Computation of Molecular Spectra on a Quantum Processor with an Error-Resilient Algorithm

Harnessing the full power of nascent quantum processors requires the efficient management of a limited number of quantum bits with finite coherent lifetimes. Hybrid algorithms, such as the variational quantum eigensolver (VQE), leverage classical resources to reduce the required number of quantum gates. Experimental demonstrations of VQE have resulted in calculation of Hamiltonian ground states, and a new theoretical approach based on a quantum subspace expansion (QSE) has outlined a procedure for determining excited states that are central to dynamical processes. We use a superconducting-qubit-based processor to apply the QSE approach to the ${\mathrm{H}}_2$ molecule, extracting both ground and excited states without the need for auxiliary qubits or additional minimization. Further, we show that this extended protocol can mitigate the effects of incoherent errors, potentially enabling larger-scale quantum simulations without the need for complex error-correction techniques.

Popular Summary

While universal quantum computers promise significant societal impact through their ability to outperform the best classical devices, the realization of such technology requires significant advances in state-of-the-art quantum hardware. Nascent quantum processors with finite coherence times, modest gate fidelities, and a limited number of qubits may still be leveraged to perform useful tasks by using so-called hybrid quantum-classical algorithms, which subdivide a given computational task into quantum and classical parts and allocate quantum resources only where necessary. We implement one such hybrid protocol and use it to calculate the complete energy spectrum of the hydrogen molecule.

This protocol, the variational quantum eigensolver (VQE), was developed to calculate the ground-state energy of complex chemical systems. It uses a classical optimization routine to minimize the expected energy of candidate wave functions, leveraging the quantum hardware to evaluate the energy. We realize this algorithm using a quantum processor comprised of two superconducting qubits with real-time classical optimization. With this architecture, we demonstrate, for the first time, the ability to calculate the full energy spectrum of a given Hamiltonian (in this case, the ${\mathrm{H}}_2$ molecule) beyond just the ground state.

Our new approach to implementing a VQE suppresses the effects of certain types of errors, making it an attractive choice for future larger-scale calculations.

Figure 1

Description of the variational quantum eigensolver algorithm and associated quantum subspace expansion. (a) Mapping between qubit states and electronic states of the hydrogen molecule in the STO-3G basis. As the molecular Hamiltonian preserves total spin projection, the four states with $s_z=0$ (dashed box) decouple from the other two. These four are mapped to the four two-qubit basis states. (b) Flow chart outline of the algorithm with classical resources colored in blue and quantum resources colored in yellow. Inset shows a cartoon example of the swarm minimization process with equal-energy contours shown in gray scale and 5 swarm particle trajectories (colored). (c) Cartoon of the QSE protocol; operators from $O_i$ are used to expand about the variational solution provided by the VQE, allowing for the mitigation of incoherent errors that otherwise render the true ground state inaccessible. (d) Typical qubit preparation and measurement sequence consisting of a projective heralding measurement (on which we later postselect so that the qubits start in the state $|00⟩$), single-qubit and two-qubit pulses, tomography pulses, and finally a projective readout.

Figure 2

Control parameter convergence as a function of classical optimizer iteration. (a) Median (solid line) and standard deviation (shaded region) for all six normalized parameter values as a function of swarm iteration number at an internuclear bond distance of 1.55 Å. A swarm of 20 particles demonstrates convergence after approximately 12 iterations or equivalently 240 function evaluations. See SM, VQE and Coherent Errors, for convergence details [28]. (b) Median energy (solid line) and standard deviation (shaded region) of swarm particles as a function of iteration number for the corresponding data in (a). Monotonic convergence of median energy towards the theoretical value is observed followed shortly thereafter by a rapid decrease in swarm energy variance. Trace distance (dots) of the variationally prepared ground state as a function of swarm iteration shows reasonable convergence to the theoretically expected ground state. The small remaining discrepancy is most likely due to the flat nature of the energy landscape and unavoidable decoherence during state preparation and readout.

Figure 3

${\mathrm{H}}_2$ energy spectrum as a function of internuclear distance. Swarm particle energies for each bond length are histogrammed after application of a linear-response expansion and Gaussian filter. Energy estimates obtained by a peak finding routine are indicated by dots with theoretically predicted energy levels shown as solid lines. An unphysical spurious state emerges at internuclear distances greater than $\sim 1.2\AA$ due to uncorrected incoherent errors. Inset shows errors in the estimated ground- and excited-state energies as compared to chemical accuracy ($1.6\times {10}^{-3}\mathrm{Ha}$).

Figure 4

Comparison of errors in the ground-state energy estimate when applying the QSE protocol using various combinations of expansion operators. The linear response expansion (dark blue dots) provides an improvement of more than an order of magnitude over the bare VQE estimate (yellow dots) for the majority of internuclear distances computed.