Calculating Properties of Many-Electron Systems via Sample-Based Quantum Bootstrap Embedding

One of the leading candidate problems for which quantum computers may demonstrate an advantage is the electronic structure problem in molecular chemistry and materials science. Short coherence times and limited qubit counts pose a challenge for scaling up such simulations to a size where quantum advantage becomes plausible. Quantum bootstrap embedding (QBE) is one method which attempts to extend the reach of what systems can be simulated on quantum computers by fragmenting a large system into overlapping fragments, running an eigensolver on each of the fragments, and imposing matching conditions on overlapping regions. By doing this, the number of qubits needed to simulate the entire system is reduced, but introduces the challenge of reconstructing the global wavefunction from the fragments. If left unaddressed, this limits the utility of QBE to the calculation of properties which do not require explicit construction of the global wavefunction. To-date, work on QBE has focused on using the variational quantum eigensolver (VQE) as its eigensolver subroutine. While VQE addresses the qubit count and coherence challenges of near-term hardware, it is known to be challenging to scale up due to the large number of measurements needed to estimate the energy. Here, we investigate the potential for the combination of the recently proposed sample-based diagonalization (SQD) method and QBE to extend the size of simulatable systems by running simulations on IBM quantum computers. SQD is compelling because it simultaneously addresses how to avoid the measurement problem in VQE and how to reconstruct the global QBE wavefunction. It avoids the long energy estimation runtimes of VQE by sampling an ansatz on a quantum computer, projecting the Hamiltonian onto a subspace spanned by the sample states, and classically diagonalizing the projected Hamiltonian. The global QBE wavefunction can then be reconstructed by combining sampled fragment states which match on overlapping regions and performing a diagonalization step on the resulting combined state space. We further investigate how the reconstructed wavefunction could be used to extend the utility of quantum computers by calculating properties which require explicit construction of the wavefunction.