Design and stability of NOR and NAND logic gates constructed with three quantum states

By encoding the digital input of a classical logic gate on the Hamiltonian of a quantum system and driving the logic operation by preparing it in the same nonstationary state whatever the input, NOR and NAND classical logic gates are designed with a minimum of three quantum states. The outputs of one gate are obtained either by measuring the distance between the $\rho \left(t\right)$ periodic quantum trajectory and the output target state ${\rho }_b$, or by measuring the secular frequency of the $\rho \left(t\right)$ almost periodic trajectory in the ${\rho }_b$ direction. A comparison of the stability to noise between the two approaches demonstrates that the frequency approach is more immune to random fluctuations in the Hamiltonian than a distance control approach, opening the way to determine the logic output using a tunneling current passing through the gate.

Figure 1

The 3-state system chosen to embody the NOR and the NAND gates. The two parameters $\alpha$ and $\beta$ are used to encode the input information, $e$ and $\mu$ being the structural parameters of the gate. The system is initially prepared in the ${\rho }_a=\mid {\varphi }_a⟩⟨{\varphi }_a\mid$ state and the output is either in the distance between $\rho \left(t\right)=\mid \psi \left(t\right)⟩⟨\psi \left(t\right)\mid$ and the target state ${\rho }_b=\mid {\varphi }_b⟩⟨{\varphi }_b\mid$ or in the secular oscillation frequency of $\rho \left(t\right)$ in the direction of ${\rho }_b$.

Figure 2

(Color online) The $\rho \left(t\right)$ trajectories of the NOR gate plotted on the reduced Bloch sphere. ${\rho }_a$, ${\rho }_b$, and ${\rho }_m$ are, respectively, the initial, the target, and the measured states. Only the {$\alpha =0$, $\beta =0$} input configuration ${\rho }_m$ reaches exactly ${\rho }_b$. In the three other cases ${\rho }_m$ avoid ${\rho }_b$ and as confirmed by the detailed ${\rho }_{01}\left(t\right)$ trajectory (see Fig. 1 inset), stays in this configuration near ${\rho }_a$. For those trajectories the structural parameters of the Hamiltonian are $\mu =e=0.0336717\mathrm{eV}$, while $\alpha$ and $\beta$ take the value 0 or $1\mathrm{eV}$, giving a $\tau =2\pi \times {10}^{-14}\mathrm{s}$ period for the two ${\rho }_{00}\left(t\right)$ and ${\rho }_{11}\left(t\right)$ periodic trajectories.

Figure 3

(Color online) The $\rho \left(t\right)$ trajectories [grey (red)] and the trajectories driven by the effective Hamiltonian [black (blue)], both plotted, for one pseudoperiod, in each input configuration of the NAND gate, on the reduced Bloch sphere. ${\rho }_a$ and ${\rho }_b$ are, respectively, the initial and the target states. The secular frequencies of the {$\alpha =0$, $\beta =0$}, {$\alpha =0$, $\beta =1$}, and {$\alpha =1,\beta =0$} input configurations are equal: $w_{00}=w_{01}=w_{10}=\frac{1}{4}\mathrm{PHz}$. For the {$\alpha =1$, $\beta =1$} input configuration, even if the dominant frequency is $w_{11}=0\mathrm{Hz}$, the two other frequencies generate a residual oscillation of weak amplitude. Nevertheless, the effective trajectory stays at ${\rho }_a$ and would reach ${\rho }_b$ only for an infinite time (see Fig. 1 inset). The structural parameters of the Hamiltonian are $\mu =0.5184854\mathrm{eV}$ and $e=1.4102094\mathrm{eV}$, while $\alpha$ and $\beta$ take the value 0 or $1\mathrm{eV}$. Notice that ${\rho }_{01}\left(t\right)$ and ${\rho }_{10}\left(t\right)$ never pass by ${\rho }_b$. But what is important here is not the distance between $\rho \left(t\right)$ and the target state but how long $\rho \left(t\right)$ will take to be at a minimum distance from ${\rho }_b$.

Figure 4

Analytical fidelities of the NAND and NOR gates using a commutative noise source as a function of the scaling parameter $\eta$ (eV). The NAND rate controlled gate is less sensitive to this noise source than the NOR distance controlled gate. For $\eta =0.02\mathrm{eV}$ the fidelity of the NOR distance controlled gate reaches its minimum whereas the fidelity of the NAND gate remains high enough reaching its minimum for a much higher value of $\eta$.

Figure 5

Fidelity of the NOR (1) and NAND (2) gates as a function of the scaling parameters $a$ and $b$ (eV) of two random Hamiltonians, one diagonal and the other off diagonal. The scaling parameters, $a$ and $b$, of the two noise sources goes from $-0.01\mathrm{eV}$ up to $0.5\mathrm{eV}$. At each $(a,b)$ point, the fidelity of the system has been computed over 1000 different random Hamiltonians. The fidelity of the NOR distance controlled gate in the off-diagonal perturbation direction decreases extremely quickly reaching almost zero for an amplitude of $0.05\mathrm{eV}$. In general the distance controlled NOR gate is very sensitive to the noise. The fidelity is acceptable in only a small region around the unperturbed Hamiltonian. On the contrary, the rate controlled NAND gate presents a large region where the fidelity is acceptable. In the direction where the fidelity is the most sensitive, that is the diagonal perturbation part, the fidelity decreases slowly to reach 0.7 for a noise amplitude of $\pm 0.1\mathrm{eV}$.