Hidden Subgroup Problem(HSP) seeks to identify an unknown subgroup H of a group G for a given injective function f defined on cosets of H. Here we present an initialization-free quantum algorithm for solving HSP in the case where G is a finite abelian group. Our algorithm can adopt an arbitrary unknown mixed state as the auxiliary register and removes the need for initialization while preserving computational cost comparable to existing methods. Our algorithm also restores the state of the auxiliary register to its original form after completing the computations. Since the recovered state can be utilized for other operations, a single preparation of the auxiliary register in an arbitrarily unknown mixed state is sufficient to execute the iterative procedure in solving hidden subgroup problems. This approach provides a promising direction for improving quantum algorithm efficiency by reducing operational time of initialization.
Hidden Subgroup Problem(HSP) seeks to identify an unknown subgroup H of a group G for a given injective function f defined on cosets of H. Here we present an initialization-free quantum algorithm for solving HSP in the case where G is a finite abelian group. Our algorithm can adopt an arbitrary unknown mixed state as the auxiliary register and removes the need for initialization while preserving computational cost comparable to existing methods. Our algorithm also restores the state of the auxiliary register to its original form after completing the computations. Since the recovered state can be utilized for other operations, a single preparation of the auxiliary register in an arbitrarily unknown mixed state is sufficient to execute the iterative procedure in solving hidden subgroup problems. This approach provides a promising direction for improving quantum algorithm efficiency by reducing operational time of initialization.
Many exponential speedups that have been achieved in quantum computing are obtained via hidden subgroup problems (HSPs). We show that the HSP over Weyl-Heisenberg groups can be solved efficiently on a quantum computer. These groups are well-known in physics and play an important role in the theory of quantum error-correcting codes. Our algorithm is based on non-commutative Fourier analysis of coset states which are quantum states that arise from a given black-box function. We use Clebsch-Gordan decompositions to combine and reduce tensor products of irreducible representations. Furthermore, we use a new technique of changing labels of irreducible representations to obtain low-dimensional irreducible representations in the decomposition process. A feature of the presented algorithm is that in each iteration of the algorithm the quantum computer operates on two coset states simultaneously. This is an improvement over the previously best known quantum algorithm for these groups which required four coset states.
We study quantum algorithms for the hidden subgroup problem (HSP). The HSP provides a unified framework for many important problems, including factoring integers, graph isomorphism problem, etc. It has been known that the problem for abelian groups can be efficiently solved. However, for non-abelian groups, it remains widely open. In this dissertation we study the problem for some specific non-abelian groups, called q-hedral groups.
We start with a quantum algorithm, proposed by Moore et al. (2003), for finding some hidden conjugate subgroups of affine groups, a special case of q-hedral groups for large q. We then adapt their algorithm to construct an algorithm for solving the hidden conjugate problem for q-hedral groups for large q and point out a difficulty in complexity analysis that involves certain exponential sums. We discuss certain bounds on these sums and present some computational data on their values to make a conjecture.
For small q, Regev (2003) gave an efficient algorithm for solving the HSP for dihedral groups (q = 2) using an oracle that solves a certain subset sum problem (SSP). A crucial step of his algorithm relies on the probability that elements in a cyclic group ZN can be expressed as sums of random elements in the group. We generalize this problem to any finite group and prove the following theorem: for any finite group G and r ≥ 32 log2 |G| + 5, if r elements from G are chosen randomly, then the 2r possible sums (or products) of the r elements will cover all the elements in G with high probability. Also, by making a further generalization of the SSP, we show that Regev's algorithm can be generalized for q-hedral groups for any fixed q. We also study this generalized subset sum problem.
In addition to the algorithm developing for small q, we give an efficient quantum algorithm for solving the coset problem and an efficient quantum algorithm for solving the hidden conjugate problem. Furthermore, we also give a survey of an algorithm for finding the largest normal subgroup in a hidden subgroup of any finite group, introduced by Hallgren et al. (2000) and provide a simplified and unified proof.
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⋊φZpk, with A finite Abelian, p prime, k∈N and the action of Zpk is generated by the scalar automorphism a↦μa, for some μ∈ZExp(A)×, where Exp(A) is the exponent of A. Our algorithm is efficient when A has bounded generator rank and Exp(A)/p=polylog(∣G∣). This includes the case A=ZN for N∈N and k=1, studied by Bacon, Childs and van Dam (FOCS 2005), and A=Zqr with q prime and 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) has been widely studied in the context of quantum computing and is known to be efficiently solvable for Abelian groups, yet appears to be difficult for many non-Abelian ones. An efficient algorithm for the HSP over a group \f G\ runs in time polynomial in \f{n\deq\log|G|.} For any subgroup \f H\ of \f G, let \f{N(H)} denote the normalizer of \f H. Let \MG\ denote the intersection of all normalizers in \f G (i.e., \f{\MG=\cap_{H\leq G}N(H)}). \MG\ is always a subgroup of \f G and the index \f{[G:\MG]} can be taken as a measure of ``how non-Abelian'' \f G is (\f{[G:\MG] = 1} for Abelian groups). This measure was considered by Grigni, Schulman, Vazirani and Vazirani, who showed that whenever \f{[G:\MG]\in\exp(O(\log^{1/2}n))} the corresponding HSP can be solved efficiently (under certain assumptions). We show that whenever \f{[G:\MG]\in\poly(n)} the corresponding HSP can be solved efficiently, under the same assumptions (actually, we solve a slightly more general case of the HSP and also show that some assumptions may be relaxed).
The hidden subgroup problem~(HSP) is one of the most important problems in quantum computation. Many problems for which quantum algorithm achieves exponential speedup over its classical counterparts can be reduced to the Abelian HSP. However, there is no efficient quantum algorithm for solving the non-Abelian HSP. We find that the HSP can be reduced to a nested structured search problem that is solved efficiently by using a quantum algorithm via multistep quantum computation. Then we solve the HSP and problems that can be reduced to both the Abelian and the non-Abelian HSP in polynomial time by using this algorithm.
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.