Nonlinear and non-CP gates for Bloch vector amplification

Abstract

Abstract Any state r = (x, y, z) of a qubit, written in the Pauli basis and initialized in the pure state r = (0, 0, 1), can be prepared by composing three quantum operations: two unitary rotation gates to reach a pure state r = x 2 + y 2 + z 2 − 1 2 × ( x , y , z ) on the Bloch sphere, followed by a depolarization gate to decrease ∣ r ∣. Here we discuss the complementary state-preparation protocol for qubits initialized at the center of the Bloch ball, r = 0, based on increasing or amplifying ∣ r ∣ to its desired value, then rotating. Bloch vector amplification increases purity and decreases entropy. Amplification can be achieved with a linear Markovian completely positive trace-preserving (CPTP) channel by placing the channel’s fixed point away from r = 0, making it nonunital, but the resulting gate suffers from a critical slowing down as that fixed point is approached. Here we consider alternative designs based on linear and nonlinear Markovian PTP channels, which offer benefits relative to linear CPTP channels, namely fast Bloch vector amplification without deceleration. These gates simulate a reversal of the thermodynamic arrow of time for the qubit and would provide striking experimental demonstrations of non-CP dynamics.