How are gates implemented in a continuous-variable quantum computer?

Question

I've mostly worked with superconducting quantum computers I am not really familiar with the experimental details of photonic quantum computers that use photons to create continuous-variable cluster states such as the one that the Canadian startup Xanadu is building. How are gate operations implemented in these types of quantum computers? And what is the universal quantum gate set in this case?

Answer 1

Taking an $n$-mode simple harmonic oscillator (SHO) in a (Fock) space $\mathcal F = \bigotimes_k\mathcal H_k$, where $\mathcal H_k$ is the Hilbert space of a SHO on mode $k$.

This gives the usual annihilation operator $a_k$, which act on a number state as $a_k\left|n\right> = \sqrt n\left|n-1\right>$ for $n\geq 1$ and $a_k\left|0\right> = 0$ and the creation operator on mode $k$ as $a_k^\dagger$, acting on a number state as $a_k^\dagger\left|n\right> = \sqrt{n+1}\left|n+1\right>$.

The Hamiltonian of the SHO is $H = \omega\left(a_k^\dagger a_k+\frac 12\right)$ (in units where $\hbar = 1$).

We can then define the quadratures $$X_k = \frac{1}{\sqrt 2}\left(a_k + a_k^\dagger\right)$$ $$P_k = -\frac{i}{\sqrt 2}\left(a_k - a_k^\dagger\right)$$ which are observables. At this point there are various operations (Hamiltonians) that can be performed. The effect of such an operation on the quadratures can be found by using the time evolution of an operator $A$ as $\dot A = i\left[H, A\right]$. Applying these for time $t$ gives: $$X:P\mapsto P-t$$ $$P:X\mapsto X+t$$ $$\frac 12\left(X^2 + P^2\right): X\mapsto \cos t X - \sin t P,\, P\mapsto \cos t P + \sin t X,$$ which is just the Hamiltonian of a SHO with $\omega = 1$ and gives a phase shift. $$\pm S = \pm\frac 12\left(XP+PX\right): X\mapsto e^{\pm t}X,\, P\mapsto e^{\mp t}P,$$ which is known as the squeezing operator, where $+S\,\left(-S\right)$ squeezes $P\,\left(X\right)$.

Any Hamiltonian of the form $aX+bP+c$ can be built by applying $X$ and $P$. Adding $S$ and $H$ allows for any quadratic Hamiltonian to be built. Further adding the (nonlinear) Kerr Hamiltonian $$\left(X^2 + P^2\right)^2$$ allows for any polynomial Hamiltonian to be created.

Finally, including the beamsplitter operation (on two modes $j$ and $k$) $$\pm B_{jk} = \pm\left(P_jX_k - X_jP_k\right): A_j\mapsto \cos tA_j + \sin tA_k,\, A_k\mapsto \cos tA_k - \sin tA_j$$ for $A_j = X_j, P_j$ and $A_k = X_k, P_k$, which acts as a beamsplitter on the two modes.

The above operations form the universal gate-set for continuous variable quantum computing. More details can be found in e.g. here

To implement these unitaries:

Applying these operations is generally hinted at in the name: Coupling a current is acting as the displacement operator $D\left(\alpha\left(t\right)\right)$ where, for an electric field $\varepsilon$ and current $j$, $\alpha\left(t\right) = i\int_{t_0}^t\int j\left(r, t'\right)\cdot\varepsilon e^{-i\left(k\cdot r - w_kt'\right)} dr\, dt'$. The displacement operator shifts $X$ by the real part of $\alpha$ and $P$ by the imaginary part of $\alpha$.

A phase shift can be applied by simply letting the system evolve by itself, as the system is a harmonic oscillator. It can also be performed by using a physical phase shifter.

Squeezing is the hard bit and is something that needs to experimentally be improved. Such methods can be found in e.g. here and here is one experiment using a limited amount of squeezed light. One possible way of squeezing is using a Kerr $\left(\chi^{\left(3\right)}\right)$ nonlinearity.

This same nonlinearity also allows for the Kerr Hamiltonian to be implemented.

The Beamsplitter operation is, unsurprisingly, performed using a beamsplitter.