Due to required reversibility, classical function [itex](f(a)=y^a \mod N)[/itex] in Shor's algorithm needs a lot of auxiliary qubits. I was afraid that their later treatment might influence the computation - and(adsbygoogle = window.adsbygoogle || []).push({}); just got confirmation from Peter Shorhimself: that we need to "uncompute" these auxiliary qubits.

Let's look at quantum subroutine of Shor's algorithm:

However, calculation of classical function requires huge number of

- Hadamard gates create superposition of all (exponential number) values for input qubits.
- Then we perform a classical function on them, which is here: [itex]f(a)=y^a \mod N[/itex], where [itex]N[/itex] is the number we would like to factorize, [itex]y[/itex] is some chosen (usually random) number smaller than [itex]N[/itex]
- Then we perform measurement of value of this function [itex]f(a)=m[/itex] (random).
This measurement restricts the original ensembleto only input values [itex]a[/itex], such that [itex]f(a)=m[/itex].- Mathematics says that this restricted ensemble has to be periodic, this period can be concluded from value of (quantum) Fourier transform, and allows to conclude the factors.
auxiliary qubits(~square of number of other qubits) - they are usually ignored in considerations, but there is a crucial questionwhat finally happens with them?

As measurement of value qubits has restricted the ensemble,measuring/collapse of auxiliary qubits should also affect the ensemble - this time in unwanted way.

I thought maybe we could "erase them" by measuring in orthogonal basis, butPeter Shor has replied:

"In order to make the factoring algorithm work properly, you need to reset all the auxiliary qubits, which started as |0⟩ at the beginning of the computation, to |0⟩ at the end of the computation. This is called "uncomputing" these qubits. (Actually, you can set them to anything you please as long as it is a constant independent of the workings of the algorithm.) Theorems about reversible classical computation ensure that it is possible to do this."

Asking for literature, he replied: "Measuring the qubits in the "orthogonal basis" isn't going to work. You need to "uncompute" them. This is discussed briefly on page 9 of my paper, and in much more detail in C. H. Bennett (1973),Logical reversibility of computation."

However, reversible computation of bits is just a permutation - I don't see how to "uncompute" most of them (auxiliary) to a fixed value such that the total process is still a permutation?

Can it be done? How? Have you seen addressed/solved this problem in a literature?

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# A Shor's algorithm - need to uncompute auxiliary qubits?

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