Figure 1

(Color online) Two schemes to couple different registers. (a) Deterministic state transfer from one register to the other [21]. (b) Probabilistic entanglement generation, heralded by distinct detector click patterns [3, 31, 32]. In the text, we argue that probabilistic, heralded entanglement generation is sufficient for deterministic distributed quantum computation.Reuse & Permissions

Figure 2

(Color online) Distributed quantum computer. (a) Illustration of distributed quantum computer based on many quantum registers. Each register has five physical qubits, including one communication qubit $\left(c\right)$, one storage qubit $\left(s\right)$, and three auxiliary qubits $\left(a_{1,2,3}\right)$. Local operations for qubits within the same register have high fidelity. Entanglement between nonlocal registers can be generated probabilistically [3, 31, 32]. Devices of optical microeletromechanical systems (MEMSs) [38] can efficiently route photons and couple arbitrary pair of registers. The detector array can simultaneously generate entanglement for many pairs of registers. (b) An ion trap coupled to a cavity also provides a promising candidate for distributed quantum computation. A single ion is resonantly coupled to the cavity and serves as the communication qubit, while the others can be storage or auxiliary qubits. (c) Nitrogen-vacancy (NV) defect center in a photonic crystal microcavity. The inset shows the atomic structure of the NV center [10], which forms a quantum register. The electronic spin localized at the vacancy is optically accessible (measurement and initialization) and can play the role of the communication qubit. The nuclear spins from proximal ${}^{13}\mathrm{C}$ atoms constitute the storage and auxiliary qubits, which are coherently controlled via hyperfine interaction and rf pulses.Reuse & Permissions

Figure 3

(Color online) Quantum circuit for a nonlocal CNOT gate between two registers $R^1$ and $R^2$. The circuit starts with the Bell state ${\mid {\Phi }^{+}⟩}_{c^1,c^2}=(\mid 00⟩+\mid 11⟩)∕\sqrt{2}$ for the communication qubits $c^1$ and $c^2$ (the left blue box). After local unitary operations within each register (the middle orange boxes), qubits $c^1$ and $c^2$ are projectively measured in the $Z$ and $X$ bases, respectively (the right green box). According to Eq. (1), up to some local unitary gates (the right green box), this circuit implements the nonlocal CNOT gate to qubits $s^1$ and $s^2$ from two quantum registers.Reuse & Permissions

Figure 4

(Color online) General circuit for a nonlocal controlled-U gate [27], with the storage qubit $S^1$ as control and the storage qubit $S^2$ as target. One Bell pair is consumed for this operation.Reuse & Permissions

Figure 5

The robust measurement scheme based on repeated quantum nondemolition measurements and majority vote. Each QND measurement consists of initializing, coupling, and measuring the communication qubit. The QND measurements are repeated $2m+1$ times using the same communication qubit.Reuse & Permissions

Figure 6

(Color online) Two-level entanglement pumping between registers $R^i$ and $R^j$ [circled by rounded rectangles in (a)]. (a),(b) Generate and store one unpurified Bell pair. (c) Generate another unpurified Bell pair to pump (purify) the previously stored pair. (d) If the purification is successful, we obtain a purified Bell pair (level-1 pair) with higher fidelity; otherwise, we discard the stored Bell pair and start the entire pumping process from the beginning. (e),(f) The second level of entanglement pumping uses previously purified pairs to purify a stored Bell pair, to obtain a Bell pair with higher fidelity (level-2 pair).Reuse & Permissions

Figure 7

(Color online) Bit-phase two-level entanglement pumping scheme to create high-fidelity entangled pairs between two registers $R^i$ and $R^j$. (a) Circuit for the first-level pumping to purify bit errors, corresponding to Fig. 6c. (b) Circuit for the second-level pumping to purify phase errors, corresponding to Fig. 6g. The arrows indicate the time direction for each register. Robust measurements are used here. If the two outcomes are the same, it is a successful attempt at pumping; otherwise, we generate new pairs and restart the pumping operation from the beginning.Reuse & Permissions

Figure 8

(Color online) Entanglement fidelity $F$ as a function of the number of successful pumping steps. $n_b$ and $n_p$ are the number of pumping steps used to purify bit errors and phase errors, respectively. We assume fixed measurement and local two-qubit gate error rates ${\epsilon }_M=p_L={10}^{-4}$. (a) For bit-flip error ${\rho }_{\text{dephasing}}=\mathrm{diag}[F,1-F,0,0]$ in the Bell basis $\{\mid {\Phi }^{+}⟩,\mid {\Phi }^{-}⟩,\mid {\Psi }^{+}⟩,\mid {\Psi }^{-}⟩\}$, with $F=0.95$, one-level entanglement pumping is sufficient. High fidelity of $F_{\mathit{fin}}=99.98\%$ can be achieved by $n_p=3$. (b) For depolarizing error ${\rho }_{\text{depolarizing}}=\mathrm{diag}[F,\frac{1-F}{3},\frac{1-F}{3},\frac{1-F}{3}]$, with $F=0.95$, two-level entanglement pumping is needed (see text for more details). The first level pumping only purifies the bit-error, but accumulates the phase error at the same time, and therefore the (thin blue) fidelity curve for the first level pumping drops for $n_b>1$. The second level (thick red curve) uses the purified level-1 pair $(n_b=1)$ to pump another stored pair. High fidelity of $F_{\mathit{fin}}=99.97\%$ can be achieved by $n_p=3$. Note that we consider the full density matrix for all numerical calculation of entanglement fidelities [57], with the error models given in Eqs. (2, 3, 4).Reuse & Permissions

Figure 9

(Color online) The contours of the infidelity ${\epsilon }_{E,\mathrm{infid}}^{(n_b,n_p)}$ as a function of $n_b$ and $n_p$ for depolarizing error. We use ${\log }_{10}{\epsilon }_{E,\mathrm{infid}}^{(n_b,n_p)}$ to label the contours. The other parameters are $F=0.90$ (left) and $F=0.95$ (right) and ${\tilde{\epsilon }}_M=p_L={10}^{-4}$ (up) and ${\tilde{\epsilon }}_M=p_L={10}^{-6}$ (down). With optimal choice of $(n_b^{*},n_p^{*})$, the minimal infidelity is comparable to the corresponding value of $p_L$.Reuse & Permissions

Figure 10

(Color online) Markov chain model for one-level entanglement pumping. We use 0 to denote the initial state with no Bell pairs, 1 for the state with one stored unpurified pair, $(j+1)$ for the state with one purified pair surviving $j$ steps of pumping, and ∗ for the final state with the purified pair surviving $n$ steps of pumping. The (success) transition probability from state $j$ to state $j+1$ is $q_j$, while the (failure) transition probability from state $j$ to state 0 is $1-q_j$, for $j=0,1,\dots ,n$. Here $q_0\equiv 1$ and $q_{j⩾1}$ can be calculated according to the density matrix of the purified Bell pair surviving $j$ steps of pumping [36]. The final state is self-trapped and goes back to itself with unit probability. Each transition attempt consumes one unpurified Bell pair.Reuse & Permissions

Figure 11

(Color online) Markov chain model for two-level entanglement pumping. The required pumping steps are $n_b$ and $n_p$ for the two levels, respectively. We use 0, 0 to denote the initial state with no Bell pairs, $0,j+1$ for the state with one purified pair surviving $j$ steps of pumping at the first level, $k+1,j+1$ for the state with one purified pair surviving $k$ steps of pumping at the second level and one purified pair surviving $j$ steps of pumping at the first level, $k+1,\ast$ for the state with one purified pair surviving $k$ steps of pumping at the second level and one purified pair surviving $n_b$ steps of pumping at the first level, and ∗, 0 for the final state with one purified pair surviving $n_p$ steps of pumping at the second level. For the first-level pumping, the (success) transition probability from state $k,j$ to state $k,j+1$ is $q_j$, while the (failure) transition probability from state $k,j$ to state $k,0$ is $1-q_j$, for $j=0,1,\dots ,n_b$. For the second-level pumping, the (success) transition probability from state $k,\ast$ to state $k+1,0$ is $Q_k$, while the (failure) transition probability from state $k,\ast$ to state 0, 0 is $1-Q_k$, for $k=0,1,\dots ,n_p$. The final state is self-trapped and goes back to itself with unit probability.Reuse & Permissions

Figure 12

(Color online) Failure probability ${\epsilon }_{E,\mathrm{fail}}$ as a function of $N_{\mathrm{tot}}$. We assume a depolarizing error with $F=0.95$, $p_L={\tilde{\epsilon }}_M={10}^{-4}$. We choose $(n_b,n_p)=(2,3)$ for the lower curve and (3, 4) for the upper curve. For large $N_{\mathrm{tot}}$, the failure probability ${\epsilon }_{E,\mathrm{fail}}$ decreases exponentially with $N_{\mathrm{tot}}$.Reuse & Permissions

Figure 13

(Color online) The optimized total error probability ${\epsilon }_E$ [Eq. (21)] (upper plots) and the optimized average infidelity ${\delta }_E$ [Eq. (22)] (lower plots) as a function $N_{\mathrm{tot}}$. The error probability for the local coupling gates is $p_L={10}^{-4}$ (left plots) and $p_L={10}^{-6}$ (right plots). One-level pumping is used for dephasing error (thin blue curves); two-level pumping is used for depolarizing error (thick red curves). The other parameters $F=0.95$, $p_I=p_M=5\%$ are the same for all plots. Both ${\epsilon }_E$ and ${\delta }_E$ saturate for large $N_{\mathrm{tot}}$. In each plot, we also show the initial infidelity $1-F$ (upper blue dashed lines) and the local error probability $p_L$ (lower violet dashed lines).Reuse & Permissions

Figure 14

(Color online) Contours of the total error probability ${\epsilon }_E$ (or average infidelity ${\delta }_E$) after purification (left), the total number of unpurified Bell pairs, $N_{\mathrm{tot}}$, associated with ${\epsilon }_E$ [Eq. (26)] (middle), and $N_{\mathrm{tot}}$, associated with ${\delta }_E$ [Eq. (27)] (right). The contours are drawn with respect to the imperfection parameters $p_L$ (horizontal axis) and $F$ (vertical axis). Two-level pumping (up) is used for depolarizing error and one-level pumping (down) for dephasing error. $p_I=p_M=5\%$ is assumed.Reuse & Permissions

Figure 15

(Color online) Markov chain model for one-level entanglement pumping with non-post-selective (NPS) scheme. The key difference from the previous Markov chain model with the post-selective (PS) pumping scheme (see Fig. 10) is that here the transition for unsuccessful pumping reduces the chain label (score) by 1, while in the previous model the transition for unsuccessful pumping goes back to state 0 (restart of the entire pumping scheme).Reuse & Permissions

Figure 16

(Color online) Markov chain model for two-level entanglement pumping with NPS scheme. We still use the PS entanglement pumping scheme at the first level to have minimal accumulation of phase errors. Only at the second level, does the NPS scheme work more efficiently than the original PS scheme (see Fig. 11).Reuse & Permissions

Figure 17

(Color online) The contours for the ratio between $N_{\mathrm{tot}}$ associated with the post-selective (PS) scheme and $N_{\mathrm{tot}}$ associated with the non-post-selective (NPS) scheme as a function of $p_L$ and $F$. For both schemes, we use the same formula [Eq. (27)], but different Markov chain models (Fig. 10, 15). The improvement from the NPS scheme becomes significant (more than a factor of 3), for $F<0.9$ and $p_L<{10}^{-4}$.Reuse & Permissions

Figure 18

(Color online) Plots of clock cycle time $t_C$ and effective error probability $\gamma$ for two-level pumping against depolarizing errors. Upper plots: contours of the normalized clock cycle time $t_C∕t_L$ (upper) as a function of $p_L$ and $\frac{\tau }{t_{L}C}$ (the normalized effective radiative lifetime). Lower plots: the effective error probability $\gamma$ as a function of $p_L$. We assume $1-F=p_I=p_M=5\%$ (left) and 1% (right) and $\eta =0.2$.Reuse & Permissions

Figure 19

(Color online) Plots of clock cycle time $t_C$ and effective error probability $\gamma$ for one-level pumping against dephasing error. Upper plots: contours of the normalized clock cycle time $t_C∕t_L$ (upper) as a function of $p_L$ and $\frac{\tau }{t_{L}C}$ (the normalized effective radiative lifetime). Lower plots: the effective error probability $\gamma$ as a function of $p_L$. We assume $1-F=p_I=p_M=5\%$ (left) and 1% (right) and $\eta =0.2$.Reuse & Permissions

Figure 20

(Color online) Contours for $t_{\mathit{mem}}∕t_L$ (shaded contours) and $\gamma$ (red contours) as a function of $p_L$ and $1-F$. The label for the contour values are in logarithmic scale with base 10. We consider both situations of (a) depolarizing error and (b) dephasing error. The other parameters are $p_M=p_I=1-F$ and $\tau ∕C=0$.Reuse & Permissions

Figure 21

(Color online) Circuits for fault-tolerant preparation of the GHZ state. (a) Conventional circuit uses eight quantum registers to generate a four-qubit GHZ state. The state ∣+⟩ is $(\mid 0⟩+\mid 1⟩)∕\sqrt{2}$. (b) New circuit requires only four registers by using partial Bell measurement (PBM) between registers, detailed in (c).Reuse & Permissions

Figure 22

Circuit for fault-tolerant preparation of the eight-qubit GHZ state with only eight quantum registers.Reuse & Permissions

Figure 23

(Color online) The contours of the infidelity ${\epsilon }_{E,\mathrm{infid}}^{(n_b,n_p)}$ as a function of $n_b$ and $n_p$. We use ${\log }_{10}{\epsilon }_{E,\mathrm{infid}}^{(n_b,n_p)}$ to label the contours. We assume a depolarizing error with initial fidelity $F=0.95$ and ${\tilde{\epsilon }}_M=p_L=0$. The final infidelity can be arbitrarily small for sufficiently large $n_b$ and $n_p$. This indicates that the bit-phase two-level entanglement pumping scheme can create pairs with arbitrarily high fidelity.Reuse & Permissions