Simulating Models of Challenging Correlated Molecules and Materials on the Sycamore Quantum Processor

Simulating complex molecules and materials is an anticipated application of quantum devices. With the emergence of hardware designed to target strong quantum advantage in artificial tasks, we examine how the same hardware behaves in modeling physical problems of correlated electronic structure. We simulate static and dynamical electronic structure on a superconducting quantum processor derived from Google’s Sycamore architecture for two representative correlated electron problems: the nitrogenase iron-sulfur molecular clusters and $\alpha$-ruthenium trichloride, a proximate spin-liquid material. To do so, we simplify the electronic structure into low-energy spin models that fit on the device. With extensive error mitigation and assistance from classical recompilation and simulated data, we achieve quantitatively meaningful results deploying about one fifth of the gate resources used in artificial quantum advantage experiments on a similar architecture. This increases to over half of the gate resources when choosing a model that suits the hardware. Our work serves to convert artificial measures of quantum advantage into a physically relevant setting.

Popular Summary

Simulating complex molecules and materials is an anticipated application of quantum devices and part of Feynman's original vision for quantum computers. Recent years have seen a rapid development of quantum computers, with some, such as Google's Sycamore processor, claiming advantage over classical computers, albeit for highly artificial computational tasks. A natural question is whether this generation of quantum computers can also demonstrate advantage in the more practically relevant setting of chemical and materials simulation. In other words, how relevant are demonstrations of quantum advantage in artificial problems to quantum advantage in “real-world” problems? Using a device based on Google's Sycamore quantum architecture, we examine the capabilities of today's quantum hardware for simulations of the behavior of electrons. In particular, we use the quantum processor to simulate spectroscopy experiments on two representative molecule and materials problems: the nitrogenase iron-sulfur molecular clusters and alpha-ruthenium trichloride, an exotic quantum material.

Using extensive error mitigation to account for the noise on the quantum processor, we find that we can obtain quantitatively meaningful spectra so long as our simulations are not so large that noise overwhelms the calculation. In particular, we see that we can successfully deploy about 1/5 of the quantum resources used in artificial quantum advantage experiments and, if we fine-tune parameters in the simulated model, up to 1/2 of the resources. This serves as a benchmark to convert the current capabilities of quantum processors from the artificial advantage setting to that of practical quantum simulations in chemistry and materials.

Figure 1

(a) The molecular structures of the [4$\mathrm{Fe}$-4$\mathrm{S}$] and [8$\mathrm{Fe}$-7$\mathrm{S}$] P clusters. (b) The topology of the four-site and eight-site Heisenberg spin models of the (top) [4$\mathrm{Fe}$-4$\mathrm{S}$] and (bottom) [8$\mathrm{Fe}$-7$\mathrm{S}$] P clusters. (c) The crystal structure of $\alpha$-$\mathrm{Ru}\mathrm{Cl}$${}_3$: trigonal space group P3${}_{1}12$. (d) The topology of the six-site and ten-site Kitaev-Heisenberg spin models of $\alpha$-$\mathrm{Ru}\mathrm{Cl}$${}_3$. The adjacent ancilla qubits are indicated as “A.” (e) Illustrative examples of qubit patches (blue) used for simulations of four-site and eight-site models. The ancilla qubits (dotted boxes) are shown in red. The color scheme of the qubits and connectivities represents the “single-qubit randomized benchmarking (RB) average error per gate” and “two-qubit $\sqrt{\mathrm{i}\mathrm{SWAP}}$ gate cross-entropy benchmarking (XEB) average error per cycle,” respectively.

Figure 2

(a) The quantum circuit to calculate a finite-temperature dynamic correlation function $⟨\hat{U}(t)\hat{V}{⟩}_{\beta }$ using QITE and one ancilla qubit. Measuring $X$ and $Y$ on the ancilla yields the real and imaginary parts of the correlation function, respectively. The thermal average over the initial states is obtained according to Eq. (4). (b) The circuit-recompilation ansatz. Each gate round after the base PhXZ layer includes a layer of two-qubit ${\sqrt{\mathrm{i}\mathrm{SWAP}}}^{†}$ and single-qubit PhXZ gates, as shown in the box.

Figure 3

(a) The finite-temperature $⟨Z_i{Z}_j⟩$ ($\beta$ = 2) correlation functions computed for the [4$\mathrm{Fe}$-4$\mathrm{S}$] model. The simulation values are in good agreement with the exact results in parentheses. (b) The finite-temperature dynamical correlation function $⟨Z_1(t)Z_2(0){⟩}_{\beta }$ ($\beta$ = 2) and its Fourier transform for the [4$\mathrm{Fe}$-4$\mathrm{S}$] model. (c) The spin-coupling pattern (derived from $⟨Z_i{Z}_j⟩$ correlation functions) in the P-cluster and/or $\mathrm{Fe}$-$\mathrm{Mo}$–cofactor model and the simulated finite-temperature $⟨Z_1{Z}_8⟩$ and $⟨Z_2{Z}_3⟩$ for the model, showing the correct magnetic coupling pattern in the low-temperature regime. The deviation between the exact and no-noise results shows the effects of recompilation. (d) The real and imaginary parts of the finite-temperature dynamical correlation function $⟨Z_1(t)Z_2(0){⟩}_{\beta }$ ($\beta$ = 1) and its Fourier transform for the P-cluster and/or $\mathrm{Fe}$-$\mathrm{Mo}$–cofactor model. Error-mitigation acronyms: SE, spin echoes; F, Floquet calibration; RO, readout-error mitigation; PS, postselection; HW, hardware.

Figure 4

(a) The thermal energy $⟨E⟩$ and heat capacity ($c/n=\frac{1}{n}\mathrm{\partial }⟨E⟩/\mathrm{\partial }T$) for the $\alpha$-$\mathrm{Ru}\mathrm{Cl}$${}_3$ six-site model ($n=6$). The two-peak structure is indicative of the proximate spin-liquid character [29, 30, 31]. (b) The dynamical correlation function $⟨Z_3(t)Z_4(0){⟩}_{\beta }$ and its transform $S(\omega )$ at $\beta$ = 1 for the $\alpha$-$\mathrm{Ru}\mathrm{Cl}$${}_3$ six-site model. The three plots for the real part show the successive improvement obtained by standard error mitigation (Floquet, readout error, postselection, orange); the introduction of spin echoes (light green); and rescaling (dark green). The same is seen in the plots for the imaginary part and for $S(\omega )$, where the results successively including these three classes of techniques are superposed. Error-mitigation acronyms: SE, spin echoes; F, Floquet calibration; RO, readout-error mitigation; PS, postselection; HW, hardware.

Figure 5

(a) The dynamical correlation function $⟨Z_3(t)Z_4(0){⟩}_{\beta }$ and its transform $S(\omega )$ at $\beta =1$ for the $\alpha$-$\mathrm{Ru}\mathrm{Cl}$${}_3$ ten-site model. Even after all error-mitigation strategies (light green), the data in this case are too degraded to rescale. (b) The dynamical correlation function $⟨Z_3(t)Z_4(0){⟩}_{\beta }$ and its transform $S(\omega )$ at $\beta =1$ for strongly anisotropic Kitaev-Heisenberg model parameters: $J=0.4$, $K_x=-8$, and $K_y=K_z=\frac{K_x}{6}$. Reasonable frequencies are obtained after rescaling (dark green). Error-mitigation acronyms: SE, spin echoes; F, Floquet calibration; RO, readout-error mitigation; PS, postselection; HW, hardware.

Figure 6

The spectra of the $S=1/2$ and $S=5/2$ Heisenberg models for the [4$\mathrm{Fe}$-4$\mathrm{S}$] cluster.

Figure 7

The spectra of the $S=1/2$ and $S=5/2$ Heisenberg models for the P and/or $\mathrm{Fe}$-$\mathrm{Mo}$–cofactor cluster.

Figure 8

Noisy emulation of the [4$\mathrm{Fe}$-4$\mathrm{S}$] model spectrum.