An information geometric viewpoint of algorithms in quantum computing

Physica A: Statistical Mechanics and its Applications

2017

The art of quantum algorithm design is highly nontrivial. Grover's search algorithm constitutes a masterpiece of quantum computational software. In this article, we use methods of geometric algebra (GA) and information geometry (IG) to enhance the algebraic efficiency and the geometrical significance of the digital and analog representations of Grover's algorithm, respectively. Specifically, GA is used to describe the Grover iterate and the discretized iterative procedure that exploits quantum interference to amplify the probability amplitude of the target-state before measuring the query register. The transition from digital to analog descriptions occurs via Stone's theorem which relates the (unitary) Grover iterate to a suitable (Hermitian) Hamiltonian that controls Schrodinger's quantum mechanical evolution of a quantum state towards the target state. Once the discrete-to-continuos transition is completed, IG is used to interpret Grover's iterative procedure as a geodesic path on the manifold of the parametric density operators of pure quantum states constructed from the continuous approximation of the parametric quantum output state in Grover's algorithm. Finally, we discuss the dissipationless nature of quantum computing, recover the quadratic speedup relation, and identify the superfluity of the Walsh-Hadamard operation from an IG perspective with emphasis on statistical mechanical considerations.

Section: B Rotationsmentioning

confidence: 99%

“…We observe that for α = β = π we recover the matrix representation of the original Grover iterate (see Eq. (43) in Section III).…”

Section: θ) a Limitation Of This Search Scheme Is That Successmentioning

confidence: 99%

Geometric algebra and information geometry for quantum computational software

Physica A: Statistical Mechanics and its Applications

2017

“…Depending on both the parametric and temporal form of the two interpolation functions used to define the time-dependent Hamiltonian, they presented two search algorithms exhibiting different features. In particular, while both of them exhibited the typical O √ N Grover-like scaling behavior in finding the target state in an N -dimensional search space, one of the algorithms required a specific time to make the measurement and find the target state (periodic oscillatory behavior and original Grover search algorithm, [9][10][11]). The other one, instead, caused the source state to evolve asymptotically into a quantum state that overlapped with the target state with high probability (monotonic behavior and fixed-point Grover search algorithm, [12,13]).…”

Section: Introductionmentioning

confidence: 99%

Continuous-time quantum search and time-dependent two-level quantum systems

Int. J. Quantum Inform.

Alsing

2019

It was recently emphasized by Byrnes, Forster, and Tessler [Phys. Rev. Lett. 120, 060501 (2018)] that the continuous-time formulation of Grover's quantum search algorithm can be intuitively understood in terms of Rabi oscillations between the source and the target subspaces.In this work, motivated by this insightful remark and starting from the consideration of a timeindependent generalized quantum search Hamiltonian as originally introduced by Bae and Kwon [Phys. Rev. A 66, 012314 (2002)], we present a detailed investigation concerning the physical connection between quantum search Hamiltonians and exactly solvable time-dependent two-level quantum systems. Specifically, we compute in an exact analytical manner the transition probabilities from a source state to a target state in a number of physical scenarios specified by a spin-1/2 particle immersed in an external time-dependent magnetic field. In particular, we analyze both the periodic oscillatory as well as the monotonic temporal behaviors of such transition probabilities and, moreover, explore their analogy with characteristic features of Grover-like and fixed-point quantum search algorithms, respectively. Finally, we discuss from a physics standpoint the connection between the schedule of a search algorithm, in both adiabatic and nonadiabatic quantum mechanical evolutions, and the control fields in a time-dependent driving Hamiltonian.Recently, Dolzell, Yoder, and Chuang investigated in detail the possibility for adiabatic quantum search algorithms to exhibit the fixed-point property in addition to quadratic quantum speedup [14]. From their theoretical analysis, supported by a number of illustrative examples, they concluded that depending on the choice of the interpolation schedule of the algorithm, it is possible to construct quantum search Hamiltonians with a variety of features: Grover-

“…For recent discussions on the transition from the digital to analog quantum computational setting for Grover's algorithm, we refer to Ref. [4][5][6]. The formulation of the analog algorithm proposed by Farhi and Gutmann can be briefly described as follows.…”

Section: Introductionmentioning

confidence: 99%

Theoretical analysis of a nearly optimal analog quantum search

Alsing

2019

Phys. Scr.

We analyze the possibility of modifying the original Farhi-Gutmann Hamiltonian algorithm in order to speed up the procedure for producing a suitably distributed unknown normalized quantum mechanical state. Such a modification is feasible provided only a nearly optimal fidelity is sought.We propose to select the lower bounds of the nearly optimal fidelity values such that their deviations from unit fidelity are less than the minimum error probability characterizing the optimum ambiguous discrimination scheme between the two nonorthogonal quantum states yielding the chosen nearly optimal fidelity values. Departing from the working assumptions of perfect state overlap and uniform distribution of the target state on the unit sphere in N -dimensional complex Hilbert space, we determine that the modified algorithm can indeed outperform the original analog counterpart of a quantum search algorithm. This performance enhancement occurs in terms of speed for a convenient choice of both the ratio γ = E /E between the energy eigenvalues E and E of the modified search Hamiltonian and the quantum mechanical overlap x between the source and the target states. Finally, we briefly discuss possible analytical improvements of our investigation together with its potential relevance in practical quantum engineering applications.