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Quantum Super Entangled Language (qsel)

What makes quantum computers more powerful than classical computers? Ask any expert, or watch any public lecture on the subject and the answer is clear. Entanglement and Superposition are what give quantum computers their power over classical computers. If these two ideas are at the heart of the power of quantum computing, it therefore makes sense that any programing language for quantum computing should make these two ideas front and center. This is where Quantum Super Entangled Language (qsel) comes in. It is a quantum programming language made entirely from superposition and entanglement.

Getting started

The qsel compiler is written in python because who needs strong typing when you are writing research code that will only ever be seen by yourself and maybe your graduate students. We recommend you work in a virtual environment and pip install the requirements listed in requirements.txt. Or don't. Yolo. A sample program is provided in example.qsel. To run this program simply run the following from your command line

python run.py example.qsel

which should produce something like

Measured 1 on qubit 0.
Measured 0 on qubit 1.

where your measurement results may be different.

The Language

Typically quantum computers are described by the abstract quantum circuit model. In this model the circuits can create entanglement and superposition. Here we want to use entanglment and superposition to describe a quantum computation. So our first choice is easy. All of our tokens in the language should be either entanglement or superposition.

A qsel program is specified by a series of lines in a text file. Each line corresponds to a single <command>. The language is case sensitive and extremely sensitive to whitespace. This sensitivity is sort of like a qubit's sensitivity to dephasing. For dephasing the issue comes from interacting with uncontrolled degrees of freedom. For qsel the whitespace sensitivity comes interacting with a lazy programmer.

Back to <command>s. Commands are one of three types

<command> -> <entaglement gate> | <superposition gate> | <measurement>

Now wait, you say, isn't measurement something beyond entanglement and superposition? This is a philosophical and religious question. Subscribers to it being entanglement belong to the Church of the Higher Hilbert Space. Those who believe it is superposition find themselves reading through the many books of the Many Worlds religion. We don't believe that programming languages should take sides in religion or philosophy debates, so we will call it <measurement>, and you can substitute your personal preference as needed.

Qubits

Before we describe the three commands, we run into another problem. If quantum computing is made entirely of entanglement in superposition, what about the actual objects of a quantum computer, the qubits? One elegant thought is that maybe we should make the two computational basis states of our qubit entanglement and superposition? Unfortunately this solution appears to cause a recursion which the universe cannot handle.

Religious aside: I think I may have generated some parallel universes in which this recursion seg faulted the universe. However if you are reading this, then you exist in one of the parallel universe branches were I didn't do this. I guess I'd warn you not to do such experiments, but really I should just make sure you use superposition to make sure one of the parallel universes doesn't do the experiment. If your religion is not of the Many World's variety, however, your view on this may be different and we are very very lucky.

The important thing about qubits is that we need to be able to distinguish one qubit from another qubit. Humans have invented an elegant solution to this problem. They label the qubits. Semi-modern humans found away around the problem of having as many labels as objects, by noting that the label could be a number. Modern humans completed this journey by insisting that the number be written in as a binary number so that machines could understand it. With this insight concerning how to label qubits, we define a qubit id made out of one or more binary bits

<qubit id> := <bit>+

And our bits need to be made out of entanglement and superposition.

<bit> := (<superposition> <superposition>) | (<entanglement> <entanglement>)

Note that we do not make the mistake of using a single entanglement as the opposite value of superposition. Entanglement can only exist between two objects. From excellent symmetry considerations, we also double the superposition. Finally note some semantics, we interpret the bits as a binary string, and compare their values based on this binary value. So <superposition> <superposition> <superposition> superposion> is the same as <superposition> <superposition>.

Commands

We now describe the quantum commands.

The superposition gate starts with a superposition op code (duh), followed by the id of the gate upon which to act.

<superposition gate> := <superposition> <superposition> <qubit id>  

What are the semantics of this gate? This corresponds to the Hadamard gate. Because nothing says a modern compiler like using a gate named after an dead mathematician who wrote down matrices. This is the gate with matrix

[[1/sqrt{2},  1/sqrt{2}],
 [1/sqrt{2}, -1/sqrt{2}]]

in the computational basis.

The entanglement gate starts with an entanglment op code (duh), followed by the ids of the gates upon which to act. Unfortunately because we need to specify two qubits, we now need some way to keep these qubits separated from each other. Otherwise, as we all know, the qubits interact via an exchange interaction. And it is certainly not possible to use exchange interactions only to build a quantum computer. So how do we keep qubits separated? For this I propose we create an odd kind of monster, a superposition entanglement divider (since each qubit is one of either type). Of course this could also be (by symmetry again) an entanglement superposition divider. So we define

<comma> := <entanglement> <superposition> | <superposition> <entanglement>

Then we can define the entanglement gate

<entanglement gate> := <entanglement> <entanglement> <qubit id> <comma> <qubit id>

What are the semantics of this gate? This corresponds to the gate that I need to make it, along with the Hamadard universal. Luckily, Kitaev is a time traveler, and he went to 2027 to find out the answer and came back and delivered it to us in 1997. This is the controlled-P gate, which is the diagonal gate in the computational basis with diagonal values

diag(1, 1, 1, i)

There is probably a potty joke here, but one has to have at least some standards. The good thing is that we don't have to tell you which qubit is which because controlled-P is symmetric (technically: bi-urinal).

We next turn to the measurement gate. Measurement, like we said, is something that, depending on your philosophical or religious leanings, is either superposition or measurement. So, following the divider above, we define it as either, followed by the qubit upon which the measurement is done

<measurement> := (<entanglement> <superposition> | <superposition> <entanglement>) <qubit id>

Measurement is done in the computational basis.

Finally we need to define <superposition> and <entanglement>. But like we said, qsel is all about writing programs in superposition and entanglement. So these are the words superposition and entanglement, respectively.

One final semantic meaning: we need to define the initial state. We stick with the boring and state that every qubit used in the program starts in the computation 0 state.

Example

A two qubit circuit. We want to apply Hadamard to the first and second qubit, then apply the controlled-P gate between the qubits, finally apply a Hadmard to the second qubit, and measure both qubits. This is simply

superposition superposition superposition superposition
superposition superposition entanglement entanglement
entanglement entanglement superposition superposition entanglement superposition entanglement entanglement
superposition superposition entanglement entanglement
entanglement superposition superposition superposition
entanglement superposition entanglement entanglement

Creative Mode

There are some people who do not belive that entanglement and superposition are at the core of quantum computing. For these people, it seem that they would like to be able to use different primitives. To support this we have creative mode. By supplying two strings after the filename of the program, we can use different symbols for superposition and entanglement. For example, if one wants to use the tab character for superposition and two tab characters for entanglement (again it would be an absurdity to have only one symbol for entanglement), then one could run

python run.py vacuum.qsel $'\t' $'\t\t'

We have supplied a program that is written in this form, called vacuum.qsel. Please be aware that despite the name of this file, and the empty seeming content, executing this program does not cause us to transition into a new vacuum state for the universe. Or at least it hasn't yet.

FAQ

Q: Wouldn't understanding the difference betwen quantum and classical polynomial time computing mean proving a major complexity theoretic breakthrough like P not equal to PSPACE?

A: Yes. Programming languages are regularly places where we present such proofs. Unlike posting these solutions on the arXiv, putting them in a programming language insures that only minds who can grok monads can check the brilliance of the proof.

Q: What abou contextuality? Isn't contextuality an important part of what powers quantum computers?

A: Go ahead and ask me that question another time, when you can see that I am busy or not busy.

Q: Isn't entanglement a kind of superposition?

A: This is refuted by the fact that such a programming language would require very inefficient unary encodings.