Out-of-time-ordered correlators and quantum walks

Out-of-time-ordered correlators (OTOCs) have recently attracted significant attention, finding applications in disparate areas, from the physics of many-body systems to quantum black holes, with an exponential growth of the OTOCs indicating quantum chaos. Here we consider OTOCs in the context of coined discrete quantum walks, a well-studied model of quantization of classical random walks with applications to quantum algorithms. Three separate cases of operators, variously localized in the coin and walker spaces, are discussed in this context and it is found that the approximated behavior of the OTOC is well described by simple algebraic functions in all three cases with different timescales of growth. The quadratic increase of OTOCs signals the absence of quantum chaos in these simplest forms of quantum walks.

Figure 1

Growth of OTOCs with time for the coin-coin case for $\theta =\pi /4$. The line with the square symbol is the $F_{\mathrm{CC}}$ and the line with the circle (red color) is the one obtained using the approximations; the inset gives the long time behavior of $F_{\mathrm{CC}}\left(t\right)$ for a lattice of size $N=200$. The blue line is $y=0.78$, which shows the algebraic damping.

Figure 2

(a) Growth of the walker-walker OTOC $F_{\mathrm{WW}}\left(t\right)$, when $\theta =\pi /4$. The black line is the exact quantum calculation for $N=1000$, while the large dots are the ones obtained using the numerical integration from Eq. (33) (plotted for every 50 time steps). The inset figure gives the same (dotted line) on a double-log scale and the unbroken line is one with slope 2 given for comparison. (b) Dependence of the $F_{\mathrm{WW}}\left(t\right)$ on time $t$ and angle $\theta$ for a lattice with size $N=100.$

Figure 3

(a) Growth of the coin-walker OTOC $F_{\mathrm{CW}}\left(t\right)$ with time for $\theta =\pi /4$; black line is the $F_{\mathrm{CW}}$ and the large dots are the ones obtained using the numerical integration of Eq. (41) (plotted for every 50 time steps); the inset figure gives the same in a double-log scale (dotted line) and the unbroken line is one with slope 2 given for comparison, the lattice size is $N=1000$. (b) Dependence of $F_{\mathrm{CW}}\left(t\right)$ on time $t$ and angle $\theta$ for a lattice of dimension $N=100$.