Quantum Reversibility

A Thought Experiment

Let’s imagine a box of red and blue marbles. When we pull out a marble from the box it is in an “superposition” state as we don’t know if it is red or blue until we look at it. How easy is it for the superposition to change? Well there is a probability associated with the likelihood of seeing either a red or blue marble which is determined by the number of red or blue marbles in the box so that will fix the probability of observing one or the other. So because this is a closed system it will not be easy for it to change, meaning that the state of our system is relatively stable.

Decoherence

But what would happen if a hole appeared in the box? Well, we could lose some of the marbles which would cause the likelihood of observing one or the other to change, meaning that the state of the system is no longer stable. This brings us to an important topic in quantum computing known as decoherence(loss or modification of a qubit state of information). The quantum state within our qubits is very fragile, and susceptible to decoherence.

We can think about this in terms of energy lost through heat dissipation, so some of that energy that we might have(heat) is lost to the environment.

Decoherence causes state change. For example it can cause a state of 1 to become a state of 0. It can also cause a superimposed state to change its phase information.

Reversibility

This now brings us to reversibility which is a requirement for quantum operations. We don’t want to lose any information before measurement so prior to measurement any quantum computation must be able to be reversed. Let’s say that the matrix U represents a quantum state and it is reversible. This means that we can have an input to our circuit recovered using the output of the function and the function inverse(U^-1).

For example if we have an initial quantum state of psi zero that is transformed by a quantum state U and we get a resulting state psi one then we can recover our original input by taking psi one and multiplying it by the inverse operation.

NOT and SWAP Gate Reversibility

Now lets look at some gates and see if they are reversible. We will first start with NOT and SWAP operations.

So if we take our output and send it back through the gate, we see that in both of these examples, we recover our original input. This shows that these functions are not only reversible but also their own inverse functions.

Now lets see if CNOT is reversible!

Here we have all the input combinations for two qubits. We get our outputs which are determined by the transformation matrix. If our control has a ket one value then we flip or not the value of our target. Let take the one one output and send it back through the CNOT, well we get an output of one zero, which is the original input! In total there are four unique input combinations and four unique output combinations.

Is the H Gate Reversible?

• ket zero = white ball
• ket one = black ball

If we have a ket zero or a white ball going into the H gate we get our super position state which has no phase and if we pass that back through the H gate, we recover a white ball.

The same thing happens when instead we send in the black ball. This brings us to the conclusion that the H gate is also reversible and is its own inverse function.

Reversibility and Superposition

Reversibility of quantum gates also applies to qubits in superposition.

For example, if we have a NOT gate when the output from the first time we transformed state we can pass that back through the NOT gate and recover our original input state.