Adiabatic quantum state preparation in integrable models

Abstract

We propose applying the adiabatic algorithm to prepare high-energy eigenstates of integrable models on a quantum computer. We first review the standard adiabatic algorithm to prepare ground states in each magnetization sector of the prototypical XXZ Heisenberg chain. Based on the thermodynamic Bethe ansatz, we show that the algorithm circuit depth is polynomial in the number of qubits $N$, outperforming previous methods explicitly relying on integrability. Next, we propose a protocol to prepare arbitrary eigenstates of integrable models that satisfy certain conditions. For a given target eigenstate, we construct a suitable parent Hamiltonian written in terms of a complete set of local conserved quantities. We propose using such Hamiltonian as an input for an adiabatic algorithm. After benchmarking this construction in the case of the non-interacting XY spin chain, where we can rigorously prove its efficiency, we apply it to prepare arbitrary eigenstates of the Richardson-Gaudin models. In this case, we provide numerical evidence that the circuit depth of our algorithm is polynomial in $N$ for all eigenstates, despite the models being interacting.

Figures (10)

• Figure 1: The RG quadratic Bethe equations are solved numerically for all eigenstates while dynamically adapting $\Delta g$, the pairwise $\Vert\cdot\Vert_2$-distance is computed and minimized over the adiabatic path $g\in[0,10]$ and all eigenstate pairs SM. This equals the minimal gap of $H^{RG}_v$ along the adiabatic path for all eigenstates $v$ and is shown against system size. Inset shows deviations of the data point from the $N^{-1}$ curve shown in black; inset legend shows fitted exponents with standard deviation error. It can be seen that this fits the gap closing for all models.
• Figure S1: Exact diagonalization results for the gap above magnetization sector lowest-energy states in the XXZ model.
• Figure S2: Extrapolated slopes for data shown in Fig. \ref{['fig:xxz_gap_scaling']}. Large dots corresponds to system sizes restricted to $N > 22$, smaller dots include all smaller system sizes. Black line corresponds to thermodynamic Bethe ansatz $N^{-1}$ gap scaling. The right hand figure is a magnification for y-values around $-1$.
• Figure S3: Numerical results of iterated solution of integral equations for different $(h, \Delta)$ as circled dots; background color shows cubic interpolation for entire phase. Black line marks phase boundary $h_\textrm{crit} = \frac{1-\Delta}{2}$ (slight offset) from para- to ferromagnetic phase. The phase is symmetric for $h \rightarrow -h$, so only the positive part is shown. The inset shows a magnification of the plot near $\Delta = 1$, only at which point the numerical solution can seen to be unstable. Bottom lines show analytic solution for limit case $h=0$, it agrees with the numerical data.
• Figure S4: ED results for magnetization sector gap for fixed $N=28$ in comparison with iterative solutions of the TBA integral equations of $v_F, \mathcal{Z}^2$. The TBA large system size limit and small size ED agree (up to finite-size effects and instability of TBA numerics near $\Delta=1$SM), the gap closes only at $\Delta=1$ or $M/N=0$. As TBA numerics is limited near $\Delta = 1$, $\Delta=0.9$ is not shown and, in conjunction with finite size effects, the deviations occur SM.