Mathematicians utilize ‘overlooked’ particles that could revitalize quantum computation

Quantum computers hold the promise of tackling problems far exceeding the capabilities of today’s fastest supercomputers. However, current machines are exceptionally delicate. The quantum bits, or “qubits,” responsible for storing and processing data are easily disturbed by their surroundings, resulting in errors that swiftly accumulate.

One of the most promising strategies to overcome this hurdle is topological quantum computing, which seeks to safeguard quantum information by encoding it in the geometric properties of unique particles known as anyons. These particles, theorized to exist within certain two-dimensional materials, are anticipated to be significantly more resistant to noise and interference when compared to traditional qubits.

“Among the frontrunners for constructing such a computer are Ising anyons, which are currently under intense investigation in condensed matter laboratories due to their potential realization in unconventional systems like the fractional quantum Hall state and topological superconductors,” stated Aaron Lauda, professor of mathematics, physics, and astronomy at the USC Dornsife College of Letters, Arts and Sciences and the senior author of the study. “Ising anyons alone cannot perform all operations required for a versatile quantum computer. The computations they support depend on ‘braiding,’ which involves physically maneuvering anyons around each other to perform quantum logic. For Ising anyons, this braiding facilitates only a limited array of operations known as Clifford gates, which do not meet the complete requirements for universal quantum computation.”

However, in a recent study published in Nature Communications, a team of mathematicians and physicists led by USC researchers demonstrated an unexpected workaround. By introducing a single new type of anyon, previously discarded in conventional approaches to topological quantum computation, the team revealed that Ising anyons can be rendered universal, capable of executing any quantum computation solely through braiding. The team named these recovered particles “neglectons,” a term that signifies both their formerly ignored status and their newly discovered significance. This new anyon naturally arises from a broader mathematical framework and supplies precisely the missing element necessary to finalize the computational toolkit.

From mathematical waste to quantum wealth

The solution lies within a novel classification of mathematical theories known as non-semisimple topological quantum field theories (TQFTs). These theories broaden the conventional “semisimple” frameworks physicists typically employ to characterize anyons. Traditional models simplify the foundational mathematics by eliminating objects with what is termed “quantum trace zero,” effectively branding them as redundant.

“Yet, those discarded entities turn out to be the crucial component,” Lauda elaborated. “It’s akin to discovering treasure amidst what everyone else deemed mathematical refuse.”

The new framework includes these overlooked aspects and uncovers the novel type of anyon — the neglecton — which, in combination with Ising anyons, facilitates universal computation via braiding.

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solitary. Importantly, only a single neglecton is sufficient, and it remains fixed while the calculations are executed by intertwining Ising anyons around it.

A residence with unstable chambers

The finding was not devoid of its mathematical obstacles. The non-semisimple framework introduces anomalies that breach unitarity, a core principle that guarantees quantum mechanics maintain probability. Most physicists would have perceived this as a critical defect.

However, Lauda’s team discovered a sophisticated solution. They structured their quantum encoding to detach these mathematical anomalies from the actual computation. “Envision it as constructing a quantum computer in a residence with some unstable chambers,” Lauda clarified. “Rather than remedying every chamber, you ensure all of your computations occur in the structurally secure zones while keeping the troubling areas off-limits.

“We’ve effectively isolated the peculiar aspects of the theory,” Lauda remarked. “By meticulously arranging where the quantum information resides, we ensure it remains in the sections of the theory that function correctly, allowing the computation to proceed even if the overarching structure is mathematically peculiar.”

From pure mathematics to quantum reality

This breakthrough demonstrates how abstract mathematics can address tangible engineering challenges in unforeseen ways.

“By accepting mathematical structures that were once deemed trivial, we have opened a whole new chapter for quantum information science,” Lauda noted.

The investigation paves new avenues in both theoretical and practical contexts. From a mathematical perspective, the team is striving to broaden their framework to additional parameter values and elucidate the role of unitarity in non-semisimple TQFTs. On the experimental front, they aspire to pinpoint specific material platforms where the stationary neglecton might emerge and devise protocols that transition their braiding-based methodology into implementable quantum operations.

“What’s especially thrilling is that this work brings us closer to universal quantum computing with particles we already know how to manipulate,” Lauda stated. “The mathematics provides a clear objective: If experimentalists can uncover a means to realize this additional stationary anyon, it could unleash the full potential of Ising-based systems.”

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About the research: Besides Lauda, other contributors include the study’s primary author, Filippo Iulianelli, along with Sung Kim from USC, and Joshua Sussan from Medgar Evers College of The City University of New York.

This study received support from the National Science Foundation (NSF) Grants (DMS-1902092, DMS-2200419, DMS-2401375), Army Research Office (W911NF-20-1-0075), Simons Foundation Collaboration Grant on New Structures in Low-Dimensional Topology, Simons Foundation Travel Support Grant, NSF Graduate Research Fellowship (DGE- 1842487), and PSC CUNY Enhanced Award (66685-00 54).