Introduction

Section author: Nathan Killoran <nathan@xanadu.ai>

Many physical systems are intrinsically continuous, with light being the prototypical example. Such systems reside in an infinite-dimensional Hilbert space, offering a paradigm for quantum computation which is distinct from the qubit model. This continuous-variable model takes its name from the fact that the quantum operators underlying the model have continuous spectra.

The CV model is a natural fit for simulating bosonic systems (electromagnetic fields, harmonic oscillators, phonons, Bose-Einstein condensates, or optomechanical resonators) and for settings where continuous quantum operators – such as position & momentum – are present. Even in classical computing, recent advances from deep learning have demonstrated the power and flexibility of a continuous picture of computation [51][52].

From qubits to qumodes

A high-level comparison of CV quantum computation with the qubit model is depicted in the following table:

 

CV

Qubit

Basic element Qumodes Qubits
Information unit 1 nat (\(\log_2e\) bits) 1 bit
Relevant operators

Quadrature operators \(\x,\p\)

Mode operators \(\a, \ad\)

Pauli operators \(\hat{\sigma}_x, \hat{\sigma}_y, \hat{\sigma}_z\)
Common states

Coherent states \(\ket{\alpha}\)

Squeezed states \(\ket{z}\)

Number states \(\ket{n}\)

Pauli eigenstates \(\ket{0/1}, \ket{\pm}, \ket{\pm i}\)
Common gates Rotation, Displacement, Squeezing, Beamsplitter, Cubic Phase Phase Shift, Hadamard, CNOT, T Gate
Common measurements Homodyne \(\hat{x}_\phi\), Heterodyne \(Q(\alpha)\), Photon-counting \(\ketbra{n}{n}\) Pauli-basis measurements \(\ketbra{0/1}{0/1}, \ketbra{\pm}{\pm}, \ketbra{\pm i}{\pm i}\)

Note

Qubit-based computations can be embedded into the CV picture, e.g., by using the Gottesman-Knill-Preskill (GKP) embedding [53][54][55], so the CV model is as computationally powerful as its qubit counterparts.

The most elementary CV system is the bosonic harmonic oscillator, defined via the canonical mode operators \(\a\) and \(\ad\). These satisfy the well-known commutation relation \([\a,\ad]=\I\). It is also common to work with the quadrature operators

\[\begin{split}\begin{aligned} \hat{x} := \sqrt{\frac{\hbar}{2}}(\a + \ad), \\ \hat{p} := -i\sqrt{\frac{\hbar}{2}}(\a - \ad),\end{aligned}\end{split}\]

with \([\x,\p]=i \hbar\). These self-adjoint operators are proportional to the hermitian and antihermitian parts of the operator \(\a\).

Note

Strawberry Fields uses the convention that \(\hbar=2\) by default, but this can be changed as required.

We can picture a fixed harmonic oscillator mode (say, within an optical fibre or waveguide on a photonic chip) as a single ‘wire’ in a quantum circuit. These qumodes are the fundamental information-carrying units of CV quantum computers. By combining multiple qumodes (each with corresponding operators \(\a_i\) and \(\ad_i\)) and interacting them via sequences of suitable quantum gates, we can implement a general CV quantum computation.

CV states

The dichotomy between qubit and CV systems is perhaps most evident in the basis expansions of quantum states:

\[\begin{split}\begin{aligned} &\rm{Qubit} &\ket{\phi} & = \phi_0 \ket{0} + \phi_1 \ket{1}, \\ &\rm{Qumode} &\ket{\psi} & = \int dx~\psi(x) \ket{x}. \end{aligned}\end{split}\]

For qubits, we use a discrete set of coefficients; for CV systems, we can have a continuum. The states \(\ket{x}\) are the eigenstates of the \(\x\) quadrature, \(\x\ket{x}=x\ket{x}\), with \(x\in\mathbb{R}\). These quadrature states are special cases of a more general family of CV states, the Gaussian states, which we now introduce.

Gaussian states

Our starting point is the vacuum state \(\ket{0}\). Other states can be created by evolving the vacuum state according to

\[\ket{\psi} = \exp(-itH)\ket{0},\]

where \(H\) is a bosonic Hamiltonian and \(t\) is the evolution time. States where the Hamiltonian \(H\) is at most quadratic in the operators \(\x\) and \(\p\) are called Gaussian.

For a single qumode, Gaussian states are parameterized by two continuous complex variables: a displacement parameter \(\alpha\in\mathbb{C}\) and a squeezing parameter \(z\in\mathbb{C}\) (often expressed as \(z=r\exp(i\phi)\), with \(r \geq 0\)). Gaussian states are so-named because we can identify each Gaussian state, through its displacement and squeezing parameters, with a corresponding Gaussian distribution. The displacement gives the centre of the Gaussian, while the squeezing determines the variance and rotation of the distribution.

Note

Many important pure states in the CV model are special cases of the pure Gaussian states; these are summarized in the following table.

State family

Displacement

Squeezing

Coherent states \(\ket{\alpha}\)

\(\alpha\in\mathbb{C}\)

\(z=0\)

Squeezed states \(\ket{z}\)

\(\alpha=0\)

\(z\in\mathbb{C}\)

Displaced squeezed states \(\ket{\alpha,z}\)

\(\alpha\in\mathbb{C}\)

\(z\in\mathbb{C}\)

\(\x\) eigenstates \(\ket{x}\)

\(\alpha\in\mathbb{C}\), \(x\propto\mathrm{Re}(\alpha)\)

\(\phi=0\), \(r\rightarrow\infty\)

\(\p\) eigenstates \(\ket{p}\)

\(\alpha\in\mathbb{C}\), \(p\propto\mathrm{Im}(\alpha)\)

\(\phi=\pi\), \(r\rightarrow\infty\)

Vacuum state \(\ket{0}\)

\(\alpha=0\)

\(z=0\)

Number states

Complementary to the continuous Gaussian states are the discrete number states (or Fock states) \(\ket{n}\), \(n\in\mathbb{N}\). These are the eigenstates of the number operator \(\n=\ad\a\). The number states form a discrete countable basis for the states of a single qumode. Thus, each of the Gaussian states considered in the previous section can be expanded in the number state basis. For example, coherent states have the form

\[\ket{\alpha} = \exp\left(-\tfrac{|\alpha|^2}{2}\right) \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}}\ket{n},\]

while (undisplaced) squeezed states only have even number states in their expansion:

\[\ket{z} = \frac{1}{\sqrt{\cosh r}}\sum_{n=0}^\infty\frac{\sqrt{(2n)!}}{2^n n!}[-e^{i\phi}\tanh (r)]^n\ket{2n}.\]

Mixed states

Mixed Gaussian states are also important in the CV picture, for instance, the thermal state

\[\rho(\nbar) := \sum_{n=0}^\infty\frac{\nbar^n}{(1+\nbar)^{n+1}}\ketbra{n}{n},\]

which is parameterized via the mean photon number \(\nbar:=\tr{(\rho(\nbar)\hat{n})}\). Starting from this state, we can consider a mixed-state-creation process, similar to above, namely

\[\rho = \exp(-itH)\rho(\nbar)\exp(itH).\]

Analogously to pure states, by applying quadratic-order Hamiltonians to thermal states, we generate the family of Gaussian mixed states.

CV gates

Unitary operations can always be associated with a generating Hamiltonian \(H\) via the recipe

\[U = \exp{(-itH)}.\]

For convenience, we can classify unitaries by the degree of their generating Hamiltonians. We can build an N-mode unitary by applying a sequence of gates from a universal gate set, each which acts only on one or two modes.

Note

A CV quantum computer is said to be universal if it can implement, to arbitrary precision and with a finite number of steps, any unitary which is polynomial in the mode operators [56].

We focus on a universal gate set which contains the following two components:

Gaussian gates
One-mode and two-mode gates which are quadratic in the mode operators, e.g., displacement, rotation, squeezing, and beamsplitter gates. These are equivalent to the Clifford group of gates from the qubit model.
Non-Gaussian gates
A single-mode gate which is degree 3 or higher, e.g., the cubic phase gate. These are equivalent to the non-Clifford gates in the qubit model.

A number of fundamental CV gates are presented in the following table:

Gate

Unitary

Symbol

Displacement

\(D_i(\alpha)=\exp{(\alpha\ad_i - \alpha^*\a_i)}\)

_images/Dgate.svg
Rotation

\(R_i(\phi)=\exp{(i\phi\hat{n}_i)}\)

_images/Rgate.svg
Squeezing

\(S_i(z)=\exp{(\frac{1}{2}(z^* \a_i^2 - z \a_i^{\dagger 2}))}\)

_images/Sgate.svg
Beamsplitter

\(BS_{i,j}(\theta,\phi)=\exp{(\theta(e^{i\phi}\ad_i\a_j - e^{-i\phi}\a_i\ad_j))}\)

_images/BSgate.svg
Cubic Phase

\(V_i(\gamma)=\exp{(i\frac{\gamma}{6}\x_i^3)}\)

_images/Vgate.svg

Note

We often also use the position displacement (\(X(x)=D(x/2)=\exp(-ix\p/2)\) where \(x\in\mathbb{R}\)) and momentum displacement (\(Z(p)=D(ip/2)=\exp(ip\x/2)\) where \(p\in\mathbb{R}\)) gates, specific cases of displacement defined above.

Together, these are known as the Weyl-Heisenberg group, satisfying the relation \(X(x)Z(p)=e^{-ixp/2}Z(p)X(x)\), and are analogous to the Pauli group operators in the discrete-variable formulation.

We can see that many of the Gaussian states from the previous section are connected to a corresponding Gaussian gate. Any multimode Gaussian gate can be implemented through a suitable combination of Displacement, Rotation, Squeezing, and Beamsplitter Gates [57], making these gates sufficient for quadratic unitaries. The cubic phase gate is presented as an exemplary higher-order gate, but any other non-Gaussian gate could also be used to achieve universality.

CV measurements

As with CV states and gates, we can distinguish between Gaussian and non-Gaussian measurements. The Gaussian class consists of two (continuous) types: homodyne and heterodyne measurements, while the key non-Gaussian measurement is photon counting.

Measurement

Measurement operator

Measurement values

Homodyne

\(\ket{\x_\phi}\bra{\x_\phi}\)

\(q\in\mathbb{R}\)

Heterodyne

\(\frac{1}{\pi}|\alpha\rangle\langle\alpha|\)

\(\alpha\in\mathbb{C}\)

Photon counting

\(\ketbra{n}{n}\)

\(n\in\mathbb{N},n\geq 0\)

Homodyne measurements

Ideal homodyne detection is a projective measurement onto the eigenstates of the quadrature operator \(\x\). These states form a continuum, so homodyne measurements are inherently continuous, returning values \(x\in\mathbb{R}\). More generally, we can consider projective measurement onto the eigenstates of the Hermitian operator

\[\x_\phi:=\cos\phi~\x + \sin\phi~\p,\]

which is equivalent to rotating the state by \(-\phi\) and performing an \(\x\)-homodyne measurement. If we have a multimode Gaussian state and we perform a homodyne measurement on one of the modes, the conditional state on the remaining modes stays Gaussian.

Heterodyne measurements

Whereas homodyne is a measurement of \(\x\), heterodyne can be seen as a simultaneous measurement of both \(\x\) and \(\p\). Because these operators do not commute, they cannot be simultaneously measured without some degree of uncertainty.

The output of a heterodyne measurement is usually phrased in terms of the Q function \(Q(\alpha)=\langle\alpha|\rho|\alpha\rangle\) (where \(\{|\alpha\rangle\}_{\alpha\in\mathbb{C}}\) are the coherent states). Like homodyne, heterodyne measurements preserve the Gaussian character of Gaussian states.

Photon counting

Photon counting is a complementary measurement method to the “-dyne” measurements, revealing the particle-like, rather than the wave-like, nature of qumodes. This measurement projects onto the number eigenstates \(\ket{n}\), returning non-negative integer values \(n\in\mathbb{N}\) [1].

Except for the outcome \(n=0\), a photon-counting measurement on a single mode of a multimode Gaussian state will cause the remaining modes to become non-Gaussian. Thus, photon-counting can be used as an ingredient for implementing non-Gaussian gates.

Footnotes

[1]A related process is photodetection, where a detector only resolves the vacuum state from non-vacuum states. This process has only two measurement projectors, namely \(\ketbra{0}{0}\) and \(\I - \ketbra{0}{0}\).