Threshold quantum gate

Question

I would like to know if it is possible to construct a set of quantum gates to do the following:

Start with a set of N qubits representing a number expressed in binary notation.
The values in the various qubits are entangled in unknown ways with each other, and have unknown ratios of likelihood of being 1 or 0.
calculate a new qubit whose probability of being 1 is the probability that the number represented by the N qubits exceeds $delta$, where $delta$ is an externally provided (classical) parameter.

The objective is to create a quantum algorithm for general purpose optimization.

Craig Gidney · Accepted Answer

You need to disambiguate what you mean by "calculate a qubits whose probability of being 1 is...". Is the qubit allowed to be entangled with the others? What phase is it supposed to have?
It sounds like you're just describing an arithmetic less-than comparison. A simple way to do it is to use an adder. Take your two n-qubit inputs and run them through an n+1 qubit subtractor, then run the bottom n qubits through an n bit adder. The extra qubit that came out of the subtractor is the carry overflow and contains the result of the comparison.

Threshold quantum gate

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