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arXiv:2609.38085

Learning quantum symmetries

Isaac Holt, Sathyawageeswar Subramanian

quant-phSubmitted Sep 29, 2026• Announced Sep 30, 2026• 0 citations
v1 · Sep 29, 2026v2 · Sep 30, 2026
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Abstract

Quantum algorithms are powerful tools for finding symmetries of classical objects, most famously through Shor's algorithm and the framework of the Hidden Subgroup Problem (HSP). In this work, we study quantum algorithms for learning symmetries of quantum objects. Our starting point is the recently introduced State Hidden Subgroup Problem (StateHSP), a quantum generalisation of HSP in which the task is to learn the symmetry subgroup of a quantum state. We obtain positive results for non-abelian StateHSP, giving efficient quantum algorithms whenever the hidden subgroup is normal and for ambient groups belonging to a broad class of non-abelian groups, extending the previous general theory beyond the abelian setting. StateHSP learns \emph{Bose} symmetries, associated with linear representations, whereas physically equivalent pure states are defined only up to global phase. Motivated by this, we introduce a new \emph{Anyonic} state symmetry learning problem based on a more physically natural notion of symmetry than that considered by StateHSP. We give an efficient quantum algorithm by reducing the problem to StateHSP, where the reduction relies on a new connection between linearisations of projective representations and linear error-correcting codes. As an application, we obtain an improved algorithm for learning stabiliser groups that applies to mixed qudit states of arbitrary local dimension. Finally, we introduce symmetry learning problems for other quantum objects, including unitaries, Hamiltonians, and finite collections of states, and obtain efficient quantum algorithms by reducing them to instances of StateHSP.

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50 pages, 1 figure

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Quantum algorithms are powerful tools for finding symmetries of classical objects, most famously through Shor's algorithm and the framework of the Hidden Subgroup Problem (HSP). In this work, we study quantum algorithms for learning symmetries of quantum objects. Our starting point is the recently introduced State Hidden Subgroup Problem (StateHSP), a quantum generalisation of HSP in which the task is to learn the symmetry subgroup of a quantum state. We obtain positive results for non-abelian StateHSP, giving efficient quantum algorithms whenever the hidden subgroup is normal and for ambient groups belonging to a broad class of non-abelian groups, extending the previous general theory beyond the abelian setting.
StateHSP learns \emph{Bose} symmetries, associated with linear representations, whereas physically equivalent pure states are defined only up to global phase. Motivated by this, we introduce a new \emph{Anyonic} state symmetry learning problem based on a more physically natural notion of symmetry than that considered by StateHSP. We give an efficient quantum algorithm by reducing the problem to StateHSP, where the reduction relies on a new connection between linearisations of projective representations and linear error-correcting codes. As an application, we obtain an improved algorithm for learning stabiliser groups that applies to mixed qudit states of arbitrary local dimension. Finally, we introduce symmetry learning problems for other quantum objects, including unitaries, Hamiltonians, and finite collections of states, and obtain efficient quantum algorithms by reducing them to instances of StateHSP.
Identifying the symmetry properties of quantum states is a central theme in quantum information theory and quantum many-body physics. In this work, we investigate quantum learning problems in which the goal is to identify a hidden symmetry of an unknown quantum state. Building on the recent formulation of the state hidden subgroup problem (StateHSP), we focus on abelian groups and develop an efficient quantum algorithm that learns any hidden symmetry subgroup using a generalized form of Fourier sampling. We showcase the versatility of the approach in three concrete applications: These are learning (i) qubit and qudit stabilizer groups, (ii) cuts along which a state is unentangled, and (iii) hidden translation symmetries. Through these applications, we reveal that well-known quantum learning primitives, such as Bell sampling and Bell difference sampling, are, in fact, special cases of Fourier sampling. Our results highlight the broad potential of the StateHSP framework for symmetry-based quantum learning tasks and provide protocols that are easier to implement on near-term quantum devices.
We advocate a new approach for addressing hidden structure problems and finding efficient quantum algorithms. We introduce and investigate the hidden symmetry subgroup problem (HSSP), which is a generalization of the well-studied hidden subgroup problem (HSP). Given a group acting on a set and an oracle whose level sets define a partition of the set, the task is to recover the subgroup of symmetries of this partition inside the group. The HSSP provides a unifying framework that, besides the HSP, encompasses a wide range of algebraic oracle problems, including quadratic hidden polynomial problems. While the HSSP can have provably exponential quantum query complexity, we obtain efficient quantum algorithms for various interesting cases. To achieve this, we present a general method for reducing the HSSP to the HSP, which works efficiently in several cases related to symmetries of polynomials. The HSSP therefore connects in a rather surprising way certain hidden polynomial problems with the HSP. Using this connection, we obtain the first efficient quantum algorithm for the hidden polynomial problem for multivariate quadratic polynomials over fields of constant characteristic. We also apply the new methods to polynomial function graph problems and present an efficient quantum procedure for constant degree multivariate polynomials over any field. This result improves in several ways the currently known algorithms.
Several early quantum algorithms, including Simon's algorithm and Shor's period-finding are instances of the hidden subgroup problem (HSP) over finite abelian groups. No polynomial-time quantum algorithm is known for the HSP over arbitrary non-abelian finite groups. The non-Abelian case is of particular interest because some instances, such as the dihedral and symmetric group HSPs, are connected to lattice problems and graph isomorphism, respectively. In this work, we give polynomial-time quantum algorithms for two further families containing non-Abelian groups. First, we consider groups of the form G=A⋊φZpkG=A\rtimes_{\varphi} \mathbb{Z}_{p^k}G=A⋊φ​Zpk​, with AAA finite Abelian, ppp prime, k∈Nk\in \mathbb{N}k∈N and the action of Zpk\mathbb Z_{p^k}Zpk​ is generated by the scalar automorphism a↦μaa\mapstoμaa↦μa, for some μ∈ZExp⁡(A)×μ\in\mathbb Z_{\operatorname{Exp}(A)}^\timesμ∈ZExp(A)×​, where Exp(A)\mathrm{Exp}(A)Exp(A) is the exponent of AAA. Our algorithm is efficient when AAA has bounded generator rank and Exp(A)/p=polylog(∣G∣)\mathrm{Exp}(A)/p=\mathrm{polylog}(|G|)Exp(A)/p=polylog(∣G∣). This includes the case A=ZNA=\mathbb{Z}_NA=ZN​ for N∈NN\in\mathbb{N}N∈N and k=1k=1k=1, studied by Bacon, Childs and van Dam (FOCS 2005), and A=ZqrA=\mathbb{Z}_{q^r}A=Zqr​ with qqq prime and r∈Nr\in \mathbb{N}r∈N studied by van Dam and Dey (TQC 2014). Second, we give a polynomial-time quantum algorithm for finite quasi-Hamiltonian groups under a mild assumption on the input structure. Quasi-Hamiltonian groups are finite nilpotent groups with modular subgroup lattice, or equivalently the finite groups in which every subgroup is permutable. As far as we know, this is the first quantum algorithm to exploit the modularity of the subgroup lattice for solving the HSP. This extends, under the aforementioned structured input assumption, the quantum algorithm for Dedekind groups given by Hallgren, Russell, and Ta-Shma (SIAM J. Comput. 32, 2003).
The Hidden Subgroup Problem (HSP) provides the fundamental framework for most quantum algorithms. Until very recently, all known problems where quantum computation provides a super-polynomial speedup over classical algorithms have been variants of the HSP for particular abelian groups. Examples include factoring integers and computing the discrete log, for which Shor found efficient quantum algorithms. The algorithm for these problems may be viewed as consisting of the main HSP solution plus an ad hoc method of dealing with the particular variant. In this dissertation, we give a systematic way of dealing with all these variants. The key component of our solution is a new theorem about the robustness of Fourier sampling. The purpose of this theorem is to better understand the structure underlying existing algorithms as well as to provide an algorithmic tool for the construction of future quantum algorithms. In addition, we also derive a new algorithm for computing the quantum Fourier transform which is asymptotically faster than any previously known algorithm. By contrast, the nonabelian HSP is wide open. It includes as a special case the longstanding open question of graph isomorphism, where the group is the symmetric group, S n . It is natural to carry over the abelian HSP algorithm to the nonabelian case. We show that in the case that the hidden subgroup is normal, this algorithm succeeds in reconstructing the subgroup in polynomial time. On the other hand, we give evidence that this algorithm is inadequate to even distinguish a trivial subgroup from an involution in t...
We study the computational complexity of quantum state isomorphism problems under group actions: given two quantum circuits that prepare pure or mixed states, decide whether the two states are related by a group action. This can be seen as a quantum state version of the Hidden Shift Problem, in much the same way that the State Hidden Subgroup Problem is a quantum version of the ordinary Hidden Subgroup Problem. We prove several results for this computational problem: - For the pure-state version, we show that the problem is BQP-hard for all nontrivial groups, and contained in QCMA ∩\cap∩ QCSZK. We further obtain refined results for specific groups of interest: for abelian groups we show that the problem reduces to the state hidden subgroup problem over the generalized dihedral group; for the Clifford group, the problem is at least as hard as Graph Isomorphism under polynomial-time reductions; for the Pauli group it is BQP-complete. - For the mixed-state version, for nontrivial, finite and efficiently representable groups, the problem is QSZK-complete. - We also study a variant of this problem over an infinite group, in particular, the bosonic linear optical unitaries. We show that in the setting where the classical description of the quantum state is given in a suitable wave function representation known as the stellar representation, the problem is at least as hard as Graph Isomorphism, and is contained in NP ∩\cap∩ SZK. Prior to our work, state isomorphism problems had only been studied for the symmetric group [LG17]. As a consequence of our results, we resolve an open question posed in [HEC25] about the existence of a quantum algorithm for the abelian state hidden subgroup problem on mixed states. We show that this problem is QSZK-hard in the worst case, thereby ruling out an efficient quantum algorithm unless QSZK = BQP.