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  • Open Access

End-to-End Quantum Algorithms for the Jones Polynomial

Tuomas Laakkonen1,*, Enrico Rinaldi1, Chris N. Self1, Eli Chertkov2, Matthew DeCross2, David Hayes2, Brian Neyenhuis2, Marcello Benedetti1, and Konstantinos Meichanetzidis1,†

  • 1Quantinuum, Partnership House, Carlisle Place, London SW1P 1BX, United Kingdom
  • 2Quantinuum, 303 S. Technology Ct., Broomfield, Colorado 80021, USA

  • *Contact author: tsrl@mit.edu
  • †Contact author: k.mei@quantinuum.com
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PRX Quantum 7, 020355 – Published 9 June, 2026

DOI: https://doi.org/10.1103/jgv8-l3j1

Abstract

We present an end-to-end algorithmic pipeline where a noisy digital quantum computer is used to approximate the value of the Jones polynomial at the fifth root of unity for any input link, i.e., a closed braid. This problem is DQC1-complete for Markov-closed braids and BQP-complete for Plat-closed braids, and we accommodate both versions of the problem. Even though it is widely believed that DQC1 is strictly contained in BQP, and so is “less quantum,” the resource requirements of classical algorithms for the DQC1 version are at least as high as for the BQP version, and so we potentially gain “more advantage” by focusing on Markov-closed braids in our exposition. We demonstrate our quantum algorithm on Quantinuum’s H2-2 quantum computer and show the effect of problem-tailored error-mitigation techniques. Further, leveraging that the Jones polynomial is a link invariant, we construct an efficiently verifiable benchmark to characterize the effect of noise present in a given quantum processor. In parallel, we develop and benchmark the state-of-the-art tensor-network-based classical algorithms for computing the Jones polynomial. The reconfigurable tools provided in this work allow for precise resource estimation to identify minimum link sizes for near-term quantum advantage in practice for a meaningful quantum-native problem in knot theory, if a candidate set of links are provided.

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Popular Summary

The fundamental problem in knot theory is distinguishing two given knots. To this end, mathematicians have invented quantities called invariants. The reason for their name is that they do not change if the knot is represented differently. A famous invariant is the Jones polynomial, which is, however, hard to compute using classical computers. Remarkably, there exists an efficient quantum algorithm for evaluating it, and so this problem is a quantum-native problem, which is meaningful for the mathematical community, that promises a quantum computational advantage. In this work, we heavily optimize this quantum algorithm and introduce a novel benchmark for quantum computers. The benchmark is problem-specific and, importantly, efficiently verifiable, and so it allows us to characterize the effect that noise in the quantum processor has on the algorithm’s output. In parallel, we develop the best classical algorithm for this problem, improving on the previous state of the art. Having defined quantum advantage rigorously in terms of the wall-clock time necessary to obtain the value of the Jones polynomial within some precision, we put it all together and can identify the knot sizes for which quantum advantage is expected, given the specifications of quantum and classical computers available. Showing how one should quantify the minimum quantum advantage system requirements is an important task on the way toward quantum utility, when quantum computers are used to advance scientific and mathematical discovery.

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Outline

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PRX Quantum 7, 020355– Published 9 June, 2026
  • Received 11 April 2025
  • Revised 2 February 2026
  • Accepted 19 February 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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