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.
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.
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.
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 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.
Nielsen et al. have introduced a Riemannian metric on the space of n-qubit unitary operators [M. A. Nielsen, M. R. Dowling, M. Gu and A. C. Doherty, Science 311 (2006), 1133]. The length of the shortest curve connecting the identity I and a desired unitary operator W defined by this metric is essentially equivalent to the quantum gate complexity of W. This metric, and thus, the Riemannian geodesic equation for this metric, has a parameter q called “penalty” that must be large enough. In this paper, we investigate the sub-Riemannian geodesic equation obtained by taking the limit of the Riemannian geodesic equation as q →∞, and show that the Riemannian geodesics for finite q can be explicitly constructed from the sub-Riemannian geodesics. We also present a numerical algorithm for finding the (sub-)Riemannian geodesics connecting I and W, which is based on the Krotov method in optimal control theory. As an example, we give an exact sub-Riemannian geodesic connecting I and the controlled-controlled-Z gate, which is obtained by guessing from the numerical results of the algorithm.
In this thesis we study the roles played by symplectic geometry in quantum mechanics,in particular quantum dynamics and quantum information theory treated as two separate parts. The common ground for both parts is the geometrical formulation of quantum mechanics. In Chapter 2, we review the associated complex projective Hilbert space of quantum pure states, with symplectic and Riemannian structures and their roles in quantum dynamics and kinematics. In Chapter 3, we motivate the idea of information-theoretic constraint on the differentiable manifold of probability distributions through the maximum uncertainty principle and the linear Schrodinger equation by introduction of the wave function. In Chapter 4, we review both regular and singular symplectic reduction of a symplectic manifold, which is acted upon properly and symplectically by a compact Lie group. Chapter 5 contains the author's original contributions to the first part of the thesis. In this chapter, by using the same information-theoretic discussion of the Chapter 3 we propose a non-relativistic, spin-less, non-linear quantum dynamical equation,with the Fisher information metric replaced by the Jensen-Shannon distance information. Furthermore, we show that the non-linear Schrodinger equation is in fact a Hamiltonian dynamics, namely it preserves the symplectic structure of the complex Hilbert space. The projected dynamics on the corresponding projective Hilbert space is derived and its properties are highlighted in further details. Chapter 6 contains the author's primary contributions to the second part of the thesis. In particular, by using the singular symplectic reduction of the Chapter 4 we explicitly construct the space of entanglement types of three-qubit pure states with a specific (shifted) spectra of single-particle reduced density matrices. Moreover, we obtain the image of the symplectic quotient under the induced Hilbert map, by using local unitary invariant polynomials. Then the symplectic structure on the principal stratum of the symplectic quotient is derived. Finally, it is discussed that other lower dimensional strata are relative equilibria on the original manifold and their stability properties are investigated under compact subgroups of the local unitary transformations.