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IEEE Access

Quantum Algorithm for Solving the Continuous Hidden Symmetry Subgroup Problem

Eunok Bae, Soojoon Lee

Published Jan 1, 2021• 0 citations
Publisher

Abstract

Quantum computers are expected to be able to outperform classical computers. In fact, some computational problems such as integer factorization can be solved on quantum computers substantially faster than classical computers. Interestingly, these problems can be cast in a framework of the hidden symmetry subgroup problem. However, only a few of quantum algorithms for efficiently solving this problem have been known, and the approaches used in all previous results can be applied to particular groups with specific group actions. In this paper, we introduce new technique for solving the hidden symmetry subgroup problem which can be applicable for any groups and any group actions with a certain condition. In addition, we define the continuous hidden symmetry subgroup problem on a group by employing a continuous oracle function, and prove that if the group is a metric space and the group action satisfies some condition, then the continuous hidden symmetry subgroup problem can be efficiently reduced to the continuous hidden subgroup problem. In particular, we show that there exists an efficient quantum algorithm to solve the continuous hidden symmetry subgroup problem on$\mathbb {R}^{n}$, while it has not yet been shown that the original hidden symmetry subgroup problem on$\mathbb {R}^{n}$can be efficiently solved by a quantum computer.

DOI
10.1109/access.2021.3092723

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Quantum computers are expected to be able to outperform classical computers. In fact, some computational problems such as integer factorization can be solved on quantum computers substantially faster than classical computers. Interestingly, these problems can be cast in a framework of the hidden symmetry subgroup problem. However, only a few of quantum algorithms for efficiently solving this problem have been known, and the approaches used in all previous results can be applied to particular groups with specific group actions. In this paper, we introduce new technique for solving the hidden symmetry subgroup problem which can be applicable for any groups and any group actions with a certain condition. In addition, we define the continuous hidden symmetry subgroup problem on a group by employing a continuous oracle function, and prove that if the group is a metric space and the group action satisfies some condition, then the continuous hidden symmetry subgroup problem can be efficiently reduced to the continuous hidden subgroup problem. In particular, we show that there exists an efficient quantum algorithm to solve the continuous hidden symmetry subgroup problem onRn\mathbb {R}^{n}Rn, while it has not yet been shown that the original hidden symmetry subgroup problem onRn\mathbb {R}^{n}Rncan be efficiently solved by a quantum computer.
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.
The hidden subgroup problem (HSP) plays an important role in quantum computation, because many quantum algorithms that are exponentially faster than classical algorithms are special cases of the HSP. In this paper we show that there exist a new efficient quantum algorithm for the HSP on groups ZN⋊Zqs\Z_{N}\rtimes\Z_{q^s}ZN​⋊Zqs​ where NNN is an integer with a special prime factorization, qqq prime number and sss any positive integer.
We give an overview of the Hidden Subgroup Problem (HSP) as of July 2010, including new results discovered since the survey of arXiv:quant-ph/0411037v1. We recall how the problem provides a framework for efficient quantum algorithms and present the standard methods based on coset sampling. We study the Dihedral and Symmetric HSPs and how they relate to hard problems on lattices and graphs. Finally, we conclude with the known solutions and techniques, describe connections with efficient algorithms as well as miscellaneous variants of HSP. We also bring various contributions to the topic. We show that in theory, we can solve HSP over a given group inductively: the base case is solving HSP over its simple factor groups and the inductive step is building efficient oracles over a normal subgroup N and over the factor group G/N. We apply this analysis to the Dedekindian HSP to get an alternative abelian HSP algorithm based on a change of the underlying group. We also propose a quotient reduction by the normal group obtained via Weak Fourier Sampling. We compute the exact expression of Strong Fourier Sampling over the dihedral group, showing how the previous reduction is natural and matches the standard one. We also give a reduction of rigid graph isomorphism problem to HSP over the alternating group. For this group and other simple groups, we propose maximal subgroup reduction as a possible approach. We also analyse Regev's algorithm for the poly(n)-uniqueSVP, prove how the degree of the polynomial is related to the oracle complexity used and we suggest several variants.
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.
Quantum algorithms are powerful tools for finding symmetries of classical objects, most famously through Shor's algorithm and the Hidden Subgroup Problem (HSP). In this work, we study quantum algorithms for learning symmetries of \emph{quantum} objects. Existing work in this area centres on 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 develop efficient quantum algorithms for non-abelian StateHSP when the hidden subgroup is normal and the ambient group belongs to a broad class of non-abelian groups, extending the previous general theory beyond the abelian setting. StateHSP learns \emph{Bose} symmetries, under which a state must be invariant exactly under the action of the symmetry group. In quantum mechanics, however, physically equivalent pure states are defined only up to global phase. Motivated by this, we introduce a natural notion of \emph{Anyonic} state symmetry learning, based on invariance up to global phase. We give an efficient quantum algorithm by reducing the problem to StateHSP, where the reduction rests on a new correspondence between linearisations of projective representations and linear error-correcting codes. As an application, we obtain an improved algorithm for learning stabiliser groups of 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 give efficient algorithms for them by reduction to state symmetry learning. Together, these results broaden the scope of state symmetry learning as a common algorithmic primitive for learning quantum symmetries.