Figure 1

(a) Schematic circuit of an NTCP gate with qubit 1 simultaneously controlling $n$ target qubits $(2,3,\dots ,n+1)$. The NTCP gate is equivalent to $n$ two-qubit control phase (CP) gates, each having a shared control qubit (qubit 1) but a different target qubit (qubit $2,3,\dots ,$ or $n+1$). Here, $\mathrm{Z̃}$ represents a controlled-phase flip on each target qubit, with respect to the computational basis formed by the two eigenstates $|+\rangle$ and $|-\rangle$ of the Pauli operator ${\sigma }_x.$ If the control qubit 1 is in the state $|-\rangle$, then the state $|-\rangle$ at each $\mathrm{Z̃}$ is phase-flipped as $|-\rangle$ $\to -|-\rangle$ while the state $|+\rangle$ remains unchanged. However, if the control qubit 1 is in the state $|+\rangle$, nothing happens to either of the states $|+\rangle$ and $|-\rangle$ at each $\mathrm{Z̃}$. (b) The relationship between an $n$-target-qubit cnot gate and an NTCP gate. The circuit on the left side of (b) is equivalent to the circuit on the right side of (b). For the circuit on the left side, the symbol $\oplus$ represents a not gate on each target qubit, with respect to the computational basis formed by the two eigenstates $|0\rangle$ and $|1\rangle$ of the Pauli operator ${\sigma }_z$. If the control qubit 1 is in the state $|1\rangle$, then the state at $\oplus$ is bit flipped as $|1\rangle$ $\to |0\rangle$ and $|0\rangle$ $\to |1\rangle$. However, when the control qubit 1 is in the state $|0\rangle$, the state at $\oplus$ remains unchanged. On the other hand, for the circuit on the right side, the part enclosed in the (red) dashed-line box represents an NTCP gate. The element containing H corresponds to a Hadamard transformation described by $|0\rangle \to |+\rangle =(1/\sqrt{2})(|0\rangle +|1\rangle )$ and $|1\rangle \to |-\rangle =(1/\sqrt{2})(|0\rangle -|1\rangle ).$Reuse & Permissions

Figure 2

Illustration of different detunings between the cavity mode frequency ${\omega }_c$ and the qubit transition frequency ${\omega }_0$. Here, ${\omega }_0$, ${\omega }_c$, and $\omega$ are the qubit transition frequency, the cavity mode frequency, and the pulse frequency, respectively. In (a), the detuning is $\delta ={\omega }_0-{\omega }_c<0.$ In (b), the detuning is ${\delta }^{\prime }={\omega }_0-{\omega }_c>0.$ To implement an NTCP gate, we propose two alternative approaches (see Sec. 4), each one with three steps. The first step for either method requires a negative detuning ($\delta <0$), as shown in (a). For either method, the second step requires a positive detuning (${\delta }^{\prime }>0$), as shown in (b).Reuse & Permissions

Figure 3

Change of the qubit transition frequency ${\omega }_0$ (or the qubit level spacings) of each qubit during a three-step NTCP gate. The three figures on the left side correspond to the control qubit $1$ while the three figures on the right side correspond to each of the target qubits ($2,3,\dots ,n+1$). Figures (a) and (a${}^{\prime }$), figures (b) and (b${}^{\prime }$), and figures (c) and (c${}^{\prime }$) correspond to the operations of step (i), step (ii), and step (iii), respectively. In each figure, the two horizontal solid lines represent the qubit energy levels for the states $|0\rangle$ and $|1\rangle ;$ ${\omega }_0$ is the qubit $|0\rangle \leftrightarrow |1\rangle$ transition frequency; ${\omega }_c$ is the cavity mode frequency; $\omega$ is the pulse frequency; $\phi$ is the initial phase of the pulse; and $\Omega$, ${\Omega }^{\prime },$ ${\Omega }_1$, and ${\Omega }_r$ are the Rabi frequencies of various applied pulses. In addition, $\delta$ and ${\delta }^{\prime }$ are the small detunings of the cavity mode with the $|0\rangle \leftrightarrow |1\rangle$ transition, which are given by $\delta ={\omega }_0-{\omega }_c<0$ and ${\delta }^{\prime }={\omega }_0-{\omega }_c>0$; $\Delta ={\omega }_0-{\omega }_c$ represents the large detuning of the cavity mode with the $|0\rangle \leftrightarrow |1\rangle$ transition. Note that the cavity mode frequency ${\omega }_c$ is kept fixed during the entire process, but the qubit transition frequency ${\omega }_0$ is adjusted to achieve a different detuning $\delta ,{\delta }^{\prime }$, or $\Delta$ for each step.Reuse & Permissions

Figure 4

Change of the cavity mode frequency ${\omega }_c$ and the transition frequency ${\omega }_0$ (or the level spacings) of qubit $1$ during a three-step NTCP gate. This second method is an alternative way to implement the NTCP gate, which is different from the steps shown in Fig. 3. Note that the transition frequency ${\omega }_0$ for qubits $2,3,\dots ,n+1$ remains unchanged during this method. The three figures on the left side correspond to the control qubit $1,$ which (from top to bottom) are for the operations of step (i), step (ii), and step (iii), respectively. The three figures on the right side correspond to each of the qubits $2,3,\dots ,n+1$, which (from top to bottom) are, respectively, for the operations of step (i), step (ii), and step (iii). In each figure, the two horizontal solid lines represent the qubit levels $|0\rangle$ and $|1\rangle ;$ ${\omega }_0$ is the qubit transition frequency; ${\omega }_c$ is the cavity mode frequency; $\omega$ is the pulse frequency; $\phi$ is the initial phase of the pulse; and $\Omega$, ${\Omega }^{\prime },$ ${\Omega }_1$, and ${\Omega }_r$ are the Rabi frequencies of the pulses (for various steps). In addition, $\delta$ and ${\delta }^{\prime }$ are the small detunings of the cavity mode with the $|0\rangle \leftrightarrow |1\rangle$ transition, which are given by $\delta ={\omega }_0-{\omega }_c<0$ and ${\delta }^{\prime }={\omega }_0-{\omega }_c>0$; $\Delta ={\omega }_0-{\omega }_c$ represents the large detuning of the cavity mode with the $|0\rangle \leftrightarrow |1\rangle$ transition.Reuse & Permissions

Figure 5

(a) Schematic diagram of a superconducting charge qubit. Here, $E_{J0}$ is the Josephson coupling energy, $C_{J0}$ is the Josephson capacitance, $C_g$ is the gate capacitance, $V_g$ is the gate voltage, and $\Phi$ is the external magnetic flux applied to the SQUID loop through a control line (not shown). (b) $N$ superconducting qubits are placed into a (gray) quasi-one-dimensional transmission line resonator. Each qubit is placed at an antinode of the electric field, yielding a strong coupling between the qubit and the resonator mode. The two blue curves represent the standing-wave electric field, along the $y$ direction. A subset of qubits from the $N$ qubits, selected for the gate, are coupled to each other via the resonator mode, while the remaining qubits, which are not controlled by the gate, are decoupled from the resonator mode by setting their $\Phi ={\Phi }_0/2,V_g^{\mathrm{dc}}=e/C_g,$ and $V_g^{\mathrm{ac}}=0$ to have their free Hamiltonian equal to zero.Reuse & Permissions

Figure 6

Change of the qubit transition frequency ${\omega }_0$ and the ac gate-voltage frequency $\omega$ during a three-step NTCP gate with charge qubits coupled to a resonator. The three figures on the left side correspond to the charge control qubit $1,$ which (from top to bottom) are for the operations of step (i), step (ii), and step (iii), respectively. The three figures on the right side correspond to the charge target qubits ($2,3,\dots ,n+1$) (from top to bottom) for the operations of step (i), step (ii), and step (iii), respectively. In each figure, the two horizontal solid lines represent the qubit levels $|0\rangle$ and $|1\rangle$; ${\omega }_c$ is the resonator mode frequency; ${\omega }_0$ is the qubit transition frequency; $\omega$ is the frequency of the ac gate voltage $V_g^{\mathrm{ac}}$ (i.e., the pulse); $\Omega (V_0)$, ${\Omega }^{\prime }(V_0^{\prime }),$ ${\Omega }_1(V_0^1)$, or ${\Omega }_r(V_0^{\mathrm{r}})$ is a function of the amplitude $V_0,$ $V_0^{\prime },$ $V_0^1,$ or $V_0^r$ of the ac gate voltage, which can be adjusted by changing the ac gate-voltage amplitude; and each circle with a symbol $~$ represents an ac gate voltage. In (b), the ac gate voltage for qubit $1$ is set to zero (i.e., $V_g^{\mathrm{ac}}=0$). In addition, $\delta$ and ${\delta }^{\prime }$ are the small detunings of the resonator mode with the $|0\rangle \leftrightarrow |1\rangle$ transition, which are given by $\delta ={\omega }_0-{\omega }_c<0$ and ${\delta }^{\prime }={\omega }_0-{\omega }_c>0$ ; $\Delta ={\omega }_0-{\omega }_c$ represents the large detuning of the resonator mode with the $|0\rangle \leftrightarrow |1\rangle$ transition. Note that the resonator mode frequency ${\omega }_c$ is kept fixed during the entire operation, but the qubit transition frequency ${\omega }_0$ is adjusted to achieve a different detuning $\delta ,{\delta }^{\prime }$, or $\Delta$ for each step.Reuse & Permissions

Figure 7

Proposed setup for three qubits and a (gray) standing-wave quasi-one-dimensional coplanar waveguide cavity (not drawn to scale). Each qubit is placed at an antinode of the electric field. The two blue curves represent the standing-wave electric field, along the $y$ direction. $D$ is the distance between any two nearest SQUID loops, and $d$ is the linear dimension of each SQUID loop.Reuse & Permissions

Figure 8

Proposed setup for an NTCP gate with $n+1$ identical neutral atoms and a cavity. For simplicity, only five atoms are drawn here. Each atom can be either loaded into the cavity or moved out of the cavity by one-dimensional translating optical lattices [20, 45, 46]. The atom in the middle represents atom 1 (the control qubit), while the remaining atoms play the role of target qubits.Reuse & Permissions

Figure 9

Large detuning of the cavity mode with the transition between energy levels. Figure (a) corresponds to the case $L$, while figure (b) corresponds to case $S.$ The dots above level $|2\rangle$ in (a) and level $|3\rangle$ in (b) represent other energy levels.Reuse & Permissions