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NP-hard problems are not believed to be exactly solvable through generalpolynomial time algorithms. Hybrid quantum-classical algorithms to address suchcombinatorial problems have been of great interest in the past few years. Suchalgorithms are heuristic in nature and aim to obtain an approximate solution.Significant improvements in computational time and/or the ability to treatlarge problems are some of the principal promises of quantum computing in thisregard. The hardware, however, is still in its infancy and the current NoisyIntermediate Scale Quantum (NISQ) computers are not able to optimizeindustrially relevant problems. Moreover, the storage of qubits andintroduction of entanglement require extreme physical conditions. An issue withquantum optimization algorithms such as QAOA is that they scale linearly withproblem size. In this paper, we build upon a proprietary methodology whichscales logarithmically with problem size - opening an avenue for treatingoptimization problems of unprecedented scale on gate-based quantum computers.In order to test the performance of the algorithm, we first find a way to applyit to a handful of NP-hard problems: Maximum Cut, Minimum Partition, MaximumClique, Maximum Weighted Independent Set. Subsequently, these algorithms aretested on a quantum simulator with graph sizes of over a hundred nodes and on areal quantum computer up to graph sizes of 256. To our knowledge, theseconstitute the largest realistic combinatorial optimization problems ever runon a NISQ device, overcoming previous problem sizes by almost tenfold.

Solving various NP-Hard problems using exponentially fewer qubits on a Quantum Computer