  One of the major problems of most quantum computing applications is that the
required number of qubits to solve a practical problem is much larger than that
of today's quantum hardware. We propose an algorithm, called large-system
sampling approximation (LSSA), to solve Ising problems with sizes up to
$N_{\rm{gb}}2^{N_{\rm{gb}}}$ by an $N_{\rm{gb}}$-qubit gate-based quantum
computer, and with sizes up to $N_{\rm{an}}2^{N_{\rm{gb}}}$ by a hybrid
computational architecture of an $N_{\rm{an}}$-qubit quantum annealer and an
$N_{\rm{gb}}$-qubit gate-based quantum computer. By dividing the full-system
problem into smaller subsystem problems, the LSSA algorithm then solves the
subsystem problems by either gate-based quantum computers or quantum annealers,
optimizes the amplitude contributions of the solutions of the different
subsystems with the full-problem Hamiltonian by the variational quantum
eigensolver (VQE) on a gate-based quantum computer, and determines the
approximated ground-state configuration. We apply the level-1 approximation of
LSSA to solving fully-connected random Ising problems up to 160 variables using
a 5-qubit gate-based quantum computer, and solving portfolio optimization
problems up to 4096 variables using a 100-qubit quantum annealer and a 7-qubit
gate-based quantum computer. We demonstrate the use of the level-2
approximation of LSSA to solve the portfolio optimization problems up to 5120
($N_{\rm{gb}}2^{2N_{\rm{gb}}}$) variables with pretty good performance by using
just a 5-qubit ($N_{\rm{gb}}$-qubit) gate-based quantum computer. The
completely new computational concept of the hybrid gate-based and annealing
quantum computing architecture opens a promising possibility to investigate
large-size Ising problems and combinatorial optimization problems, making
practical applications by quantum computing possible in the near future.
