  In fault-tolerant quantum computing systems, realising (approximately)
universal quantum computation is usually described in terms of realising
Clifford+T operations, which is to say a circuit of CNOT, Hadamard, and
$\pi/2$-phase rotations, together with T operations ($\pi/4$-phase rotations).
For many error correcting codes, fault-tolerant realisations of Clifford
operations are significantly less resource-intensive than those of T gates,
which motivates finding ways to realise the same transformation involving
T-count (the number of T gates involved) which is as low as possible.
Investigations into this problem [arXiv:1206.0758, 1303.2042, 1308.4134,
1601.07363, 1606.01904, 1701.00140] has led to observations that this problem
is closely related to NP-hard tensor decomposition problems [arXiv:1712.01557]
and is tantamount to the difficult problem of decoding exponentially long
Reed-Muller codes [arXiv:1601.07363]. This problem then presents itself as one
for which must be content in practise with approximate optimisation, in which
one develops an array of tactics to be deployed through some pragmatic
strategy. In this vein, we describe techniques to reduce the T-count, based on
the effective application of "spider nest identities": easily recognised
products of parity-phase operations which are equivalent to the identity
operation. We demonstrate the effectiveness of such techniques by obtaining
improvements in the T-counts of a number of circuits, in run-times which are
typically less than the time required to make a fresh cup of coffee.
