Quantum computation of multifractal exponents through the quantum wavelet transform

We study the use of the quantum wavelet transform to extract efficiently information about the multifractal exponents for multifractal quantum states. We show that, combined with quantum simulation algorithms, it enables to build quantum algorithms for multifractal exponents with a polynomial gain compared to classical simulations. Numerical results indicate that a rough estimate of fractality could be obtained exponentially fast. Our findings are relevant, e.g., for quantum simulations of multifractal quantum maps and of the Anderson model at the metal-insulator transition.

Figure 1

(Color online) Example of one eigenfunction of Eq. (4) (top) and its wavelet transform (bottom). The red (gray) dashed vertical lines on the bottom panel separate the different scales in the wavelet transform.

Figure 2

(Color online) Average of ${\text{log}}_2{Z}_{{\left|\psi \right|}^2}\left(a,q\right)$ as a function of the scale $a$ for $q=2$ (◻), $q=3$ (○), and $q=4$ (△). The slope of the fitting straight lines (dashed) gives $-{\tau }_q$. Averaging is done over $\sim 25000$ eigenvectors of Eq. (4) of size $N=2^{14}$ for $n_2=3$ $\left(\gamma =1/3\right)$.

Figure 3

(Color online) Average of ${\text{log}}_2{Z}_{\psi }\left(a,q\right)$ as a function of the scale $a$ for $q=2$ (◻), $q=3$ (○), and $q=4$ (△). The slope of the fitting straight lines (dashed) gives $-{\tau }_q$. Averaging is done over $\sim 25000$ eigenvectors of Eq. (4) of size $N=2^{14}$ for $n_2=3$ $\left(\gamma =1/3\right)$.

Figure 4

(Color online) ${\tau }_2$ for the multifractal cascade computed with $Z_{{\left|\psi \right|}^2}\left(a,q\right)$ [Eq. (9)] (filled circles) and $Z_{\psi }\left(a,q\right)$ [Eq. (10)] (empty circles). The values of $p$ are from left to right and top to bottom, $p=0.1,0.2,0.3,0.4$. The straight horizontal lines are the theoretical values ${\tau }_2=-{\text{log}}_2\left[p^2+{\left(1-p\right)}^2\right]$.

Figure 5

(Color online) ${\tau }_2$ for eigenvectors of intermediate map (4) computed with $Z_{{\left|\psi \right|}^2}\left(a,q\right)$ [Eq. (9)] (filled symbols) and $Z_{\psi }\left(a,q\right)$ [Eq. (10)] (empty symbols) for $n_2=3$ (squares) and $n_2=5$ (circles) $\left(n_1=1\right)$. Average is done over $\sim 25000$ eigenvectors. The green/gray area shows the corresponding range of values obtained in [38] (top $n_2=5$, down $n_2=3$).

Figure 6

${\tau }_2$ for iterates of column vectors corresponding to time $t=1000$ of intermediate map (4) computed with $Z_{{\left|\psi \right|}^2}\left(a,q\right)$ [Eq. (9)] (filled symbols) and $Z_{\psi }\left(a,q\right)$ [Eq. (10)] (empty symbols) for $n_2=3$ (squares) and $n_2=5$ (circles) $\left(n_1=1\right)$. The number of vectors averaged for each case is $\sim 25000$.

Figure 7

(Color online) ${\tau }_1^{\prime }$ as a function of $p$ for the multifractal cascade; vectors have size $N=2^n$ with $n$ varying from bottom to top from $n=6$ to $n=20$.

Figure 8

${\tau }_q^{\prime }$ for eigenvectors of Eq. (4) as a function of $q$ for $n_2=3$ (solid), $n_2=5$ (dashed), $n_2=7$ (dotted), $n_2=11$ (small-dot), and $n_2=13$ (dash-dot) $\left(n_1=1\right)$. Average is done over $\sim 25000$ eigenvectors of size $2^{12}$.

Figure 9

${\tau }_1^{\prime }={\sum }_b{\left|T_{\psi }\right|}^2$ for eigenvectors of intermediate map (4) as a function of $n_2$ (with $n_1=1$) for different values of $n$ (number of qubits) bottom to top $n=6,7,8,9,10,11,12$. The number of vectors averaged for each case is $\sim 25000$.