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Hidden symmetry detection on a quantum computer

Ralf Schützhold, William George Unruh

Published Apr 12, 2003• 0 citations
OpenAlex

Abstract

The fastest quantum algorithms (for the solution of classical computational problems) known so far are basically variations of the hidden subgroup problem f(U[x])=f(x). By means of some simple examples it is demonstrated that the detection of more general hidden (two-point) symmetries V{f(x),f(U[x])}=0 by a quantum computer can also admit an exponential speed-up. PACS: 03.67.Lx, 03.67.-a, 89.70.+c.

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Hidden symmetry detection on a quantum computer

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The fastest quantum algorithms (for the solution of classical computational problems) known so far are basically variations of the hidden subgroup problem f(U[x])=f(x). By means of some simple examples it is demonstrated that the detection of more general hidden (two-point) symmetries V{f(x),f(U[x])}=0 by a quantum computer can also admit an exponential speed-up. PACS: 03.67.Lx, 03.67.-a, 89.70.+c.
The fastest quantum algorithms (for the solution of classical computational tasks) known so far are basically variations of the hidden subgroup problem with {f(U[x])=f(x)f(U[x])=f(x)f(U[x])=f(x)}. Following a discussion regarding which tasks might be solved efficiently by quantum computers, it will be demonstrated by means of a simple example, that the detection of more general hidden (two-point) symmetries {V{f(x),f(U[x])}=0V\{f(x),f(U[x])\}=0V{f(x),f(U[x])}=0} by a quantum algorithm can also admit an exponential speed-up. E.g., one member of this class of symmetries {V{f(x),f(U[x])}=0V\{f(x),f(U[x])\}=0V{f(x),f(U[x])}=0} is discrete self-similarity (or discrete scale invariance). PACS: 03.67.Lx, 89.70.+c.
The fastest quantum algorithms (for the solution of classical computational tasks) known so far are basically variations of the hidden subgroup problem with {f(U[x])=f(x)f(U[x])=f(x)f(U[x])=f(x)}. Following a discussion regarding which tasks might be solved efficiently by quantum computers, it will be demonstrated by means of a simple example, that the detection of more general hidden (two-point) symmetries {V{f(x),f(U[x])}=0V\{f(x),f(U[x])\}=0V{f(x),f(U[x])}=0} by a quantum algorithm can also admit an exponential speed-up. E.g., one member of this class of symmetries {V{f(x),f(U[x])}=0V\{f(x),f(U[x])\}=0V{f(x),f(U[x])}=0} is discrete self-similarity (or discrete scale invariance).
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 search for practical problems that can be solved exponentially faster on a quantum computer is one of the primary goals in research into algorithms for quantum computers. It has long been known that such speedups are only possible for highly structured problems. We consider two problems with algebraic structure: (1) the hidden subgroup problem, where we have an unknown subgroup, a function that is constant on cosets of the subgroup, and the task is to find the unknown subgroup; (2) the hidden shift problem, where we have an unknown element of a group, a function that has been shifted by that group element, and the task is to find the unknown group element. We show that the Fourier transform is intimately connected to both these problems. This is because the Fourier transform respects group symmetries. For the hidden subgroup problem, we introduce the idea of using information theoretic techniques to design quantum algorithms. This consists of first finding an optimal set of measurements for distinguishing between different subgroups, followed by attempts to implement this set of measurements. We show that the standard algorithm for the abelian hidden subgroup problem is optimal. For the nonabelian hidden subgroup problem, the optimal measurements respect the Fourier transform in that they can be implemented by first performing a Fourier transform on each coset state, measuring the name of the irreducible representation for each state, followed by a joint measurement over the resulting lower dimensional space. This shows that the “strong standard method” cannot be used to implement the optimal set of measurements. For some instances of the hidden shift problem, we give a quantum algorithms that are exponentially faster than the best known classical algorithm. The algorithms make use of the Fourier transform in a way that may be interpreted as solving a deconvolution problem. As evidence of hardness, we give a reduction from algebraically homomorphic cryptosystems to the shifted Legendre symbol problem. This is an example of a problem which admits an exponential quantum speedup yet is not an instance of the abelian hidden subgroup problem.
Attempts to find new quantum algorithms that outperform classical computation have focused primarily on the nonAbelian hidden subgroup problem, which generalizes the central problem solved by Shor's factoring algorithm. We suggest an alternative generalization, namely to problems of finding hidden nonlinear structures over finite fields. We give examples of two such problems that can be solved efficiently by a quantum computer, but not by a classical computer. We also give some positive results on the quantum query complexity of finding hidden nonlinear structures.