Resource estimations for the Hamiltonian simulation in correlated electron materials

Correlated electron materials, such as superconductors and magnetic materials, are regarded as fascinating targets in quantum computing. However, the quantitative resources, specifically the number of quantum gates and qubits, required to perform a quantum algorithm to simulate correlated electron materials remain unclear. In this study, we estimate the resources required for the Hamiltonian simulation algorithm for correlated electron materials, specifically for organic superconductors, iron-based superconductors, binary transition-metal oxides, and perovskite oxides, using the fermionic swap network. The effective Hamiltonian derived using the ab initio downfolding method is adopted for the Hamiltonian simulation, and a procedure for the resource estimation by using the fermionic swap network for the effective Hamiltonians including the exchange interactions is proposed. For example, in the system for the ${10}^2$ unit cells, the estimated numbers of gates per Trotter step and qubits are approximately ${10}^7$ and ${10}^3$, respectively, on average for the correlated electron materials. Furthermore, our results show that the number of interaction terms in the effective Hamiltonian, especially for the Coulomb interaction terms, is dominant in the gate resources when the number of unit cells constituting the whole system is up to ${10}^2$, whereas the number of fermionic swap operations is dominant when the number of unit cells is more than ${10}^3$.

Figure 1

Procedure of the fswap network for a one-dimensional array of ten qubits (= ten spin orbitals, five orbitals). $p$ and $p\sigma$ ($q\sigma$) in the boxes are the indices of the orbital and spin orbital, respectively, where $p,q\in [1,5]$ and $\sigma =\uparrow ,\downarrow$. (a) Representation of the pair of the spin-orbitals. (b) The pair swapping operation. (c) Depiction of the procedure.

Figure 2

Resource estimation results for the number of gates for target compounds. (a) The number of gates $N_{\mathrm{gates}}\left[N_{\text{cells}}\right]$ for ${10}^2, {10}^3$, and ${10}^4 N_{\text{cells}}$. (b) The number of interaction terms per unit cell, $N_{\mathrm{terms}/\mathrm{cell}}^{i_{\mathrm{int}}}$. (c) The number of gates of the interaction term per unit cell, $N_{\mathrm{terms}/\mathrm{cell}}^{i_{\mathrm{int}}}{N}_{\mathrm{gates}}^{i_{\mathrm{int}}}$. The values of (b) and (c) are shown for $\left|t\right|, U$, and $J$.

Figure 3

$N_{\text{cells}}$ dependence for the number of gates in the interaction term (blue line), fswap term (red line), and the total (black dash line) in Eq. (8). The values are averaged over all the compounds.

Figure 4

Quantum gates corresponding to each interaction operator of $i_{\mathrm{int}}$ and the fswap operator in our procedure. $T$ in the box of the circuit is the one-qubit gate as $\left(\begin{array}{cc}1& 0\\ 0& e^{-i\theta }\end{array}\right), G$ is the global phase gate as $e^{-i\theta }\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right), H$ is the Hadamard gate, $Y$ is the $R_y(-\frac{\pi }{2})$ gate, and $R_z$ is the $R_z\left(\theta \right)$ gate, where $\theta$ is the parameter determined with the interaction coefficient (see Ref. [80] for details). (a) Quantum gates for each interaction operator of $i_{\mathrm{int}}$. The spin-orbital indices are indicated on the left-hand side of each circuit. The part enclosed by a dashed line in the circuit of the second row is assigned to a circuit with $g=H$ substituted, followed by a circuit with $g=Y$ in succession. The similar operation is performed for {$g_1,g_2,g_3,g_4$} in the last row. (b) A quantum gate for the fswap operator between the neighboring qubits.