geodesic in (2 n+1 -1)-dimensional sphere.Further, the reduced search sequence is given rise to the complex projective space CP 2 n -1 through a geometric reduction, which is also shown to be on a geodesic in CP 2 n -1 .Roughly speaking, the reduction in [15] is made through the phase-factor elimination from quantum states, so that CP 2 n -1 is thought of as the space of rays.Note that the geodesics above are associated with the standard metric on (2 n+1 -1)-dimensional sphere and with the Fubini-Study metric on CP 2 n -1 , respectively.The Fubini-Study metric on CP 2 n -1 is utilized also in [15] to measure the minimum distance from each state involved in the search sequence to the submanifold consisting of non-entangled states, which characterizes the entanglement of the states along the search.As expected benefits of geometric and dynamical views on quantum algorithmic studies like [15], the following would be worth listed;1 By revealing underlying geometry of quantum algorithms (not necessarily universal), numbers of results in geometry are expected to be applied to make advances in quantum computation and information.2 On looked upon the iterations made in algorithms as (discrete) time-evolutions of states, numbers of results in dynamical systems are expected to be applied to make advances in quantum computation and information.3 In view of a close connection between geometry and dynamical systems, geometric and dynamical-systems studies on quantum algorithms may provide interesting examples of dynamical systems.It would be worth noting here that there exists another approach to quantum searches using adiabatic evolution [16][17][18].That approach, however, is outside the scope of this chapter since the search dealt with in this chapter is organized on the so-called amplitude magnification technique [8] which differs from the adiabatic evolution.
geodesic in (2 n+1 -1)-dimensional sphere.Further, the reduced search sequence is given rise to the complex projective space CP 2 n -1 through a geometric reduction, which is also shown to be on a geodesic in CP 2 n -1 .Roughly speaking, the reduction in [15] is made through the phase-factor elimination from quantum states, so that CP 2 n -1 is thought of as the space of rays.Note that the geodesics above are associated with the standard metric on (2 n+1 -1)-dimensional sphere and with the Fubini-Study metric on CP 2 n -1 , respectively.The Fubini-Study metric on CP 2 n -1 is utilized also in [15] to measure the minimum distance from each state involved in the search sequence to the submanifold consisting of non-entangled states, which characterizes the entanglement of the states along the search.As expected benefits of geometric and dynamical views on quantum algorithmic studies like [15], the following would be worth listed;1 By revealing underlying geometry of quantum algorithms (not necessarily universal), numbers of results in geometry are expected to be applied to make advances in quantum computation and information.2 On looked upon the iterations made in algorithms as (discrete) time-evolutions of states, numbers of results in dynamical systems are expected to be applied to make advances in quantum computation and information.3 In view of a close connection between geometry and dynamical systems, geometric and dynamical-systems studies on quantum algorithms may provide interesting examples of dynamical systems.It would be worth noting here that there exists another approach to quantum searches using adiabatic evolution [16][17][18].That approach, however, is outside the scope of this chapter since the search dealt with in this chapter is organized on the so-called amplitude magnification technique [8] which differs from the adiabatic evolution.
Quantum computation and quantum information are known well to be very hot interdisciplinary fields [1], where a number of approaches have been taken. In a series of papers [2, 3, 4, 5], the author takes geometric and dynamical-system approaches to differential equations on the quantum information space (QIS) raised from algorithms in engineering. One of the keys to the papers is the fact that the SLD-Fisher metric of the QIS is obtained from the natural Riemannian metric of the space of ordered tuples of multiqubits through a geometric reduction [2]. On the QIS, three types of geodesics are of much interest; the m-geodesics, the e-geodesics and the Levi-Civita geodesics. As a continuation of the paper [5] showing that the Grover search geodesic for a given ordered tuple of multiqubits is reduced to an m-geodesic on the QIS, this talk presents presents the classes curves which are reduced to the m-, the e-, and the Levi-Civita geodesics on the QIS. In particular, the curves related with the m-geodesics are discussed in detail. Since the QIS consists of positive-(semi)definite matrices with unit trace, computer algebra would be worth applying to matrix computations on the QIS. Applicability of computer algebra will be discussed also in this talk.
Determining the quantum circuit complexity of a unitary operation is closely related to the problem of finding minimal length paths in a particular curved geometry [Nielsen et al, Science 311, 1133-1135 (2006)]. This paper investigates many of the basic geometric objects associated to this space, including the Levi-Civita connection, the geodesic equation, the curvature, and the Jacobi equation. We show that the optimal Hamiltonian evolution for synthesis of a desired unitary necessarily obeys a simple universal geodesic equation. As a consequence, once the initial value of the Hamiltonian is set, subsequent changes to the Hamiltonian are completely determined by the geodesic equation. We develop many analytic solutions to the geodesic equation, and a set of invariants that completely determine the geodesics. We investigate the problem of finding minimal geodesics through a desired unitary, U, and develop a procedure which allows us to deform the (known) geodesics of a simple and well understood metric to the geodesics of the metric of interest in quantum computation. This deformation procedure is illustrated using some three-qubit numerical examples. We study the computational complexity of evaluating distances on Riemmanian manifolds, and show that no efficient classical algorithm for this problem exists, subject to the assumption that good pseudorandom generators exist. Finally, we develop a canonical extension procedure for unitary operations which allows ancilla qubits to be incorporated into the geometric approach to quantum computing.
Determining the quantum circuit complexity of a unitary operation is closely related to the problem of finding minimal length paths in a particular curved geometry [Nielsen et al, Science 311, 1133-1135 (2006)]. This paper investigates many of the basic geometric objects associated to this space, including the Levi-Civita connection, the geodesic equation, the curvature, and the Jacobi equation. We show that the optimal Hamiltonian evolution for synthesis of a desired unitary necessarily obeys a simple universal geodesic equation. As a consequence, once the initial value of the Hamiltonian is set, subsequent changes to the Hamiltonian are completely determined by the geodesic equation. We develop many analytic solutions to the geodesic equation, and a set of invariants that completely determine the geodesics. We investigate the problem of finding minimal geodesics through a desired unitary, U, and develop a procedure which allows us to deform the (known) geodesics of a simple and well understood metric to the geodesics of the metric of interest in quantum computation. This deformation procedure is illustrated using some three-qubit numerical examples. We study the computational complexity of evaluating distances on Riemmanian manifolds, and show that no efficient classical algorithm for this problem exists, subject to the assumption that good pseudorandom generators exist. Finally, we develop a canonical extension procedure for unitary operations which allows ancilla qubits to be incorporated into the geometric approach to quantum computing.
We propose a fundamentally new paradigm for quantum computation based on the intrinsic geometry of quantum state space. Rather than specifying computation as a sequence of discrete gates applied over external time, Geometric Quantum Computing (GQC) defines computation as geodesic evolution through Hilbert space, where the computational 'time' emerges from the Fubini-Study metric itself. This framework inverts the standard relationship between quantum dynamics and time: instead of viewing decoherence as degradation of quantum information over time, we recognize that decoherence is time—the environment's information processing. We derive the theoretical foundations of GQC, demonstrate that it naturally circumvents barren plateaus by operating directly in state space rather than parameter space, and propose experimental protocols for implementation. The framework suggests that optimal quantum computation is achieved not by fighting decoherence but by engineering the effective geometry of state space to make desired computations correspond to natural geodesic flow.
Nielsen, et al. [1, 2] proposed a view of quantum computation where determining optimal algorithms is equivalent to extremizing a geodesic length or cost functional. This view of optimization is highly suggestive of an action principle of the space of N-qubits interacting via local operations. The cost or action functional is given by the cost of evolution operators on local qubit operations leading to causal dynamics, as in Blute et. al. [3] Here we propose a view of information geometry for quantum algorithms where the inherent causal structure determines topology and information distances [4, 5] set the local geometry. This naturally leads to geometric characterization of hypersurfaces in a quantum cellular automaton. While in standard quantum circuit representations the connections between individual qubits, i.e. the topology, for hypersurfaces will be dynamic, quantum cellular automata have readily identifiable static hypersurface topologies determined via the quantum update rules. We demonstrate construction of quantum cellular automata geometry and discuss the utility of this approach for tracking entanglement and algorithm optimization.
We show that quantum information geometry can be used to characterize Grover's searching algorithm. Specifically, quantifying the notion of quantum distinguishability between parametric density operators by means of the Wigner-Yanase quantum information metric, we uncover that the quantum searching problem can be recast in an information geometric framework where Grover's dynamics is characterized by a geodesic on the manifold of the parametric density operators of pure quantum states constructed from the continuos approximation of the parametric quantum out-put state in Grover's algorithm.