Previous work argued that efficient quantum circuits can be obtained from special curves called geodesics, and they studied geodesics in Riemannian manifolds equipped with a penalty metric where the penalty was taken to infinity. Taking such limits seems problematic, because it is not clear that all extremals of a limiting optimal control problem can be arrived at as limits of solutions. To rectify this we use the Pontryagin Maximum Principle to construct equations for normal and abnormal subRiemannian geodesics and cubics. We also investigate whether neural networks can be trained to help generate quantum circuits.
Previous work argued that efficient quantum circuits can be obtained from special curves called geodesics, and they studied geodesics in Riemannian manifolds equipped with a penalty metric where the penalty was taken to infinity. Taking such limits seems problematic, because it is not clear that all extremals of a limiting optimal control problem can be arrived at as limits of solutions. To rectify this we use the Pontryagin Maximum Principle to construct equations for normal and abnormal subRiemannian geodesics and cubics. We also investigate whether neural networks can be trained to help generate quantum circuits.
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.
A review is given of some aspects of the Riemannian geometry of quantum computation in which the quantum evolution is represented in the tangent space manifold of the special unitary unimodular group SU(2n) for n qubits. The Riemannian right-invariant metric, connection, curvature, geodesic equation for minimal complexity quantum circuits, Jacobi equation and the lifted Jacobi equation for varying penalty parameter are reviewed. Sharpened tools for calculating the geodesic derivative are presented. The geodesic derivative may facilitate the numerical investigation of conjugate points and the global characteristics of geodesic paths in the group manifold, the determination of optimal quantum circuits for carrying out a quantum computation, and the determination of the complexity of particular quantum algorithms.
What is the minimal size quantum circuit required to exactly implement a specified n-qubit unitary operation, U, without the use of ancilla qubits? We show that a lower bound on the minimal size is provided by the length of the minimal geodesic between U and the identity, I, where length is defined by a suitable Finsler metric on the manifold SU(2^n). The geodesic curves on these manifolds have the striking property that once an initial position and velocity are set, the remainder of the geodesic is completely determined by a second order differential equation known as the geodesic equation. This is in contrast with the usual case in circuit design, either classical or quantum, where being given part of an optimal circuit does not obviously assist in the design of the rest of the circuit. Geodesic analysis thus offers a potentially powerful approach to the problem of proving quantum circuit lower bounds. In this paper we construct several Finsler metrics whose minimal length geodesics provide lower bounds on quantum circuit size. For each Finsler metric we give a procedure to compute the corresponding geodesic equation. We also construct a large class of solutions to the geodesic equation, which we call \emph{Pauli geodesics}, since they arise from isometries generated by the Pauli group. For any unitary U diagonal in the computational basis, we show that: (a) provided the minimal length geodesic is unique, it must be a Pauli geodesic; (b) finding the length of the minimal Pauli geodesic passing from I to U is equivalent to solving an exponential size instance of the closest vector in a lattice problem (CVP); and (c) all but a doubly exponentially small fraction of such unitaries have minimal Pauli geodesics of exponential length.
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.
The main goal of this work is to make use of sub-Riemannian geometry and Pontryagins Maximum Principle to investigate optimal control for computing one- and two-qubit quantum gates, as well as to analyze how the presence of uncontrollable interactions, such as noise and cross-talk operators, influence on the energetic cost of performing such unitary operations. Initially, we study the case of a single noisy qubit. We aim to identify sub-Riemannian geodesics that connect the identity to a unitary operation, sufficiently close to the ideal case, despite the presence of decoherence noise. Next, we expand the study to the case of two qubits, initially only under a time-constant cross-talk interaction, and then, under both: cross-talk interaction and decoherence noise. For all case studies, the main challenge lies in determining the unknown parameters in the geodesic equation for the desired unitary operator. We present a novel method, based on random sampling, for determining such parameters and how they can lead to high-fidelity quantum gates which are synthesized through smooth external control fields.