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I‘m just starting out with this topic and it’s already a bit confusing: I was just wondering, if I could take a really, really big number, let’s say a 256-bit number and add another number to it on a quantum computer (or simulation). There’s no problem doing it on a regular CPU, but on a quantum computer I‘m not sure. Do I need a 256 Qubit register to handle the computation, or is there another way to represent big Integers on a quantum computer? Considering the fact that most providers give access to anywhere between 20-100 Qubits per job, a 256 Qubit register would not be practical.

Ted Met
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There are constructions for very efficient adders (in terms of circuit depth, T count, T depth, etc), but generally the limitation of $n$-bit numbers in $n$ qubits is due to some application of Holevo's Theorem. You need $n+1$ qubits to be able to reliably encode the outcome of an addition of two $n$ bit numbers using qubits.

Addition is also classically very easy, its pseudopolynomial time. You don't really get a speed up at all when running it on a quantum device. The only time you would want to add two numbers on a quantum device instead of on a classical one is if the algorithm benefits somehow from adding the registers in superposition.

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