Figure 1

(a) CNOT gate simulation for the operation $|00⟩\to |00⟩$, realized by the Hamiltonian (1) with $H_0=H_{\mathrm{CNOT}}+\delta H$ [see Eq. (3) and below]. Some levels cross due to symmetry. The small orange diamonds on the vertical axis of the main panel show the results of direct diagonalization of $H_0$ and coincide with the results of the evolution (2) to four significant figures. (b) Evolution of 50 energy levels in Eq. (2) as a function of the parameter $\lambda$, for a single GUE realization of the Hamiltonian (see text). The rms amplitude of the fluctuations in $E_n$ and $⟨m|Z{H}_b|n⟩$ is $0.1⟨\Delta E⟩$, where $⟨\Delta E⟩=⟨E_{n+1}-E_n⟩$ is the average level spacing. The ground state (red) and the highest excited state (orange) are labeled. It is clearly seen in the insets (which enlarge the green boxes) that avoided level crossings appear at small values of $\lambda$ (e.g., green vertical arrow in the top inset). In the left inset notice the significant separation between the ground state and the first excited state.Reuse & Permissions

Figure 2

(a) Probability $P(n|n)$ for the system to remain in the initial state $|n⟩$, as a function of $1/T$, during the adiabatic evolution (i.e., decreasing the external field with constant speed $|\dot{\lambda }|=1/T$). The data are averaged over 400 GUE realizations with $N=50$. The power-law dependence $P\propto T^{\gamma }$ is clearly seen, with two distinct exponents: (i) $\gamma =\frac{1}{2}$ for all the bulk states and (ii) $\gamma \lesssim \frac{1}{3}$ for the edge states. For example, for $N=50$: $\gamma =\frac{1}{2}$ for $n=2(49)$, solid (empty) purple circles; $n=5(45)$, solid (empty) green triangles; $n=10(40)$, solid (empty) upturned magenta triangles; $n=25$, black diamonds; but $\gamma =\frac{1}{4}$ for the ground state ($n=1$, red squares) and the highest excited state ($n=50$, empty blue squares). For $N=150$ (left inset): $\gamma =\frac{1}{2}$ for $n=2$ (purple stars), $n=75$ (black pluses); but $\gamma \approx \frac{1}{3}$ for the ground state ($n=1$, red crosses). However, the average number of avoided level crossings (right inset) is a smooth function of the level number for both $N=50$ (red) and $N=150$ (black). As expected in the absence of external noise, the probability $P(n|n)$ saturates at 1 as $1/T\to 0$ (when $1/T<0.01$). (b) The standard deviation $⟨(n-n_0{)}^2⟩$ of the system from the initial state $n=n_0$ during the adiabatic evolution, as a function of $1/T$ $(N=50)$. The scaling is approximately power law, but with the exponent smoothly dependent on the initial state. (Inset) Probability $P(n|n_0)$ to occupy level $|n⟩$ at the end of the evolution, starting from level $n_0=1$, 25, 50 ($N=50$). Different curves correspond (top to bottom peaks) to ${10}^3\times |\dot{\lambda }|=1$, 2.5, 5, 10, 25, 50, 75.Reuse & Permissions