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Demonstration of Algorithmic Quantum Speedup for an Abelian Hidden Subgroup Problem

Phattharaporn Singkanipa1,2, Victor Kasatkin3,2, Zeyuan Zhou4, Gregory Quiroz4,5, and Daniel A. Lidar6,7

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Phys. Rev. X 15, 021082 – Published 5 June, 2025

DOI: https://doi.org/10.1103/PhysRevX.15.021082

Abstract

Simon’s problem is to find a hidden period (a bitstring) encoded into an unknown 2-to-1 function. It is one of the earliest problems for which an exponential quantum speedup was proven for ideal, noiseless quantum computers, albeit in the oracle model. Here, using two different 127-qubit IBM Quantum superconducting processors, we demonstrate an algorithmic quantum speedup for a variant of Simon’s problem where the hidden period has a restricted Hamming weight w. For sufficiently small values of w and for circuits involving up to 58 qubits, we demonstrate an exponential speedup, albeit of a lower quality than the speedup predicted for the noiseless algorithm. The speedup exponent and the range of w values for which an exponential speedup exists are significantly enhanced when the computation is protected by dynamical decoupling. Further enhancement is achieved with measurement error mitigation. This case constitutes a demonstration of a bona fide quantum advantage for an Abelian hidden subgroup problem.

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Popular Summary

One of the central goals of quantum computing is to demonstrate a clear algorithmic advantage over classical computers. This validation is essential not only for quantum theory but also for enabling future breakthroughs in fields such as cryptography, chemistry, and optimization. Yet, proving such an advantage on today’s noisy and imperfect quantum hardware remains a challenge. To tackle this issue, we implement a variation of Simon’s problem, a well-known example where quantum algorithms can, in theory, solve a task exponentially faster than any classical counterpart.

Simon’s problem involves finding a hidden repeating pattern in a mathematical function. For this study, we use IBM’s 127-qubit superconducting quantum processors. To adapt the problem for real quantum devices, we modify it and incorporate practical techniques such as error suppression via dynamical coupling and measurement error mitigation. These efforts help maintain quantum coherence and improve result accuracy despite hardware limitations. Our experiments successfully demonstrate an exponential speedup in solving the adapted Simon’s problem on actual quantum hardware—an important milestone for the field.

This result not only bridges the gap between theory and practice but also highlights the growing capabilities of current quantum processors. As hardware continues to improve, our approach paves the way for even more powerful demonstrations of quantum advantage in the near future.

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Phys. Rev. X 15, 021082– Published 5 June, 2025
  • Received 15 January 2024
  • Revised 12 January 2025
  • Accepted 28 April 2025
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