Suppression of quantum chaos in a quantum computer hardware

We present numerical and analytical studies of a quantum computer proposed by the Yamamoto group in Phys. Rev. Lett. 89, 017901 (2002). The stable and quantum chaos regimes in the quantum computer hardware are identified as a function of magnetic field gradient and dipole-dipole couplings between qubits on a square lattice. It is shown that a strong magnetic field gradient leads to suppression of quantum chaos.

Figure 1

(Color online) Disordered spin lattice: Local deviation from the vertex position $(n_{x_i}b,n_{y_i}b)$ is given by $\delta x_i={\delta }_{x_i}b$ and $\delta y_i={\delta }_{y_i}b$ (see text). The index $i$ labels the spins and runs from 1 to $n_q=n_x{n}_y$. The magnetic field gradient has angle $\theta$ with $x$ axis (shown by arrow).

Figure 2

The level spacing distribution $P\left(s\right)$ for a case with disorder fluctuations amplitude $\delta =0.1$, for $\theta =0.3$, and for ${\omega }_g∕d=1$ $(●)$, 4 $(∎)$, 7 $(◻)$, and 10 $(엯)$. The solid curves show the Poissonian and the Wigner-Dyson distributions. The number of disorder realizations is $N_d=10$. There are $n_q=16$ qubits placed on the lattice of size $4\times 4$.

Figure 3

(Color online) (a) Parameter $\eta$ as a function of ${\omega }_g∕d$ for different lattice sizes $5\times 2$ $(◻)$, $6\times 2$ $(▵)$, $7\times 2$ $(\circ )$, $4\times 3$ $(엯)$, $5\times 3$ $(●)$, $6\times 3$ $(*)$, and $4\times 4$ $(◇)$. (b) Parameter $\eta$ as a function of the rescaled frequency shift ${\omega }_g∕{\omega }_g^c$, where ${\omega }_g^c$ is defined by $\eta ({\omega }_g^c∕d)=0.3$. For clarity only some curves presented in (a) are shown rescaled in (b). Upper inset: critical frequency shift ${\omega }_g^c$ as a function of $n_y$, the straight line shows the linear fit (see text). Lower inset: critical frequency shift as a function of the total number of qubits $n_q=n_x{n}_y$, where circles corresponds to $n_y=2$, triangles to $n_y=3$, and the square to $n_y=4$. In all cases $\delta =0.1$ and $\theta =0.3$. The number of disorder realization is $N_d=10$ ($N_d=4$ for the case $6\times 3$).

Figure 4

(Color online) Entropy ${\mathcal{S}}_n$ as a function of ${\omega }_g∕d$, colors are proportional to the entropy. Red (gray) and blue (black) colors correspond to ${\mathcal{S}}_n\approx 13$ and ${\mathcal{S}}_n=0$, respectively; there are no eigenstates in the region with white color. Horizontal axis gives the value of ${\omega }_g∕d$ from ${\omega }_g∕d=0$ (left) to ${\omega }_g∕d=20$ (right). Vertical axis gives the energies of the central band eigenstates (arbitrary units). The size of the lattice is $4\times 4$, $n_q=16$. Disorder corresponds to $\delta =0.1$ and $\theta =0.3\mathrm{rad}$. All the points on the figure has been computed for the same realization of the disorder.

Figure 5

(Color online) Entropy ${\mathcal{S}}_n$ for ${\omega }_g∕d=4$ as a function of the magnetic field gradient angle $\theta$. Colors are as in Fig. 4. Horizontal axis gives the value of $\theta$ from $\theta =0$ (left) to $\theta =\pi ∕4$ (right). Vertical axis gives the energies of the central band eigenstates (arbitrary units). Disorder strength is $\delta =0.1$. All the points on the figure have been computed for the same realization of the disorder as in Fig. 4 at the lattice size $4\times 4$.

Figure 6

Parameter $\eta$ as a function of the magnetic field gradient angle $\theta$ for ${\omega }_g∕d=3$ $(•)$, 4 $(∎)$, and 7 $(▴)$. The disorder strength is $\delta =0.1$. The lattice size is $5\times 3$, $N_d=10$.

Figure 7

Parameter $\eta$ as a function of the disorder strength parameter $\delta$ for ${\omega }_g∕d=2$ $(●)$, 4 $(∎)$, and 6 $(▴)$. The magnetic field gradient angle is $\theta =0.3\mathrm{rad}$. The lattice size is $5\times 3$, $N_d=10$.

Figure 8

Dependence of the width $\Gamma$ of the local density of states ${\rho }_W$ on the frequency shift ${\omega }_g$ for a disorder $\delta =0.1$ at a magnetic field gradient angle $\theta =0.3\mathrm{rad}$ and a number of qubit $n_q=n_x{n}_y=5\times 3$. The dashed line shows the Gaussian regime with $\Gamma \propto d$ and the full line shows the Breit-Wigner regime with $\Gamma \propto d^2∕{\omega }_g$. Lower inset: an example of the local density of states ${\rho }_W$ for ${\omega }_g∕d=3$; the full curve shows the best fit of the Breit-Wigner form with $\Gamma =2.1d$. Upper inset: an example of the local density of states ${\rho }_W$ for ${\omega }_g∕d=1$; the full curve shows the best Gaussian fit of width $\Gamma =2.52d$. In both insets $ϵ∕d$ is rescaled energy, crosses give numerical data. The transition towards integrability takes place near ${\log }_{10}({\omega }_g∕d)\approx 0.7$ that corresponds to the quantum chaos border (11).