The reachable set of single-mode quadratic Hamiltonians

Abstract

The question of open-loop control in the Gaussian regime may be cast by asking which Gaussian unitary transformations are reachable by turning on and off a given set of quadratic Hamiltonians. For compact groups, including finite dimensional unitary groups, the well known Lie algebra rank criterion provides a sufficient and necessary condition for the reachable set to cover the whole group. Because of the non-compact nature of the symplectic group, which corresponds to Gaussian unitary transformations, this criterion turns out to be still necessary but not sufficient for Gaussian systems. If the control Hamiltonians are unstable, in a sense made rigorous in the main text, the peculiar situation may arise where the rank criterion is satisfied and yet not all symplectic transformations are reachable. Here, we address this situation for one degree of freedom and study the properties of the reachable set under unstable control Hamiltonians. First, we provide a partial analytical characterisation of the reachable set and prove that no orthogonal ('energy-preserving' or 'passive' in the literature) symplectic operations may be reached with such controls. Then, we apply numerical optimal control algorithms to demonstrate a complete characterisation of the set in specific cases.

Holomorphic representation of quantum computations

“…Recall that Gaussian unitary operations act on the vector of quadrature operators through their symplectic representation [48]. The elements of the single-mode symplectic Lie algebra are classified as parabolic/elliptic/hyperbolic according to their trace [98]. This categorisation has a geometrical interpretation where parabolics are related to shears (shearing), elliptics to rotations (phase-shifting), and hyperbolics to squeezes (squeezing), and the symplectic term in Eq.…”

Numerical estimation of reachable and controllability sets for a two-level open quantum system driven by coherent and incoherent controls

“…where δ x T = 0. For the problems (26), (27), control u is considered in the class of controls ( 5), ( 6) satisfying only the constraint (4).…”

“…with δ x T = 1/(M z). If the regularizer (8) or (10) or ( 14) is considered, then the estimation C(T, x target , U([0, T ], Q)) is defined by such nodes {x s } of the grid G(M ) that each of them is an initial point of the trajectory representing the solution, correspondingly, of (25) or (26) or (27) with δ x T = 1/(M z).…”

Locally optimal control of continuous-variable entanglement

“…Each of the single-mode symplectic matrices S 1 and S 2 admits a singular value decomposition as S j = Q j Z j R j , where Q j and R j are orthogonal symplectic matrices and Z j = diag(z j , 1/z j ) is a local squeezing operation [17,33]. This fact simplifies our optimisation considerably: first, notice that all the terms depending on S j in Eqs.…”

“…Initial state preparation versus repeated control A few preliminary considerations concerning the optimal control strategy emerge directly from Eqs. (32)(33)(34)(35)(36). If the magnitudes of the xand p-correlations in the normal form c + and c − are the same, i.e.…”