  Recent research has shown that wavefunction evolution in real- and
imaginary-time can generate quantum subspaces with significant utility for
obtaining accurate ground state energies. Inspired by these methods, we propose
combining quantum subspace techniques with the variational quantum eigensolver
(VQE). In our approach, the parameterized quantum circuit is divided into a
series of smaller subcircuits. The sequential application of these subcircuits
to an initial state generates a set of wavefunctions that we use as a quantum
subspace to obtain high-accuracy groundstate energies. We call this technique
the circuit subspace variational quantum eigensolver (CSVQE) algorithm. By
benchmarking CSVQE on a range of quantum chemistry problems, we show that it
can achieve significant error reduction compared to conventional VQE,
particularly for poorly optimized circuits, greatly improving convergence
rates. Furthermore, we demonstrate that when applied to circuits trapped at a
local minima, CSVQE can produce energies close to the global minimum of the
energy landscape, making it a potentially powerful tool for diagnosing local
minima.
