The exact story is contested, but in every version, it has a grim ending. A man discovers a new mathematical truth and loses his life for it. In every version of the narrative, the ancient Greek philosopher Hippasus of Metapontum drowns at sea, punished either by the gods or by his peers. His sin? In most tellings, it is discovering that there are numbers that can’t be expressed as an exact fraction, their decimal places running into oblivion without repeating. These irrational numbers – like the square root of 2 or pi – greatly upset Hippasus’s peers, the cultish Pythagoreans, who may have tossed him overboard for bringing them to light.
More than 2000 years later, irrational numbers were embedded in the mathematical foundations of quantum mechanics. The physicists who wrote the theory’s early formulas didn’t have to face stormy seas, but their peers called their work shocking, bordering on heretical. We have made our peace with irrational numbers for a century, even though the quantum realm that includes them is bizarre, hosting particles that may not exist if we aren’t looking at them, or entanglement that can tie two particles together across the cosmos – what Albert Einstein called “spooky action at a distance”.
But one physicist says it doesn’t have to be this way. “Nobody’s entirely happy with what goes on [in quantum mechanics], but most physicists say: ‘Look, I just use the theory because it predicts the right things’,” says at the University of Oxford. In a bold new proposal, he has drafted a way out of that unhappiness: banishing the irrational numbers that cost Hippasus his life.
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Palmer argues that this new theory wouldn’t suffer any oddness – gone is entanglement and its spookiness, away with the strangeness of superpositions that allow two contradictory possibilities at once. Crucially, his rational quantum mechanics (RaQM) would put a fundamental limit on the power of quantum computers, which means it could not only be tested in the near term, but may also imperil a billion-dollar industry. In fact, if Palmer’s proposal is correct, quantum computers will eventually fail to run the famous Shor’s algorithm – a quantum recipe for factoring very large numbers that could break widely used encryption – dispelling decades of dreams that have motivated and justified their development since the 1990s.
“This would, for sure, have a big impact on quantum computing, but it would pale against the impact it would have on physics, which would be the biggest revolution since quantum mechanics itself,” says at Griffith University in Australia.
Stamping out strangeness
To be clear, Palmer is no Pythagorean. He has no problem with irrational numbers, per se. For him, the true mathematical villain of the story of quantum mechanics is John von Neumann, who set out its first rigorous framework. Quantum mechanics classes still teach von Neumann’s maths, starting with the Hilbert space, the mathematical arena where all quantum action happens.
To analyse the behaviour of something quantum like an electron, or to predict its future behaviour, physicists assign it a mathematical object that lives in Hilbert space. In a three-dimensional Hilbert space, you can think of it as an arrow pointing to the spot specified by three numbers for depth, height and length, respectively. The coordinate system that you use in Hilbert space depends on what it is that you want to measure – finding an electron’s spin or position will require different coordinate systems. Once you have selected one, you can draw an arrow and use its length to calculate the probability of your measurement.
Traditionally, anything goes in Hilbert space: arrows of all sorts are welcome. That’s because Hilbert space is continuous. It’s similar to a number line in that way – zoom in on any spot, and you always find more numbers. But Palmer argues that this tolerance of all numbers is at the root of quantum mechanics’ problems.
In RaQM, not all arrows are allowed, leaving mathematical fissures in the abstract space they occupy. Any arrow whose length would square to an irrational number is absolutely banished. It is as if you went looking for the square root of 2 on a number line, zoomed in on the space between 1 and 1.5 where it would normally sit, and found only a gap. “Nature abhors a continuum,” says Palmer.
Run-of-the-mill quantum mechanics is stunningly successful, having withstood a century of strenuous experimental tests. Importantly, Palmer says RaQM is consistent with those findings. The RaQM sieve is so fine that, in all those experiments, it can be mistaken for the meshless, solid vessel of traditional quantum mechanics. But the gifts of accepting RaQM are stunning and many – and deeply consequential.
First, RaQM doesn’t suffer the oddities that come with conventional quantum mechanics. To exemplify this, Palmer focuses on the Bell test. Named after physicist John Stewart Bell, it is a famous experiment with particles connected via quantum entanglement. Imagine pairs of entangled particles that are produced repeatedly, with one particle from each pair sent to an experimenter called Alice, while its partner particle goes to a faraway experimenter called Bob.
Alice and Bob independently choose to measure one of their particle’s properties. Alice might check whether her first particle has “spin up” or “spin down” with respect to the vertical direction. For the next particle, she might measure spin with respect to the horizontal direction, and so on. Simultaneously, Bob is independently choosing how to measure his particles, but the two don’t communicate. Only at the experiment’s conclusion do they plug their respective data into a so-called inequality equation that Bell derived in the 1960s, which tests for correlations between Alice’s and Bob’s measurements.
Some may arise by chance, but Bell determined a threshold that demonstrates that the particles are correlated in a non-local way, somehow influencing each other across large distances. Decades of experiments, including those awarded the 2022 Nobel prize in physics, have consistently found that quantum effects persist across vast distances. But what to make of this apparent non-local connection is an open question.
Some interpretations of standard quantum mechanics simply accept non-locality. Others go as far as to suggest that some law of physics predetermines the measurements – apparent correlations arise not from spookiness, but constraints on Alice and Bob’s free will. One of the strangest parts of the Bell test, according to some interpretations, is that a particle possesses only the properties we measure – if Alice measures vertical spin, her particle doesn’t have a definite value for spin in the horizontal direction until she observes it. Palmer’s theory cuts through all this.
In RaQM, free will remains intact, but there are simply some measurements that are impossible to make. Abandoning the continuum for a more mesh-like picture has seemingly made some forbidden measurements fall through mathematical cracks in reality.
Palmer says a careful analysis of the Bell test shows that these missing measurements are the ones that arise counterfactually – that is, they are the measurements Alice could have taken, but didn’t. “The whole interpretation of Bell’s test utterly, utterly, utterly changes. You don’t need any more non-locality, the weird ‘spooky action at a distance’.” In his view, carefully rethinking which quantum states are even theoretically possible can resolve all manner of odd quantum scenarios, including Erwin Schrödinger’s famous thought experiment in which a cat is in a bizarre mix of two states, alive and dead, while unobserved.
Escaping Q-Day
Abandoning irrational numbers and the continuum of Hilbert space in RaQM is also practically significant for quantum computing. Within a quantum computer, information is encoded in the quantum states of objects called qubits. To manipulate information in a quantum computer, you make qubits interact or entangle them. These steps of quantum computation can then be used to run calculations that are intractable for all conventional computers.
Importantly, there is just a small number of problems that mathematicians have proved can be solved only by a quantum computer. One stands out: the algorithm for factoring large numbers first developed by mathematician Peter Shor in 1994. Because most modern encryption keys are based on the mathematics of factoring, a quantum computer capable of running Shor’s algorithm would be a terrifyingly powerful decryption machine. That is why governments, firms and researchers around the world are preparing for Q-Day, the moment when a quantum computer will vanquish all defences currently safeguarding our digital communications and transactions.
However, researchers’ best estimates say that this device would need about 500,000 qubits, hundreds of times more than the largest existing quantum computers. Current qubits are also noisy and error-prone, so they are insufficient for running complex computations. It is a formidable challenge to build a quantum computer capable of running Shor’s algorithm, and a tremendous amount of resources is being put towards overcoming it. But this is all in vain if our physical reality matches RaQM.
In RaQM, qubits have a finite information capacity because their quantum states follow a new mathematical restriction. If the maximum amount of information that can be encoded in a qubit’s quantum state were represented as a string of 0s and 1s, that bit string would be incredibly long, but still finite.
Palmer has estimated the maximum length RaQM would allow, and it is absolutely enormous: 2^400, or more than 10 billion quadrillion quadrillion times larger than the estimated number of all atoms in the universe. It is also a number that could get a quantum computer with more than 400 perfectly error-proof qubits – which is still many, many more than we have now – into trouble. Running an algorithm like Shor’s, which assumes traditional quantum mechanics, would require quantum states that can contain more information than bit strings with 2^400 entries. As this exceeds the maximum string length possible in RaQM, Palmer predicts that this quantum computer would fail.
So, there would be no large numbers factored and no encryption broken. But if a future quantum computer can indeed do this, RaQM goes up in smoke. “If they can factor some large number, my theory is wrong, and I’m happy to walk away and do something different,” says Palmer. Yet he is confident. The resolution of long-time mysteries like the Bell test assures him that he is on the right path, he says.
Reality’s make-up
These are tremendous stakes that invite careful scrutiny of RaQM as a theory that explains the quantum foundations of our world. Two questions become unavoidable. The first is why Hilbert space would be replaced with something that isn’t continuous to begin with.
Physicists may be used to working with continuous, rather than discrete, theories – smooth spaces rather than sieve-like ones – but these are more contested among mathematicians and information theory experts, says at Michigan State University. There is a very strong case that aspects of mathematics that rely on the continuum aren’t as rigorous as those built on top of finite, discrete parts, he says.
This is because continuous numbers inevitably lead to infinity. Some mathematicians consider infinity a threat to the sensible foundations of their discipline, and it is also debated whether infinity can ever be physical, says Hsu. “If it takes an infinite number of non-repeating digits for me to represent this [irrational] number… I’m not sure what axiom system actually contains that number, and how you would generate it, because you can’t generate it with a finite number of steps.”
The second question is why irrational numbers, and not some other set, would be the ones to fall through the gaps in Palmer’s mathematical space. Though he focused on removing irrationals, he didn’t pluck the idea of discretisation out of nowhere. In 2005, that if space-time were discrete, then no experiment could exclude the possibility of something like a discrete Hilbert space. Their analysis of such a space led them to conclusions similar to Palmer’s, including the fact that quantum computers won’t ever have the capacity to run Shor’s algorithm.
This adds weight to RaQM, but it also invites caveats. Just because the possibility of a mesh-like mathematical space for quantum theory can’t be excluded, that doesn’t mean this is exactly what it must be like. And what about space-time? Could it really be discrete? The key quantity to keep in mind here is a number called the Planck length. All our best theories of physics break down for objects smaller than this, which is about 100 billion billion times smaller than a proton, leading some physicists to suggest that the Planck length is something like the pixel size of the universe.
Whether this is the case will remain unclear until physicists formulate a theory that successfully combines quantum physics, the laws of all things tiny, with gravity, which governs the cosmically large. “The discretisation of space-time is just a hypothesis, as we don’t have a theory of quantum gravity,” says Wiseman.
But Palmer thinks gravity ought to play a role in the mathematical space that dictates the reality of quantum objects. He built this assumption directly into RaQM. Quantum objects, such as qubits, are fragile in the sense that they easily lose their quantumness. A qubit put into a mix of two mutually exclusive states will eventually collapse into just one if left alone, losing its distinct quantum qualities. The exact mechanism behind this collapse is hotly contested, but in the 1980s, at the Wigner Research Centre for Physics in Hungary and at the University of Oxford proposed that it could be caused by gravity. Palmer applied this idea to his qubit with finite information capacity, and that crucial number of 400 perfect qubits popped out, spurring his hypothesis about the limits of quantum computing. “It’s a reasonable assumption. Maybe a wild assumption that not everybody will agree with, but it’s a rational argument,” says at the University of Geneva in Switzerland.
For Gisin, invoking gravity doesn’t add clarity to RaQM; it only raises more questions. For example, qubits can be made from massless particles, and gravity affects only objects with mass, he says. He also isn’t convinced that RaQM can fully save quantum physics from the non-local spookiness revealed by the Bell test. In Gisin’s view, if experiments with future quantum computers and Shor’s algorithm confirm that quantum theory needs a discrete space, rather than the continuous Hilbert space, it would take additional investigations to show that its gaps are exactly those that the oddities of standard quantum mechanics can fall through.
The way forward
For RaQM to replace quantum mechanics, it will have to address a lot more than qubits, says Wiseman. It currently doesn’t. There ought to be a system by which RaQM would allow us to account for everything that can exist, including large objects, he says. This would be a true remodel of how we think about reality. Overall, Wiseman’s stance is that Palmer’s proposal is radical.
at the University of Seville in Spain is similarly concerned. Traditional quantum mechanics is built on equations and formulas that only work in the Hilbert space that physicists have come to know and love. For example, the momentum and position of quantum objects must satisfy a very specific relationship that, according to Cabello, breaks down once the quantum states can no longer be represented by an infinite string of bits.
“This relation has numerous experimentally verified consequences, including the Heisenberg uncertainty principle and the quantised energy levels of atoms, all tested to extraordinary precision,” says Cabello. If someone built a quantum computer that could test RaQM by running Shor’s algorithm tomorrow, Cabello would bet on standard quantum mechanics winning out.
For his part, Palmer isn’t worried about fundamental rules, or axioms, of his theory. He says that, unlike traditional quantum theory, where there are several basic axioms, RaQM has only one, and everything else emerges from it. This is another big advantage of RaQM, he says.
Despite this dissonance, Cabello is supportive of Palmer’s work. “There are zillions of theories that approximately look like quantum theory, but they aren’t completely developed, and they don’t make experimentally testable predictions,” says Cabello. RaQM has promise in that it puts forward a specific experimental test, one for which devices are already, serendipitously, being built. Cabello argues that some more complex versions of the Bell test experiment may also be able to indicate whether quantum physics needs the Hilbert space that physicists have championed for a century, or actually plays by the holey rules of RaQM.
Gisin has similar mixed feelings. He doesn’t shy away from calling some of Palmer’s ideas wild, and he wouldn’t bet on RaQM persevering through an experimental test. “But I think it’s good physics, because you can test it,” he says.
In June, just a few months after Palmer’s paper outlining RaQM was published, quantum-computing firm QuEra announced plans to build a device with more than 1000 error-free qubits by 2029. Several other companies have put forward similar timelines, and the race to build a truly powerful quantum computer is heating up. At the finish line, there awaits an answer to one of the most fundamental questions of all: have we been thinking about quantum physics all wrong for a century?
