Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Modelling optimal generation of an arbitrary N-qubit quantum gate within the generalized Bloch vectors formalism due to the Pontryagin principle

Sergey Kuznetsov, Elena R. Loubenets

TL;DR

The paper tackles the problem of optimally generating arbitrary $N$-qubit quantum gates for closed quantum systems. It develops a universal model based on the generalized Bloch-vector formalism and applies the Pontryagin maximum principle to derive a boundary-value problem for the unitary evolution $U(t)$, with a cost functional that combines a terminal fidelity term and an energy-penalty integral. The authors provide analytic necessary conditions for optimality and validate the approach numerically for $N=1,2,3$ qubits, including the Toffoli gate, demonstrating high-precision gate synthesis that is independent of the system’s initial state. This framework offers a state-independent, energy-aware method for synthesizing quantum gates using standard numerical solvers and is applicable to a broad class of $N$-qubit Hamiltonians.

Abstract

This paper is devoted to the problem of optimal generation of N-qubit gates for closed quantum systems -- a key task for the practical implementation of various quantum applications. Based on the generalized Bloch vectors formalism for a finite-dimensional quantum system, we develop a new optimal gates generation model, which is universal in the sense that it is applicable for an arbitrary N-qubit gate, any Hamiltonian of a closed N-qubit system and within this model an optimal control is determined only by N-qubit system parameters and does not depend on its initial state. Within the developed model, the synthesis of optimal control, carried out via the Pontryagin principle, leads to the boundary value problem for the system of ODEs, which can be explored by various computational methods. Numerical experiments conducted for generation of a variety of one/two/three qubit gates demonstrate viability of the developed optimal model which allows one to generate N-qubit quantum gates with a high degree of precision.

Modelling optimal generation of an arbitrary N-qubit quantum gate within the generalized Bloch vectors formalism due to the Pontryagin principle

TL;DR

The paper tackles the problem of optimally generating arbitrary -qubit quantum gates for closed quantum systems. It develops a universal model based on the generalized Bloch-vector formalism and applies the Pontryagin maximum principle to derive a boundary-value problem for the unitary evolution , with a cost functional that combines a terminal fidelity term and an energy-penalty integral. The authors provide analytic necessary conditions for optimality and validate the approach numerically for qubits, including the Toffoli gate, demonstrating high-precision gate synthesis that is independent of the system’s initial state. This framework offers a state-independent, energy-aware method for synthesizing quantum gates using standard numerical solvers and is applicable to a broad class of -qubit Hamiltonians.

Abstract

This paper is devoted to the problem of optimal generation of N-qubit gates for closed quantum systems -- a key task for the practical implementation of various quantum applications. Based on the generalized Bloch vectors formalism for a finite-dimensional quantum system, we develop a new optimal gates generation model, which is universal in the sense that it is applicable for an arbitrary N-qubit gate, any Hamiltonian of a closed N-qubit system and within this model an optimal control is determined only by N-qubit system parameters and does not depend on its initial state. Within the developed model, the synthesis of optimal control, carried out via the Pontryagin principle, leads to the boundary value problem for the system of ODEs, which can be explored by various computational methods. Numerical experiments conducted for generation of a variety of one/two/three qubit gates demonstrate viability of the developed optimal model which allows one to generate N-qubit quantum gates with a high degree of precision.

Paper Structure

This paper contains 11 sections, 1 theorem, 45 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries: the generalized Bloch vectors formalism
  3. Generalized Bloch vectors
  4. Unitary evolution
  5. Optimal realization of $N$-qubit gates
  6. Numerical study
  7. One-qubit gates
  8. Two-qubit gates
  9. Three-qubit gate
  10. Conclusion
  11. Acknowledgments

Key Result

Theorem 1

The solution of the optimal model BFCS_model_statement satisfies the following necessary optimality conditions where $u^{(0)}, \, p^{(0)} \in\mathbb{C}$; $u, \, p\in\mathbb{C}^{d^{2}-1}$; $\nu_{\text{ctr}} \in \mathbb{R}^{s}$ and In Eqs. BFCS_Pontryagins_equations, the derivatives of the Hamiltonian function $\mathrm{H}(\cdot )$ over the complex variables are in the sense of the formal derivativ

Figures (3)

  • Figure 1: Results of the gate generation for (\ref{['fig:BFCS_1a']}) NOT, (\ref{['fig:BFCS_1b']}) H, (\ref{['fig:BFCS_1c']}) S and (\ref{['fig:BFCS_1d']}) T via system \ref{['BFCS_one_qubit_hamiltonian']} with $\omega = 2$, $\alpha = 1$. Considered gate generation time is $T = 1$ for NOT and H gates, $T = 0.6$ for S gate and $T = 0.3$ for T gate.
  • Figure 2: Results of the gate generation for (\ref{['fig:BFCS_2a']}) CNOT via the system \ref{['BFCS_subsection_2qgates']} with $\omega_{1} = 3$, $\omega_{2} = 4$, $\beta = 1.25$, $\alpha = 1$ and $T = 4.75$; (\ref{['fig:BFCS_2b']}) CZ with $\omega = 2$, $\beta_{1} = 0.5$, $\beta_{2} = 0.75$, $\alpha = 1$ and $T = 9.8$.
  • Figure 3: Results of the CCNOT gate generation via the system \ref{['BFCS_three_qubit_hamiltonian']} with $\omega_{1} = 1$, $\omega_{2} = 2$, $\omega_{3} = 3$, $\beta_{12}^{y} = 1$, $\beta_{12}^{z} = 3$, $\beta_{23}^{y} = 5$, $\beta_{23}^{z} = 1.5$ and $T = 7.44$.

Theorems & Definitions (1)

  • Theorem 1