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One of the major problems of most quantum computing applications is that therequired number of qubits to solve a practical problem is much larger than thatof today's quantum hardware. We propose an algorithm, called large-systemsampling 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 quantumcomputer, and with sizes up to $N_{\rm{an}}2^{N_{\rm{gb}}}$ by a hybridcomputational architecture of an $N_{\rm{an}}$-qubit quantum annealer and an$N_{\rm{gb}}$-qubit gate-based quantum computer. By dividing the full-systemproblem into smaller subsystem problems, the LSSA algorithm then solves thesubsystem problems by either gate-based quantum computers or quantum annealers,optimizes the amplitude contributions of the solutions of the differentsubsystems with the full-problem Hamiltonian by the variational quantumeigensolver (VQE) on a gate-based quantum computer, and determines theapproximated ground-state configuration. We apply the level-1 approximation ofLSSA to solving fully-connected random Ising problems up to 160 variables usinga 5-qubit gate-based quantum computer, and solving portfolio optimizationproblems up to 4096 variables using a 100-qubit quantum annealer and a 7-qubitgate-based quantum computer. We demonstrate the use of the level-2approximation of LSSA to solve the portfolio optimization problems up to 5120($N_{\rm{gb}}2^{2N_{\rm{gb}}}$) variables with pretty good performance by usingjust a 5-qubit ($N_{\rm{gb}}$-qubit) gate-based quantum computer. Thecompletely new computational concept of the hybrid gate-based and annealingquantum computing architecture opens a promising possibility to investigatelarge-size Ising problems and combinatorial optimization problems, makingpractical applications by quantum computing possible in the near future.

Hybrid Gate-Based and Annealing Quantum Computing for Large-Size Ising Problems