We propose a novel hybrid classical-quantum algorithm to solve the MQO on a gate-based quantum computer. In this paper we tackle the multiple query optimization problem (MQO) which is an important NP-hard problem in the area of data-intensive problems. To overcome errors introduced by today's quantum computers, hybrid algorithms combining classical and quantum computers are used. Quantum computing promises to solve difficult optimization problems in chemistry, physics and mathematics more efficiently than classical computers, but requires fault-tolerant quantum computers with millions of qubits. The answers to these open research questions will determine the routes for the VQE to achieve quantum advantage as the quantum computing hardware scales up and as the noise levels are reduced.

We identify four main areas of future research:(1) optimal measurement schemes for reduction of circuit repetitions (2) large scale parallelization across many quantum computers (3) ways to overcome the potential appearance of vanishing gradients in the optimization process, and how the number of iterations required for the optimization scales with system size (4) the extent to which VQE suffers for quantum noise, and whether this noise can be mitigated. All the different components of the algorithm are reviewed in detail including representation of Hamiltonians and wavefunctions on a quantum computer, the optimization process, the post-processing mitigation of errors, and best practices are suggested. This review aims to provide an overview of the progress that has been made on the different parts of the algorithm. Despite strong theoretical underpinnings suggesting excellent scaling of individual VQE components, studies have pointed out that their various pre-factors could be too large to reach a quantum computing advantage over conventional methods.

Finding a path to navigate the relevant literature has rapidly become an overwhelming task, with many methods promising to improve different parts of the algorithm. The VQE may be used to model complex wavefunctions in polynomial time, making it one of the most promising near-term applications for quantum computing. Conventional computing methods are constrained in their accuracy due to the computational limits. The variational quantum eigensolver (or VQE) uses the variational principle to compute the ground state energy of a Hamiltonian, a problem that is central to quantum chemistry and condensed matter physics.

Steinspring quantum error correction