# What are the postulates of Continuous Variable Quantum Computing?

I am a computer scientist. When I learned quantum information for the first time (in the circuit model, with qubits), I was presented four postulates that described mathematically (i) the possible states of a single qubit, (ii) the possible states of multiple qubits, (iii) the reversible transformations of quantum registers; and (iv) measurements that can be performed and the outcome probabilities (Born's Rule).

I am trying to find a similar set of postulates for continuous variable quantum computation. My questions:

• What are the possible states of a qumode?
• What are the possible states of multiple qumodes?
• What reversible transformations are possible on registers of qumodes?
• What measurements can I perform on qumodes?

There is quite a bit to unpack here, so I provide a brief overview of things with links to relevant wikis and papers wherever necessary.

## Qumode States vs Qubits

To start with, it is important we first clarify what a qumode really is, in contrast to a qubit. A pure quantum state is an element of a Hilbert space defined over the field of complex numbers $$(| \psi \rangle \in \mathcal{H}(\mathbb{C}))$$. One can choose a basis for a Hilbert space, given it is a vector space. The cardinality of this basis corresponds to the dimension of the vector space. This is usually a Schauder basis as the Hilbert space may be infinite dimensional.

Now, a qubit is an element $$| \psi \rangle \in \mathcal{H}^2(\mathbb{C})$$, where the $$2$$ indicates the dimension of the Hilbert space. A typical basis known as the computational basis is $$\mathfrak{B} = \{| 0 \rangle, | 1 \rangle\}$$. Thus any qubit is a linear combination of these basis vectors upto a unitary transformation. A physical example of a qubit would be a state in any 2-level system such as a single spin-$$1/2$$ particle (eg. an electron).

On the other hand, a qumode state $$| \phi \rangle \in \mathcal{H}(\mathbb{C})$$ is typically expressed in an uncountably infinite basis. A particularly common and useful way to analyse qumode states (and observables) is by looking at their phase space representations using the so-called Wigner quasiprobability distributions. The quadrature operators $$\hat{q}$$ and $$\hat{p}$$ provide us with a continuous basis $$| q \rangle$$ and even though these elements do not technically lie in $$\mathcal{H}(\mathbb{C})$$, the following expansion can be rigorously justified - $$| \phi \rangle = \int_{\mathbb{R}} | q \rangle \langle q | \phi \rangle \, dq = \int_{\mathbb{R}} \phi (q) | q \rangle \, dq,$$ where, $$\phi (q) \in L^2(\mathbb{R})$$, $$\langle q | q' \rangle = \delta(q-q')$$ and $$\int_{\mathbb{R}} | q \rangle \langle q | \, dq = \mathbb{1}$$. (Similarly for $$p$$).

Physically such representations are particularly useful and natural for unbounded systems such as the Quantum Harmonic Oscillator which has an infinite number of energy levels $$(\mathfrak{B} = \{| n \rangle \, | \, n \in \mathbb{N} \})$$, each of which can be represented as a qumode state and so can their linear combinations. The coherent state of a photon is another common example of a single qumode state. A multimode state $$| \phi \rangle \in \mathcal{H}^{\otimes n}$$ can not only be product states $$\bigotimes_k | \phi_k \rangle$$, but also entangled states. Typically, such states, particularly for many qumodes is formally dealt with Fock spaces and Second quantisation.

So in a crude sense, qumode states are uncountably infinite dimensional analogues of qubits. However, notice that so far I have been referring to qumode states as opposed to a qumode. A qumode is in fact, a mode of a quantized field, usually the electromagnetic field.

## Observables in Continuous Variable (CV) Quantum Computing (QC)

All CV observables are unitaries defined in terms of the quadrature operators $$\hat{q}$$ and $$\hat{p}$$ or equivalently in terms of the ladder operators $$\hat{a}$$ and $$\hat{a}^\dagger$$. Any conceivable unitary in CV can be expressed in terms of a complete set of Gaussian unitaries which generate all possible affine symplectic transformations on the phase space and at least one non-Gaussian unitary that generates non-linear transformations on the phase space. This is the crux of universality in CV QC, analogous to the Clifford and non-Clifford unitaries in Discrete Variable (DV) QC. The name Gaussian stems from the fact that when such operations act on Gaussian states (whose Wigner quasiprobabilities are Gaussian distributions), the Gaussian behavior is preserved.

An example of a complete set of Gaussian unitaries are $$\{\mathcal{D}(\alpha), \mathcal{S}(r), \mathcal{R}(\varphi), \mathcal{BS}(\theta, \phi)\}$$, which are parametrized displacement, squeezing, rotation (all act on single qumode states) and beam-splitter (acts on two-qumode states) transformations or gates in QC terminology. The Gaussian unitaries are all generated by at most quadratic Hamiltonians. An example of a non-Gaussian gate is the cubic phase gate $$\mathcal{V}(\gamma)$$. These gates are usually generated by cubic or higher degree Hamiltonians. Definitions of the aforementioned gates and some computational details may be found in this paper.

## Measurements in CV QC

Similar to DV QC, where one typically makes measurments in the computational basis, in CV QC, one makes measurements in a quadrature basis. This is called Homodyne Measurement. A general Homodyne measurement is a projection onto the eigenbasis $$| q_\phi \rangle$$ of the operator - $$\hat{q}_\phi = \hat{q} \cos{\phi} + \hat{p} \sin{\phi}.$$ For $$\phi = 0$$, the measurement is along the "position" quadrature and the outcomes are quadratures $$q \in \mathbb{R}.$$ In a real CV quantum computer, this is likely the Homodyne measurement of choice.

On the other hand, we also have heterodyne measurments which can be thought of as a simultaneous measurement in $$\hat{q}$$ and $$\hat{p}$$. Since they are non-commuting observables, there is an inherent uncertainty associated with such a measurement. Equivalently, one could think of this as a projection onto the set of all coherent states with measurement operators $$\pi^{-1}| \alpha \rangle \langle \alpha |$$ with outcomes $$\alpha \in \mathbb{C}$$. Since coherent states are not orthogonal, there is a lack of sharpness in such measurements.

The above two measurements are Gaussian in the sense that if we measure a multimode Gaussian state, the conditional state on the remaining modes is still Gaussian. Furthermore, the CV gates and measurements all heavily draw from ideas in quantum optics. Practically, the main platform for CV QC is an optical quantum computer after all. As such, this leads us to a third measurement option, one that is non-Gaussian - photon counting. While the dyne measurements relied on the wave-like nature of light, this one relies on the particle nature. The measurements are on the Fock basis $$| n \rangle$$ and give us the number of photons $$n \in \mathbb{N}$$ associated to the mode.

PS: In simulations and even in practice of course, we always consider a cutoff dimension to indicate the highest energy level we can access or require. This leads to us working with a truncated Hilbert space and the truncation errors that come along with it.