Controlled-NOT logic with nonresonant Josephson phase qubits

We establish theoretical bounds on qubit detuning for some of the previously proposed controlled-NOT (CNOT) logic gate implementations with weakly coupled Josephson phase qubits. It is found that in the two-step, $\sqrt{i\text{SWAP}}$-based case the value of the detuning during the entangling operations must not exceed $2g$, where $g$ is the characteristic coupling constant. In the single-step case we consider two practical, physically distinct implementations, in which one of the qubits is driven by a concurrent rf pulse of fixed frequency. We find that when the local drive is applied to the “reference” qubit (with which it is in resonance), the detuning should not exceed $g$. If the drive is applied to the “detuned” qubit, generation of the perfect CNOT gate is possible at any value of detuning provided that the amplitude of the pulse can be made arbitrarily large.

Figure 1

(Color online) Makhlin invariants $-G_1\left(\delta ,t\right)=-\left[G_2\left(\delta ,t\right)-1\right]/2$ as functions of qubit detuning $\delta$ and the gate time $t$ for the two-step control sequence of Sec. 3. CNOT class corresponds to $G_1\left(\delta ,t\right)=0$, $G_2\left(\delta ,t\right)=1$. Notice that in order to implement perfect controlled-NOT logic, qubit detuning must be restricted to $\left|\delta \right|\le 2g$, in accordance with Eqs. (24, 26).

Figure 2

Two-step CNOT implementation of entangling duration $t_{\left(2\right)}=\sqrt{2}\left(\pi /2g\right)$ for capacitively coupled Josephson phase qubits at maximally allowed detuning, $\delta =2g$. Panel (a): time dependence of the steering parameters $c_1\left(t\right)$ and $c_2\left(t\right)$. In the middle of the sequence, a fast $\pi$ pulse is applied to one of the qubits. Dashed curves represent the resonant case, $\delta =0$. Panel (b): the corresponding steering trajectory on the Weyl chamber. Here, $c_3=0$ at all times.

Figure 3

(Color online) Graph of $\left[-d^{1/4}\left(\Omega ,t\right)\right]$ as a function of entangling time $t$ and Rabi frequency $\Omega$ at different values of detuning $\delta$ for the single-step implementation of Sec. 4B1 in which a resonant local drive $\Omega X_1$ is applied to the reference qubit no. 1. Singularities at $\left(-d^{1/4}\right)=0$ correspond to various CNOT gates. Panels (a)–(c): as $\left|\delta \right|/g\to 1^{-}$, the two sets of CNOTs having $1<t/\left(\pi /2g\right)<3$ gradually approach each other while maintaining separation even at $\left|\delta \right|=g$. Panel (d): at larger values of detuning the graph flattens out and the distance function tends to its maximal constant value $d_{\text{max}}=\sqrt{5}$. In this regime $\left(\left|\delta \right|>g\right)$ perfect CNOT generation is no longer possible.

Figure 4

Single-step CNOT implementation of entangling duration $t_{\left(1\right)}=1.2753\left(\pi /2g\right)$ and Rabi frequency $\Omega =3.7781g$ at maximally allowed detuning $\delta =g$ for implementation of Sec. 4B1 in which the local drive $\Omega X_1$ is applied to the reference qubit no. 1. Panel (a): time dependence of the steering parameters $c_1\left(t\right)$ and $c_2\left(t\right)$. Panel (b): the corresponding steering trajectory on the Weyl chamber. Here, $c_3=0$ at all times. Dashed curves represent the resonant case, $\delta =0$.

Figure 5

Single-step implementation generating the gate closest to CNOT in terms of Eq. (2) at $\delta =1.5g$ for implementation of Sec. 4B1 in which the local drive $\Omega X_1$ is applied to the reference qubit no. 1. Panel (a): time dependence of the steering parameters $c_1\left(t\right)$ and $c_2\left(t\right)$. Closest class is reached at $t_{\left(1\right)}=1.0961\left(\pi /2g\right)$ and has $\Omega =3.7152g$. Panel (b): the corresponding steering trajectory on the Weyl chamber. Here, $c_3=0$ at all times. Dashed curves represent the resonant case, $\delta =0$.

Figure 6

(Color online) Graph of $\left[-d^{1/4}\left(\Omega ,t\right)\right]$ as a function of entangling time $t$ and Rabi frequency $\Omega$ at different values of detuning $\delta$ for the single-step implementation of Sec. 4B2, in which a nonresonant local drive $\Omega X_2$ is applied to the detuned qubit no. 2. The peaks at $\left(-d^{1/4}\right)=0$ correspond to various controls that generate the CNOT equivalence class. Panels (a)–(c): notice how a pair of CNOTs having $\Omega /g<\sqrt{15}$ and $1<t/\left(\pi /2g\right)<2$ gradually disappears as detuning increases. (This process repeats itself for other pairs of CNOTs at larger detuning.) Panel (d): minimal-time CNOTs $\left(t\approx \pi /2g\right)$ can be generated at any $\delta$ provided $\Omega$ is chosen to be sufficiently large.

Figure 7

Single-step CNOT implementation with $t_{\left(1\right)}=1.1794\left(\pi /2g\right)$, $\Omega =2.7131g$, at subcritical detuning $\delta =1.5g$ for implementation of Sec. 4B2 in which the local drive $\Omega X_2$ is applied to the detuned qubit no. 2. Panel (a): time dependence of the steering parameters $c_1\left(t\right)$ and $c_2\left(t\right)$. Panel (b): the corresponding steering trajectory on the Weyl chamber. Here, $c_3=0$ at all times. Dashed curves represent the resonant case, $\delta =0$.

Figure 8

Single-step implementation generating the gate closest to CNOT in the sense of Eq. (2) at $\delta =2.5g$ for implementation of Sec. 4B2 in which the local drive $\Omega X_2$ is applied to the detuned qubit no. 2. Panel (a): time dependence of the steering parameters $c_1\left(t\right)$ and $c_2\left(t\right)$. The gate has $t_{\left(1\right)}=1.0865\left(\pi /2g\right)$ and $\Omega =3.2368g$. Panel (b): the corresponding steering trajectory on the Weyl chamber. Here, $c_3=0$ at all times. Dashed curves represent the resonant case, $\delta =0$.