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root_not - Demonstration of a root-not logic gate.




 $foo is true!


This is an implementation of a root_not gate using Quantum::Entanglement.

Logic Gates

The simplist possible logic gate is one which maps a single input {0,1} to a single output {0,1}. This can be illustrated using the following diagram:

 Possible Inputs Gate    Possible Outputs


  0  -----------|----|-------  0

                |\b /|

                | \/ |

                | /\ |

                |/c \|

  1  -----------|----|-------  1


The constants a,b,c,d represent the probability with which a certain input will map onto a given output. For instance, a=d=1, b=c=0 is a pass through gate and a=d=0, b=c=1 is a convential NOT gate.

We can also use this gate as a random number generator, if we set a=b=c=d=0.5 then the output of this machine will be 0 half of the time and 1 the other half of the time, the output will also be uncorrelated with the input. It is also easy to see that if we were to chain two of these gates together we would still get a random stream as our output.

All the above is entirely classical. Things get a little wierd if instead of using straight probabilities for a,b,c,d we instead use probability amplitudes which we allow to be complex numbers. We need them to be normalised so that (a**2 + b**2) == 1 and (c**2+d**2)==1 rather than (a+b)==1 as before.

Now, if we let a=d=i/root(2) and b=c=1/root(2). With one gate, we transform an input state of |0> into an output state of i|0> + 1|1> and an input state of 1 into an output state of 1|1> + i|0> (without normalisation). If we look at the results of this single gate, we will measure 0 with a probability of |(i)|/2 == 0.5 and 1 with a probability of (1)/2 == 0.5 (for an input of 0). In this case, we have the same output as we did with the classical random number generator.

If we do not look at the results of this gate and feed it into a second gate we see the following with an input of 0 (normalization delayed until end):

 state at start        = |0>

 state after one gate  = i|0>            + 1|1>

 state after two gates = i*i|0> + 1*i|1> + 1*i|1> + 1*1|0>

                       = (-1+1)|0> + (i+i)|1>

Which if observed collapses to |1>. If an input of 1 is used on the first gate, then it is easy to show that an output of 0 will come from the second gate. Here we have a device where if only one is used, behaves as a random number generator, but if two are used in series, acts as a NOT gate.