Encoding qubits into oscillators with atomic ensembles and squeezed light

The Gottesman-Kitaev-Preskill (GKP) encoding of a qubit within an oscillator provides a number of advantages when used in a fault-tolerant architecture for quantum computing, most notably that Gaussian operations suffice to implement all single- and two-qubit Clifford gates. The main drawback of the encoding is that the logical states themselves are challenging to produce. Here we present a method for generating optical GKP-encoded qubits by coupling an atomic ensemble to a squeezed state of light. Particular outcomes of a subsequent spin measurement of the ensemble herald successful generation of the resource state in the optical mode. We analyze the method in terms of the resources required (total spin and amount of squeezing) and the probability of success. We propose a physical implementation using a Faraday-based quantum nondemolition interaction.

Figure 1

Example wave functions [Eqs. (10) and (11)] for low-quality (8-dB) and high-quality (15-dB) target states. These states all have equal spacing in momentum and position $\left(g=\sqrt{\pi }\right)$ and symmetric embedded error, allowing us to express their quality using a single value—the squeezing of the state, as defined in Eq. (12). This parameter determines the width of the spikes and envelopes through Eq. (13). The red and cyan (darker and lighter) in ${\psi }_{-}\left(q\right)$ indicate a relative phase of $\pm 1$, respectively, alternating between adjacent spikes, while the phase is uniform for ${\psi }_{+}\left(q\right)$. Notice that when the position-space envelope (thin, gray) is not centered (i.e., $q_0\ne 0)$, the momentum-space representation has a nonconstant phase, indicated by color (grayscale variation) as shown in the legend on the right. This variation is nontrivial only where the magnitude of the wave function is negligible and is therefore largely immaterial to the error-correction properties of the state because the spikes themselves, i.e., the high-amplitude parts of the state, all have the appropriate location and phase in both position and momentum.

Figure 2

Circuit that creates a resource state. The optical mode is prepared in the vacuum state ${\left|0\right.〉}_O$, and an atomic ensemble, with total collective spin $J$, is prepared in a spin-coherent state polarized along $x,{\left|J,m_x=J\right.〉}_A$. The optical mode is first squeezed via the standard squeezing operator $\widehat{S}\left(r\right)$ and is then coupled to the collective spin via a controlled position shift ${\widehat{D}}_c\left(g\right)$. The spin is then projectively measured yielding outcome $x$, conditionally preparing the optical mode in state ${\left|{\psi }_d^x\right.〉}_O$. When certain conditions are met (see Sec. 4), this state may serve as a resource state, as defined at the end of Sec. 2c.

Figure 3

Position- and momentum-space wave functions, Eqs. (29), for the conditional optical state, ${\left|{\psi }_d^x\right.〉}_O$, with total angular momentum $J=4$ and $\frac{9}{2}$, for each of the $2J+1$ measurement outcomes $x$. Phase is indicated by color (grayscale) as in Fig. 1. When $x=\pm J$, the resulting state approximates the target state ${\left|\pm \right.〉}_{\text{L}}$ [after a position displacement, Eq. (41), in the case of half-integer $J]$. To see this, compare ${\psi }_d^{\pm J}\left(q\right)$ with the position-space wave functions ${\psi }_{\pm }\left(q\right)$ shown in the first row (for integer $J)$ and second row (for half-integer $J)$ of Fig. 1. We choose the initial squeezing of the optical mode and the coupling strength $g$ to match the desired target state identified in Sec. 5. Specifically, $g=\sqrt{\pi }$ and ${\sigma }^2=2/\pi J$, corresponding to squeezing of 8.0 dB $\left(J=4\right)$ and 8.5 dB $\left(J=\frac{9}{2}\right)$, where $\left(\text{\#dB}\right)=10{log}_{10}{e}^{2r}$. Also notice that the momentum-space wave function ${\tilde{\psi }}_d^0\left(p\right)$, which only exists in the integer-$J$ case, represents the logical state ${\left|0\right.〉}_{\text{L}}$ after a momentum displacement is applied. (Compare with ${\psi }_{+}\left(q\right)$ in the second row of Fig. 1, and see Sec. 4 for more details.) [Note for readers viewing in grayscale: The relevant plots are the ones in the left-hand column of both tables, which are purely real, with red (darker) and cyan (lighter) indicating a relative phase of $\pm 1$, respectively. Also relevant is the indicated ${\left|0\right.〉}_{\text{L}}$ in the right-hand column of the $J=4$ table. It has a uniform phase of $+1$. The other plots are presented merely for completeness and can be viewed in color online.]

Figure 4

(a) Required total spin $J$ [Eq. (40)] and (b) success probability ${\mathcal{P}}_s$ [Eq. (52)] for generating a resource state, i.e., an approximate GKP state satisfying the three conditions identified at the beginning of Sec. 4, with a given amount of squeezing. The resource state's squeezing level is defined in Eq. (12) and evaluated in Eqs. (39).

Figure 5

The probability $\mathcal{P}\left(x\right)$ [Eq. (44)] of a particular spin-measurement outcome $x$. The plot uses $J=50$ and a symmetric encoding with balanced embedded error (see the beginning of Sec. 5). Of the $2J+1$ possible outcomes, two of them, namely, $x=\pm J$, which also just happen to be the most probable, yield useful resource states. (In the case of integer $J$, the $x=0$ case also produces a useful state, albeit with slightly different properties; see Sec. 4.)