Highly efficient encoding for job-shop scheduling problems and its application on quantum computers

Abstract Combinatorial optimization problems are considered to be an application, where quantum computing can have transformative impact. In the industrial context, job shop scheduling problems that aim at finding the optimal schedule for a set of jobs to be run on a set of machines are of immense interest. Here we introduce an efficient encoding of job shop scheduling problems, which requires much fewer bit-strings for counting all possible schedules than previously employed encodings. For problems consisting of N jobs with N operations, the number of required bit-strings is at least reduced by a factor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>log</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> as compared to time indexed encodings. This is particularly beneficial for solving job shop scheduling problems on quantum computers, since much fewer qubits are needed to represent the problem. Our approach applies to the large class of flexible and usual job-shop scheduling problems, where operations can possibly be executed on multiple machines. Using variational quantum algorithms, we show that the encoding we introduce leads to significantly better performance of quantum algorithms than previously considered strategies. Importantly, the encoding we develop also enables significantly more compact classical representations and will therefore be highly useful even beyond applicability on quantum hardware.