Parent Hamiltonian Reconstruction via Inverse Quantum Annealing

Finding a local Hamiltonian $\widehat{\mathcal{H}}$ that has a given many-body wave function $|\psi ⟩$ as its ground state, i.e., a parent Hamiltonian, is a challenge of fundamental importance in quantum technologies. Here we introduce a numerical method, inspired by quantum annealing, that efficiently performs this task through an artificial inverse dynamics: a slow deformation of the states $|\psi \mathbf{(}\lambda (t)\mathbf{)}⟩$, starting from a simple state $|{\psi }_0⟩$ with a known ${\widehat{\mathcal{H}}}_0$, generates an adiabatic evolution of the corresponding Hamiltonian. We name this approach inverse quantum annealing. The method, implemented through a projection onto a set of local operators, only requires the knowledge of local expectation values, and, for long annealing times, leads to an approximate parent Hamiltonian whose degree of locality depends on the correlations built up by the states $|\psi (\lambda )⟩$. We illustrate the method on two paradigmatic models: the Kitaev fermionic chain and a quantum Ising chain in longitudinal and transverse fields.

Figure 1

IQA with fermionic Gaussian states. Panel (a): maximum value ${\max }_{\lambda }[R_{T,\mathrm{\Delta }T}(\lambda )]$ of the relative distance $R_{T,\mathrm{\Delta }T}(\lambda )$ between solutions of Eq. (6) with different annealing times $T$ and $T+\mathrm{\Delta }T=2T$, for $l=6$, as a function of the annealing time, for systems of different sizes. Panel (b): fidelity between the target state $|\psi (\lambda )⟩$ and the ground state $|{\psi }_{\mathrm{GS}}^{(l)}(\lambda )⟩$ of the adiabatic $l$-local Hamiltonian ${\widehat{\mathcal{H}}}^{(l)}(\lambda )$ obtained via IQA. We consider a system of 50 sites and interaction ranges from $l=1$ to $l=26$.

Figure 2

IQA with fermionic Gaussian states. Panels (a) and (b): fidelity between the target state $|\psi (\lambda )⟩$ and the ground state $|{\psi }_{\mathrm{GS}}^{(l)}(\lambda )⟩$ of the adiabatic $l$-local Hamiltonian ${\widehat{\mathcal{H}}}^{(l)}(\lambda )$ as a function of the interaction range $l$ and for different system sizes $N$ [legend in panel (a)]. Panel (a) $\lambda$ is just before the phase transition, i.e., at $\lambda =0.9{\lambda }_c$, in panel (b) just after the phase transition, i.e., at $\lambda =1.1{\lambda }_c$. Panel (c): minimal interaction range $l_{\epsilon }$ required to ensure a fidelity $F\ge 1-\epsilon$ ($\epsilon =0.005$) between the target state and the ground state of the Hamiltonian obtained via the IQA, as a function of the system size and for different values of $\lambda$.

Figure 3

IQA with a nonintegrable Ising chain ($B_z=0.8$). Panel (a): fidelity between the target state $|\psi (\lambda )⟩$ and the ground state $|{\psi }_{\mathrm{GS}}^{(l)}(\lambda )⟩$ of the adiabatic two-local Hamiltonian, for a system of 10 spins and different annealing times. Panel (b): final infidelity at $\lambda =1$ as a function of the annealing time, for different system sizes.