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 infinitedimensional Hilbert space, offering a paradigm for quantum computation which is distinct from the qubit model. This continuousvariable 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, BoseEinstein 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 [58][59].
From qubits to qumodes¶
A highlevel 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 \(Q(\alpha)\), Photoncounting \(\ketbra{n}{n}\)  Paulibasis measurements \(\ketbra{0/1}{0/1}, \ketbra{\pm}{\pm}, \ketbra{\pm i}{\pm i}\) 
Note
Qubitbased computations can be embedded into the CV picture, e.g., by using the GottesmanKnillPreskill (GKP) embedding [60][61][62], 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 wellknown commutation relation \([\a,\ad]=\I\). It is also common to work with the quadrature operators
with \([\x,\p]=i \hbar\). These selfadjoint 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 informationcarrying 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:
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
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 sonamed 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
while (undisplaced) squeezed states only have even number states in their expansion:
Mixed states¶
Mixed Gaussian states are also important in the CV picture, for instance, the thermal state
which is parameterized via the mean photon number \(\nbar:=\tr{(\rho(\nbar)\hat{n})}\). Starting from this state, we can consider a mixedstatecreation process, similar to above, namely
Analogously to pure states, by applying quadraticorder 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
For convenience, we can classify unitaries by the degree of their generating Hamiltonians. We can build an Nmode 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 [63].
We focus on a universal gate set which contains the following two components:
 Gaussian gates
 Onemode and twomode 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.
 NonGaussian gates
 A singlemode gate which is degree 3 or higher, e.g., the cubic phase gate. These are equivalent to the nonClifford gates in the qubit model.
A number of fundamental CV gates are presented in the following table:
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 WeylHeisenberg 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 discretevariable 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 [64], making these gates sufficient for quadratic unitaries. The cubic phase gate is presented as an exemplary higherorder gate, but any other nonGaussian gate could also be used to achieve universality.
CV measurements¶
As with CV states and gates, we can distinguish between Gaussian and nonGaussian measurements. The Gaussian class consists of two (continuous) types: homodyne and heterodyne measurements, while the key nonGaussian measurement is photon counting.
Measurement  Measurement operator 
Measurement values 

Homodyne  \(\ket{\x_\phi}\bra{\x_\phi}\) 
\(q\in\mathbb{R}\) 
Heterodyne  \(\frac{1}{\sqrt{\pi}}\ket{\alpha}\bra{\alpha}\) 
\(\alpha\in\mathbb{C}\) 
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
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 particlelike, rather than the wavelike, nature of qumodes. This measurement projects onto the number eigenstates \(\ket{n}\), returning nonnegative integer values \(n\in\mathbb{N}\) [1].
Except for the outcome \(n=0\), a photoncounting measurement on a single mode of a multimode Gaussian state will cause the remaining modes to become nonGaussian. Thus, photoncounting can be used as an ingredient for implementing nonGaussian gates.
Footnotes
[1]  A related process is photodetection, where a detector only resolves the vacuum state from nonvacuum states. This process has only two measurement projectors, namely \(\ketbra{0}{0}\) and \(\I  \ketbra{0}{0}\). 