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We have taken significant steps towards the realization of a practical quantum computer: using nuclear spins and magnetic resonance techniques at room temperature, we provided proof of principle of quantum computing in a series of…
We employ concepts and tools from the theory of finite permutation groups in order to analyse the Hidden Subgroup Problem via Quantum Fourier Sampling (QFS) for the symmetric group. We show that under very general conditions both the weak…
We give an algorithm to solve the quantum hidden subgroup problem for maximal cyclic non-normal subgroups of the affine group of a finite field (if the field has order $q$ then the group has order $q(q-1)$) with probability $1-\varepsilon$…
The prime numbers have been a source of fascination for millenia and continue to surprise us. Motivated by the hyperuniformity concept, which has attracted recent attention in physics and materials science, we show that the prime numbers in…
Following the example of Shor's algorithm for period-finding in the integers, we explore the hidden subgroup problem (HSP) for discrete infinite groups. On the hardness side, we show that HSP is NP-hard for the additive group of rational…
It has been shown that for a certain special type of quantum graphs the random-matrix form factor can be recovered to at least third order in the scaled time \tau using periodic-orbit theory. Two types of contributing pairs of orbits were…
Graded rings provide a natural algebraic framework for encoding symmetry via decompositions into homogeneous components indexed by a group, together with multiplication rules reflecting the group operation. Among graded rings, strongly…
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…
A precise estimation of the computational complexity in Shor's factoring algorithm under the condition that the large integer we want to factorize is composed by the product of two prime numbers, is derived by the results related to number…
The theory of finite simple groups is a (rather unexplored) area likely to provide interesting computational problems and modelling tools useful in a cryptographic context. In this note, we review some applications of finite non-abelian…
Grover's quantum algorithm can find a marked item from an unstructured database faster than any classical algorithm, and hence it has been used for several applications such as cryptanalysis and optimization. When there exist multiple…
A quantum computer can efficiently find the order of an element in a group, factors of composite integers, discrete logarithms, stabilisers in Abelian groups, and `hidden' or `unknown' subgroups of Abelian groups. It is already known how to…
In this review article, we discuss connections between the physics of disordered systems, phase transitions in inference problems, and computational hardness. We introduce two models representing the behavior of glassy systems, the spiked…
Quantum algorithms use the principles of quantum mechanics, as for example quantum superposition, in order to solve particular problems outperforming standard computation. They are developed for cryptography, searching, optimisation,…
We study planted problems---finding hidden structures in random noisy inputs---through the lens of the sum-of-squares semidefinite programming hierarchy (SoS). This family of powerful semidefinite programs has recently yielded many new…
To accelerate the algorithms for the dihedral hidden subgroup problem, we present a new algorithm based on algorithm SV(shortest vector). A subroutine is given to get a transition quantum state by constructing a phase filter function, then…
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.…
(Abridged abstract.) In this thesis we introduce new models of quantum computation to study the emergence of quantum speed-up in quantum computer algorithms. Our first contribution is a formalism of restricted quantum operations, named…
In condensed matter systems, the atoms, electrons or spins can sometimes arrange themselves in ways that result in unexpected properties but that cannot be detected by conventional experimental probes. Several historical and contemporary…
We describe how orbital graphs can be used to improve the practical performance of many algorithms for permutation groups, including intersection and stabilizer problems. First we explain how orbital graphs can be integrated in partition…