  Distributed quantum computation is often proposed to increase the scalability
of quantum hardware, as it reduces cooperative noise and requisite connectivity
by sharing quantum information between distant quantum devices. However, such
exchange of quantum information itself poses unique engineering challenges,
requiring high gate fidelity and costly non-local operations. To mitigate this,
we propose near-term distributed quantum computing, focusing on approximate
approaches that involve limited information transfer and conservative
entanglement production. We first devise an approximate distributed computing
scheme for the time evolution of quantum systems split across any combination
of classical and quantum devices. Our procedure harnesses mean-field
corrections and auxiliary qubits to link two or more devices classically,
optimally encoding the auxiliary qubits to both minimize short-time evolution
error and extend the approximate scheme's performance to longer evolution
times. We then expand the scheme to include limited quantum information
transfer through selective qubit shuffling or teleportation, broadening our
method's applicability and boosting its performance. Finally, we build upon
these concepts to produce an approximate circuit-cutting technique for the
fragmented pre-training of variational quantum algorithms. To characterize our
technique, we introduce a non-linear perturbation theory that discerns the
critical role of our mean-field corrections in optimization and may be suitable
for analyzing other non-linear quantum techniques. This fragmented pre-training
is remarkably successful, reducing algorithmic error by orders of magnitude
while requiring fewer iterations.
