Finite approximation of unitary operators for conditional analog simulators

Simulating unitary dynamics unavailable in nature by finite perturbative expansion of its Hamiltonian is a straightforward and intuitive approach which often fails when continuous-variable systems are considered. We propose a solution to this problem, which lies in constructing these expansions from suitably bounded operators. We demonstrate this by suggesting an elementary bounded gate based on Rabi coupling and showing its optimality and suitability for approximations of unitary operators.

Figure 1

Bounding functions for approximate expansions of cubic gate with $t=1$ and $z=1$. Green denotes truncated Taylor expansion constructed with $X$ gates, red denotes truncated Taylor expansion constructed from Rabi gates, solid lines represent first-order expansion with $N=1$, squares show second-order expansion with $N=2$, and circles indicate third-order expansion with $N=3$.

Figure 2

The valid regions for approximations of cubic operation with effective strength $t$ acting on coherent states $|\alpha \rangle$. The filled areas correspond to parameters for which the resulting state has fidelity $F>0.99$. Different colors relate to different orders of expansion $N$. The considered composing gates were as follows: (a) the Rabi gate with the optimal parameter $z$ obtained as $z=0.36$ for $N=1, z=0.20$ for $N=5$, and $z=0.15$ for $N=9$; (b) the X gate.