Index theory of one dimensional quantum walks and cellular automata

TL;DR

The paper develops a comprehensive index theory for one-dimensional quantum walks and quantum cellular automata, defining ind to quantify the net flow of quantum information across a cut. It proves that crossovers and homotopy classes of dynamics are completely characterized by the index, with ind(U) ∈ \mathbb{Z} for walks and ind(\alpha) ∈ \mathbb{Q}_{+} for QCAs, and shows that the index is additive under composition and stable under local deformations. The work provides explicit formulations of the index, connects to translation-invariant cases via winding numbers, and establishes a robust link to classical reversible automata through the Welch index. It also extends the theory to notions of local implementability, crossovers, and one-cut quotient formulas using support algebras, and outlines avenues for approximate causality and higher-dimensional generalizations. Collectively, the results offer a unifying topological framework for 1D quantum information dynamics with potential for broad application in quantum information and many-body physics.

Abstract

If a one-dimensional quantum lattice system is subject to one step of a reversible discrete-time dynamics, it is intuitive that as much "quantum information" as moves into any given block of cells from the left, has to exit that block to the right. For two types of such systems - namely quantum walks and cellular automata - we make this intuition precise by defining an index, a quantity that measures the "net flow of quantum information" through the system. The index supplies a complete characterization of two properties of the discrete dynamics. First, two systems S_1, S_2 can be pieced together, in the sense that there is a system S which locally acts like S_1 in one region and like S_2 in some other region, if and only if S_1 and S_2 have the same index. Second, the index labels connected components of such systems: equality of the index is necessary and sufficient for the existence of a continuous deformation of S_1 into S_2. In the case of quantum walks, the index is integer-valued, whereas for cellular automata, it takes values in the group of positive rationals. In both cases, the map S -> ind S is a group homomorphism if composition of the discrete dynamics is taken as the group law of the quantum systems. Systems with trivial index are precisely those which can be realized by partitioned unitaries, and the prototypes of systems with non-trivial index are shifts.

Figures (4)

• Figure 1: Absolute value squared of a vector initially localized at site $6$ (of $10$) under the action of the time evolution $e^{\mathrm{i}tH}$ for $t\in\{0,.25,.5.,.75,1\}$. The Hamiltonian is given by Eq. (\ref{['eqn:fourierHopping']}). The dynamics is not causal for non-integer values of $t$.
• Figure 2: Combining a crossover from $U_1$ to $U_2$ with a crossover from $U_2$ to $U_3$ to obtain a crossover from $U_1$ to $U_3$. The shaded double chains can be fused to a single cell by Prop. \ref{['ANW']}.
• Figure 3: Example for eigenvalues of $\widehat{U}(p)$, with $\dim{\mathcal{K}}=3$, $L=5$, ${\mathop{\rm ind}\nolimits}\, U=-1$. In this case the spectrum is a single curve on the torus. The index can be computed by the signed number of crossings of any horizontal line, or as the sum of the derivatives of all branches.
• Figure 4: Cell structure with support algebras. (Read top to bottom) If the width of cells is taken as log dimension, the index can be read off the slant in the boxes representing mapping by $\alpha$.

Theorems & Definitions (14)

• Proposition 1
• Definition 2
• Theorem 3
• Lemma 4
• Proposition 5
• Proposition 6
• Lemma 7
• Lemma 8
• Theorem 9
• Proposition 10