Resource estimate for quantum many-body ground-state preparation on a quantum computer

We estimate the resources required to prepare the ground state of a quantum many-body system on a quantum computer of intermediate size. This estimate is made possible using a combination of quantum many-body methods and analytic upper bounds. Our routine can also be used to optimize certain design parameters for specific problem instances. Lastly, we propose and benchmark an improved quantum state preparation procedure. We find that it reduces the circuit $T$-depth by a factor as large as ${10}^6$ for intermediate-size lattices.

Figure 1

The cost of the rewind procedure for one step will be the sum of all paths that start at the top-right state and end at the top-left one. The probability to follow an arrow is given by the fidelity $F$ and its cost is given by the gap $\mathrm{\Delta }$ where the arrow ends.

Figure 2

Gain obtained from the rewind procedure to prepare the ground state of the 10% doped Hubbard model on $m\times k$ lattices. The units of the TTS are set by the natural units of the Hubbard model ($T_{\text{hub}}=1$).

Figure 3

Gain obtained from qubitization with rewind compared with a direct rewind procedure.

Figure 4

$T$ depth of the circuit presented is this work. Discrete adiabatic evolution with qubitization and rewind procedure (filled symbols) vs high-order products using sixth-order formulas with empirical error bound [46] and rewind procedures only (empty symbols).

Figure 5

MPS of the $m\times k$ lattice ($m=5,k=4$).

Figure 6

TTS obtained with and without qubitization (empty and filled symbols reps.) on chain, $m\times 1$, lattices. The units of the TTS are set by the natural units of the Hubbard model ($T_{\text{hub}}=1$). The $x$ axis corresponds to the minimal gap of the original problem also in the natural units of the Hubbard model (before qubitization).

Figure 7

TTS obtained with and without qubitization (empty and filled symbols reps.) on ladder, $m\times 2$, lattices. The units of the TTS are set by the natural units of the Hubbard model ($T_{\text{hub}}=1$). The $x$ axis corresponds to the minimal gap of the original problem also in the natural units of the Hubbard model (before qubitization).