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This paper presents a unified mathematical framework for optimizing arbitrary N-qubit gate implementations using generalized Bloch vectors and optimal control theory. By formulating the problem as a two-point boundary value problem, the authors provide a numerically solvable model that achieves high accuracy across 1 to 3 qubit systems. This theoretical advancement directly addresses core challenges in quantum hardware control and algorithm synthesis.
Modelling optimal implementation of an arbitrary $N$-qubit quantum gate within the generalized Bloch vectors formalism
arXiv:2509.18059v4 Announce Type: replace Abstract: 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.
Abstract: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 [quant-ph]
(or arXiv:2509.18059v4 [quant-ph] for this version)
From: Elena R. Loubenets [view email] [v1] Mon, 22 Sep 2025 17:36:58 UTC (1,987 KB) [v2] Wed, 19 Nov 2025 09:42:20 UTC (1,987 KB) [v3] Sat, 23 May 2026 11:21:07 UTC (1,988 KB) [v4] Sat, 17 Oct 2026 06:04:10 UTC (1,991 KB)
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