Efficient encoding of the weighted MAX k-CUT on a quantum computer using QAOA

TL;DR

This paper introduces a quantum algorithm for the weighted MAX k-CUT problem that uses a binary encoding to optimize qubit resources and is suitable for implementation on noisy intermediate-scale quantum devices, with promising simulation results.

Contribution

It presents a new binary encoding formulation for MAX k-CUT suitable for NISQ devices, including a novel decomposition of the phase separation operator and resource analysis.

Findings

Numerical simulations compare different encodings for MAX k-CUT.

The binary encoding reduces qubit requirements to |V|log_2(k).

Potential for quantum advantage on NISQ devices for small k.

Abstract

The weighted MAX k-CUT problem consists of finding a k-partition of a given weighted undirected graph G(V,E) such that the sum of the weights of the crossing edges is maximized. The problem is of particular interest as it has a multitude of practical applications. We present a formulation of the weighted MAX k-CUT suitable for running the quantum approximate optimization algorithm (QAOA) on noisy intermediate scale quantum (NISQ)-devices to get approximate solutions. The new formulation uses a binary encoding that requires only |V|log_2(k) qubits. The contributions of this paper are as follows: i) A novel decomposition of the phase separation operator based on the binary encoding into basis gates is provided for the MAX k-CUT problem for k >2. ii) Numerical simulations on a suite of test cases comparing different encodings are performed. iii) An analysis of the resources (number of qubits, CX gates) of the different encodings is presented. iv) Formulations and simulations are extended to the case of weighted graphs. For small k and with further improvements when k is not a power of two, our algorithm is a possible candidate to show quantum advantage on NISQ devices.