On encoded quantum gate generation by iterative Lyapunov-based methods

• URL: http://arxiv.org/abs/2409.01153v1
• Date: Mon, 2 Sep 2024 10:41:15 GMT
• Title: On encoded quantum gate generation by iterative Lyapunov-based methods
• Authors: Paulo Sergio Pereira da Silva, Pierre Rouchon,
• Abstract summary: The problem of encoded quantum gate generation is studied in this paper. The emphReference Input Generation Algorithm (RIGA) is generalized in this work.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The problem of encoded quantum gate generation is studied in this paper. The idea is to consider a quantum system of higher dimension $n$ than the dimension $\bar n$ of the quantum gate to be synthesized. Given two orthonormal subsets $\mathbb{E} = \{e_1, e_2, \ldots, e_{\bar n}\}$ and $\mathbb F = \{f_1, f_2, \ldots, f_{\bar n}\}$ of $\mathbb{C}^n$, the problem of encoded quantum gate generation consists in obtaining an open loop control law defined in an interval $[0, T_f]$ in a way that all initial states $e_i$ are steered to $\exp(\jmath \phi) f_i, i=1,2, \ldots ,\bar n$ up to some desired precision and to some global phase $\phi \in \mathbb{R}$. This problem includes the classical (full) quantum gate generation problem, when $\bar n = n$, the state preparation problem, when $\bar n = 1$, and finally the encoded gate generation when $ 1 < \bar n < n$. Hence, three problems are unified here within a unique common approach. The \emph{Reference Input Generation Algorithm (RIGA)} is generalized in this work for considering the encoded gate generation problem for closed quantum systems. A suitable Lyapunov function is derived from the orthogonal projector on the support of the encoded gate. Three case-studies of physical interest indicate the potential interest of such numerical algorithm: two coupled transmon-qubits, a cavity mode coupled to a transmon-qubit, and a chain of $N$ qubits, including a large dimensional case for which $N=10$.

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