Noise, approximation, and reserved error headroom

For a finite circuit, replacing individual complete gates or instruments with half-diamond errors ${\gamma }_v$ gives a total allowance at most ${\sum }_v{\gamma }_v$, by replacing them one at a time. One must include preparation, controls, stored memory, propagation, and readout where they are actually approximated.

Finite-dimensional entropy and the channel diagnostics are continuous in the device matrix. In particular,

The bound follows by expanding $U\rho U^{†}-U^{\prime }\rho (U^{\prime }{)}^{†}$ into two differences, also with references. A successful device chosen with strict leakage headroom and a strict entropy gap therefore has sufficiently accurate finite approximations preserving those inequalities.

This is an existence and stability statement. No Haar circuit, decoder matrix, gate approximation, or successful compiler execution is supplied.