Gate-Efficient Simulation of Molecular Eigenstates on a Quantum Computer

A key requirement to perform simulations of large quantum systems on near-term quantum hardware is the design of quantum algorithms with a short circuit depth that finish within the available coherence time. A way to stay within the limits of coherence is to reduce the number of gates by implementing a gate set that matches the requirements of the specific algorithm of interest directly in the hardware. Here, we show that exchange-type gates are a promising choice for simulating molecular eigenstates on near-term quantum devices since these gates preserve the number of excitations in the system. We report on the experimental implementation of a variational algorithm on a superconducting qubit platform to compute the eigenstate energies of molecular hydrogen. We utilize a parametrically driven tunable coupler to realize exchange-type gates that are configurable in amplitude and phase on two fixed-frequency superconducting qubits. With gate fidelities around 95%, we are able to compute the eigenstates to within an accuracy of 50 mHa (milliHartree) on average, a limit set by the coherence time of the tunable coupler.

Figure 1

The circuit depth required to achieve chemical accuracy for the ground-state energy with a VQE algorithm for the ${\mathrm{H}}_2$, LiH, ${\mathrm{BeH}}_2$, and ${\mathrm{H}}_2\mathrm{O}$ molecules. Non-excitation-conserving circuits based on cnot gates (red squares) are compared to excitation-conserving circuits based on exchange-type gates (blue circles) and a decomposition thereof into cnot gates (yellow triangles). In some cases, only a lower boundary to the circuit depth can be estimated (empty symbols). Bounded by the $T_1$ time in the currently available hardware, only circuits within the gray region can be implemented in practice without error mitigation or reduction schemes (see text).

Figure 2

An (a) optical micrograph and (b) the circuit scheme of the device consisting of two fixed-frequency transmons $(Q1,Q2)$ capacitively coupled to a flux-tunable transmon acting as tunable coupler (TC). The tunable coupler is controlled by a flux line (FL) providing a current $I(t)$ and a consequent flux $\Phi (t)={\Phi }_{\mathrm{D}C}+\delta \cos ({\omega }_{\mathrm{\Phi }}t+{\phi }_{\mathrm{\Phi }})$ threading the superconducting quantum-interference device (SQUID) loop of the coupler. Each of the fixed-frequency qubits is coupled to an individual readout resonator (R1, R2). (c) The level diagram of the device. Here, $|n_1{n}_2⟩$ denotes the state of the combined system with the qubit excitation number $n_{1,2}$. Modulation of the magnetic flux $\Phi (t)$ at the difference frequency ${\omega }_{\Phi }={\omega }_1-{\omega }_2$ of the qubits drives the transition between $|10⟩$ and $|01⟩$.

Figure 3

Quantum-process tomography of the chemistry gate ${\hat{U}}_{\mathrm{EX}}(\theta ,\phi )$. (a) The gate fidelities $\mathcal{F}$ as a function of $\phi$ for $\theta =\pi$. The bottom panel shows the gate fidelities calculated from the overlap of the measured process matrices ${\chi }_{\mathrm{meas}}(\phi )$ with the ideal process matrix ${\chi }_{\mathrm{ideal}}$ of a ${\hat{U}}_{\mathrm{EX}}(\pi ,\phi )$ (blue dots), iswap (orange triangles), ${\hat{U}}_{\mathrm{EX}}(\pi ,\pi /2)$ (red squares), and ${\hat{U}}_{\mathrm{EX}}(\pi ,\pi )$ (green diamonds) gate operation. The top panel shows the gate fidelities with respect to ${\hat{U}}_{\mathrm{EX}}(\pi ,\phi )$. The black dashed lines depicts the average gate fidelity for ${\hat{U}}_{\mathrm{EX}}(\pi ,\phi )$ (see text). The colored dashed lines are a fit to Eq. (3). (b) The gate fidelities $\mathcal{F}$ as a function of $\theta$, where the phase ${\phi }_{\mathrm{opt}}$ is tuned to maximize the QPT fidelity. The dashed line is a fit with an exponential decay function, with a decay time of $6.7\mu \mathrm{s}$.

Figure 4

The experimental VQE solution for the ground state and the EOM solution for the excited states of molecular hydrogen using a tunable-couple architecture. (a) The ground (G) and excited-state ($E_1$, $E_2$, $E_3$) energies as a function of the bond length. The symbols depict the experimental VQE solution, the solid lines represent the exact solution from the diagonalization of ${\hat{H}}_{{\mathrm{H}}_2}$, and the dashed line represent the solution including decoherence effects. (b) The accuracy for the ground and excited-state energies as a function of the bond length. The symbols correspond to the accuracy of the measured ground and excited-state energies determined with respect to the exact solution, while the dashed lines correspond to the expected accuracy including decoherence effects (see text). The depicted ground-(excited-)state energy is the minimum (median) value from a set of five measurements. The error bars depict the range between the first and third quantile (excited states only). The blue shaded area represents the region of chemical accuracy from 0 to 6.5 mHa.