Adaptive variational quantum computing approaches for Green's functions and nonlinear susceptibilities

TL;DR

This work develops and benchmarks an adaptive variational quantum computing pipeline for real-time quantum dynamics aimed at Green's functions $G^{\mathrm{R}}_{p,q}(t)$ and nonlinear susceptibilities $\chi^{(n)}_{\alpha\beta\gamma\delta}(t,\tau,0)$. By combining AVQDS for time propagation with AVQITE for ground-state preparation in a controlled-unitaries-liberated (CUL) framework, the authors achieve compact, problem-specific circuits capable of long-time evolution and accurate spectral analyses via Padé, Prony, or compressive sensing. They validate the approach on Fermi-Hubbard chains, the LiH molecule, and a two-site spin-1 model with Dzyaloshinskii–Moriya interaction, demonstrating faithful reproduction of time-domain correlators and 2D spectra while providing detailed resource estimates (CNOT counts, circuit depth) for near-term devices. The results indicate that real-time adaptive quantum dynamics can feasibly access both linear and nonlinear response functions with shallower circuits than conventional fixed-ansatz or CUR/HVA methods, offering a practical route to quantum simulations of dynamical properties in materials and molecules.

Abstract

We present and benchmark quantum computing approaches for calculating real-time single-particle Green's functions and nonlinear susceptibilities of Hamiltonian systems. The approaches leverage adaptive variational quantum algorithms for state preparation and propagation. Using automatically generated compact circuits, the dynamical evolution is performed over sufficiently long times to achieve adequate frequency resolution of the response functions. We showcase accurate Green's function calculations using a statevector simulator on classical hardware for Fermi-Hubbard chains of 4 and 6 sites, with maximal ansatz circuit depths of 65 and 424 layers, respectively, and for the molecule LiH with a maximal ansatz circuit depth of 81 layers. Additionally, we consider an antiferromagnetic quantum spin-1 model that incorporates the Dzyaloshinskii-Moriya interaction to illustrate calculations of the third-order nonlinear susceptibilities, which can be measured in two-dimensional coherent spectroscopy experiments. These results demonstrate that real-time approaches using adaptive parameterized circuits to evaluate linear and nonlinear response functions can be feasible with near-term quantum processors.

Figures (10)

• Figure 1: Hadamard test circuit to compute the Green's function. (a) Circuit to measure the Green's function component $I_{\alpha,\beta}^{p,q}$, eq \ref{['eq:Ipq']}, using the exact time-evolution operator $\mathrm{e}^{-\mathrm{i}\hat{\mathcal{H}} t}$. The ancillary qubit, initially in the state $\vert0\rangle$, is represented by the upper horizontal line. $X$ and $H$ denote Pauli-$X$ and Hadamard gates on the ancilla, respectively. A register of qubits for the physical system of interest, initially in a reference product state $\vert\varphi_0\rangle$, is denoted by the lower horizontal line. The application of the general multi-qubit Pauli gates $P_\alpha$ and $P_\beta$ is controlled by the ancilla qubit, while the unitary operator $U_\mathrm{G}$ prepares the ground state, $\vert\mathrm{G}\rangle=U_\mathrm{G}\vert\varphi_0\rangle$. (b) Measurement circuit for $I_{\alpha,\beta}^{p,q}$ using a variational state evolution circuit. The state propagation circuit highlighted with the dashed rectangle in (a) is replaced by the (adaptive) variational circuit in (b), where the angles in the parameterized unitaries $U_\mathrm{G}[\boldsymbol{\theta}^1]$ and $U_t[\boldsymbol{\theta}^2]$ evolve with time. The results are obtained from $Z$-basis measurements on the ancilla qubit.
• Figure 2: Quantum circuit for measuring the third-order susceptibility eq \ref{['eq:chi3_trans']}. (a) Circuit utilizing exact time-evolution gates. The upper horizontal line represents the ancillary qubit, initially in the state $\vert0\rangle$. $X$, $H$, and $S$ correspond to the Pauli-$X$, Hadamard rotation, and $S$-gate operations on the ancilla. The lower horizontal line denotes the qubit register representing the quantum spin system, initially in a reference product state $\vert\varphi_0\rangle$, which is rotated to the ground state $\vert G\rangle=U_\mathrm{G}\vert\varphi_0\rangle$ by a unitary circuit $U_\mathrm{G}$. Multiple controlled Pauli gates are used to measure different terms in eq \ref{['eq:chi3_trans']}. The expectation value $\mathrm{Im}[\langle G\vert P_0 P_1 \mathrm{e}^{\mathrm{i}\hat{\mathcal{H}}(t+\tau)} P_2\, \mathrm{e}^{-\mathrm{i}\hat{\mathcal{H}}t} P_3\, \mathrm{e}^{-\mathrm{i}\hat{\mathcal{H}}\tau} P_4 P_5\vert G\rangle]$ is obtained by Pauli-$Z$ measurement on the ancilla qubit. (b) The CUL circuit to measure the third-order susceptibility, which is an adaptive variational circuit equivalent to (a). The ground state is prepared using a parameterized circuit $U_\mathrm{G}[\boldsymbol{\theta}^1]$, the state propagation by $\mathrm{e}^{-\mathrm{i}\hat{\mathcal{H}} \tau}$ is achieved by evolving the angles in $U_\tau [\boldsymbol{\theta}^3]$ and $U_\mathrm{G}[\boldsymbol{\theta}^1]$, and state propagation by $\mathrm{e}^{-\mathrm{i}\hat{\mathcal{H}} t}$ is achieved by evolving angles in $U_t [\boldsymbol{\theta}^2]$, $U_\tau [\boldsymbol{\theta}^3]$, and $U_\mathrm{G}[\boldsymbol{\theta}^1]$. The parameterized unitaries $U_\mathrm{G}[\boldsymbol{\theta}^1]$, $U_t [\boldsymbol{\theta}^2]$, and $U_\tau [\boldsymbol{\theta}^3]$ are automatically generated using adaptive variational algorithms.
• Figure 3: Numerical simulation of the AVQDS approach for computing the single-particle Green's function of Fermi-Hubbard model. Examples of $I^{p,q}_{\alpha,\beta}(t)$ dynamics for six different combinations of $p$, $q$, $\alpha$, and $\beta$, obtained by evaluating the quantum circuit in Figure \ref{['fig1']}\ref{['fig1']} for Fermi-Hubbard chains with (a) $N=4$- and (b) $N=6$-sites. The results obtained with the AVQDS approach (solid lines) are compared with those of the exact simulations (black dashed lines) obtained via exact diagonalization eq \ref{['eq:ED']}. The corresponding infidelities $1-f$ in (c) and (d) demonstrate the high accuracy of AVQDS in calculating the Green's function components $I^{p,q}_{\alpha,\beta}(t)$, achieving a fidelity of at least 99.93 $\%$ for $N=4$ and 99.64 $\%$ for $N=6$. The corresponding number of CNOT gates in (e) and (f) increases from an initial count of 270 (2402) to a maximum of 610 (6148) at the final simulation time of $t=10$ for $N=4$ ($N=6$). The circuit depth in (g) and (h) grows from 26 (147) to a maximum circuit depth of 65 (424) at $t=10$ for $N=4$ ($N=6$). Note that the pair $I^{(0,\uparrow), (0,\uparrow)}_{1,1}$ (green line) and $I^{(1,\uparrow), (0,\uparrow)}_{1,1}$ (yellow line) as well as the pair $I^{(0,\uparrow), (0,\uparrow)}_{2,2}$ (dark blue line) and $I^{(0,\downarrow), (0,\uparrow)}_{1,2}$ (orange line) have the same $P_\beta$ but different $P_\alpha$ in the circuit of Figure \ref{['fig1']}\ref{['fig1']}. As a result, the circuits for measuring these pairs involve exactly the same parameterized unitaries $U_\mathrm{G}[\boldsymbol{\theta}^1]$ and $U_t[\boldsymbol{\theta}^2]$ in Figure \ref{['fig1']}\ref{['fig1']}, and therefore the same evolution of the number of CNOTs, circuit depth, and infidelity. However, the dynamics of these pairs in (a) and (b) are distinct due to different $P_\alpha$.
• Figure 4: Single-particle Green's function in momentum space and spectral function. Dynamics of the real and imaginary parts of $G^\mathrm{R}_{k}(t)$ at momentum $k=0$ for Fermi-Hubbard model with (a) $N=4$- and (b) $N=6$-sites. The real part (red circles) and imaginary part (cyan diamonds) of $G^\mathrm{R}_{k=0}(t)$ obtained with the AVQDS approach agree well with the corresponding results of the exact simulations (solid black lines). (c), (d) Spectral function $A_{k=0}(\omega)$ obtained by Fourier transforming the dynamics presented in (a) and (b) using the Padé approximation and calculating eq \ref{['eq:Ak']}. The result is shown for three different $t_\mathrm{max}$ used within the Padé approximation. As a comparison, the exact result for the spectral function based on eq \ref{['eq:GL']} is plotted as a shaded area. $t_\mathrm{max}=7$ is sufficient to accurately reproduce the main features in $A_{k=0}(\omega)$. (e), (f) $|S_{\nu,0}|^2$ (black dots) and $|S_{0,\nu}|^2$ (red dots) as a function of the energy differences $E_0-E_\nu$ and $E_\nu - E_0$, respectively, for (e) $N=4$ and (f) $N=6$. Only the dominant transition amplitudes with $|S_{\nu,\mu}|^2 > 0.02$ are shown. The peaks in the spectral functions originate from transitions between the ground state with energy $E_0$ to the excited states with energies $E_\nu$, and vice versa.
• Figure 5: Comparison of different signal processing techniques for spectral function calculation. (a), (b) Spectral function $A_{k=0}(\omega)$ obtained by Fourier transforming the dynamics presented in Figure \ref{['fig3']}\ref{['fig3']} and \ref{['fig3']}\ref{['fig3']}, using a damping factor of $\varepsilon=0.3$ in the Fourier transformation. The result of the Padé approximation (shaded area) is shown together with the results of Prony approximation (blue line) and compressive sensing (orange line). (c), (d) The corresponding results without broadening ($\varepsilon=0$) for (c) $N=4$ and (d) $N=6$.