Fast Quantum State Preparation and Bath Dynamics Using Non-Gaussian Variational Ansatz and Quantum Optimal Control

Fast preparation of quantum many-body states is essential for myriad quantum algorithms and metrological applications. Here, we develop a new pathway for fast, nonadiabatic preparation of quantum many-body states that combines quantum optimal control with a variational Ansatz based on non-Gaussian states. We demonstrate our approach on the spin-boson model, a single spin interacting with the bath. We use a multipolaron Ansatz to prepare near-critical ground states. For one mode, we achieve a reduction in infidelity of up to $\approx 60$ ($\approx 10$) times compared to linear (optimized local adiabatic) ramps; for many modes, we achieve a reduction in infidelity of up to $\approx 5$ times compared to nonadiabatic linear ramps. Further, we show that the typical control quantity, the leakage from the variational manifold, provides only a loose bound on the state’s fidelity. Instead, in analogy to the bond dimension of matrix product states, we suggest a controlled convergence criterion based on the number of polarons. Finally, motivated by the possibility of realizations in trapped ions, we study the dynamics of a system with bath properties going beyond the paradigm of (sub- and/or super-) Ohmic couplings.

Figure 1

Single mode: (a) Order parameter $⟨{\sigma }_x⟩$ of the $P_{\mathrm{ex}}=1$ ground state and (c) fidelity $\mathcal{F}=|⟨{\mathrm{\Psi }}_{\mathrm{gs}}|{\psi }_{\mathrm{gs}}⟩{|}^2$ for $\epsilon /\mathrm{\Delta }=0.15/1.0$. The white and gray regions indicate the phase boundary between the normal and superradiant phase in the thermodynamic limit with the critical point $g_c=\sqrt{\epsilon \mathrm{\Delta }}$. (b) Time evolution of $⟨{\sigma }_x⟩$ from initial state $|0⟩|+⟩$ for $\epsilon /\mathrm{\Delta }=1.0/1.1$, $g/\mathrm{\Delta }=2.0/1.1$. (d) The fidelity $\mathcal{F}=|⟨\mathrm{\Psi }(t)|\psi (t)⟩{|}^2$ (solid) with the lower bound obtained from the leakage (dotted), cf. Eq. (5). Many modes: (e) Order parameter $⟨{\sigma }_x⟩$ for $N=10$ modes. For illustration, we also include the perturbative result with critical point ${\alpha }_c\approx 1+\mathrm{\Delta }/2{\omega }_c$ separating the delocalized (white) and localized (gray) phases [39, 41]. The inset plots $\mathrm{var}(H)$, which is largest near ${\alpha }_c$ due to the absence of squeezing in the Ansatz. (f) Real-time dynamics from initial state $|0⟩|+⟩$, with the purple line $N_p=16$. Parameters are ${\omega }_c/\mathrm{\Delta }=1.0/1.1$ and $\alpha =4.0$.

Figure 2

(a) Bath-mode position quadrature of the mid-$k$ ($k=6$) mode for a quench from a state $|\psi (t=0)⟩=|0⟩|+⟩$ for the coupling profiles $g_k^{+}$ (blue, dashed) and $g_k^{-}$ (orange, solid), see Eq. (7) and the inset in (b). The vertical dashed line indicates the scrambling time $t^{*}$ corresponding to the first maximum of $\mathrm{var}(x_6)$. The inset shows $t^{*}$ vs coupling amplitude $\overline{g}$ for the two profiles. (b) The occupation of the bath modes at the scrambling time. Parameters are ${\omega }_c/\mathrm{\Delta }=1/1.1$, $\overline{g}/\mathrm{\Delta }=1/1.1$ and $N=11$, $N_p=10$.

Figure 3

Single mode: (a) The infidelity $1-\mathcal{F}$, $\mathcal{F}=|⟨{\psi }_{\mathrm{gs}}|\psi (t_f)⟩{|}^2$, where $|{\psi }_{\mathrm{gs}}⟩$ is the target ground state with $\alpha =7$, and $|\psi (t_f)⟩$ is the state prepared with CRAB, linear, and LA ramps. An example of the three ramps for $t_f=70$ is shown in (b). In (c) we verify the accuracy of the CRAB simulation (purple) and its nonadiabaticity (red) by computing the overlap with the instantaneous ground state $|{\mathrm{\Psi }}_{\mathrm{gs}}(t)⟩$. (d) Minimum ramp times required to prepare a target ground state at $\alpha$ with fidelity $\mathcal{F}>0.99$. The right axis shows the ground state boson number (gray dashed line). Many modes: (e) Infidelity for CRAB and linear ramp protocols vs ramp time for $N=10$ modes with finite-size scaling of the minimum gap (inset). For comparison we show the gray dashed line obtained by extrapolating the linear ramp data from (a), see Supplemental Material [63]. (f) Ramp times required to prepare the target state with fidelity $\mathcal{F}>0.90$ vs $\alpha$, see text for details. Parameters used, $N_p=5$, ${\omega }_c/\mathrm{\Delta }=0.15$.