Relation between random walks and quantum walks

Abstract

Based on studies of four specific networks, we conjecture a general relation between the walk dimensions d w of discrete-time random walks and quantum walks with the (self-inverse) Grover coin. In each case, we find that d w of the quantum walk takes on exactly half the value found for the classical random walk on the same geometry. Since walks on homogeneous lattices satisfy this relation trivially, our results for heterogeneous networks suggest that such a relation holds irrespective of whether translational invariance is maintained or not. To develop our results, we extend the renormalization-group analysis (RG) of the stochastic master equation to one with a unitary propagator. As in the classical case, the solution ρ(x,t) in space and time of this quantum-walk equation exhibits a scaling collapse for a variable x dw /t in the weak limit, which defines d w and illuminates fundamental aspects of the walk dynamics, e.g., its mean-square displacement. We confirm the collapse for ρ (x,t) in each case with extensive numerical simulation. The exact values for d w themselves demonstrate that RG is a powerful complementary approach to study the asymptotics of quantum walks that weak-limit theorems have not been able to access, such as for systems lacking translational symmetries beyond simple trees.

Universal Phase Diagram and Scaling Functions of Imbalanced Fermi Gases

“…The precise value unfortunately depends on the criterion applied for the vanishing of the inverse vertex at finite Q. Our extrapolated value (h/µ) c = 1.09 ± 0.05 for the CC-transition of the unitary gas is considerably larger than the one obtained by Lobo et al [58], who find (h/µ) c = 0.96, and also the result (h/µ) c = 0.83 in the more recent work by Boettcher et al [59]. Within leading order in a 1/N -expansion, the result is (h/µ) c = 0.807 .…”

Superfluid-normal quantum phase transitions in an imbalanced Fermi gas

“…which is between the values found by Frank et al [38], which obtained (δµ/µ) c = 1.09, Lobo et al [34], that found (δµ/µ) c = 0.96, and Boettcher et al, which found (δµ/µ) c = 0.83 [37]. The (δµ/µ) c from Eq.…”

“…The "effective" fields and order of the quantum phase transitions are h c /µ = 0.79, a first-order phase transition from the unpolarized SF to a PP normal phase, h s /µ = 1.06, a second-order phase transition from the PP to the FP normal phase [22,40], and the maximum field which sets the transition from FP normal phase to the vacuum, h m /µ = 1.41, which is also of second-order. [35] 0.808 FRG [37] 0.83 This work, SPM and DFE 0.88 0.79 QMC [34] 0.96 ǫ = 4 − d expansion [36] 1.15 QMC [30] 1.22 LW formalism [38] 1.09 Experiments [22] 0.95 Experiments [18] 0.878…”

Spectral dimension controlling the decay of the quantum first-detection probability

“…[59,60] for the return probability, and also in Ref. [61] where it manifested in a halving of the walk dimension for certain discrete-time quantum walks. However, only for ordinary states the relevant spectral dimension d S agrees with the one found in the DOS, d DOS S .…”