We study a generalization of entanglement testing which we call the “hidden cut problem.” Taking as input copies of an n-qubit pure state which is product across an unknown bipartition, the goal is to learn precisely where the state is unentangled, i.e. to determine which of the exponentially many possible cuts separates the state. We give a polynomial-time quantum algorithm which can find the cut using O(n/є2) many copies of the state, which is optimal up to logarithmic factors. Our algorithm also generalizes to learn the entanglement structure of arbitrary product states. In the special case of Haar-random states, we further show that our algorithm requires circuits of only constant depth. To develop our algorithm, we introduce a state generalization of the hidden subgroup problem (StateHSP) which might be of independent interest, in which one is given a quantum state invariant under an unknown subgroup action, with the goal of learning the hidden symmetry subgroup. We show how the hidden cut problem can be formulated as a StateHSP with a carefully chosen Abelian group action. We then prove that Fourier sampling on the hidden cut state produces similar outcomes as a variant of the well-known Simon’s problem, allowing us to find the hidden cut efficiently. Therefore, our algorithm can be interpreted as an extension of Simon’s algorithm to entanglement testing. We discuss possible applications of StateHSP and hidden cut problems to cryptography and pseudorandomness.
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.
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
Rn, while it has not yet been shown that the original hidden symmetry subgroup problem on
Rncan be efficiently solved by a quantum computer.
Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which an unknown subgroup H of a group G must be determined from a quantum state
ψ over G that is uniformly supported on a left coset of H. These hidden subgroup problems are typically solved by Fourier sampling: the quantum Fourier transform of
ψ is computed and measured. When the underlying group is nonabelian, two important variants of the Fourier sampling paradigm have been identified: the weak standard method, where only representation names are measured, and the strong standard method, where full measurement (i.e., the row and column of the representation, in a suitably chosen basis, as well as its name) occurs. It has remained open whether the strong standard method is indeed stronger, that is, whether there are hidden subgroups that can be reconstructed via the strong method but not by the weak, or any other known, method. In this article, we settle this question in the affirmative. We show that hidden subgroups H of the q-hedral groups, i.e., semidirect products
Zq⋉Zp, where
q∣(p−1), and in particular the affine groups
Ap, can be information-theoretically reconstructed using the strong standard method. Moreover, if
∣H∣=p/polylog(p), these subgroups can be fully reconstructed with a polynomial amount of quantum and classical computation. We compare our algorithms to two weaker methods that have been discussed in the literature—the “forgetful” abelian method, and measurement in a random basis—and show that both of these are weaker than the strong standard method. Thus, at least for some families of groups, it is crucial to use the full power of representation theory and nonabelian Fourier analysis, namely, to measure the high-dimensional representations in an adapted basis that respects the group's subgroup structure. We apply our algorithm for the hidden subgroup problem to new families of cryptographically motivated hidden shift problems, generalizing the work of van Dam, Hallgren, and Ip on shifts of multiplicative characters. Finally, we close by proving a simple closure property for the class of groups over which the hidden subgroup problem can be solved efficiently.
Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which a subgroup H of a group G must be determined from a quantum state y uniformly supported on a left coset of H. These hidden subgroup problems are then solved by Fourier sampling: the quantum Fourier transform of y is computed and measured. When the underlying group is non-Abelian, two important variants of the Fourier sampling paradigm have been identified: the weak standard method, where only representation names are measured, and the strong standard method, where full measurement occurs. It has remained open whether the strong standard method is indeed stronger, that is, whether there are hidden subgroups that can be reconstructed via the strong method but not by the weak, or any other known, method. In this article, we settle this question in the affirmative. We show that hidden subgroups of semidirect products of Z_p by Z_q, where q divides (p-1) and q = p / polylog(p), can be efficiently determined by the strong standard method. Furthermore, the weak standard method and the ``forgetful'' Abelian method are insufficient for these groups. We extend this to an information-theoretic solution for the hidden subgroup problem over semidirect products of Z_p by \Z_q where q divides (p-1) and, in particular, the Affine groups A_p. Finally, we prove a closure property for the class of groups over which the hidden subgroup problem can be solved efficiently.
After Shor's discovery of an efficient quantum algorithm for integer factoring, hidden subgroup problems play a central role in developing efficient quantum algorithms. In spite of many intensive studies, no efficient quantum algorithms are known for hidden subgroup problems for many non-Abelian groups. Of particular interest are the hidden subgroup problems for the symmetric group and for the dihedral group, because an efficient algorithm for the former implies an efficient solution to the graph isomorphism problem, and that for the latter essentially solves a certain lattice-related problem whose hardness is assumed in cryptography. This paper focuses on the latter case and gives a comprehensive survey of known facts related to the dihedral hidden subgroup problem.