I want to create an operator $A$ which, given three binary numbers, $a_1$, $a_2,a_3$, will detect whether $a_1a_2a_3$ (as a binary number) is in certain set of numbers (for example, detect whether $a_1a_2a_3$ is any integer number between 3 and 5, or if it is an odd number, etc). For that I introduce an ancilla qubit, which will be 1 if $a_1a_2a_3$ is in the desired interval and 0 otherwhise. It is easy to build up the matrix of such linear operator, but it has of course linearly dependent columns. Is there away to express this "detection operation" in terms of unitary transformations, possibly adding more ancillary qubits?For example, I would have $q_0,q_1,q_2,$ these are the qubits where all the relevant info for the information is contained (the binary number $a_1a_2a_3$), I would then add $q_3$, the qubit that yields the output (1 if $a_1a_2a_3$ is in the desired subset of numbers) and then some other qubits $q_4,\cdots$ which transform in some way so as to make the overall transformation unitary. I have tried with up to two auxiliary qubits, but I always get a singular matrix.
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