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Progress of Theoretical Physics

Quantum Computational Riemannian and Sub-Riemannian Geodesics

Kosuke Shizume, Takao Nakajima, Ryo Nakayama, Yutaka Takahashi

Published Jun 1, 2012• 10 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.127.997

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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.
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.
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.
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.
Computational complexity is a quantum information concept that recently has found applications in the holographic understanding of the black hole interior. We consider quantum computational complexity for nnn qubits using Nielsen's geometrical approach. In the definition of complexity there is a big amount of arbitrariness due to the choice of the penalty factors, which parametrizes the cost of the elementary computational gates. In order to reproduce desired features in holography, such as ergodicity and exponential maximal complexity for large number of qubits nnn, negative curvatures are required. With the simplest choice of penalties, this is achieved at the price of singular sectional curvatures in the large nnn limit. We investigate a choice of penalties in which we can obtain negative curvatures in a smooth way. We also analyze the relation between operator and state complexities, framing the discussion with the language of Riemannian submersions. This provides a direct relation between geodesics and curvatures in the unitaries and the states spaces, which we also exploit to give a closed-form expression for the metric on the states in terms of the one for the operators. Finally, we study conjugate points for a large number of qubits in the unitary space and we provide a strong indication that maximal complexity scales exponentially with the number of qubits in a certain regime of the penalties space.
We formulate and investigate the simplest version of time-optimal quantum computation theory (TO-QCT), where the computation time is defined by the physical one and the Hamiltonian contains only one- and two-qubit interactions. This version of TO-QCT is also considered as optimality by sub-Riemannian geodesic length. The work has two aims: One is to develop a TO-QCT itself based on a physically natural concept of time, and the other is to pursue the possibility of using TO-QCT as a tool to estimate the complexity in conventional gate-optimal quantum computation theory (GO-QCT). In particular, we investigate to what extent is true the following statement: Time complexity is polynomial in the number of qubits if and only if gate complexity is also. In the analysis, we relate TO-QCT and optimal control theory (OCT) through fidelity-optimal computation theory (FO-QCT); FO-QCT is equivalent to TO-QCT in the limit of unit optimal fidelity, while it is formally similar to OCT. We then develop an efficient numerical scheme for FO-QCT by modifying Krotov's method in OCT, which has a monotonic convergence property. We implemented the scheme and obtained solutions of FO-QCT and of TO-QCT for the quantum Fourier transform and a unitary operator that does not have an apparent symmetry. The former has a polynomial gate complexity and the latter is expected to have an exponential one which is based on the fact that a series of generic unitary operators has an exponential gate complexity. The time complexity for the former is found to be linear in the number of qubits, which is understood naturally by the existence of an upper bound. The time complexity for the latter is exponential in the number of qubits. Thus, both the targets seem to be examples satisfyng the preceding statement. The typical characteristics of the optimal Hamiltonians are symmetry under time reversal and constancy of one-qubit operation, which are mathematically shown to hold in fairly general situations.
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.