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Exponential algorithmic speedup by quantum walk

Andrew M. Childs, Richard Cleve, Enrico Deotto, Edward Farhi, Sam Gutmann, Daniel A. Spielman

TL;DR

The paper constructs an explicit oracular graph problem on which a quantum computer achieves exponential speedup using a continuous-time quantum walk, contrasting with all classical subexponential strategies. It develops a concrete implementation of the quantum walk within the circuit model by encoding the graph via edge colorings and an oracle, and proves that the walk reaches the exit from the entrance in polynomial time with high probability. A robust classical lower bound is established, showing that no subexponential classical algorithm can solve the problem with high probability in this oracle setting. Collectively, the results demonstrate a novel quantum algorithmic approach beyond quantum Fourier transform techniques and indicate a potential broader applicability of quantum walks to non-oracular and algorithmic problems.

Abstract

We construct an oracular (i.e., black box) problem that can be solved exponentially faster on a quantum computer than on a classical computer. The quantum algorithm is based on a continuous time quantum walk, and thus employs a different technique from previous quantum algorithms based on quantum Fourier transforms. We show how to implement the quantum walk efficiently in our oracular setting. We then show how this quantum walk can be used to solve our problem by rapidly traversing a graph. Finally, we prove that no classical algorithm can solve this problem with high probability in subexponential time.

Exponential algorithmic speedup by quantum walk

TL;DR

The paper constructs an explicit oracular graph problem on which a quantum computer achieves exponential speedup using a continuous-time quantum walk, contrasting with all classical subexponential strategies. It develops a concrete implementation of the quantum walk within the circuit model by encoding the graph via edge colorings and an oracle, and proves that the walk reaches the exit from the entrance in polynomial time with high probability. A robust classical lower bound is established, showing that no subexponential classical algorithm can solve the problem with high probability in this oracle setting. Collectively, the results demonstrate a novel quantum algorithmic approach beyond quantum Fourier transform techniques and indicate a potential broader applicability of quantum walks to non-oracular and algorithmic problems.

Abstract

We construct an oracular (i.e., black box) problem that can be solved exponentially faster on a quantum computer than on a classical computer. The quantum algorithm is based on a continuous time quantum walk, and thus employs a different technique from previous quantum algorithms based on quantum Fourier transforms. We show how to implement the quantum walk efficiently in our oracular setting. We then show how this quantum walk can be used to solve our problem by rapidly traversing a graph. Finally, we prove that no classical algorithm can solve this problem with high probability in subexponential time.

Paper Structure

This paper contains 11 sections, 9 theorems, 42 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The problem
  3. Quantum algorithm
  4. Quantum walk
  5. Implementing the quantum walk
  6. Propagation on a line
  7. Upper bound on the hitting time
  8. Classical lower bound
  9. Discussion
  10. An efficient classical algorithm to traverse the hypercube
  11. Implementation of the quantum walk starting from a particular vertex of a bipartite graph

Key Result

Lemma 1

Consider the quantum walk in $G_{n-1}'$ starting at the ${\textsc{entrance}}$. Let the walk run for a time $t$ chosen uniformly in $[0,\tau]$ and then measure in the computational basis. If $\tau \ge {4n \over \epsilon \, \Delta E}$ for any constant $\epsilon>0$, where $\Delta E$ is the magnitude of

Figures (5)

  • Figure 1: The graph $G_4$.
  • Figure 2: A typical graph $G^{\prime}_4$.
  • Figure 3: A circuit for simulating $e^{-iTt}$.
  • Figure 4: Quantum walks on lines. (a) Reduction of the quantum walk on $G_n'$ to a quantum walk on a line. (b) Quantum walk on an infinite, translationally invariant line. (c) Quantum walk on an infinite line with a defect. (d) Quantum walk on a finite line without a defect.
  • Figure 5: The transmission probability $|{\cal T}(p)|^2$ as a function of momentum for the infinite line with a defect (with $\alpha=\sqrt 2$).

Theorems & Definitions (9)

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