  We extend molecular bootstrap embedding to make it appropriate for
implementation on a quantum computer. This enables solution of the electronic
structure problem of a large molecule as an optimization problem for a
composite Lagrangian governing fragments of the total system, in such a way
that fragment solutions can harness the capabilities of quantum computers. By
employing state-of-art quantum subroutines including the quantum SWAP test and
quantum amplitude amplification, we show how a quadratic speedup can be
obtained over the classical algorithm, in principle. Utilization of quantum
computation also allows the algorithm to match -- at little additional
computational cost -- full density matrices at fragment boundaries, instead of
being limited to 1-RDMs. Current quantum computers are small, but quantum
bootstrap embedding provides a potentially generalizable strategy for
harnessing such small machines through quantum fragment matching.
