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arXiv:2409.01153

On encoded quantum gate generation by iterative Lyapunov-based methods

Paulo Sérgio Pereira da Silva, Pierre Rouchon

Published Sep 2, 2024• 0 citations
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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 $\exp(\jmath ϕ) f_i, i=1,2, \ldots ,\bar n$ up to some desired precision and to some global phase $ϕ\in \mathbb{R}$. 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 $N=10$.

DOI
10.48550/arxiv.2409.01153

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A new approach called Reference Input Generation Algorithm (RIGA) is introduced for quantum gate generation. Let X¯ℓ−1(t) be the trajectory that is obtained in step ℓ−1, with X¯ℓ−1(t)=I. In step ℓ, X¯ℓ−1 is right-translated in order to displace X¯ℓ−1(Tf) to Xgoal. The translated trajectory is used as a reference for a Lyapunov-based tracking control law, generating X¯ℓ(T), and so on. A proof of the exponential convergence of X¯ℓ−1(Tf) to zero is provided for Tf large enough. Two examples present numerical experiments regarding N coupled qubits. The first example considers N = 3 with a known minimum time T∗. It presents excellent results for Tf=T∗. The second example is a benchmark for the comparison between RIGA and GRAPE, considering a Hadamard gate for the systems with N = 2, 3, …, 10 qubits. The runtime of RIGA could be improved, and GRAPE was implemented in a faster CPU. However, RIGA presents results that are similar to GRAPE, with faster runtime in some cases, showing that the RIGA is indeed a promising algorithm.
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