Fastest scrambler

Abstract

Scrambling is a unitary Hamiltonian evolution that evolves known initial states such that at time t*, called the scrambling time, it is not possible to distinguish different initial states without measuring a large fraction of the degrees of freedom of the system. The variable that describes how far a system is from an idealistic scrambled system is called the order of scrambling. For a quantum system, we can compare the scrambling time of well-normalized Hamiltonians to the same order of scrambling and find the fastest scrambler. For normalization of Hamiltonian, we assume the energy of the system grows extensively with the number of qubits. Because the fastest scrambler is different for various orders of scrambling, we choose the smallest possible order of scrambling in what follows. The questions that we answered are finding the Hamiltonian of the fastest scrambler and its scrambling time, especially for large numbers of qubits. Also because scrambling and entanglement have a close correspondence, we were able to find a diagram that produces the Hamiltonian which evolves any computational basis to a highly symmetric entangled state in the fastest way. Such states are highly useful in quantum information processing, for example in quantum error correction codes.