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

Abstract

The weighted MAX k-CUT problem aims at finding a k-partition of a given weighted undirected graph G(V, E) so as to maximize the sum of the weights of the crossing edges. 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. This is an exponential improvement with respect to k on the number of qubits when compared to previously known encodings. We provide a decomposition of the phase separation operator into basis gates and we show that the circuit depth is very shallow if k is a power of two. The resulting circuits are implemented using Qiskit [2], which is used to benchmark our algorithm on a suite of test cases. The simulations on random graphs with up to 10 vertices show that we are able to achieve competitive approximation ratios for all cases. We provide an analysis of the circuit depth, which shows that the circuit depth is very shallow if k is a power of two. 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.

mat2qubit: A lightweight pythonic package for qubit encodings of vibrational, bosonic, graph coloring, routing, scheduling, and general matrix problems

“…Notably, there are many scientifically and industrially relevant problems that are not most naturally defined over qubit, fermionic, or classical binary degrees of freedom. Such problems, for which the subsystems often have cardinality greater than 2, include bosonic [6,7], vibrational/phononic [8][9][10], spin-s [11], graph coloring [12][13][14][15], routing [16,17], and scheduling [18][19][20][21][22][23] problems, as well as more general classical linear algebra problems. mat2qubit is a Pythonic software package for encoding these Hamiltonians and cost functions into qubit operators; previously, software had been developed mainly for compiling fermionic problems [24][25][26] and classical problems over binary variables [25][26][27][28].…”