A post-selected linear-optical controlled- gate succeeds with probability ; when it…
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Problem
A post-selected linear-optical controlled-Z gate succeeds with probability 1/9; when it succeeds, its average gate fidelity (squared-type, as defined for benchmarking in Chapter 16) is Fˉ=0.98. Because a failed attempt is simply discarded and never becomes part of the output state, it cannot make the surviving, heralded output any less entangled: discarding results, unlike a noise process acting on the kept state, does not by itself add error. So the effective error of a heralded photonic CZ, defined as r=1−Fˉ, is set only by the fidelity of the successful branch, not by the success probability 1/9. A superconducting CZ gate (Chapter 17's datasheet figures) succeeds on essentially every attempt but has average gate fidelity Fˉ=0.995.
((a)) Compute the gate error r=1−Fˉ for each platform.
((b)) For a circuit using N=50 sequential CZ gates, with no error correction and each gate's error acting independently, estimate the overall circuit fidelity (1−r)N for each platform.
((c)) State which platform's heralded output is more trustworthy for this circuit, and note that this conclusion is about fidelity, separate from the very different attempt rates the two gates need.