Come vengono implementate le porte in un computer quantistico a variabile continua?

Ho lavorato principalmente con i computer quantistici superconduttori. Non ho molta familiarità con i dettagli sperimentali dei computer quantistici fotonici che usano i fotoni per creare stati cluster a variabile continua come quello che la startup canadese Xanadu sta costruendo. Come vengono implementate le operazioni di gate in questi tipi di computer quantistici? E qual è la porta quantistica universale impostata in questo caso?

— Mark Fingerhuth
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Tim Ralph ha anche descritto una serie di porte in arxiv.org/abs/1103.6071

— M. Stern,

Prendere un mode semplice oscillatore armonico (SHO) in uno spazio (Fock) , dove è lo spazio di Hilbert di un SHO in modalità on $n$ $\mathcal F = \bigotimes_k\mathcal H_k$ $\mathcal H_k$ $k$ .

Questo dà al solito operatore di annientamento , che agisce su uno stato numerico come $a_k$ peree l'operatore creazione sul modocome , agendo su uno stato numero come $a_k\left|n\right> = \sqrt n\left|n-1\right>$ $n\geq 1$ $a_k\left|0\right> = 0$ $k$ $a_k^\dagger$ . $a_k^\dagger\left|n\right> = \sqrt{n+1}\left|n+1\right>$

L'Hamiltoniano della SHO è (in unità dove). $H = \omega\left(a_k^\dagger a_k+\frac 12\right)$ $\hbar = 1$

Possiamo quindi definire le quadrature

X_{k} = \frac{1}{\sqrt{2}} (a_{k} + a_{k}^{†})

$X_k = \frac{1}{\sqrt 2}\left(a_k + a_k^\dagger\right)$

P_{k} = - \frac{i}{\sqrt{2}} (a_{k} - a_{k}^{†})

$P_k = -\frac{i}{\sqrt 2}\left(a_k - a_k^\dagger\right)$

A

$A$

\dot{A} = i [H, A]

$\dot A = i\left[H, A\right]$

t

$t$

X : P \mapsto P - t

$X:P\mapsto P-t$

P : X \mapsto X + t

$P:X\mapsto X+t$

\frac{1}{2} (X^{2} + P^{2}) : X \mapsto \cos t X - \sin t P, P \mapsto \cos t P + \sin t X,

$\frac 12\left(X^2 + P^2\right): X\mapsto \cos t X - \sin t P,\, P\mapsto \cos t P + \sin t X,$ which is just the Hamiltonian of a SHO with

ω = 1

$\omega = 1$ and gives a phase shift.

\pm S = \pm \frac{1}{2} (X P + P X) : X \mapsto e^{\pm t} X, P \mapsto e^{\mp t} P,

$\pm S = \pm\frac 12\left(XP+PX\right): X\mapsto e^{\pm t}X,\, P\mapsto e^{\mp t}P,$ which is known as the squeezing operator, where

+ S (- S)

$+S\,\left(-S\right)$ squeezes

P (X)

$P\,\left(X\right)$ .

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

{(X^{2} + P^{2})}^{2}

$\left(X^2 + P^2\right)^2$ allows for any polynomial Hamiltonian to be created.

Finally, including the beamsplitter operation (on two modes $j$ and $k$ )

\pm B_{j k} = \pm (P_{j} X_{k} - X_{j} P_{k}) : A_{j} \mapsto \cos t A_{j} + \sin t A_{k}, A_{k} \mapsto \cos t A_{k} - \sin t A_{j}

$\pm B_{jk} = \pm\left(P_jX_k - X_jP_k\right): A_j\mapsto \cos tA_j + \sin tA_k,\, A_k\mapsto \cos tA_k - \sin tA_j$ for

A_{j} = X_{j}, P_{j}

$A_j = X_j, P_j$ and

A_{k} = X_{k}, P_{k}

$A_k = X_k, P_k$ , 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.

— Mithrandir24601
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