Photonic graph state generation from quantum dots and color centers for quantum communications

Highly entangled “graph” states of photons have applications in universal quantum computing and in quantum communications. In the latter context, they have been proposed as the key ingredient in the establishment of long-distance entanglement across quantum repeater networks. Recently, a general deterministic approach to generate repeater graph states from quantum emitters was given. However, a detailed protocol for the generation of such states from realistic systems is still needed in order to guide experiments. Here, we provide such explicit protocols for the generation of repeater graph states from two types of quantum emitters: NV centers in diamond and self-assembled quantum dots. A crucial element of our designs is an efficient controlled-$Z$ gate between the emitter and a nuclear spin, used as an ancilla qubit. Additionally, a fast protocol for using pairs of exchange-coupled quantum dots to produce repeater graph states is described. Our focus is on near-term experiments feasible with existing experimental capabilities.

Figure 1

Graph states. Each vertex represents a qubit, and edges represent entanglement between qubits. (a) Cluster state used for measurement-based quantum computation [19], and (b) all-photonic repeater state introduced in Ref. [17].

Figure 2

Quantum emitter level structure needed to produce graph states. Two ground states each separately couple to one excited state via circularly polarized light of opposite polarization.

Figure 3

Graph states and their construction. We use the notation $\left|\pm \right.〉=(\left|0\right.〉\pm \left|1\right.〉)/\sqrt{2}$, and remark that $\text{C}\text{Z}\left|\pm +\right.〉=(\left|0+\right.〉\pm \left|1-\right.〉)/\sqrt{2}=(\left|\pm 0\right.〉+\left|\mp 1\right.〉)/\sqrt{2}$. (a) A graph state and stabilizer $K_a=X_a{Z}_b{Z}_c{Z}_d$, with edges shown in zigzag pattern. (b) Generation of linear cluster states: the emitter is initialized in $\left|{\psi }_1\right.〉=\left|+\right.〉$; after one pumping cycle, the emitter is entangled with the emitted photon such that the total state is $\left|{\psi }_2\right.〉=\left(\left|0+\right.〉+\left|1-\right.〉\right)/\sqrt{2}$; after the second pumping cycle (including Hadamards), the second photon is entangled with both the first photon and the emitter so that the total state is $\left|{\psi }_3\right.〉=\left(\left|00+\right.〉+\left|01-\right.〉+\left|10+\right.〉-\left|11-\right.〉\right)/2$, and so on. (c) A ladder graph state, a simple two-dimensional cluster state. (d) Fully connected, or “complete,” graphs.

Figure 4

A joint measurement is performed on photons $a$ and $b$ by passing them through a polarizing beam splitter, which reflects only one mode (say horizontal) of the photons. When only a single photon $c$ emerges, a new graph is formed, with $c$ inheriting all the edges of $a$ and $b$.

Figure 5

(a) An emitter entangled with an ancilla is pumped twice, interspersed with Hadamard gates. (b) The emitter is detached by measuring in the $z$ direction [see Fig. 3 for simplified wave function]. (c) The ancilla is entangled with the emitter by a $\text{C}\text{Z}$ gate. (d) Process is repeated to create six arms. (e) “Local complementation” is performed on the ancilla and core photons creating (f) a completely connected subgraph. (g) The ancilla is measured, removing it from the graph, creating the final repeater graph state.

Figure 6

(a) A stacked double quantum dot with one electron in each dot. The two electron spins are coupled via exchange interaction $J$ and subject to a magnetic field applied orthogonally to the growth direction. (b) Free evolution under the exchange interaction can generate the $\text{C}\text{Z}$ gate necessary to produce RGSs. The magnetic field implements the single-spin Hadamard gates.

Figure 7

A modified RGS that is equivalent to the original RGS of [17]: the missing connections are local to each secondary node and are thus not required for the repeater protocol.

Figure 8

(a) Single-rung ladder state produced from two coupled QDs. Performing two $y$ measurements on the central photons produces (b) an “almost” RGS. The missing edge does not affect the functionality of the state for quantum repeaters.

Figure 9

(a) Two quantum dots (yellow and red) are pumped to produce a pair of entangled photons each. The yellow dot is pumped a third time, and a $\text{C}\text{Z}$ gate (red line) is applied between the dots. (b) Hadamards are performed on the last photon and yellow dot, realizing two local complementations: one on the quantum dot, and another on the last photon. (c) The yellow dot is alternatingly pumped and Hadamard-rotated five times to produce a linear cluster state, and the third (top) photon is measured in the $z$ basis to separate the “loop” into two arms; an additional sixth pumping, without Hadamard, is performed. [The Hadamard performed in step (d) can be performed at this point instead, because the $\text{C}\text{Z}$ gate does not affect the photon. The visualization, however, is simpler. Notice that the Hadamard on the quantum dot must be performed in the stated order.] The $\text{C}\text{Z}$ gate between dots is then applied a second (and final) time. (d) Hadamards are performed on the last photon and yellow dot, again creating two local complementations as in (b). (e) The yellow dot is pumped to produce a linear cluster of length two, and then disconnected (by either direct measurement or measurement of another emitted photon). The red dot is pumped once. (f) Hadamards are performed on the last photon and the red dot, again realizing two local complementations as in (b). (g) Local complementations (realized by wave plate operations) are performed on the two “corner” photons, and then measured in the $z$ basis. Simultaneously, the red dot is pumped to produce a linear cluster state of two photons, and then disconnected. (h) A local complementation is performed on the “core” photon (realized by local operations on the core photon and six attached interior photons) and then measured in the $z$ basis. The resulting RGS is only missing two edges, and can be used in the repeater protocol without any limitation.

Figure 10

(a) Energy-level diagram for NV centers in diamond. Hyperfine coupling $A$ is much smaller than the zero-field splitting $E_{\mathrm{ZFS}}$. Selection rules guarantee that decays to the $\left|1,\uparrow \right.〉$ or $\left|1,\downarrow \right.〉$ states have different polarizations. (b) A nearby ${}^{13}\mathrm{C}$ nuclear spin can be used as the ancilla qubit in the RGS generation protocol. Microwave driving (blue arrows) between the $S_z=0$ and $|S_z|=1$ states can be used to implement Hadamard gates and entangling $\text{C}\text{Z}$ gates.

Figure 11

NV-nuclear spin entangling $\text{C}\text{Z}$ gate implemented with right-circularly polarized sech pulse, $\mathrm{\Omega }\left(t\right)={\mathrm{\Omega }}_0\text{sech}\left[\sigma (t-t_0)\right]$. (a) Frequency-selective, resonant $2\pi$ pulse (${\mathrm{\Omega }}_0=\sigma$) induces a minus sign on the target transition $\left|0\downarrow \right.〉\leftrightarrow \left|1\downarrow \right.〉$ while approximately avoiding the competing transition $\left|0\uparrow \right.〉\leftrightarrow \left|1\uparrow \right.〉$. (b) Designed $4\pi$ pulse (${\mathrm{\Omega }}_0=2\sigma$) that provides a 20-fold speedup compared to (a) by driving both transitions and inducing a $-1$ phase factor to the target transition and a $+1$ phase factor to the competing transition (with which the pulse is resonant). The other logical states $\left|\overline{1}\downarrow \right.〉$ and $\left|\overline{1}\uparrow \right.〉$ are not affected by the pulse. Here, we have taken $A=50\mathrm{kHz}$, $E_{\mathrm{ZFS}}=2.88\mathrm{GHz}$, $\sigma =1.9\mathrm{kHz}$ (a) and $\sigma =28.9\mathrm{kHz}$ (b), and gate durations $2t_0=1.1\mathrm{ms}$ (a) and $2t_0=58\mu \mathrm{s}$ (b), respectively.

Figure 12

Fastest NV-nuclear spin $\text{C}\text{Z}$ gate, obtained by setting the pulse frequency halfway between the $\left|0,\downarrow \right.〉\leftrightarrow \left|1,\downarrow \right.〉$ and $\left|0,\uparrow \right.〉\leftrightarrow \left|1,\uparrow \right.〉$ transitions and using a right-circularly polarized $4\pi$ sech pulse. The other logical states $\left|\overline{1}\downarrow \right.〉$ and $\left|\overline{1}\uparrow \right.〉$ are not affected by the pulse. Here, we have taken $A=50\mathrm{kHz}$, $E_{\mathrm{ZFS}}=2.88\mathrm{GHz}$, $\sigma =38.7\mathrm{kHz}$, and gate time $2t_0=30\mu \mathrm{s}$.

Figure 13

NV-nuclear spin $\text{C}\text{Z}$ gates implemented with linearly polarized microwave sech pulses, achieving $99.9\%$ fidelity. (a) Gate produced by frequency-selective $4\pi$ sech pulse that induces a net minus sign between the transitions $\left|0\downarrow \right.〉\leftrightarrow \left|1\downarrow \right.〉$ and $\left|0\uparrow \right.〉\leftrightarrow \left|\overline{1}\uparrow \right.〉$ while approximately avoiding the competing transitions $\left|0\uparrow \right.〉\leftrightarrow \left|1\uparrow \right.〉$ and $\left|0\downarrow \right.〉\leftrightarrow \left|\overline{1}\downarrow \right.〉$. (b) Gate produced by numerically identified $4\pi$ sech pulse that provides a 40-fold speedup compared to (a) by driving all transitions. Here, we have taken $A=50\mathrm{kHz}$, $E_{\mathrm{ZFS}}=2.88\mathrm{GHz}$, and the pulse bandwidth $\sigma$ is $1.8\mathrm{kHz}$ and $12.8\mathrm{kHz}$ for (a) and (b), respectively. Pulse (b) drives at the detuned frequency $\mathrm{\Delta }=-5.2\mathrm{kHz}$ for a total time of $2t_0=84\mu \mathrm{s}$. All fidelity calculations assume the hyperbolic secant is truncated around its maximum, but, unlike the other gates, this gate is sensitive to increased driving duration; if the drive is applied for longer, the fidelity drops.