# What is the most economical and preferred basis for the qudit?

In Classical Simulation of Quantum Error Correction in a Fibonacci Anyon Code, the authors state on page 2 in section I. BACKGROUND - A. Topological model:

We consider a system supporting nonabelian Fibonacci anyon excitations, denoted by $\tau$. Two such anyons can have total charge that is either $\tau$ or $\mathbb I$ (vacuum), or any superposition of these, and so the fusion space in this case is 2-dimensional. We can represent basis states for this space using diagrams of definite total charge for the Wilson loops, and arbitrary states as linear combinations of these diagrams:

For $n$ anyons of type $\tau$, the dimension of the fusion space grows asymptotically as $\varphi^n$, where $\varphi$ = $\frac{1+\sqrt 5}{2}$ is the golden ratio.

Observables associated with non-intersecting loops commute, and so a basis for the space can be built from a maximal set of disjoint, nested loops.

Finally, in "Coherence Frame, Entanglement Conservation, and Einselection" the authors state on page 3:

Preferred basis problem. From the method of CF, we discuss the preferred basis problem (PBP), which has been studied via the einselection approach [7, 8]. We will show, yet, the method of einselection is incomplete.

This method is described via the Stern-Gerlach experiment, as shown in the Fig. 1 of Ref. 7. The system is represented by the spin states (up and down) along some directions. One atom is put near one channel to serve as the apparatus to interact with the spin, causing entanglement. In this measurement, the PBP means that there is no physical difference between states

We also demonstrated that the preferred basis problem can be resolved more naturally by the method of coherence frame than the einselection method.

Question: Are there different bases for each part of the model. Is there a preferred basis for a qudit or does it depend upon the underlying technology used to implement the qudits. How is the basis integral to the control and measurement?

Note: This may be related to, but is not a duplicate of: What is the Computational Basis?, nor is one of the answers currently offered there an answer to this question.

Initially, efforts where made to address some comments but this only succeeded in making the question longer and less clear, I've stripped it down but the edits can be reviewed by the curious in the edit history.

• There's a lot going on here; it's a bit difficult to sift through. Note that in your remarks on radix economy, the result that the most economical integer base is ternary is a mathematical result: it's difficult to see what you are getting at when you ask about "what radix to use" for qubits, which would seem to bake in the radix of 2. I also don't see what any of your linked references have to do with the question, with the (possible?) exception of the one on Fibonacci anyons, depending on what exactly you are trying to find out. Could you clarify what precisely you would like to know? – Niel de Beaudrap Apr 5 '18 at 9:27
• Can you explain why you expect the radix economy to be different in the quantum case? You encode the digits x into $|x\rangle$, so I don't see why it should be different. Or are you thinking of a different representation of the numbers? – M. Stern Apr 5 '18 at 10:20
• @Rob: that helps, but I'm still confused. Are you asking "what dimension of qudit would be most effective to compute with", as a theoretical optimum, in some sense analogous to the efficiency of a radix (motivated similarly, though perhaps taking into account more than just the typical efficiency of expressing integers)? Are you interested in choices motivated by a specific model of computation (your reference to anyons)? Or are you interested in practical issues related to what is possible due to decoherence in some architecture (your other references)? – Niel de Beaudrap Apr 5 '18 at 13:59
• @Rob: Higher dimensions are not necessarily useful. It is more important to have the dynamics that you want, to perform the applications that you want, than to have high dimension. That motivates concerning yourself not with wondering what "the right dimension of qudit to use", but "what systems seem practical to control, and what dimensions of qudits are natural for that system". That is, you put the system first, and then ask questions about the qudits relative to that choice of system. Doing it in the opposite order is like asking what is the best number of apples. – Niel de Beaudrap Apr 5 '18 at 17:23
• @Rob: In many circumstances it is better to carry more than just apples: it depends on what you want to do with the apples. Even if all you want is apples, what variety of apple also matters, not everyone likes to eat Granny Smith, for example. Sheer number is not what is important --- suitability for a given purpose is a higher consideration. The dimensionality of the qudit is just one factor for that purpose. – Niel de Beaudrap Apr 5 '18 at 18:41

The preferred basis problem is essentially something from the many worlds interpretation: If we are to interpret a superposition as representing many universes, what basis should we choose? Since this comes from the foundations of QM, this aspect of your question is perhaps better suited to the physics stack exchange.

Is there a preferred basis for a qudit or does it depend upon the underlying technology used to implement the qudits.

For qudits (and qubits) the only distinction between bases is in the physical implementation. At the abstract mathematical level, all are equivalent.

This not only means that you are free to choose your computational basis. You also have some freedom in how to generalize the Pauli matrices. For a $d$ level qudit, for example, you could choose to label your basis states with the elements of a group with order $d$. Your can then define generalizations of Pauli $X$ that implement the group multiplication, and generalization of $Z$ that depend on the representations. See here for some examples.

How you choose to do this might depend on the physics of the qudits (perhaps the interactions naturally implement such operations) or it might depend on what you want to do with the qudits (such as create exotic topological error correcting codes). But other than concerns like these, nothing is forcing you to make any particular decisions.

Usually we choose the basis that is easiest for us to measure. Superconducting qubits/qudits for examples are made from the lowest $d$ levels of an anharmonic oscillator. These energy eigenstates are what we typically measure, and that is the reason they are used as the computational basis.

For Fibonacci anyons, we have to deal with a Hilbert space that isn’t really built for being carved into qubits. Typically we take a subspace for which measurements aren’t too convoluted. But then we also need to worry about braiding leaking the state out of the subspace. This gives us a whole bunch of practical concerns to think about when choosing our basis. But nevertheless, it is only these practical concerns that lead to a preferred choice, and different authors may very well choose different conventions.

You may be confusing two uses of the word "base". One definition of "base" has to do with how many digits are used to represent a number. For example, base two uses the digits 0 and 1, and the number five is written as 101 in base two. But in quantum mechanics there is another use of the word "base" which has to do with basis vectors for a vector space. This is almost entirely unrelated to the "base" of a number system.

I see that once you start talking about qubits versus qudits there is further confusion. Perhaps you could try thinking of a qubit as a two dimensional space, and the basis within that space as giving a preferred "direction", or coordinate axes. Similarly, a qutrit is a three dimensional space, etc. (This is a geometric intuition that might help you get started with thinking about quantum states, it needs some more work before it is precise.)

• This answer should probably be a couple of comments --- but it seems like a plausible theory about what the OP is thinking about. – Niel de Beaudrap Apr 5 '18 at 11:53
• @NieldeBeaudrap Note that you need 50 reputation to comment. It's ok to post this answer, and maybe it can be expanded. – M. Stern Apr 5 '18 at 12:11
• @SimonBurton - I made the question shorter, and tried not to invalidate your answer. You can edit your answer if you wish or leave it as-is. Thanks for responding to the initial questions. – Rob Apr 7 '18 at 16:00
• Mods can put answers in comment as this question should be in comment (-50rep) – DIDIx13 Apr 11 '18 at 7:02
• @DIDIx13 - Agreed. He ought to have asked for clarification before answering. The word "bases" is plural of the word "basis": english.stackexchange.com/questions/274268/… . I flagged it to be converted to a comment to the question. – Rob Apr 11 '18 at 16:20