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

arXiv:quant-ph/0406038 · Journal of Mathematical Physics

Exact solutions of the isoholonomic problem and the optimal control problem in holonomic quantum computation

Shogo Tanimura, Mikio Nakahara, Daisuke Hayashi

Published Jan 14, 2005• 26 citations
arXivHTMLPDF

Abstract

The isoholonomic problem in a homogeneous bundle is formulated and solved exactly. The problem takes a form of a boundary value problem of a variational equation. The solution is applied to the optimal control problem in holonomic quantum computer. We provide a prescription to construct an optimal controller for an arbitrary unitary gate and apply it to a k-dimensional unitary gate which operates on an N-dimensional Hilbert space with N⩾2k. Our construction is applied to several important unitary gates such as the Hadamard gate, the CNOT gate, and the two-qubit discrete Fourier transformation gate. Controllers for these gates are explicitly constructed.

DOI
10.1063/1.1835545

Related papers

Exact solutions of the isoholonomic problem and the optimal control problem in holonomic quantum computation

Tanimura, S, Nakahara, M, Hayashi, D

The isoholonomic problem in a homogeneous bundle is formulated and solved exactly. The problem takes a form of a boundary value problem of a variational equation. The solution is applied to the optimal control problem in holonomic quantum computer. We provide a prescription to construct an optimal controller for an arbitrary unitary gate and apply it to a $ k $-dimensional unitary gate which operates on an $ N $-dimensional Hilbert space with $ N \\geq 2k $. Our construction is applied to several important unitary gates such as the Hadamard gate, the CNOT gate, and the two-qubit discrete Fourier transformation gate. Controllers for these gates are explicitly constructed.

•Jan 1, 2004•15 citations

ISOHOLONOMIC PROBLEM AND HOLONOMIC QUANTUM COMPUTATION

Shogo Tanimura

Geometric phases accompanying adiabatic processes in quantum systems can be utilized as unitary gates for quantum computation. Optimization of control of the adiabatic process naturally leads to the isoholonomic problem. The isoholonomic problem in a homogeneous fiber bundle is formulated and solved completely.

arXiv:quant-ph/0505033•International Symposium on Topology in Ordered Phases•Aug 1, 2006•0 citations

Realization of arbitrary gates in holonomic quantum computation

A. O. Niskanen, Mikio Nakahara, Martti M. Salomaa

Among the many proposals for the realization of a quantum computer, holonomic quantum computation is distinguished from the rest as it is geometrical in nature and thus expected to be robust against decoherence. Here we analyze the realization of various quantum gates by solving the inverse problem: Given a unitary matrix, we develop a formalism by which we find loops in the parameter space generating this matrix as a holonomy. We demonstrate that such a one-qubit gate as the Hadamard gate and such two-qubit gates as the controlled-NOT gate and the SWAP gate, and the discrete Fourier transformation can be obtained with a single loop.

arXiv:quant-ph/0209015•Physical Review A•Jan 29, 2003•30 citations

Isoholonomic inequality and tight implementations of holonomic quantum gates

Ole Sönnerborn

In holonomic quantum computation, quantum logic gates are realized by cyclic parallel transport of the computational space. The resulting quantum gate corresponds to the holonomy associated with the closed path traced by the computational space. The isoholonomic inequality for gates establishes a fundamental lower bound on the path length of such cyclic transports, which depends only on the spectrum of the holonomy, that is, the eigenvalues of the implemented quantum gate. The isoholonomic inequality also gives rise to an estimate of the minimum time required to execute a holonomic quantum gate, underscoring the central role of the inequality in quantum computation. In this paper, we show that when the codimension of the computational space is sufficiently large, any quantum gate can be implemented using a parallel transporting Hamiltonian in a way that saturates the isoholonomic inequality and the corresponding time estimate. We call such implementations tight. The treatment presented here is constructive and lays the foundation for the development of efficient and optimal implementation strategies in holonomic quantum computation.

arXiv:2412.12013•Physical Review A•May 16, 2025•2 citations

Estimate of the time required to perform a nonadiabatic holonomic quantum computation

Ole Sönnerborn

Nonadiabatic holonomic quantum computation has been proposed as a method to implement quantum logic gates with robustness comparable to that of adiabatic holonomic gates but with shorter execution times. In this paper, we establish an isoholonomic inequality for quantum gates, which provides a lower bound on the lengths of cyclic transformations of the computational space that generate a specific gate. Then, as a corollary, we derive a nonadiabatic execution time estimate for holonomic gates. In addition, we demonstrate that under certain dimensional conditions, the isoholonomic inequality is tight in the sense that every gate on the computational space can be implemented holonomically and unitarily in a time-optimal way. We illustrate the results by showing that the procedures for implementing a universal set of holonomic gates proposed in a pioneering paper on nonadiabatic holonomic quantum computation saturate the isoholonomic inequality and are thus time optimal. Published by the American Physical Society 2024

arXiv:2404.05844•Physical Review A•Jun 24, 2024•5 citations

Modelling optimal implementation of an arbitrary $N$-qubit quantum gate within the generalized Bloch vectors formalism

Elena R. Loubenets, Sergey Kuznetsov

The optimal implementation of quantum gates for closed $N$-qubit systems is one of the key challenges for practical realization of many quantum information processing tasks. In the present article, based on the generalized Bloch vectors formalism [J. Phys. A: Math. Theor. 54, 195301 (2021)] for a finite-dimensional quantum system, we develop a new general model for the optimal implementation of quantum gates, which is formulated in terms of the Bloch vectors for the unitary evolution operator and the system Hamiltonians, drift and control, and has the unified mathematical form, which is standard for the mathematical control theory and applicable for the implementation of an arbitrary $N$-qubit gate within any closed $N$-qubit system, satisfying the controllability conditions. Within the developed optimal model, the cost functional has both the terminal part and also, the integral part with the special scaling, and this allows us to specify the quantum optimal control synthesis via solving the two-point boundary value problem (BVP) for the system of ordinary differential equations (ODEs), which can be explored numerically by any of the known BVP solvers for ODEs. The numerical experiments, conducted for the implementation within the developed optimal model of a variety of $N=1,2,3$ qubit gates, demonstrate the high accuracy of the model-based results.

arXiv:2509.18059•Sep 22, 2025•0 citations
The isoholonomic problem in a homogeneous bundle is formulated and solved exactly. The problem takes a form of a boundary value problem of a variational equation. The solution is applied to the optimal control problem in holonomic quantum computer. We provide a prescription to construct an optimal controller for an arbitrary unitary gate and apply it to a k-dimensional unitary gate which operates on an N-dimensional Hilbert space with N⩾2k. Our construction is applied to several important unitary gates such as the Hadamard gate, the CNOT gate, and the two-qubit discrete Fourier transformation gate. Controllers for these gates are explicitly constructed.
The isoholonomic problem in a homogeneous bundle is formulated and solved exactly. The problem takes a form of a boundary value problem of a variational equation. The solution is applied to the optimal control problem in holonomic quantum computer. We provide a prescription to construct an optimal controller for an arbitrary unitary gate and apply it to a $ k $-dimensional unitary gate which operates on an $ N $-dimensional Hilbert space with $ N \\geq 2k $. Our construction is applied to several important unitary gates such as the Hadamard gate, the CNOT gate, and the two-qubit discrete Fourier transformation gate. Controllers for these gates are explicitly constructed.
Among the many proposals for the realization of a quantum computer, holonomic quantum computation is distinguished from the rest as it is geometrical in nature and thus expected to be robust against decoherence. Here we analyze the realization of various quantum gates by solving the inverse problem: Given a unitary matrix, we develop a formalism by which we find loops in the parameter space generating this matrix as a holonomy. We demonstrate that such a one-qubit gate as the Hadamard gate and such two-qubit gates as the controlled-NOT gate and the SWAP gate, and the discrete Fourier transformation can be obtained with a single loop.
In holonomic quantum computation, quantum logic gates are realized by cyclic parallel transport of the computational space. The resulting quantum gate corresponds to the holonomy associated with the closed path traced by the computational space. The isoholonomic inequality for gates establishes a fundamental lower bound on the path length of such cyclic transports, which depends only on the spectrum of the holonomy, that is, the eigenvalues of the implemented quantum gate. The isoholonomic inequality also gives rise to an estimate of the minimum time required to execute a holonomic quantum gate, underscoring the central role of the inequality in quantum computation. In this paper, we show that when the codimension of the computational space is sufficiently large, any quantum gate can be implemented using a parallel transporting Hamiltonian in a way that saturates the isoholonomic inequality and the corresponding time estimate. We call such implementations tight. The treatment presented here is constructive and lays the foundation for the development of efficient and optimal implementation strategies in holonomic quantum computation.
Nonadiabatic holonomic quantum computation has been proposed as a method to implement quantum logic gates with robustness comparable to that of adiabatic holonomic gates but with shorter execution times. In this paper, we establish an isoholonomic inequality for quantum gates, which provides a lower bound on the lengths of cyclic transformations of the computational space that generate a specific gate. Then, as a corollary, we derive a nonadiabatic execution time estimate for holonomic gates. In addition, we demonstrate that under certain dimensional conditions, the isoholonomic inequality is tight in the sense that every gate on the computational space can be implemented holonomically and unitarily in a time-optimal way. We illustrate the results by showing that the procedures for implementing a universal set of holonomic gates proposed in a pioneering paper on nonadiabatic holonomic quantum computation saturate the isoholonomic inequality and are thus time optimal. Published by the American Physical Society 2024
Modelling optimal implementation of an arbitrary NNN-qubit quantum gate within the generalized Bloch vectors formalism
The optimal implementation of quantum gates for closed NNN-qubit systems is one of the key challenges for practical realization of many quantum information processing tasks. In the present article, based on the generalized Bloch vectors formalism [J. Phys. A: Math. Theor. 54, 195301 (2021)] for a finite-dimensional quantum system, we develop a new general model for the optimal implementation of quantum gates, which is formulated in terms of the Bloch vectors for the unitary evolution operator and the system Hamiltonians, drift and control, and has the unified mathematical form, which is standard for the mathematical control theory and applicable for the implementation of an arbitrary NNN-qubit gate within any closed NNN-qubit system, satisfying the controllability conditions. Within the developed optimal model, the cost functional has both the terminal part and also, the integral part with the special scaling, and this allows us to specify the quantum optimal control synthesis via solving the two-point boundary value problem (BVP) for the system of ordinary differential equations (ODEs), which can be explored numerically by any of the known BVP solvers for ODEs. The numerical experiments, conducted for the implementation within the developed optimal model of a variety of N=1,2,3N=1,2,3N=1,2,3 qubit gates, demonstrate the high accuracy of the model-based results.