  This paper builds on the idea of simulating stabiliser circuits through
transformations of quadratic form expansions. This is a representation of a
quantum state which specifies a formula for the expansion in the standard
basis, describing real and imaginary relative phases using a degree-2
polynomial over the integers. We show how, with deft management of the
quadratic form expansion representation, we may simulate individual stabiliser
operations in $O(n^2)$ time matching the overall complexity of other simulation
techniques [arXiv:quant-ph/0406196, arXiv:quant-ph/0504117, arXiv:1808.00128].
Our techniques provide economies of scale in the time to simulate simultaneous
measurements of all (or nearly all) qubits in the standard basis. Our
techniques also allow single-qubit measurements with deterministic outcomes to
be simulated in constant time. We also describe throughout how these bounds may
be tightened when the expansion of the state in the standard basis has
relatively few terms (has low 'rank'), or can be specified by sparse matrices.
Specifically, this allows us to simulate a 'local' stabiliser syndrome
measurement in time $O(n)$, for a stabiliser code subject to Pauli noise --
matching what is possible using techniques developed by Gidney
[arXiv:2103.02202] without the need to store which operations have thus far
been simulated.
