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Exponentially decaying velocity bounds of quantum walks in periodic fields

Houssam Abdul-Rahman, Günter Stolz

TL;DR

This work analyzes 1D discrete-time quantum walks with two internal states in the presence of a periodic local field, modeled by CMV-type unitary operators. The authors derive an explicit exponential-in-period bound on the asymptotic velocity: for a fixed transmission parameter $t\in(0,1/4)$ and an $n$-periodic field $\mathcal{D}_n$ with $v_{U_{t,n}} \le (4t)^n$, velocity can be made arbitrarily small by choosing large $n$. The key technique expands the $n$-step evolution into products of $n$ unitary blocks and expresses the resulting commutators in terms of elementary symmetric polynomials of noncommuting operators $\{B^{(j)},C^{(j)}\}$; a central result shows that all lower-bandwidth symmetric polynomials collapse to a scalar multiple of the identity under the $n$-periodic field, due to destructive interference. This leads to a Step 1 bound on average velocities and, via interpolation with a linear bound in $t$, yields a bound on the full velocity. The findings illustrate localization-like effects in a disorder-free, periodically driven unitary system and suggest a conjecture that similar velocity suppression may hold for all $t\in(0,1)$.

Abstract

We consider a class of discrete-time one-dimensional quantum walks, associated with CMV unitary matrices, in the presence of a local field. This class is parametrized by a transmission parameter $t\in[0,1]$. We show that for a certain range for $t$, the corresponding asymptotic velocity can be made arbitrarily small by introducing a periodic local field with a sufficiently large period. In particular, we prove an upper bound for the velocity of the $n$-periodic quantum walk that is decaying exponentially in the period length $n$. Hence, localization-like effects are observed even after a long number of quantum walk steps when $n$ is large.

Exponentially decaying velocity bounds of quantum walks in periodic fields

TL;DR

This work analyzes 1D discrete-time quantum walks with two internal states in the presence of a periodic local field, modeled by CMV-type unitary operators. The authors derive an explicit exponential-in-period bound on the asymptotic velocity: for a fixed transmission parameter and an -periodic field with , velocity can be made arbitrarily small by choosing large . The key technique expands the -step evolution into products of unitary blocks and expresses the resulting commutators in terms of elementary symmetric polynomials of noncommuting operators ; a central result shows that all lower-bandwidth symmetric polynomials collapse to a scalar multiple of the identity under the -periodic field, due to destructive interference. This leads to a Step 1 bound on average velocities and, via interpolation with a linear bound in , yields a bound on the full velocity. The findings illustrate localization-like effects in a disorder-free, periodically driven unitary system and suggest a conjecture that similar velocity suppression may hold for all .

Abstract

We consider a class of discrete-time one-dimensional quantum walks, associated with CMV unitary matrices, in the presence of a local field. This class is parametrized by a transmission parameter . We show that for a certain range for , the corresponding asymptotic velocity can be made arbitrarily small by introducing a periodic local field with a sufficiently large period. In particular, we prove an upper bound for the velocity of the -periodic quantum walk that is decaying exponentially in the period length . Hence, localization-like effects are observed even after a long number of quantum walk steps when is large.
Paper Structure (14 sections, 9 theorems, 144 equations, 5 figures)

This paper contains 14 sections, 9 theorems, 144 equations, 5 figures.

Table of Contents

  1. Introduction
  2. General setup
  3. Main Results
  4. Field-free velocity bounds
  5. Velocity bounds in the presence of a periodic local field
  6. An upper bound for a subsequence of average velocities
  7. A quantum walk generated by $n$ unitary operators
  8. The elementary symmetric polynomial of operators $\mathcal{S}_{[0,n-1]}^{\ell,m}$
  9. Expanding in terms of $\mathcal{S}_{[0,n-1]}^{\ell,m}$
  10. A proof of Theorem \ref{['thm:main-step']}
  11. Some properties of the symmetric polynomial $\mathcal{S}_{[0,n-1]}^{\ell,m}$
  12. An induction argument on $\ell$ and $m$
  13. Auxiliary Lemmas
  14. A Proof of Lemma \ref{['lem:v<t']}

Key Result

Lemma 3.1

For the quantum walk generated by the unitary $U_{t,\mathcal{D}}=\mathcal{D}V_t W_t$ we have for any $t\in[0,1]$, $N\in{\mathord{\mathbb Z}}^+$, and any local field $\mathcal{D}$. Moreover, the two inequalities in (eq:lem:v<t) become equalities in the extreme cases $t=0$ and $t=1$.

Figures (5)

  • Figure 1: The probability distribution of the quantum walk generated by $U_t$ in (\ref{['U']}) after 100 steps starting from $|\uparrow\rangle\otimes|0\rangle$ with different values for the transmission parameter $t$.
  • Figure 2: The probability distribution of the quantum walk generated by $U_{t=0.2,n}$ after 100 steps starting from $|\uparrow\rangle\otimes|0\rangle$ with different periods $n$. The graphs are restricted to the position window $[-30,30]$.
  • Figure 3: The probability distribution of the quantum walk generated by $U_{t=0.8,n}$ after 100 steps starting from $|\uparrow\rangle\otimes|0\rangle$ with different periods $n$.
  • Figure 4: The shaded region corresponds to the nonzero terms in (\ref{['eq:exact-long-formula']}) and the sum in (\ref{['eq:Com-bound-0']})
  • Figure 5: Region $\Xi_n\subset{\mathord{\mathbb Z}}^2$ and the steps of proof by induction on $\ell$ and $m$.

Theorems & Definitions (22)

  • Lemma 3.1
  • Remark 3.2
  • Theorem 3.3
  • Conjecture 3.4
  • Remark 4.1
  • Example 4.2
  • Theorem 4.3
  • Remark 4.4
  • Lemma 4.5
  • proof : Proof of Lemma \ref{['lem:Com-bound']}
  • ...and 12 more