as FIPS 203, FIPS 204, and FIPS 205. What that migration involves, and how urgent it is, is a different subject and belongs to its own file. Here, Shor's algorithm is what it was in 1994: proof that a quantum computer could do something no classical computer is known to do efficiently.

The second famous algorithm is quieter and far more general. Grover's algorithm, published by Lov Grover in 1996, searches an unsorted database of N items in roughly the square root of N steps instead of N. That is a quadratic speedup, not an exponential one. Both are real, and they are not the same size of claim, which is worth stating plainly because the two are routinely reported under the same word.

The two algorithms designed for the machines that actually exist are less famous and more honest about their constraints. The variational quantum eigenvalue solver, VQE, from Peruzzo and colleagues in 2014, estimates the ground-state energy of a molecule. The Quantum Approximate Optimization Algorithm, QAOA, from Farhi and colleagues in the same year, produces approximate solutions to combinatorial optimization problems. Both are hybrids: a noisy quantum processor does one part of the work and a classical computer does the rest, and the division of labour is chosen specifically so that NISQ-era hardware can contribute at all. Whether QAOA delivers a genuine quantum advantage is still debated, and this file leaves that open because the field has.

Quantum computers are not universally faster than classical computers. They offer an advantage only for specific problem classes where somebody has engineered an algorithm to exploit interference; for everything else, which is nearly everything, a classical machine remains better and cheaper. The parallelism is not brute force. A quantum computer does not check every possibility simultaneously and report the winner, it arranges for the wrong possibilities to cancel, and an algorithm has to be deliberately built to make that happen. The expectation in the field is that these machines will work as specialized co-processors, the way a GPU does, sitting alongside classical computers rather than replacing them. And there is no proof that a quantum computer can solve NP-complete problems in polynomial time: BQP, the complexity class quantum computers work in, is not believed to be equal to NP.

Gil Kalai, of the Hebrew University, has argued for years that fault-tolerant quantum computing will not work at useful scale. His case is not that quantum mechanics is wrong. It is that noise in physical qubit systems may scale with the complexity of the computation in a way that prevents error correction from ever succeeding in a machine large enough to matter, so that the threshold is never actually reachable in practice. He set it out in "The Quantum Computer Puzzle" in the Notices of the American Mathematical Society, volume 63, pages 508 to 516, in 2016, and has pressed it in public debate with Scott Aaronson of the University of Texas at Austin. This is a serious, named, ongoing academic position, and filing it as a fringe claim would be a failure of this file's own standard.

Aaronson's counter-position is the one most of the field holds: no known physical principle rules out fault-tolerant quantum computing. That sentence deserves to be read as carefully as it was written. It does not say the machine will be built. It says nothing in the physics forbids it, which leaves the question sitting in engineering rather than in principle, and engineering questions get answered by hardware rather than by argument.

Mikhail Dyakonov has made a blunter version of the same objection, calling useful-scale quantum computing a mirage. His argument is that the state space of a quantum machine is exponentially large, and that the number of ways it can go wrong grows with it, so errors grow uncontrollably as systems scale. This file carries that at Tier 2 rather than Tier 1 for a specific reason: the sourcing pass could confirm the physicist and the position, but could not pin this exact claim to a particular citable paper and venue the way it could for Kalai. The argument deserves stating. The citation deserves the caveat.

The mainstream response has three parts, and only the first of them is new. Google's Willow result of December 2024 is the first real evidence against a fundamental noise-scaling barrier: the logical error rate went down exponentially as the code got larger, which is the behaviour the skeptical argument says should not survive scaling. Second, four independent hardware platforms are making simultaneous progress on genuinely different physics, so the enterprise does not depend on any single fragile technology turning out to be the right one. Third, even if universal fault-tolerant quantum computing takes twenty years or more, other near-term quantum technologies outside computing already have real value today, and this file leaves those to the articles that own them.