Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Finsler metrics in quantum circuit optimization

Howard E. Brandt

Published May 1, 2012• 0 citations
Publisher

Abstract

AbstractA brief review is given of possible Finsler metrics de ned on the SU (2 n )group tangent bundle which may be suitable for quantum circuit optimiza-tion. 1 INTRODUCTION In the Riemannian geometry of quantum computation, a Riemannian hasbeen exploited for the purpose of quantum circuit optimization [1]-[4]. Finslermetrics are a more general class of metrics of which Riemannian metrics area special case based on a quadratic form [5]-[8]. A Finsler need not bequadratic. In fact Bernard Riemann was the rst to consider a basedon a quartic form [9], but he settled on a quadratic form because of its greatersimplicity. Much later, Paul Finsler introduced Finsler geometry in his thesis[10]. The purpose of the present work is to brie‡y review possible Finsler metricsfor quantum circuit optimization [11],[1] . 2 QUANTUM CIRCUIT OPTIMIZATION For a manifold M with a so-called local metric [11], there is a function F (x;y )of coordinates x on M and y on the tangent space manifold TM . One requiresF (x;y ) = 0 for xed x, and F (x;y ) is vanishing for y = 0. F (x;y ) is positivelyhomogeneous of degree 1 in y , namely,F (x;y ) = F (x;y );  > 0; (1)and satis es the triangle inequality in the tangent space variable y , namely,F (x;y

DOI
10.1117/12.918323

Related papers

ASPECTS OF THE RIEMANNIAN GEOMETRY OF QUANTUM COMPUTATION

Howard E. Brandt

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.

International Journal of Modern Physics B•Sep 18, 2012•2 citations

A geometric approach to quantum circuit lower bounds

Michael A. Nielsen

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.

arXiv:quant-ph/0502070•Quantum Information and Computation•May 1, 2006•253 citations

Geometric Control For Analysing the Quantum Speed Limit and the Physical Limitations of Computers

Benjamin Russell

This thesis studies the role of Finsler geometry in quantum time optimal control of systems with constrained control field power and other constraints. The systems considered are all finite dimensional systems with pure states. A Finsler metric is constructed such that its geodesics are the time optimal trajectories for the quantum time evolution operator on the special unitary group. This metric is shown to be right invariant. The geodesic equation, in the form of an Euler-Poincar\\'{e} equation is found. It is also shown that the geodesic lengths of this same metric equal the optimal times for implementing any desired quantum gate. In a special case, where all are control fields are equally constrained, the desired geodesics are found in closed form. The results obtained are discussed in the general context of natural computation.

•Feb 1, 2015•0 citations

Differential geometry of quantum computation

Howard E. Brandt

This is a brief review of aspects of a recent development in the differential geometry of quantum computation. Expository descriptions are offered of the associated Riemannian geometry of the special unitary unimodular group in 2 n -dimensions, including the choice of metric, connection, and optimal geodesics for achieving minimal complexity quantum computations.

Journal of Modern Optics•Nov 10, 2008•3 citations

Riemannian optimization of isometric tensor networks

Markus Hauru, Maarten Van Damme, Jutho Haegeman

Several tensor networks are built of isometric tensors, i.e. tensors satisfying W\dagger W = \mathbb{1} W † W = 1 . Prominent examples include matrix product states (MPS) in canonical form, the multiscale entanglement renormalization ansatz (MERA), and quantum circuits in general, such as those needed in state preparation and quantum variational eigensolvers. We show how gradient-based optimization methods on Riemannian manifolds can be used to optimize tensor networks of isometries to represent e.g. ground states of 1D quantum Hamiltonians. We discuss the geometry of Grassmann and Stiefel manifolds, the Riemannian manifolds of isometric tensors, and review how state-of-the-art optimization methods like nonlinear conjugate gradient and quasi-Newton algorithms can be implemented in this context. We apply these methods in the context of infinite MPS and MERA, and show benchmark results in which they outperform the best previously-known optimization methods, which are tailor-made for those specific variational classes. We also provide open-source implementations of our algorithms.

arXiv:2007.03638•SciPost Physics•Feb 17, 2021•68 citations

Optimizing quantum circuits with Riemannian gradient flow

Roeland Wiersema, Nathan Killoran

Variational quantum algorithms are a promising class of algorithms that can be performed on currently available quantum computers. In most settings, the free parameters of a variational circuit are optimized using a classical optimizer that updates parameters in Euclidean geometry. Since quantum circuits are elements of the special unitary group, we can consider an alternative optimization perspective that depends on the structure of this group. In this work, we investigate a Riemannian optimization scheme over the special unitary group and we discuss its implementation on a quantum computer. We illustrate that the resulting Riemannian gradient-flow algorithm has favorable optimization properties for deep circuits and that an approximate version of this algorithm can be performed on near-term hardware. We highlight the connections of our work with previously proposed heuristics like ADAPT-VQE and show that they can be understood as variants of our algorithm.

arXiv:2202.06976•Physical Review A•Jun 27, 2023•33 citations
AbstractA brief review is given of possible Finsler metrics de ned on the SU (2 n )group tangent bundle which may be suitable for quantum circuit optimiza-tion. 1 INTRODUCTION In the Riemannian geometry of quantum computation, a Riemannian hasbeen exploited for the purpose of quantum circuit optimization [1]-[4]. Finslermetrics are a more general class of metrics of which Riemannian metrics area special case based on a quadratic form [5]-[8]. A Finsler need not bequadratic. In fact Bernard Riemann was the rst to consider a basedon a quartic form [9], but he settled on a quadratic form because of its greatersimplicity. Much later, Paul Finsler introduced Finsler geometry in his thesis[10]. The purpose of the present work is to brie‡y review possible Finsler metricsfor quantum circuit optimization [11],[1] . 2 QUANTUM CIRCUIT OPTIMIZATION For a manifold M with a so-called local metric [11], there is a function F (x;y )of coordinates x on M and y on the tangent space manifold TM . One requiresF (x;y ) = 0 for xed x, and F (x;y ) is vanishing for y = 0. F (x;y ) is positivelyhomogeneous of degree 1 in y , namely,F (x;y ) = F (x;y );  > 0; (1)and satis es the triangle inequality in the tangent space variable y , namely,F (x;y
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.
This thesis studies the role of Finsler geometry in quantum time optimal control of systems with constrained control field power and other constraints. The systems considered are all finite dimensional systems with pure states. A Finsler metric is constructed such that its geodesics are the time optimal trajectories for the quantum time evolution operator on the special unitary group. This metric is shown to be right invariant. The geodesic equation, in the form of an Euler-Poincar\\'{e} equation is found. It is also shown that the geodesic lengths of this same metric equal the optimal times for implementing any desired quantum gate. In a special case, where all are control fields are equally constrained, the desired geodesics are found in closed form. The results obtained are discussed in the general context of natural computation.
Several tensor networks are built of isometric tensors, i.e. tensors satisfying W\dagger W = \mathbb{1} W † W = 1 . Prominent examples include matrix product states (MPS) in canonical form, the multiscale entanglement renormalization ansatz (MERA), and quantum circuits in general, such as those needed in state preparation and quantum variational eigensolvers. We show how gradient-based optimization methods on Riemannian manifolds can be used to optimize tensor networks of isometries to represent e.g. ground states of 1D quantum Hamiltonians. We discuss the geometry of Grassmann and Stiefel manifolds, the Riemannian manifolds of isometric tensors, and review how state-of-the-art optimization methods like nonlinear conjugate gradient and quasi-Newton algorithms can be implemented in this context. We apply these methods in the context of infinite MPS and MERA, and show benchmark results in which they outperform the best previously-known optimization methods, which are tailor-made for those specific variational classes. We also provide open-source implementations of our algorithms.
Variational quantum algorithms are a promising class of algorithms that can be performed on currently available quantum computers. In most settings, the free parameters of a variational circuit are optimized using a classical optimizer that updates parameters in Euclidean geometry. Since quantum circuits are elements of the special unitary group, we can consider an alternative optimization perspective that depends on the structure of this group. In this work, we investigate a Riemannian optimization scheme over the special unitary group and we discuss its implementation on a quantum computer. We illustrate that the resulting Riemannian gradient-flow algorithm has favorable optimization properties for deep circuits and that an approximate version of this algorithm can be performed on near-term hardware. We highlight the connections of our work with previously proposed heuristics like ADAPT-VQE and show that they can be understood as variants of our algorithm.