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Quantum Data Sketches

Qin Zhang, Mohsen Heidari

TL;DR

The paper tackles the challenge of managing inherently quantum data at scale by introducing succinct classical sketches of quantum states that enable core database operations such as equality testing, search, join, selection, and sorting. It develops two sketch families—vector sketches with $L_1/L_2$ embeddings and shadow seeds based on classical shadow tomography—providing distortion guarantees that relate quantum trace distance to tractable, dimension-independent distances and enabling sublinear-time query capabilities via ANN techniques. The results include concrete bounds on sketch size $O(\log(1/\delta)/\iota^2)$, measurement-time guarantees $O(\log^8 d)$, and circuit-depth considerations, along with a hybrid quantum-classical approach to accelerate queries. This work constitutes an initial step toward a sustainable quantum data management model, outlining paths to incorporate more operations, mixed states, and integration with relational database theory. The methods have potential practical impact in quantum sensing, simulation, and experiments where large quantum datasets must be stored, indexed, and queried efficiently while respecting quantum mechanical constraints.

Abstract

Recent advancements in quantum technologies, particularly in quantum sensing and simulation, have facilitated the generation and analysis of inherently quantum data. This progress underscores the necessity for developing efficient and scalable quantum data management strategies. This goal faces immense challenges due to the exponential dimensionality of quantum data and its unique quantum properties such as no-cloning and measurement stochasticity. Specifically, classical storage and manipulation of an arbitrary n-qubit quantum state requires exponential space and time. Hence, there is a critical need to revisit foundational data management concepts and algorithms for quantum data. In this paper, we propose succinct quantum data sketches to support basic database operations such as search and selection. We view our work as an initial step towards the development of quantum data management model, opening up many possibilities for future research in this direction.

Quantum Data Sketches

TL;DR

The paper tackles the challenge of managing inherently quantum data at scale by introducing succinct classical sketches of quantum states that enable core database operations such as equality testing, search, join, selection, and sorting. It develops two sketch families—vector sketches with embeddings and shadow seeds based on classical shadow tomography—providing distortion guarantees that relate quantum trace distance to tractable, dimension-independent distances and enabling sublinear-time query capabilities via ANN techniques. The results include concrete bounds on sketch size , measurement-time guarantees , and circuit-depth considerations, along with a hybrid quantum-classical approach to accelerate queries. This work constitutes an initial step toward a sustainable quantum data management model, outlining paths to incorporate more operations, mixed states, and integration with relational database theory. The methods have potential practical impact in quantum sensing, simulation, and experiments where large quantum datasets must be stored, indexed, and queried efficiently while respecting quantum mechanical constraints.

Abstract

Recent advancements in quantum technologies, particularly in quantum sensing and simulation, have facilitated the generation and analysis of inherently quantum data. This progress underscores the necessity for developing efficient and scalable quantum data management strategies. This goal faces immense challenges due to the exponential dimensionality of quantum data and its unique quantum properties such as no-cloning and measurement stochasticity. Specifically, classical storage and manipulation of an arbitrary n-qubit quantum state requires exponential space and time. Hence, there is a critical need to revisit foundational data management concepts and algorithms for quantum data. In this paper, we propose succinct quantum data sketches to support basic database operations such as search and selection. We view our work as an initial step towards the development of quantum data management model, opening up many possibilities for future research in this direction.
Paper Structure (49 sections, 16 theorems, 78 equations, 1 table, 1 algorithm)

This paper contains 49 sections, 16 theorems, 78 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Unique Challenges in the Quantum World
  3. Our Contribution
  4. Paper Outline
  5. Related Work
  6. Quantum State Learning
  7. Preliminaries
  8. Quantum States and Qubits
  9. Quantum Operations
  10. Quantum Measurements
  11. Standard Math Notations Versus Dirac Notations
  12. Trace Distance
  13. Performance Metrics
  14. Nearest Neighbor in High Dimensions
  15. Basic Operations on Quantum Data
  16. ...and 34 more sections

Key Result

Theorem 2

For $dist(\cdot, \cdot)$ being $\ell_1$ or $\ell_2$, a database $X$ of $m$ vectors, and a $d$-dimensional vector $q$, there is an algorithm that solves $\mathtt{ANN}(q, X, r, \beta)$ using $O(dm + m^{1+\gamma})$ space and $O(d m^\gamma)$ classical time, where $\gamma \approx 1/\beta$ for $\ell_1$ di

Theorems & Definitions (28)

  • Definition 1: $(r, \beta)$-ANN-search
  • Theorem 2: IM98DII+04AI06
  • Remark 3
  • Remark 4
  • Definition 5: $(\varepsilon, \beta)$-equality-test
  • Definition 6: $(\varepsilon, \beta)$-search
  • Definition 7: $(\varepsilon, \beta)$-natural-join
  • Definition 8: $(\eta, \varepsilon)$-selection
  • Definition 9: $k$-local observable
  • Definition 10: $\varepsilon$-sorting
  • ...and 18 more