A little bit more than magic: the secret to quantum computing may lie in negativity

Researchers at the University of Cambridge have used an early 20^th century mathematical tool to help identify which quantum states are truly useful for quantum computing and which are not, reshaping our understanding of where the power of classical computers ends and the one of quantum computers begins.

“Without this kind of foundational work, we will never know whether a quantum device has achieved something genuinely beyond classical reach.”

David Arvidsson-Shukur

Quantum computers hold great promise for applications from drug discovery to cybersecurity. Yet figuring out what would give quantum computers their edge over everyday, “classical,” computers is a subtle problem. A new theoretical study led by researchers at the Cavendish Laboratory shows that quantum computers are harder to make powerful than previously assumed, while offering the clearest picture yet of what actually makes them work.

The research, published in Physical Review Letters, helps identify the precise quantum states that are genuinely useful for quantum computation. It also increases the set of quantum calculations known to be easy for classical computers to compute, meaning quantum devices face a higher bar to demonstrate an advantage.

At the heart of quantum computing is a kind of “magic”. Quantum computers typically work with quantum bits, or qubits: particles such as electrons or atoms that can exist in two states at once. To run algorithms that outperform any classical computer, those qubits must be prepared in special starting configurations known as ‘magic states’. These states act as the computational fuel: without them, a quantum computer is no better than a conventional machine.

But the new research expands scientists’ understanding that not all magic states are equal. Many states that appear ‘magic’ and were previously assumed to be useful turn out to offer no quantum advantage.

By identifying this class of useless magic states, the team redraws the boundary between calculations that need a quantum computer and those that can still be handled classically.

“We’re showing that magic is necessary but not sufficient to unlock quantum computer’s full power,” said Dr David Arvidsson-Shukur, from the Hitachi Laboratory at the Cavendish Laboratory. “If a quantum state is not magic, you can’t get a quantum advantage. But having magic alone doesn’t guarantee you have one either. The picture is more nuanced and much more interesting than that.”

To identify which quantum states have what the team classifies as “useful” magic, and which have the ‘useless’ kind, the researchers turned to a mathematical framework developed in Cambridge in 1945 by Paul Dirac, the physicist behind the relativistic quantum equation that predicted antimatter. Working at St John’s College, Dirac independently established a distribution similar to one introduced a decade earlier by MIT’s John Kirkwood, and extended the idea by building the mathematical framework in which to use it. This became known as the Kirkwood-Dirac distribution.

The framework involves a concept that sounds paradoxical: negative probabilities, which the team used to phrase the task of quantum computation. And just as Dirac once argued that a negative solution to an equation should be taken seriously (a theory that eventually led to the discovery of anti-matter), the team shows that the appearance of negative values in the Kirkwood-Dirac distribution is a meaningful signal. When that distribution stays entirely positive (or zero) throughout a computation for a given input quantum state, a classical computer can simulate the quantum computation with ease. When it goes negative, classical simulation becomes exponentially harder and a genuine quantum advantage may exist.

“We’re essentially bringing a new ingredient to the magic mix: the Kirkwood-Dirac negativity,” said J.J. Thio, lead author of the study and PhD student in Prof Crispin Barnes’ group at the Cavendish Laboratory and St John’s College, the same as Dirac’s. “The idea builds on probabilities—for example, the odds of obtaining a heads-up upon flipping a coin – but coming with a twist: they may be negative. And those negative “probabilities” are needed for the quantum computer to outperform its classical counterpart as well.

“That distribution provides a stricter, more precise framework for mapping which quantum states can be efficiently simulated by classical computers.”

To demonstrate their theory, Thio worked with fellow student Rishi Goel to complete a classical simulation programme that runs on a standard laptop, performing computations previously thought to require a quantum computer. The result is a direct demonstration that the classical frontier is larger than we previously thought.

These findings matter because billions of pounds are being invested in quantum computing by governments and private companies worldwide. Yet there is still genuine scientific uncertainty about when quantum computers will outperform classical machines on tasks of practical value.

By pushing the boundary of what classical computers can do, the Cambridge team is helping establish a clearer, more rigorous threshold for what would constitute a true quantum advantage. “Without this kind of foundational work, we will never know whether a quantum device has achieved something genuinely beyond classical reach,” said Arvidsson-Shukur.

The team hopes the results will guide both the design of quantum software and the production of magic states, which remains one of the major engineering challenges in building large-scale quantum computers.

“The better we can identify these useful magic states, the better we can produce and apply them,” said Dr Nicole Yunger Halpern, a Fellow of the Joint Center for Quantum Information and Computer Science (Maryland, USA). “Some of us have conjectured for years that Kirkwood–Dirac negativity can assist with this goal, and I’m delighted that the group has finally answered affirmatively.”

This new work brings the field a step closer to a complete understanding of what gives quantum computers their potential. A question that, despite decades of research, remains open.

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Reference:

Jonathan J. Thio, Songqinghao Yang,  Nicole Yunger Halpern, Stephan De Bièvre, Crispin H. W. Barnes and David R. M. Arvidsson-Shukur,  ‘Kirkwood-Dirac Nonpositivity Is a Necessary Resource for Quantum Computing.’ Physical Review Letters (2026). DOI: 10.1103/x819-898d

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Images:

Lead image: Futuristic central processor unit. Credit: Da-Kik  via Getty Images

Body: ‘Even Wizards Struggle in the Quantum Industry’ – Credit: J.J. Thio