Fast and virtually exact quantum gate generation in U(n) via iterative Lyapunov methods. Issue 4 (3rd April 2021) - Pereira da Silva, Paulo Sergio; Silveira, Hector Bessa; Rouchon, Pierre
ABSTRACT: 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 ( T f ) to X g o a l . 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 ( T f ) to zero is provided for T f 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 T f = 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.