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Spectra of Quantized Walks and a $\sqrt{δε}$ rule

Mario Szegedy

TL;DR

This work introduces quantized bipartite walks and a memory-enabled quantum diffusion framework that extends Grover/Ambainis-style speedups to general symmetric Markov chains. A central contribution is a spectral theorem linking the quantum walk operator to a discriminant matrix, enabling precise eigenvalue analyses and performance bounds. By incorporating a memory mechanism via a modified chain $P'$, the authors derive the $ sqrt{oldsymbol{ delta}oldsymbol{ varepsilon}}$ rule, demonstrating a square-root improvement over classical approaches for detecting large marked sets. The results yield concrete algorithmic consequences, including a re-derivation of Ambainis-type speedups and implications for search on graphs and expanders, grounded in a detailed spectral framework for products of reflections. The work also provides an explicit example based on the Grover chain to illustrate the construction and its amplification properties.

Abstract

We introduce quantized bipartite walks, compute their spectra, generalize the algorithms of Grover \cite{g} and Ambainis \cite{amb03} and interpret them as quantum walks with memory. We compare the performance of walk based classical and quantum algorithms and show that the latter run much quicker in general. Let $P$ be a symmetric Markov chain with transition probabilities $P[i,j]$, $(i ,j\in [n])$. Some elements of the state space are marked. We are promised that the set of marked elements has size either zero or at least $εn$. The goal is to find out with great certainty which of the above two cases holds. Our model is a black box that can answer certain yes/no questions and can generate random elements picked from certain distributions. More specifically, by request the black box can give us a uniformly distributed random element for the cost of $\wp_{0}$. Also, when ``inserting'' an element $i$ into the black box we can obtain a random element $j$, where $j$ is distributed according to $P[i,j]$. The cost of the latter operation is $\wp_{1}$. Finally, we can use the black box to test if an element $i$ is marked, and this costs us $\wp_{2}$. If $δ$ is the eigenvalue gap of $P$, there is a simple classical algorithm with cost $O(\wp_{0} + (\wp_{1}+\wp_{2})/δε)$ that solves the above promise problem. (The algorithm is efficient if $\wp_{0}$ is much larger than $\wp_{1}+\wp_{2}$.) In contrast,we show that for the ``quantized'' version of the algorithm it costs only $O(\wp_{0} + (\wp_{1}+\wp_{2})/\sqrt{δε})$ to solve the problem. We refer to this as the $\sqrt{δε}$ rule. Among the technical contributions we give a formula for the spectrum of the product of two general reflections.

Spectra of Quantized Walks and a $\sqrt{δε}$ rule

TL;DR

This work introduces quantized bipartite walks and a memory-enabled quantum diffusion framework that extends Grover/Ambainis-style speedups to general symmetric Markov chains. A central contribution is a spectral theorem linking the quantum walk operator to a discriminant matrix, enabling precise eigenvalue analyses and performance bounds. By incorporating a memory mechanism via a modified chain , the authors derive the rule, demonstrating a square-root improvement over classical approaches for detecting large marked sets. The results yield concrete algorithmic consequences, including a re-derivation of Ambainis-type speedups and implications for search on graphs and expanders, grounded in a detailed spectral framework for products of reflections. The work also provides an explicit example based on the Grover chain to illustrate the construction and its amplification properties.

Abstract

We introduce quantized bipartite walks, compute their spectra, generalize the algorithms of Grover \cite{g} and Ambainis \cite{amb03} and interpret them as quantum walks with memory. We compare the performance of walk based classical and quantum algorithms and show that the latter run much quicker in general. Let be a symmetric Markov chain with transition probabilities , . Some elements of the state space are marked. We are promised that the set of marked elements has size either zero or at least . The goal is to find out with great certainty which of the above two cases holds. Our model is a black box that can answer certain yes/no questions and can generate random elements picked from certain distributions. More specifically, by request the black box can give us a uniformly distributed random element for the cost of . Also, when ``inserting'' an element into the black box we can obtain a random element , where is distributed according to . The cost of the latter operation is . Finally, we can use the black box to test if an element is marked, and this costs us . If is the eigenvalue gap of , there is a simple classical algorithm with cost that solves the above promise problem. (The algorithm is efficient if is much larger than .) In contrast,we show that for the ``quantized'' version of the algorithm it costs only to solve the problem. We refer to this as the rule. Among the technical contributions we give a formula for the spectrum of the product of two general reflections.

Paper Structure

This paper contains 14 sections, 16 theorems, 94 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notations
  3. Bipartite Walks
  4. The Discriminant Matrix
  5. A Spectral Theorem
  6. Bipartite Walks from Ordinary Markov Chains
  7. Walks with "Memory"
  8. The $\sqrt{\delta\varepsilon}$ rule
  9. Consequences
  10. Acknowledgments
  11. Operator Notation in Linear Algebra
  12. Spectra of Products of two General Reflections
  13. Norms and Inner Products
  14. An Example

Key Result

Theorem 1

Let ${\bf v_{1}},\ldots,{\bf v_{n}}$ and ${\bf w_{1}},\ldots,{\bf w_{m}}$ be two orthogonal systems of unit vectors spanning spaces ${\cal A}$ and ${\cal B}$, respectively. The eigenvectors and eigenvalues of the operator $\mu = {\rm ref}_{{\cal B}}{\rm ref}_{{\cal A}}$ on ${\cal A} + {\cal B}$ are

Figures (4)

  • Figure 1: Example to a bipartite walk, and one of its instantiations
  • Figure 2: The transformation that takes the eigenvalues of $M$ into the eigenvalues of $\mu$
  • Figure 3: Two rectangular unit vectors with inner product $1/(2\sqrt{3})$.
  • Figure 4: The Markov chain associated with Grover's algorithm, where items are marked with probability $p$

Theorems & Definitions (35)

  • Definition 1
  • Definition 2: Discriminant Matrix
  • Definition 3
  • Definition 4
  • Definition 5: Discriminant Matrix (generalized)
  • Theorem 1: Spectral Theorem
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • ...and 25 more