ON states as resource units for universal quantum computation with photonic architectures

Universal quantum computation using photonic systems requires gates the Hamiltonians of which are of order greater than quadratic in the quadrature operators. We first review previous proposals to implement such gates, where specific non-Gaussian states are used as resources in conjunction with entangling gates such as the continuous-variable versions of controlled-phase and controlled-not gates. We then propose ON states which are superpositions of the vacuum and the $N\mathrm{th}$ Fock state, for use as non-Gaussian resource states. We show that ON states can be used to implement the cubic and higher-order quadrature phase gates to first order in gate strength. There are several advantages to this method such as reduced number of superpositions in the resource state preparation and greater control over the final gate. We also introduce useful figures of merit to characterize gate performance. Utilizing a supply of on-demand resource states one can potentially scale up implementation to greater accuracy, by repeated application of the basic circuit.

Figure 1

Real (left) and imaginary (right) parts of the wave function (thick red line) of a squeezed displaced vacuum state $|\psi \rangle$ and its modulation (dashed line) by the action of the cubic phase gate.

Figure 2

Probability of the homodyne detection $P\left(q\right)$ for input Fock states and target cubic gate strength set to $\gamma =0.1$, where the plots from the top to the bottom correspond to Fock states in increasing order from $n=0$ to 5. We observe that as the Fock number increases the probability distribution gets flatter.

Figure 3

Probability of the homodyne detection $P\left(q\right)$ for input squeezed states and target cubic gate strength set to $\gamma =0.1$, where the plots from the top to the bottom correspond to squeezing value in increasing order from $0$ to $9.5\mathrm{dB}$. We observe that higher input squeezing leads to a damping of the probability distribution.

Figure 4

Probability of the homodyne detection $P\left(q\right)$ for input $x$-displaced vacuum states and target cubic gate strength set to $\gamma =0.1$, where the plots from right to left correspond to $x$ displacements ranging from $-1$ to 1.5. We find that input displacements lead to proportional shifts for the probability distribution.

Figure 5

GKP circuit with the 03 resource state given by $c_a\left(\right|0\rangle +a|3\rangle )$. The circuit implements a weak cubic phase gate up to an overall Gaussian noise factor ${\widehat{T}}_q={\widehat{A}}_{q}V\left(\gamma \right)$. The feed-forward (dashed lines) Gaussian gates are the quadratic phase gate $P$ and the displacement $Z$, both of which are conditioned on the homodyne measurement outcome ${\mathrm{\Pi }}_x=q$ and are therefore dynamic optical elements. The four coherent states $|z\rangle ,|u\rangle ,|v\rangle ,\text{and}|w\rangle$ are part of the displacement beams performed on one arm of the two-mode squeezed state as given in Eq. (36). Detector 1 corresponds to the measurement $\{{\mathrm{\Pi }}_1,11-{\mathrm{\Pi }}_1\}$ and detector 0 corresponds to the measurement $\{{\mathrm{\Pi }}_0,11-{\mathrm{\Pi }}_0\}$. Gate $S$ stands for a single-mode squeeze operator. The entangling gate in the primary circuit is the $C_X^{†}$ gate.

Figure 6

Gate fidelity $F_{\gamma },F_{\text{sq}},\text{and}{F}_n$ using, respectively, (a) coherent states, as a function of target cubic gate strength $\gamma$ (since the fidelity is displacement independent for a fixed $\gamma$); (b) squeezed states, for $\gamma =0.1$; and (c) Fock states, for $\gamma =0.1$.