What is a ket? How do traits work? Quantum computing in Rust, part 1

https://walther.guru/blog/impl-quantum-part-1/

by Walther • 7 years ago

87 27 7 years ago

ivanbakel• 7 years ago

`assert_eq(A == B, true)`? `assert_eq(A, B)`, or even `assert(A == B)` by all means.

And this does not really explain what a `Ket` is, rather unfortunately. I now know it's a pair of complex numbers, but that's not very handy - and the `is_valid` definition is not explained, so I don't even get told what subset of pairs of complex numbers make up valid kets without reading code.

db48x• 7 years ago

The title of the article is "Quantum Computing fin Rust, part 1", so the implication is that it is there to teach you about Rust, not to teach Quantum Mechanics which you already know.

eden_h• 7 years ago

It's not too much to expect an article entitled "What is a ket?" to explain what a Ket is.

Your point is arguable if it was instead "How to implement a Ket in Rust", and even then it's a bit vague. I left with the same feeling that it was a bit barebones in terms of definition.

Walther5• 7 years ago

Heh, what a silly little mistake to make with the asserts. I've now improved their readability. Thank you!

You're right, I need to add a better explanation for what a ket is. Same applies to the validity, should explain a bit about normalization.

Thanks for the feedback!

reallydontask2• 7 years ago

https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation

I remember learning about this at uni in one of the more advanced QM modules.

I remember thinking: Seriously, couldn't they've come up with a better name?

eigenloss5• 7 years ago

It's basically a 90-year-old pun involving splitting the word "bracket" into "bra" and "ket".

_Microft• 7 years ago

Why bracket? The reason is that there is not only ket vectors |v> but also functionals <u| that can be applied to them, which will look like this <u|v>. So there are brackets around the construct. Think <bra|ket> if you like.

saalweachter• 7 years ago

Sometimes I don't know whether to be proud of or disappointed in our species.

perl4ever• 7 years ago

The word "quark" is kind of whimsical, but the names of the types of quark seem arbitrary and unhelpful.

reallydontask2• 7 years ago

it is strange that you have such a charming notion about the flavours of quarks. If we turn this notion upside down maybe we could then get to the bottom of this or should i say the top?

I'll see myself out

philzook• 7 years ago

Really interesting. Rust is on my radar. You might be interested in a post in Haskell that I made that has a similar feel.

http://www.philipzucker.com/functors-and-vectors/

reikonomusha4• 7 years ago

This does not present the correct definition of a ket. More specifically, it doesn’t present a reasonably general definition of a ket. It would be like asking “What is a list?” and implementing a data structure that can store only one element.

In quantum computation, a ket is a vector in a Hilbert space. A Hilbert space is just a fancy way to describe a typical space you find in linear algebra, where the space allows you to compute lengths and angles. When discussing kets, the usual vector space is the set of complex-element vectors with unit length (or “norm”). The vectors can have any number of elements (or “dimension”), but when discussing qubits, they are 2^n-dimensional for n qubits.

(It’s important to note that a ket is not distinguishable from a vector. It’s actually called so because of a notational convention, not because it has deeper underlying meaning. However, physicists will still use the word “ket” instead of “vector” or “quantum state” even if they’re not emphasizing notation.)

More interesting, though, is how kets combine with other kets via tensor products. This ingredient is as essential to QC as flour is to cake.

This article [0] informally presents a fully general definition of a ket along with the tensor product with an emphasis on why a representation and notation was chosen. But it does require a good understanding of linear algebra already.

[0] “Someone shouts |01000>! Who’s excited?” https://arxiv.org/abs/1711.02086

Walther5• 7 years ago

Thank you for feedback! Going to read the paper to improve my understanding. I'm writing this blog series in part to learn more - both about quantum computing as well as Rust. As some people say, teaching is a great way to learn (or at least check your knowledge).

What do you think would be a good correction or disclaimer to add? Would it be sufficient to e.g. say that in this first part, we only implement a two-component ket? Generalizing the ket type to an arbitrary-size vector while keeping type constraints, perhaps briefly mentioning const generics (an upcoming language feature), etc could easily make for another part in this series. Making the ket into an actual vector could then give way to doing proper linear algebra on them, cleaning up some of the manually implemented operations here, as their teaching purpose has been fulfilled already.

Thoughts?

reikonomusha4• 7 years ago

I would say you’ve implemented a state vector for a single qubit (when the norm = 1 and when equality is defined to be phase invariant, i.e., x and y are equal whenever there’s an ‘a’ such that e^(ia)x = y.), which can be written down notationally with a ket:

|s> = p|0> + q|1>

where p and q are complex, and |p|^2 + |q|^2 = 1. The ket is the notation |s>, |1>, and |0>, which could also be written purely as column vectors.

At the end it all depends on what you want to present. What will be the climax? Unitary evolution of a pure state? Superoperators on a mixed state? An introduction to linear algebra used in quantum computation and its notation?

In any case I’d present what a quantum state is on N qubits, which requires 2^N-element complex vectors that satisfy a few properties.

eigenloss5• 7 years ago

Having "quantum computing" in the title makes this genuine clickbait when the body lacks any explanation of what a state is or what a ket is. Energy or momentum or spin states are rarely (never, in reality) just numbers.

You made no attempt to explain what the two different states in a single ket mean: hint, it's superposition. Never provided even a trivial justification for why normalization is important; where is the actual physics?

Using Rust here is also quite pointless when you're just doing linear algebra... where is the actual benefit over Python or even Haskell?

You've written a very cute and inefficient math library, but calling this QC is just false advertising.

abdullahkhalids• 7 years ago

(It’s important to note that a ket is not distinguishable from a vector. It’s actually called so because of a notational convention, not because it has deeper underlying meaning. However, physicists will still use the word “ket” instead of “vector” or “quantum state” even if they’re not emphasizing notation.)

This is not correct when you start talking about continuous-variable states. In the (canonical) position basis, the state of an infinite-dimensional system is commonly expressed as \ket{\psi} = \int \psi(x)\ket{x}dx. Here {\ket{x}} are a basis set, but not square-integrable, since \bra{x}\ket{x} = deltafunction(0). Hence \ket{x} is not a valid quantum mechanical state.

reikonomusha4• 7 years ago

Kets sometimes even represent vector-valued functions (e.g., |f(t)> = e^(iHt)|0> or so) which in and of themselves aren’t states, but representations of system dynamics. We could pontificate as to whether |f(t)> is the function or is merely the image of t -> |f(t)> at some t. :)

In any event, I’m speaking in the context of my comment and of the article: (gate-based) quantum computation. Here, 9 out of 10 dentists will use a ket to denote the (computational) state of the quantum computer. Infinite dimensional states and bases aren’t pertinent at the level of logical computation.

sampo• 7 years ago

In quantum computation, a ket is a vector in a Hilbert space

The blog implements kets as 2-dimensional complex vectors, so I don't see any problem there. A 2-dimensional space is a Hilbert space.

reikonomusha4• 7 years ago

It’s woefully under-representing what a ket can be, and the data structure isn’t amenable to extension to a broader set of states. A one qubit state lives in a Hilbert space just as a one element list is still a list.

Walther5• 7 years ago

Thank you all for the feedback! I'll continue to make improvements to this post, as well as build more in the next parts.

My current understanding is mostly based on this wonderful website: https://quantum.country/qcvc - if you can recommend more good resources, I'm all ears!

Walther5• 7 years ago

Forgot to mention, I'm also getting the book "Quantum Computing Since Democritus" by Scott Aaronson soon.

Gonzih• 7 years ago

Oh thats pretty cool, wanted to subscribe to rss/atom feed for a followup blog posts, but cant find any feed link. I hope i wont miss future posts in series.

What is a ket? How do traits work? Quantum computing in Rust, part 1

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