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A typology of quantum algorithms

Pablo Arnault, Pablo Arrighi, Steven Herbert, Evi Kasnetsi, Tianyi Li

TL;DR

To map the rapidly evolving landscape of quantum algorithms, the authors assemble a large typology of roughly 134 algorithms classified by fundamental mathematical problems, applications, subroutines, and other criteria. They build and analyze a comprehensive classification table and a dependency network to extract algorithmic primitives and track historical and practical trends, particularly focusing on NISQ-era feasibility. The study identifies core primitives—QFT, Amplitude Amplification, Quantum Adiabatic methods, and VQE—and highlights the rising prominence of QSVT as a unifying framework, with clear trajectories toward first-principle quantum simulation, machine learning, and operations research. The results provide a practical reference for researchers and industry to navigate the near-term quantum landscape, and the authors make their data openly available for extension.

Abstract

We draw the current landscape of quantum algorithms, by classifying about 130 quantum algorithms, according to the fundamental mathematical problems they solve, their real-world applications, the main subroutines they employ, and several other relevant criteria. The primary objectives include revealing trends of algorithms, identifying promising fields for implementations in the NISQ era, and identifying the key algorithmic primitives that power quantum advantage.

A typology of quantum algorithms

TL;DR

To map the rapidly evolving landscape of quantum algorithms, the authors assemble a large typology of roughly 134 algorithms classified by fundamental mathematical problems, applications, subroutines, and other criteria. They build and analyze a comprehensive classification table and a dependency network to extract algorithmic primitives and track historical and practical trends, particularly focusing on NISQ-era feasibility. The study identifies core primitives—QFT, Amplitude Amplification, Quantum Adiabatic methods, and VQE—and highlights the rising prominence of QSVT as a unifying framework, with clear trajectories toward first-principle quantum simulation, machine learning, and operations research. The results provide a practical reference for researchers and industry to navigate the near-term quantum landscape, and the authors make their data openly available for extension.

Abstract

We draw the current landscape of quantum algorithms, by classifying about 130 quantum algorithms, according to the fundamental mathematical problems they solve, their real-world applications, the main subroutines they employ, and several other relevant criteria. The primary objectives include revealing trends of algorithms, identifying promising fields for implementations in the NISQ era, and identifying the key algorithmic primitives that power quantum advantage.
Paper Structure (26 sections, 7 figures, 1 table)

This paper contains 26 sections, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methodology and criteria to build the classification table
  3. Algorithm selection process
  4. Classification criteria
  5. Main subroutine
  6. Fundamental mathematical problem
  7. Applications
  8. Heuristic versus Proven algorithms
  9. NISQ versus LSQ algorithms
  10. Oracular (versus Non-oracular) algorithms
  11. Promise algorithms
  12. Sampling algorithms
  13. Partial dequantization
  14. Computational model
  15. Citations
  16. ...and 11 more sections

Figures (7)

  • Figure 1: Dependency Network.
  • Figure 2: Application-Domains Growth. Evolution over time of the number of new algorithms per application domain.
  • Figure 3: Mathematical-Classes Growth. Evolution over time of the number of new algorithms per mathematical class.
  • Figure 4: Correlations Array. Array of correlations between the Popular (non-primitive) Subroutines (of the second column of Table \ref{['tab:primitives']}) and either (i) mathematical classes for the top array, or (ii) application domains for the bottom array.
  • Figure 5: Sankey Diagram. Correlation between Popular Primitives (first column of Table \ref{['tab:primitives']}), classes of the criterion "Fundamental mathematical problem", and domains of the criterion "Applications". In this diagram, the meaning of "Q. Adiabatic" in the column of Popular Primitives is "any algorithm of the QA type", i.e., we do not limit ourselves to the original QA algorithm. The same applies to "Hamiltonian Simulations". The positions of the "connections" which "arrive" on the column "Fundamental mathematical problems" from the left, have no relationship with the positions of the "connections" which "go out" of that column towards the right. All connections are always ordered very logically from top to bottom.
  • ...and 2 more figures