QNNs with Noisy Harmonic Oscillators, Strings, and Gravitational Control

Abstract

Let [ a , a = 1. Let https://www.w3.org/1998/Math/MathML" display="block"> ρ ( t ) = Z ( t ) exp − β 1 ( t ) a ∗ a + β 2 ( t ) a 2 + β ¯ 2 ( t ) a ∗ 2 − β 3 ( t ) a − β ¯ 3 ( t ) a ∗ where β 1 ( t ) is real, Z ( t ), β 1 ( t ), β 2 ( t ), β 3 ( t ) are constrained by the conditions https://www.w3.org/1998/Math/MathML" display="block"> T r ( ρ ( t ) ) = and ρ ( t ) satisfies the GKSL equation https://www.w3.org/1998/Math/MathML" display="block"> ρ ′ ( t ) = − i [ H , ρ ( t ) ] − ( 1 / 2 ) L L ∗ ρ ( t ) + ρ ( t ) L L ∗ − 2 L ∗ ρ ( t ) L with https://www.w3.org/1998/Math/MathML" display="block"> H = ω a ∗ a + λ a + λ ¯ a ∗ L = α a + β a ∗ , α , β , λ ∈ ℂ