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Quantum Computing in Spin-Adapted Representations for Efficient Simulations of Spin Systems

Anthony Gandon1,2,*, Alberto Baiardi1, Max Rossmannek1, Werner Dobrautz3,4,5, and Ivano Tavernelli1

  • *Contact author: anthony.gandon@ibm.com
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PRX Quantum 6, 030306 – Published 14 July, 2025

DOI: https://doi.org/10.1103/dbnd-sl4j

Abstract

Exploiting inherent symmetries is a common and effective approach to speed up the simulation of quantum systems. However, efficiently accounting for non-Abelian symmetries, such as the global SU(2) spin symmetry, remains a major challenge. In fact, expressing total-spin eigenstates in terms of the computational basis can require an exponentially large number of coefficients. In this work, we introduce a novel formalism for designing quantum algorithms directly in an eigenbasis of the total-spin operator. Our strategy relies on the symmetric group approach in conjunction with a truncation scheme for the internal degrees of freedom of total-spin eigenstates. For the case of the antiferromagnetic Heisenberg model, we show that this formalism yields a hierarchy of spin-adapted Hamiltonians, for each truncation threshold, whose ground-state energy and wave function quickly converge to their exact counterparts, calculated on the full model. These truncated Hamiltonians can be encoded with sparse and local qubit Hamiltonians that are suitable for quantum simulations. We demonstrate this by developing a state-preparation schedule to construct shallow quantum-circuit approximations, expressed in a total-spin eigenbasis, for the ground states of the Heisenberg Hamiltonian in different symmetry sectors.

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Popular Summary

Many quantum systems obey global symmetries—such as conservation of total spin—that strongly influence their physical behavior. While classical simulation methods often exploit these symmetries to improve efficiency, incorporating them directly into quantum algorithms has remained a significant challenge. In this work, we present a novel approach for representing spin systems on quantum computers that explicitly respects global spin symmetry from the outset.

We introduce a novel approximate encoding of spin eigenstates based on a truncation of the permissible values for the internal spin quantum number. The proposed truncation scheme preserves the variational principle, and it is especially accurate for antiferromagnetic Hamiltonians, for which the ground-state energy converges remarkably fast, with respect to the truncated subspace size, to the exact energy. Combining the newly proposed state encoding and the truncation of height variables, we derive qubit Hamiltonians that are local and sparse and that well describe the low-energy spectrum of the antiferromagnetic Heisenberg model. Based on this framework, we derive shallow quantum circuits for preparing approximate ground-state wave functions that exactly satisfy the global symmetries of the system. The states prepared with our strategy can be sampled directly in the more compact symmetry eigenbasis, without requiring nonlocal information-theoretical-based transformations.

While this present work focuses on spin systems, it can find application also to fermionic systems that possess the respective global spin symmetry. This will be subject to future investigations.

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PRX Quantum 6, 030306– Published 14 July, 2025
  • Received 30 January 2025
  • Revised 30 May 2025
  • Accepted 2 June 2025
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