Fermionic simulators for enhanced scalability of variational quantum simulation

Near-term quantum simulators are mostly based on qubit-based architectures. However, their imperfect nature significantly limits their practical application. The situation is even worse for simulating fermionic systems, which underlie most of material science and chemistry, as one has to adopt fermion-to-qubit encodings which create significant additional resource overhead and trainability issues. Thanks to recent advances in trapping and manipulation of neutral atoms in optical tweezers, digital fermionic quantum simulators are becoming viable. A key question is whether these emerging fermionic simulators can outperform qubit-based simulators for characterizing strongly correlated electronic systems. Here we perform a comprehensive comparison of resource efficiency between qubit and fermionic simulators for variational ground-state emulation of fermionic systems in both condensed matter systems and quantum chemistry problems. We show that the fermionic simulators indeed outperform their qubit counterparts with respect to resources for quantum evolution (circuit depth) as well as classical optimization (number of required parameters and iterations). In addition, they show less sensitivity to the random initialization of the circuit. The relative advantage of fermionic simulators becomes even more pronounced as interaction becomes stronger, or tunneling is allowed in more than one dimension, as well as for spinful fermions. Importantly, this improvement is scalable, i.e., the performance gap between fermionic and qubit simulators only grows for bigger system sizes.

Figure 1

(a) Decomposition of two-qubit gates ${\mathcal{U}}_{j{j}^{\prime }}^{\text{qubit}}$ (purple) in terms of rotations and cnot gates, (b) Decomposition of one layer of qubit circuit used in this paper: first ${\mathcal{U}}_{j{j}^{\prime }}^{\text{qubit}}$ (purple) is applied on all neighboring sites and then $R_j^z\left(\theta \right)$ (red) is applied on all sites, (c) Decomposition of one layer of the fermionic circuit used in this paper: first ${\mathcal{U}}_{j{j}^{\prime }}^{\text{tun}}$ (orange) is applied on all neighboring sites, and then ${\mathcal{U}}_{j{j}^{\prime }}^{\text{int}}$ (green) is applied in the same way.

Figure 2

(a) Open chain configuration for $N=12$ spinless Fermi-Hubbard model. (b) Dependence of particle number density on the Coulomb repulsion strength $V$ for the ground state. (c) One layer of the fermionic circuit which is decomposed into four steps (orange steps denote ${\mathcal{U}}_{j{j}^{\prime }}^{\text{tun}}$, green steps denote ${\mathcal{U}}_{j{j}^{\prime }}^{\text{int}}$). (d) One layer of the qubit circuit which is decomposed into three steps (purple steps denote ${\mathcal{U}}_{j{j}^{\prime }}^{\text{qubit}}$, and red steps denote the phase rotations). (e) Performance of the fermionic circuit-based VQE (orange) vs the qubit circuit-based VQE (green) in terms of convergence to ground-state energy for Fermi-Hubbard model interaction strength $V=0$. (f) Performance of the fermionic circuit-based VQE (orange) vs the qubit circuit-based VQE (green) in terms of fidelity with ground state for Fermi-Hubbard model interaction strength $V=0$. (g) Performance of the fermionic circuit-based VQE (orange) vs the qubit circuit-based VQE (green) in terms of convergence to ground-state energy for Fermi-Hubbard model interaction strength $V=2$. (h) Performance of the fermionic circuit-based VQE (orange) vs the qubit circuit-based VQE (green) in terms of fidelity with ground state for Fermi-Hubbard model interaction strength $V=2$. Tunneling strength $t=1$ throughout.

Figure 3

(a) Ladder $6\times 2$ configuration for $N=12$ spinless Fermi-Hubbard model. (b) Dependence of particle number density on the Coulomb repulsion strength $V$ for the ground state. (c) One layer of the fermionic circuit which is decomposed into six steps (orange steps denote ${\mathcal{U}}_{j{j}^{\prime }}^{\text{tun}}$, green steps denote ${\mathcal{U}}_{j{j}^{\prime }}^{\text{int}}$). (d) One layer of the qubit circuit which is decomposed into four steps (purple steps denote ${\mathcal{U}}_{j{j}^{\prime }}^{\text{qubit}}$, and red steps denote the phase rotations). (e) Performance of the fermionic circuit-based VQE (orange) vs the qubit circuit-based VQE (green) in terms of convergence to ground-state energy for Fermi-Hubbard model interaction strength $V=2$. (f) Performance of the fermionic circuit-based VQE (orange) vs the qubit circuit-based VQE (green) in terms of fidelity with ground state for Fermi-Hubbard model interaction strength $V=2$. Tunneling strength $t=1$ throughout.

Figure 4

(a) Rectangular $3\times 4$ configuration for $N=12$ spinless Fermi-Hubbard model. (b) One layer of the qubit circuit is decomposed into five steps (purple steps denote ${\mathcal{U}}_{j{j}^{\prime }}^{\text{qubit}}$, and red steps denote the phase rotations). (c) One layer of the fermionic circuit which is decomposed into eight steps (orange steps denote ${\mathcal{U}}_{j{j}^{\prime }}^{\text{tun}}$, green steps denote ${\mathcal{U}}_{j{j}^{\prime }}^{\text{int}}$). (d) Dependence of particle number density on the Coulomb repulsion strength $V$ for the ground state. (e) Performance of the fermionic circuit-based VQE (orange) vs the qubit circuit-based VQE (green) in terms of convergence to ground-state energy for Fermi-Hubbard model interaction strength $V=2$. (f) Performance of the fermionic circuit-based VQE (orange) vs the qubit circuit-based VQE (green) in terms of fidelity with the ground state for Fermi-Hubbard model interaction strength $V=2$. Tunneling strength $t=1$ throughout.

Figure 5

(a) Open chain configuration for the $N=6$ spinful Fermi-Hubbard model as simulated by qubit (left) and fermonic (right) simulators. (b) Dependence of particle number density on the site depth $U$ and Coulomb repulsion strength $V$ for the ground state. (c) One layer of the fermionic circuit, which is decomposed into six steps (orange steps denote ${\mathcal{U}}_{j{j}^{\prime },\alpha \alpha }^{\text{tun}}$, green steps denote ${\mathcal{U}}_{j{j}^{\prime },\alpha {\alpha }^{\prime }}^{\text{int}}$). (d) One layer of the qubit circuit which is decomposed into five steps (purple steps denote ${\mathcal{U}}_{j{j}^{\prime }}^{\text{qubit}}$, and red steps denote the phase rotations). (e) Performance of the fermionic circuit-based VQE (orange) vs the qubit circuit-based VQE (green) in terms of convergence to ground-state energy for $V=0.5,U=2.5$. (f) Performance of the fermionic circuit-based VQE (orange) vs the qubit circuit-based VQE (green) in terms of fidelity with ground state for $V=0.5,U=2.5$. Tunneling strength $t=1$ throughout.

Figure 6

(a) Ladder $2\times 3$ configuration for $N=6$ spinful Fermi-Hubbard model as simulated by qubit (top) and fermonic (bottom) simulators. (b) One layer of the fermionic circuit, which is decomposed into nine steps (orange steps denote ${\mathcal{U}}_{j{j}^{\prime },\alpha \alpha }^{\text{tun}}$, green steps denote ${\mathcal{U}}_{j{j}^{\prime },\alpha {\alpha }^{\prime }}^{\text{int}}$). (c) One layer of the qubit circuit is decomposed into seven steps (purple steps denote ${\mathcal{U}}_{j{j}^{\prime }}^{\text{qubit}}$, and red steps denote the phase rotations). (d) Dependence of particle number density on the site depth $U$ and Coulomb repulsion strength $V$ for the ground state. (e) Performance of the fermionic circuit-based VQE (orange) vs the qubit circuit-based VQE (green) in terms of convergence to ground-state energy for $V=0.5,U=2.5$. (f) Performance of the fermionic circuit-based VQE (orange) vs the qubit circuit-based VQE (green) in terms of fidelity with ground state for $V=0.5,U=2.5$. Tunneling strength $t=1$ throughout.

Figure 7

(a) Required quantum resource $R_{\text{Q}}$, and (b) the corresponding classical resource $R_{\text{C}}$, as system size $N$ increases for the spinless 1D open chain. Orange lines denote fermionic-based and green lines denote qubit-based simulator performance. Inset to panel (a) depicts fitting lines for $log{R}_{\text{Q}}$ vs $logN$ (slopes $\approx 2.19$ for fermionic vs $\approx 2.81$ for qubit). Inset to panel (b) depicts fitting lines for $log{R}_{\text{C}}$ vs $logN$ (slopes $\approx 3.90$ for fermionic vs $\approx 4.48$ for qubit). Interaction strength $V=2$ in every case. Tunneling strength $t=1$ throughout.

Figure 8

(a) The molecular geometry of water with four spin orbits, (b) One layer of the qubit circuit decomposed into five steps (purple steps denote ${\mathcal{U}}_{j{j}^{\prime }}^{\text{qubit}}$, red denote phase rotation $R_j^z$). (c) One layer of the fermionic circuit decomposed into seven steps (orange steps denote ${\mathcal{U}}_{j{j}^{\prime }}^{\text{tun}}$, green steps denote ${\mathcal{U}}_{j{j}^{\prime }}^{\text{int}}$). (d) Performance of the fermionic circuit based VQE (orange) vs the qubit circuit-based VQE (green), in terms of convergence to ground-state energy. (e) Performance of the fermionic circuit-based VQE (orange) vs the qubit circuit-based VQE (green), in terms of fidelity to the actual ground state.