Ultrafast variational simulation of nontrivial quantum states with long-range interactions

State preparation protocols ideally require as minimal operations as possible in order to be implemented in near-term, potentially noisy quantum devices. Motivated by long-range interactions (LRIs) intrinsic to many present-day experimental platforms (trapped ions, Rydberg atom arrays, etc.), we investigate the efficacy of variationally simulating nontrivial quantum states using the variational quantum-classical simulation (VQCS) protocol explored recently [W. W. Ho and T. H. Hsieh, SciPost Phys. 6, 29 (2019)], in the presence of LRIs. We show that this approach leads to extremely efficient state preparation: for example, Greenberger-Horne-Zeilinger (GHZ) states can be prepared with $O\left(1\right)$ iterations of the protocol, and a quantum critical point of the long-range transverse-field Ising model (TFIM) can be prepared with $>99\%$ fidelity on a 100-qubit system with only one iteration. Furthermore, we show that VQCS with LRIs is a promising route for exploring generic points in the phase diagram of the long-range TFIM. Our approach thus provides concrete, ultrafast protocols for quantum simulators equipped with long-range interactions.

Figure 1

Fidelity $|{\langle {\psi }_{QCP}|\psi (\vec{\gamma },\vec{\beta })\rangle }_{p=1}{|}^2$ in preparation of LMG critical state as a function of system size $N$, for VQCS depth $p=1$. Remarkably, even at $N=100$, the fidelity is very close to unity ($>99\%$).

Figure 2

Left: Fidelity in preparation of GHZ state as a function of system size $N$, for $\alpha =0.2$. The fidelity decreases with increasing $N$; however, this can be compensated by going to higher $p\mathrm{s}$. Right: Fidelity in preparation of GHZ state as a function of $\alpha$, for $N=14$.

Figure 3

Heat map depicting fidelities in the state preparation of the ground state of the long-range TFIM [Eq. (3)] for system sizes $N=11$ and 12, for $p=1,2$. Blue (red) demarcates regions of high (low) fidelity. As noted before, there is an odd-even system size distinction near $\alpha \approx 0, h\approx 0$ for $p=1$, but this distinction vanishes at the next level, $p=2$.