Bases for optimising stabiliser decompositions of quantum states

Stabiliser states play a central role in the theory of quantum computation. For example, they are used to encode data in quantum error correction schemes. Arbitrary quantum states admit many stabiliser decompositions: ways of being expressed as a superposition of stabiliser states. Understanding the structure of stabiliser decompositions has applications in verifying and simulating near-term quantum computers. We introduce and study the vector space of linear dependencies of $n$-qubit stabiliser states. These spaces have canonical bases containing vectors whose size grows exponentially in $n$. We construct elegant bases of linear dependencies of constant size three. We apply our methods to computing the stabiliser extent of large states and suggest potential future applications to improving bounds on the stabiliser rank of magic states.