Abstract
In this paper we discuss Grover Adaptive Search (GAS) for Constrained Polynomial Binary Optimization (CPBO) problems, and in particular, Quadratic Unconstrained Binary Optimization (QUBO) problems, as a special case. GAS can provide a quadratic speed-up for combinatorial optimization problems compared to brute force search. However, this requires the development of efficient oracles to represent problems and flag states that satisfy certain search criteria. In general, this can be achieved using quantum arithmetic, however, this is expensive in terms of Toffoli gates as well as required ancilla qubits, which can be prohibitive in the near-term. Within this work, we develop a way to construct efficient oracles to solve CPBO problems using GAS algorithms. We demonstrate this approach and the potential speed-up for the portfolio optimization problem, i.e. a QUBO, using simulation and experimental results obtained on real quantum hardware. However, our approach applies to higher-degree polynomial objective functions as well as constrained optimization problems.
Highlights
Using the laws of quantum mechanics, quantum computers offer novel solutions for resource-intensive problems
We provide a framework for automatically generating efficient oracles for solving Constrained Polynomial Binary Optimization (CPBO)— a generalization of Quantum Unconstrained Binary Optimization (QUBO)—with Grover Adaptive Search (GAS)
Added to the objective, where λ > 0 is a large number to enforce the constraint to be satisfied. This results in a quadratic term that will again lead to a QUBO problem
Summary
Using the laws of quantum mechanics, quantum computers offer novel solutions for resource-intensive problems. There are multiple approaches to solve QUBO problems on a quantum computer, discussed in the following paragraphs. It can be used to approximate optimal solutions of QUBO problems. We provide a framework for automatically generating efficient oracles for solving Constrained Polynomial Binary Optimization (CPBO)— a generalization of QUBO—with GAS. We test our algorithm on the portfolio optimization problem [22, 23] Multiple variants of this problem have been studied in the quantum optimization literature, ranging from convex continuous formulations [24] to QUBOs [13, 25, 26].
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