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Background

I am curious to learn more about any work that has been done regarding quantum systems that deal with infinite values. I am primarily interested in photonic quantum computing; however I am open to learning about other systems where infinite values are used.

Here are a few snippets that lead me to believe this is a valid concept to consider:

In Can we process infinite matrices with a quantum computer?, there is an answer that states:

If instead of manipulating the quantum information in qubits, your quantum computer were to do operations on qu𝑑its with 𝑑 being infinity, then you'd essentially be processing infinite matrices on a quantum computer.

Additionally, in another question there is an answer that states:

To get you started, the fundamental difference between the CV model and the traditional qubit model is that in the CV model, we formally apply operations on infinite-dimensional instead of two-dimensional systems. Of course, in practice, each system can be effectively described by a large but finite-dimensional Hilbert space, but it is more mathematically convenient to describe operators and states on the full infinite-dimensional space.

In Is quantum computing limited to a superposition of only two states? there is an answer that states:

In principle, there is no limit to the dimension of the state space of a quantum system. There could even be an "infinite" dimensional separable Hilbert space (in short, separable means denumerable/countable with a one-one onto mapping to the natural numbers).

From Nonseparable Hilbert space:

Furthermore, the article on Hilbert spaces contains the following:

A Hilbert space is separable if and only if it admits a countable orthonormal basis.

In case of field theory, it states:

Even in quantum field theory, most of the Hilbert spaces are in fact separable, as stipulated by the Wightman axioms. However, it is sometimes argued that non-separable Hilbert spaces are also important in quantum field theory, roughly because the systems in the theory possess an infinite number of degrees of freedom and any infinite Hilbert tensor product (of spaces of dimension greater than one) is non-separable.

Questions

What systems make use of countable infinities? How are they used?

What systems make use of uncountable infinities? How are they used?

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  • $\begingroup$ I do not know much about "transfinite ordinals". If you were to ask a question with just "What systems make use of countable infinities? " and "What systems make use of uncountable infinities?" I would be able to give some answers. $\endgroup$ – user1271772 Apr 17 at 18:07
  • $\begingroup$ @user1271772 Updated question. Looking forward to what you have to say! $\endgroup$ – user820789 Apr 18 at 0:14
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You are right, photonic systems are described by an infinite (separable) Hilbert space---the bosonic Fock space---and their formalism makes extensive use of infinite values, both countable and uncountable. The quantum computing paradigm based on this Hilbert space is called continuous-variable (CV) quantum computing, and a lot of different protocols and algorithms have been proposed using this framework, see for instance this recent review by Xanadu (who is developing optical quantum computers with the goal of working with continuous variables). Two important points to note: 1) CV quantum computers could in principle be built with other systems than photons, such as molecular vibrations (phonons), which obey the same equations as photons ; 2) you can restrict the Hilbert space of photons in order to get qubits, for instance by considering only polarization or by encoding qubits into continuous variables. This is the approach taken by the photonic quantum computing company PsiQuantum (as far as I understand).

Where does CV quantum computing come from?

There exists tons of equivalent ways to introduce the CV paradigm. The most physical is the quantization of the electromagnetic field: you take Maxwell's equations and you turn the electric and magnetic fields into non-commuting operators. You find that your system now describes a quantum Harmonic oscillator and that the Hamiltonian have infinitely many eigenstates, forming an infinite-dimensional Hilbert space.

Another more rigorous way to define this Hilbert space is called the second quantization: you define bosonic quantum states as multi-particle states that are invariant when you permute particles, and after some steps, you find that the correct Hilbert space to describe bosons is the so-called Fock space (which is a separable Hilbert space when defined properly).

Finally, you can formalize bosonic systems in a much more mathematical/computer sciency way, that allows you to talk about complexity theory. Three examples of such formalisms are given in Section 3 of this paper.

Formalism and infinities

All those formalisms have a common point: you end up with a separable Hilbert space. And all separable Hilbert spaces are the same up to an isometric isomorphism. Moreover, separable Hilbert spaces have the amazing properties to contain an infinite countable basis, that we can note $(|n\rangle)_{n \in {\mathbb{N}}}$. Therefore, for any state $|\psi\rangle \in \mathcal{H}$, there exists $(a_n)_{n \in {\mathbb{N}}}$ such that $$|\psi\rangle = \sum_{n=0}^{\infty} a_n |n\rangle$$ Physically, $|n\rangle$ is a state that contains $n$ indistinguishable photons.

Using this basis $(|n\rangle)_{n \in \mathbb{N}}$ (called the Fock basis), we can define many important objects of the CV framework, such as the creation and annihilation operators $$\hat{a}^{\dagger}|n\rangle=\sqrt{n+1} |n+1\rangle$$ $$\hat{a}|n\rangle=\sqrt{n} |n-1\rangle,$$ the position and momentum operators (which physically correspond the amplitude of the electric and magnetic fields, not to spatial coordinates) $$\hat{X}=\frac{1}{\sqrt{2}} (\hat{a}^{\dagger} + \hat{a})$$ $$\hat{P}=\frac{1}{\sqrt{2}} i (\hat{a}^{\dagger} - \hat{a})$$ and the number operator $$\hat{N}|n\rangle = n|n\rangle$$

Now, you can verify that $\hat{X}$ and $\hat{P}$ are hermitian (infinite-dimensional) operators, and are therefore observables that you can physically measure. Their eigenstates $|x\rangle$ and $|p\rangle$ form two new bases of your Hilbert space, but this time uncountably infinite, i.e. for every state $|\psi\rangle$, there exists a function $x\mapsto \psi(x)$ and a function $p \mapsto \phi(p)$ such that $$|\psi\rangle = \int \psi(x) |x\rangle dx$$ $$|\psi\rangle = \int \phi(p) |p\rangle dp$$

Therefore, the same state can be represented both using countable infinities and uncountable infinities. Which basis you want to choose depends on your measuring device (photon detectors measure in the $|n\rangle$ basis and homodyne detectors in the $|x\rangle$ and $|p\rangle$ bases), the initial state of your algorithm (the output of a laser, called a coherent state---and more generally Gaussian states---are more easily representable with $\hat{X}$ and $\hat{P}$, while single-photons are more easily described in the Fock basis) or on the details of your algorithm (are integral or sums more convenient to analyze it?).

Algorithms

We saw what a CV state looks like, what measurements can look like, but what about gates? As usual, any unitary operator (here infinite-dimension matrix) can be seen as a gate. Elementary gates include squeezing, displacement, rotation, etc. and are very well described in the paper of the CV library Strawberry Fields. A particular representation of states called the Wigner function (roughly describing the quasi-probability to find a particle at a certain position and momentum) is often used to describe the effect of those gates.

Now, what are the applications of CV quantum computing? One of the main area where CV quantum information is used is in quantum communication. Indeed, photons can be transmitted through optical fibers and rarely interact, making it a perfect choice for communication. Moreover, communication protocols such as teleportation and QKD have been ported to CV systems.

Going back to computation, an important CV algorithm is Boson Sampling, which is mostly considered as a way to demonstrate quantum supremacy, but might have applications such as finding dense subgraphs or simulating molecular vibronic spectra

Finally, CV quantum computing has been considered in order to solve partial differential equation (porting the HHL algorithm to an infinite-dimensional system), to improve Monte-Carlo algorithms or to do quantum machine learning and variational circuits

If you are interested to go deeper in understanding continuous variables, apart from all the papers I've cited, you can also read the first section of my master's thesis, which explains all of that in more details and (I hope) in an understandable way.

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    $\begingroup$ Wow - thanks for such a thorough answer! $\endgroup$ – user820789 Apr 21 at 22:55

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