  Limited quantum memory is one of the most important constraints for near-term
quantum devices. Understanding whether a small quantum computer can simulate a
larger quantum system, or execute an algorithm requiring more qubits than
available, is both of theoretical and practical importance. In this Letter, we
introduce cluster parameters $K$ and $d$ of a quantum circuit. The tensor
network of such a circuit can be decomposed into clusters of size at most $d$
with at most $K$ qubits of inter-cluster quantum communication. We propose a
cluster simulation scheme that can simulate any $(K,d)$-clustered quantum
circuit on a $d$-qubit machine in time roughly $2^{O(K)}$, with further
speedups possible when taking more fine-grained circuit structure into account.
We show how our scheme can be used to simulate clustered quantum systems --
such as large molecules -- that can be partitioned into multiple significantly
smaller clusters with weak interactions among them. By using a suitable
clustered ansatz, we also experimentally demonstrate that a quantum variational
eigensolver can still achieve the desired performance for estimating the energy
of the BeH$_2$ molecule while running on a physical quantum device with half
the number of required qubits.
