What exactly is a logical (non-physical? error corrected?) qunit?

Can quantum systems be built exclusively w/ logical qunits?

  • $\begingroup$ Possible duplicate of What is a qubit? $\endgroup$ – bytebuster Jul 2 '18 at 19:29
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    $\begingroup$ I'm not asking about a qubit in general. I'm specifically after logical qubits. $\endgroup$ – meowzz Jul 2 '18 at 19:31

A logical qubit is made out of many physical qubits (or qudits), simply selecting a particular two-dimensional subspace. So you can’t make it “exclusively” out of logical qubits because they sit on top of real physical qubits. In fact, if you're thinking about a terminology of "virtual qubits", that is actually best thought of as a synonym for "logical qubits".

Remember what should almost be the mantra of quantum computing: "information is physical". Information doesn't exist unless it is recorded somewhere, so it must be recorded on something physical and the physical operations that can be performed on that information carrier determine the nature of the information theory. So if you want a logical qubit you'd better be using quantum information carriers at the physical level. It doesn't matter what quantum information carrier you use as your physical qubit, whether that's a spin or a photon, or any type of two-level system.

There is no particular relation between the number of physical qubits and the number of logical qubits (so long as the size of the space for logical qubits $\leq$ size of space for physical qubits). You might, however, use the relation as some sort of measure of efficiency. For example, we often talk about error correcting codes as defining a logical qubit. They’re defined on multiple physical qubits and give you a smaller number of “useful” qubits. Most critically, when you start talking about fault-tolerance, you have two parameters: $p_c$, the critical error rate above which the fault tolerant scheme cannot achieve arbitrary accuracy, and $p_a<p_c$, the actual error rate that you can achieve. The variation of the number of physical qubits you require to achieve a single error corrected logical qubit as $p_a$ approaches $p_c$ tells you a lot about the feasibility of fault-tolerance and the accuracy we need to aim for to get good quality quantum computing on modestly sized quantum computer..

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    $\begingroup$ Is there any sort of generalization as to how physical & logical qubits correlate? (eg. $n$ logical qubits requires $n^2$ physical qubits) Although they tend to be made of physical qubits, can they be made of other things? (eg. how could they be made photonicly or could they be purely virtual) $\endgroup$ – meowzz Jul 2 '18 at 20:10
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    $\begingroup$ If you did an error correcting code that embedded a logical qubit into k physical qubits $\mathbb{C}^{2^k}$, then you automatically get an encoding $(\mathbb{C}^2)^{\otimes n} \to (\mathbb{C}^{2^k})^{\otimes n}$. That gives kn physical qubits for n logical qubits. This does not scale $k$ like $n$ to get $n^2$. $k$ is up to you depending on what errors you seek to correct. $\endgroup$ – AHusain Jul 2 '18 at 21:30
  • $\begingroup$ I had indeed been thinking about "virtual qubits." Excellent addition. $\endgroup$ – meowzz Jul 3 '18 at 7:28
  • $\begingroup$ Why would is it necessary to have the underlying system be quantum? Could logical qunits be coded into logical qunits? (eg emulating a quantum environment w/ a quantum system inside of it) $\endgroup$ – meowzz Jul 16 '18 at 21:37
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    $\begingroup$ The underlying system has to be quantum because you have to be able to have superpositions. You can’t get that in the logical qubit if you don’t have it in the physical. As for qudits, sure. We always talk about qubits because they’re easier to deal with most of the time, have a far more developed theory etc. But there’s nothing special about qubits compared to qudits of other d. And there’s no need to match the d of logical and physical qubits either. $\endgroup$ – DaftWullie Jul 17 '18 at 5:16

Short answer:

Logical qubits are just an abstraction above physical qubits. A logical qubit is something (see after for examples) that acts like a qubit.

Some examples

A logical qubit can be:

  1. A single physical qubit. This is the case for most of (all?) the quantum chips currently available. In this case, the logical qubit has no advantage over the physical one (they are the same qubits).
  2. Multiple qubits used for quantum error correction code.

    In this case, the user sees the logical qubit as a single qubit and does not need to know the underlying error correction scheme because the logical qubit behaves like a physical one. This case is quite interesting for quantum computing because we could "hide" the complexity of quantum error correction algorithm behind logical qubits. The user will only see one qubit with low error rates, but the physical setup will be composed of multiple qubits.

  3. As of today, the term "logical qubit" is mostly used in quantum error correction but may be used in other fields in the future.

Side note: to answer your question in comment:

Is there any sort of generalization as to how physical & logical qubits correlate? (eg. n logical qubits requires $n^2$ physical qubits) Although they tend to be made of physical qubits, can they be made of other things? (eg. how could they be made photonicly or could they be purely virtual)

There is no generalisation on the number of physical qubits needed to encode one logical qubit. This is entirely dependant on the error correction algorithm used.

A purely virtual qubit does not exists. Logical qubits are necessarily composed of physical qubits. Note that a "qubit" is already an abstraction: it abstract the physical representation used. As a logical qubit is defined in term of "qubits", logical qubits are independant of the physical implementation of the underlying qubits.

  • $\begingroup$ Would it be fair to say that the number of physical qubits needed per logical (error corrected) qubit is a good metric for benchmarking? $\endgroup$ – meowzz Jul 2 '18 at 21:36
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    $\begingroup$ I don't think so. You can have 2 chips with the same error rates but one needs 1000 physical qubits to obtain this error rate (because its physical qubits are not very good) and the other needs only 10 physical qubits per logical one (because its physical qubits already have a low error rate). $\endgroup$ – Nelimee Jul 2 '18 at 21:46

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