# Error analysis on the approximation of an adiabatic evolution operator by a QAOA circuit

I would like to know what would be the approximation error of a QAOA circuit.

Suppose we have time-dependent Hamiltonian $$H(t) = (1 - s(t))H_{init} + s(t)H_{prob}$$ where $$H_{init}$$ in an initial Hamiltonian whose ground is $$|\psi_0\rangle$$, $$H_{prob}$$ is a Hamiltonian that encodes an optimization problem and $$s(t)$$ is an adiabatic schedule with $$t \in [0,T]$$. $$H_{init}$$ and $$H_{prob}$$ do not commute.

The solution to the Schrodinger equation $$|\psi(t)\rangle$$ is given by a unitary $$U(t,0)$$ such that $$|\psi(t)\rangle = U(t,0)|\psi_0\rangle$$ and \begin{align} U(t,0) = \lim_{p \rightarrow \infty} &\exp \left \{ -\frac{i \Delta t}{\hbar} H(p \Delta t) \right \} \exp \left \{ -\frac{i \Delta t}{\hbar} H((p-1) \Delta t) \right \} \cdots \\ &\exp \left \{ -\frac{i \Delta t}{\hbar} H(0) \right \}. \end{align} In the equation above $$\Delta t = t/p$$. We can approximate $$U(t,0)$$ with the QAOA circuit $$\hat{U}(t,0)$$ by setting $$p$$ to be finite and applying the first order Trotter approximation with error $$O(\Delta t^2)$$.

I would like to know if someone could provide analysis or refs on the error of $$||U(t,0) - \hat{U}(t,0)|| \leq \ ???$$ or alternatively $$U(t,0) = \hat{U}(t,0) + O(\textrm{???}) \textrm{ as} \ p \rightarrow \infty.$$