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Unbounded quantum-classical separation in sample complexity for sphere center finding

Guanzhong Li, Lvzhou Li

TL;DR

The paper studies the problem of finding the center $t$ of a sphere $S_r+t$ in the finite field space $\mathbb{F}_p^n$ from samples drawn on the sphere. It proves a classical lower bound of $\Omega(n)$ samples by reducing to Warning's second theorem and employing a specialized Yao's minimax principle, while presenting a quantum algorithm based on continuous-time quantum walks on a Euclidean graph that needs only $O(\log p)$ quantum samples (constant when $p$ is fixed). This yields an unbounded quantum–classical separation in the relevant notion of sample complexity for this geometric task, though the authors do not claim a quantum speedup in runtime. The results highlight the potential of quantum walk techniques for solving geometric problems and motivate further exploration of quantum advantages in geometry-informed learning tasks.

Abstract

Fast quantum algorithms can solve important computational problems more efficiently than classical algorithms. However, little is known about whether quantum computing can speed up solving geometric problems. This article explores quantum advantages for the problem of finding the center of a sphere in vector spaces over finite fields, given samples of random points on the sphere. We prove that any classical algorithm for this task requires approximately as many samples as the dimension of the vector space, by a reduction to an old and basic algebraic result -- Warning's second theorem. On the other hand, we propose a quantum algorithm based on quantum walks that needs only a constant number of samples to find the center. Thus, an unbounded quantum advantage is revealed for a natural and intuitive geometric problem, which highlights the power of quantum computing in solving geometric problems.

Unbounded quantum-classical separation in sample complexity for sphere center finding

TL;DR

The paper studies the problem of finding the center of a sphere in the finite field space from samples drawn on the sphere. It proves a classical lower bound of samples by reducing to Warning's second theorem and employing a specialized Yao's minimax principle, while presenting a quantum algorithm based on continuous-time quantum walks on a Euclidean graph that needs only quantum samples (constant when is fixed). This yields an unbounded quantum–classical separation in the relevant notion of sample complexity for this geometric task, though the authors do not claim a quantum speedup in runtime. The results highlight the potential of quantum walk techniques for solving geometric problems and motivate further exploration of quantum advantages in geometry-informed learning tasks.

Abstract

Fast quantum algorithms can solve important computational problems more efficiently than classical algorithms. However, little is known about whether quantum computing can speed up solving geometric problems. This article explores quantum advantages for the problem of finding the center of a sphere in vector spaces over finite fields, given samples of random points on the sphere. We prove that any classical algorithm for this task requires approximately as many samples as the dimension of the vector space, by a reduction to an old and basic algebraic result -- Warning's second theorem. On the other hand, we propose a quantum algorithm based on quantum walks that needs only a constant number of samples to find the center. Thus, an unbounded quantum advantage is revealed for a natural and intuitive geometric problem, which highlights the power of quantum computing in solving geometric problems.
Paper Structure (12 sections, 10 theorems, 45 equations, 2 figures)

This paper contains 12 sections, 10 theorems, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Classical lower bound
  3. Yao's minimax principle specific to randomized algorithms for sample problems
  4. Lower bound on sample complexities of deterministic strategies
  5. Finishing the proof of Theorem \ref{['thm:classical_lower']}
  6. Quantum algorithm
  7. Conclusion
  8. Numerical simulation
  9. Uniqueness of the center of a sphere
  10. Proof of Lemma \ref{['thm:minimax']}
  11. Lower bound on the quantum algorithm's success probability
  12. Invariant supspace of the CTQW

Key Result

Theorem 1

There is a quantum algorithm that solves Problem prob:main with bounded error, and uses $O(\log p)$The standard asymptotic notations $O(\cdot)$ and $\Omega(\cdot)$ are used. We say the complexity is $O(f(n))$ (resp. $\Omega(f(n))$), if for large enough $n$, it is at most (resp. at least) $cf(n)$ for

Figures (2)

  • Figure 1: The success probability $p(T_1)$ for the case where $p=3,r=1,n=5$.
  • Figure 2: Optimal CTQW time $T_\mathrm{max}$ and corresponding success probability $p(T_\mathrm{max})$. Here, $p=3, r=1$, and $n=3\sim 12$.

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Lemma 3: Warning's second theorem
  • proof : Proof of Lemma \ref{['lem:m_x']}
  • Lemma 4
  • proof : Proof of Lemma \ref{['lem:CTQW_prob']}
  • Theorem 3
  • proof
  • ...and 7 more