  We provide evidence that commonly held intuitions when designing quantum
circuits can be misleading. In particular we show that: a) reducing the T-count
can increase the total depth; b) it may be beneficial to trade CNOTs for
measurements in NISQ circuits; c) measurement-based uncomputation of relative
phase Toffoli ancillae can make up to 30\% of a circuit's depth; d) area and
volume cost metrics can misreport the resource analysis. Our findings assume
that qubits are and will remain a very scarce resource. The results are
applicable for both NISQ and QECC protected circuits. Our method uses multiple
ways of decomposing Toffoli gates into Clifford+T gates. We illustrate our
method on addition and multiplication circuits using ripple-carry. As a
byproduct result we show systematically that for a practically significant
range of circuit widths, ripple-carry addition circuits are more resource
efficient than the carry-lookahead addition ones. The methods and circuits were
implemented in the open-source QUANTIFY software.
