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Elementary Quantum Recursion Schemes That Capture Quantum Polylogarithmic Time Computability of Quantum Functions

Tomoyuki Yamakami

TL;DR

This work develops an elementary, recursion-schematic framework (EQS) to precisely capture quantum polylogarithmic-time computability, linking it to the complexity class $ ext{BQPOLYLOGTIME}$. It introduces a fast code-controlled quantum recursion that halves the input at each step, along with a robust binary-encoding scheme and a binary-search toolkit (BinSearch/Bit) to enable polylogtime operations. The paper proves that EQS exactly characterizes quantum polylogtime, while establishing separations from $ ext{NLOGTIME}$ and $ ext{PPOLYLOGTIME}$, and shows that a natural divide-and-conquer extension cannot be realized within EQS. It further analyzes how EQS relates to polylogtime quantum Turing machines through simulation and inverse-simulation results, and demonstrates that DC raises expressive power beyond EQS, thereby clarifying the boundary between elementary recursion schemes and more powerful algorithmic paradigms in quantum computing.

Abstract

Quantum computing has been studied over the past four decades based on two computational models of quantum circuits and quantum Turing machines. To capture quantum polynomial-time computability, a new recursion-theoretic approach was taken lately by Yamakami [J. Symb. Logic 80, pp.~1546--1587, 2020] by way of recursion schematic definition, which constitutes six initial quantum functions and three construction schemes of composition, branching, and multi-qubit quantum recursion. By taking a similar approach, we look into quantum polylogarithmic-time computability and further explore the expressing power of elementary schemes designed for such quantum computation. In particular, we introduce an elementary form of the quantum recursion, called the fast quantum recursion, and formulate $EQS$ (elementary quantum schemes) of ``elementary'' quantum functions. This class $EQS$ captures exactly quantum polylogarithmic-time computability, which forms the complexity class BQPOLYLOGTIME. We also demonstrate the separation of BQPOLYLOGTIME from NLOGTIME and PPOLYLOGTIME. As a natural extension of $EQS$, we further consider an algorithmic procedural scheme that implements the well-known divide-and-conquer strategy. This divide-and-conquer scheme helps compute the parity function but the scheme cannot be realized within our system $EQS$.

Elementary Quantum Recursion Schemes That Capture Quantum Polylogarithmic Time Computability of Quantum Functions

TL;DR

This work develops an elementary, recursion-schematic framework (EQS) to precisely capture quantum polylogarithmic-time computability, linking it to the complexity class . It introduces a fast code-controlled quantum recursion that halves the input at each step, along with a robust binary-encoding scheme and a binary-search toolkit (BinSearch/Bit) to enable polylogtime operations. The paper proves that EQS exactly characterizes quantum polylogtime, while establishing separations from and , and shows that a natural divide-and-conquer extension cannot be realized within EQS. It further analyzes how EQS relates to polylogtime quantum Turing machines through simulation and inverse-simulation results, and demonstrates that DC raises expressive power beyond EQS, thereby clarifying the boundary between elementary recursion schemes and more powerful algorithmic paradigms in quantum computing.

Abstract

Quantum computing has been studied over the past four decades based on two computational models of quantum circuits and quantum Turing machines. To capture quantum polynomial-time computability, a new recursion-theoretic approach was taken lately by Yamakami [J. Symb. Logic 80, pp.~1546--1587, 2020] by way of recursion schematic definition, which constitutes six initial quantum functions and three construction schemes of composition, branching, and multi-qubit quantum recursion. By taking a similar approach, we look into quantum polylogarithmic-time computability and further explore the expressing power of elementary schemes designed for such quantum computation. In particular, we introduce an elementary form of the quantum recursion, called the fast quantum recursion, and formulate (elementary quantum schemes) of ``elementary'' quantum functions. This class captures exactly quantum polylogarithmic-time computability, which forms the complexity class BQPOLYLOGTIME. We also demonstrate the separation of BQPOLYLOGTIME from NLOGTIME and PPOLYLOGTIME. As a natural extension of , we further consider an algorithmic procedural scheme that implements the well-known divide-and-conquer strategy. This divide-and-conquer scheme helps compute the parity function but the scheme cannot be realized within our system .
Paper Structure (25 sections, 23 theorems, 2 equations)

This paper contains 25 sections, 23 theorems, 2 equations.

Table of Contents

  1. Background, Motivations, and Challenges
  2. Recursion Schematic Definitions That Capture Quantum Computability
  3. Our Challenges in This Work
  4. Preparation: Notions and Notation
  5. Numbers, Strings, and Languages
  6. Quantum States and Hilbert Spaces
  7. Conventions on the Bra and the Ket Notations
  8. A Recursion Schematic Definition of EQS
  9. Skeleton EQS
  10. Binary Encoding of Various Types of Objects
  11. Code-Controlled Fast Quantum Recursion Scheme
  12. Power of Scheme IV
  13. Elementary Quantum Schemes
  14. Quantum Functions Definable within EQS
  15. Basic Properties of $\widehat{EQS}$
  16. ...and 10 more sections

Key Result

Lemma 2.1

Let $k,n\in\mathbb{N}^{+}$ be any two numbers. (1) If $n$ is in $[2^{k-1}+1,2^k]_{\mathbb{Z}}$ (i.e., $\lceil \log{n} \rceil=k$), then $LH^{k}(n)=1$ holds. Moreover, if $k\geq 2$, then $LH^{k-1}(n) =2$. (2) If $n$ is in $[2^{k},2^{k+1}-1]_{\mathbb{Z}}$ (i.e., $\lfloor \log{n} \rfloor=k$), then $RH^{

Theorems & Definitions (38)

  • Lemma 2.1
  • Definition 3.1
  • Example 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Example 3.8
  • Example 3.9
  • ...and 28 more