# Quantum computations, irreversibility, unitarity

• kdv
In summary, the conversation discusses quantum computations and the concept of unitarity and reversibility in operators. It also explains the concept of balanced and constant functions and how they relate to Deutsch's algorithm. The conversation also touches on the idea of representing a function as a quantum operator and whether it is possible. The conclusion is that while the operation used in Deutsch's algorithm is possible, the representation of a function as a quantum operator is not possible.
kdv
A stupid couple of questions...

In quantum computations, one typically starts with some initial quantum state on which an operator is applied.

This operator must be unitary , right? (I guess that otherwise, it would not corerspond to an actual physical quantum setup). And this implies that the operator must be reversible, correct?
So in quantum computing, we consider only irreversible processes, correct?

Now consider Deutsch's algorithm. It deals with a function acting on a single qubit which may be either "balanced" or "constant".

It is balanced if it may give both 0 and 1 a sresult. One possibility is f(0)=0, f(1)=1 and the other balanced possibility is f(0)=1, f(1)=0.

It is constant if it gives the samr answer no matter what the input is. So the first possibility is f(0) =f(1) =0 and the second possibility is f(0)=f(1)=1.

Now, the whole idea of Deutsch's algorithm is to able to distinguish whether f is balanced or constant with a single application of a certain operation which must be chosen judiciously. (and the input state must be chosen judiciously as well).

Now, usually people say: classically, one cannot determine if the function is constant or balanced without two runs fo teh function. (for example, observing that f(0) gives zero does not tell us if it's balanced or not). Then people say that quantum mechanically, one can tell if f is balanced or not with a single run of an operator applied to a suitable combination of two qubits.

Fine. But my question is this: if I understand correctly, the function f(x) itself cannot be represented by any quantum operator , correct? Because in the constant case it is not reversible and therefore could not be associated to a unitary operator.

This kind of surprised me when I thought about this today...It's weird in some sense because the starting point is the operator "f" and yet I have never seen anyone mentioning that f itself cannot be implemented as a quantum operation. Maybe I am missing something?

He probably means something like the following operation on two qubits:

|x>|y> ---> |x>|y + f(x)>

Hurkyl said:
He probably means something like the following operation on two qubits:

|x>|y> ---> |x>|y + f(x)>

Yes, I know this is the operation used in Deutsch's algorithm. I was wondering if I was correct in saying that the operation on a single qubit |x> ---> |f(x)> would be impossible to implement.

## 1. What is a quantum computation?

A quantum computation is a type of computation that utilizes quantum bits, or qubits, instead of classical bits. Qubits are able to exist in multiple states simultaneously, allowing for exponentially faster processing and more complex calculations compared to classical computers.

## 2. How does unitarity play a role in quantum computations?

In quantum mechanics, unitarity refers to the conservation of information during a quantum process. In quantum computations, unitarity ensures that the final state of a qubit can be traced back to the initial state, allowing for accurate and reversible calculations.

## 3. What is the concept of irreversibility in quantum computations?

Irreversibility refers to the inability to accurately reverse a quantum computation and retrieve the initial state of the qubits. This is due to the probabilistic nature of quantum mechanics, where measurements of qubits collapse their superposition of states.

## 4. How are quantum computations different from classical computations?

Quantum computations differ from classical computations in several ways. One major difference is the ability of qubits to exist in multiple states simultaneously, allowing for parallel processing and faster calculations. Additionally, quantum computations are probabilistic in nature, while classical computations are deterministic.

## 5. What are some potential applications of quantum computations?

Quantum computations have the potential to revolutionize fields such as cryptography, drug discovery, and optimization problems. They can also be used to simulate complex quantum systems that are difficult to study with classical computers. However, practical quantum computers are still in development and it may be some time before these applications become a reality.

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