Efficient Encoding of the Weighted MAX $$k$$-CUT on a Quantum Computer Using QAOA

Abstract The weighted MAX $$k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math> -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 $$k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math> -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 $$|V|\log _2k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mi>V</mml:mi><mml:mo>|</mml:mo></mml:mrow><mml:msub><mml:mo>log</mml:mo><mml:mn>2</mml:mn></mml:msub><mml:mi>k</mml:mi></mml:mrow></mml:math> 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 $$k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math> -CUT problem for $$k&gt;2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>k</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> . (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.