Bootstrap Embedding on a Quantum Computer*

Finding the ground state of interacting fermionic systems is an outstanding challenge for quantum chemistry, material science, and condensed matter physics. However, numerically solving the time-independent Schrodinger equation of a meaningfully large many-electron system in an exact fashion is a daunting task because the dimension of the underlying Hilbert space grows exponentially as the number of electrons increases whereas any practically available computational resources will be finite. Extending bootstrap embedding methods for addressing this challenge, we present a quantum bootstrap embedding theory that formulates the electronic structure problem of the total system as a constraint optimization problem for a composite Lagrangian where the constraint is constructed from matching conditions on the qubit reduced density matrices. We present an iterative algorithm to solve the optimization problem using a quantum subroutine as an eigensolver to solve each fragment Hamiltonian. An adaptive sampling scheduling and a quantum coherent matching algorithm based on a quantum SWAP test are designed to dramatically improve the efficiency of the algorithm as compared to the usual exponentially costly method of measuring every qubit on the fragment edge to construct the reduced density matrix. Moreover, by using amplitude amplification and a binary search algorithm, an additional quadratic speedup could be realized. Current quantum computers are small, but quantum bootstrap embedding proves a potentially generalizable strategy for harnessing such small machines, since it enables the stitching together of fragment solutions to solve a quantum chemistry problem that is much larger than current quantum computer capacities.