Measurements are a primitive for characterizing quantum systems. Reducing the time taken to perform a measurement may be beneficial in many areas of quantum information processing. We show that permuting the eigenvalues of the state matrix in the logical basis, using open-loop control, provides an O(n) reduction in the measurement time, where n is the number of qubits in the register. This reduction is of the same order as the (previously introduced) locally optimal feedback protocol. The advantage of the open-loop protocol is that it is far less difficult experimentally. Because the control commutes with the measured observable at all times, our rapid measurement protocol could be used for characterizing a quantum system, by state or process tomography, or to implement measurement-based quantum error correction.

I describe the role of symmetry in two quantum algorithms, with a focus on how that symmetry is made manifest by the Fourier transform. The Fourier transform can be considered in a wider context than the familiar one of functions on Rn or Z/nZ ; instead it can be defined for an arbitrary group where it is known as representation theory. The first quantum algorithm solves an instance of the hidden subgroup problem—distinguishing conjugates of the Borel subgroup from each other in groups related to PSL(2; q). I use the symmetry of the subgroups under consideration to reduce the problem to a mild extension of a previously solved problem. This generalizes a result of Moore, Rockmore, Russel and Schulman by switching to a more natural measurement that also applies to prime powers. In contrast to the first algorithm, the second quantum algorithm is an attempt to use naturally continuous spaces. Quantum walks have proved to be a useful tool for designing quantum algorithms. The natural equivalent to continuous time quantum walks is evolution with the Schrödinger equation, under the kinetic energy Hamiltonian for a massive particle. I take advantage of quantum interference to find the center of spherical shells in high dimensions. Any implementation would be likely to take place on a discrete grid, using the ability of a digital quantum computer to simulate the evolution of a quantum system. In addition, I use ideas from the second algorithm on a different set of starting states, and find that quantum evolution can be used to sample from the evolute of a plane curve. The method of stationary phase is used to determine scaling exponents characterizing the precision and probability of success for this procedure.

We reduce a case of the hidden subgroup problem (HSP) in SL(2; q), PSL(2; q), and PGL(2; q), three related families of finite groups of Lie type, to efficiently solvable HSPs in the affine group AGL(1; q). These groups act on projective space in an "almost" 3-transitive way, and we use this fact in each group to distinguish conjugates of its Borel (upper triangular) subgroup, which is also the stabilizer subgroup of an element of projective space. Our observation is mainly group-theoretic, and as such breaks little new ground in quantum algorithms. Nonetheless, these appear to be the first positive results on the HSP in finite simple groups such as PSL(2; q).

We reduce a case of the hidden subgroup problem (HSP) in several families of finite groups of Lie type, such as SL(2; q), PSL(2; q), and PGL(2; q), to efficiently solvable HSPs in the affine group AGL(1; q). These groups act on projective space in an “almost” 3-transitive way, and we use this fact in each group to distinguish conjugates of its Borel (upper triangular) subgroup. Our observation is purely grouptheoretic, and as such breaks little new ground in quantum algorithms. Nonetheless, these appear to be the first positive results on the HSP in finite simple groups such as PSL(2; q).

We reduce a case of the hidden subgroup problem (HSP) in several families of finite groups of Lie type, such as SL2(Fq), PSL2(Fq), and PGL2(Fq), to efficiently solvable HSPs in the affine group AGL 1(Fq). These groups act on projective space in an "almost" 3-transitive way, and we use this fact in each group to distinguish conjugates of its Borel (upper triangular) subgroup. Our observation is purely group- theoretic, and as such does not break new ground in quantum algorithms. Nonetheless, these appear to be the first positive results on the HSP in finite simple groups such as PSL2(Fq).