Quantum Solver of Contracted Eigenvalue Equations for Scalable Molecular Simulations on Quantum Computing Devices

The accurate computation of ground and excited states of many-fermion quantum systems is one of the most consequential, contemporary challenges in the physical and computational sciences whose solution stands to benefit significantly from the advent of quantum computing devices. Existing methodologies using phase estimation or variational algorithms have potential drawbacks such as deep circuits requiring substantial error correction or nontrivial high-dimensional classical optimization. Here, we introduce a quantum solver of contracted eigenvalue equations, the quantum analog of classical methods for the energies and reduced density matrices of ground and excited states. The solver does not require deep circuits or difficult classical optimization and achieves an exponential speed-up over its classical counterpart. We demonstrate the algorithm though computations on both a quantum simulator and two IBM quantum processing units.

Figure 1

For a 1-qubit Hamiltonian the solution of the quantum ACSE converges to the ground state, indicated by ${\overrightarrow{v}}_{-}$, in about 8 iterations on a 1-qubit IBM quantum computer.

Figure 2

For the ${\mathrm{H}}_2$ molecule the figure shows (a) the energy at each iteration in the solution of the ACSE at an internuclear distance $R$ of 2 Å and (b) the energy dissociation curve of the molecule. The error in the ACSE on the quantum computer, due to noise, is fairly uniform throughout the dissociation, indicating that the ACSE captures the spin entanglement. The sampling error at each point is indicated by the line width.

Figure 3

The errors in the potential energy curves from the equal-bond dissociation of the ${\mathrm{H}}_3$ molecule, relative to FCI, are shown for the classical and quantum ACSE algorithms with the quantum ACSE being more accurate by 6 orders of magnitude. Dotted line indicates “chemical accuracy” ($1\mathrm{kcal}/\mathrm{mol}$).