# Self-Adjoint Operators and Reversible Logic gates

1. Aug 13, 2009

### karatemonkey

Does anyone know if there is a relationship between the requirement in Quantum Computing that logic gates be reversible and the requirement in Quantum Mechanics that observables have to be self-adjoint?

2. Aug 19, 2009

### karatemonkey

If this question has revealed stupidity on my part please let me know, and give me a reference to straighten me out

3. Aug 19, 2009

### Avodyne

AFAIK, no. This is not my area of expertise, but I would guess that "reversible" implies "unitary", and symmetries in QM must be represented by unitary ops, so there may be a connection there.

4. Aug 19, 2009

### karatemonkey

Thanks for the reply.

A reversible logic gate essentailly means that if you run an input through the gate and get an output, you can apply that output to the gate and get back the input. The map is one to one, so that is a unitary operation.

I'm going to show my ignorance again and what the relationship is between self-adjoint and unitary operators?

5. Aug 19, 2009

### Avodyne

Formally, a unitary operator U can be written as U = exp(iA), where A is self-adjoint.

6. Aug 20, 2009

### RedX

Take a simple gate that takes two inputs: the possibilities are: (on,on), (on,off), (off, on), or (off,off). If there is only one output, (on) or (off), then how can such a mapping be one-to- one?

Reversible seems more like "invertible" than "unitary". They are almost synonymous if the transformations form a group, because then an inverse would exist, and also the result that every representation is equivalent to a unitary representation.

What's interesting is that if every representation is equivalent to a unitary representation, then that means every group is isomorphic to a subgroup of the unitary group? Is this useful for anything?

7. Aug 21, 2009

### karatemonkey

Thanks for the replies.

To RedX, a single output gate is not reversible when it takes two or more inputs. (See the hand wavy definition of a reversible gate I gave :) ) Actually, you have to have the same number of outputs and inputs for it to be reversible. So yes you are correct in what you stated about two inputs and one output not being one to one. For a cogent discussion look at Tofolli Gates, or Hadamard Gates.

To Avodyne
To the relation between unitary and self-adjoint, sorry for the bone head question.

I'm now thinking that a better question is "Is there a relationship between Stone's Theorem and the requirement that QC use reversible logic gates" Since Stone's theorem takes the static feature of observables being self-adjoint and allows one to talk about evolution in time.

I'm hunting around here trying to get my head around this stuff, please bear with me.

Last edited: Aug 21, 2009

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