Operational Framework for a Quantum Database

TL;DR

This work introduces quantum databases within the broader context of data structures using classical, or more precisely cloneable, and quantum data and indexing, focusing on the definition of the basic operations needed to create and manipulate data stored in a superposition state.

Abstract

Databases are an essential component of modern computing infrastructures and allow efficient manipulation of inherently structured data. The structure depends on the type and relationships of the individual data elements and on the access pattern. Extending the concept of databases to the quantum domain is expected to increase both the storage efficiency and the access parallelism through quantum superposition. In addition, quantum databases may be seen as the result of a prior state preparation ready to be used by quantum algorithms when needed. On the other hand, limiting factors exist and include entanglement creation, the impossibility of perfect copying due to the no-cloning theorem, and the impossibility of coherently erasing a quantum state. In this work, we introduce quantum databases within the broader context of data structures using classical, or more precisely cloneable, and quantum data and indexing. In particular, we are interested in quantum databases' practical implementation and usability, focusing on the definition of the basic operations needed to create and manipulate data stored in a superposition state. Specifically, we address the case of quantum indexing in combination with cloneable data. For this scenario, we define the operations for database preparation, extension, removal of indices, writing, and read-out of data, as well as index permutation. We present their algorithmic implementation and highlight their advantages and limitations. Finally, we introduce the next steps toward defining the same operations in the more general context of quantum indexing and quantum data.

Figures (11)

• Figure 1: The different types of database structured with respect to classical and quantum states as index and data ($C$: classical, $Q$: quantum). Operations defined for each possible combination of type of data and indexing for a database differ vastly. We discuss the term classical data in a more general context of cloneable data.
• Figure 2: Schematic of the quantum database extension procedure consisting of adding new qubits and creating new indices that are correlated to the empty data string. In contrast to Algorithm \ref{['alg:combined_transfer']}, Algorithm \ref{['alg:imbalanced_extend']} does not include the amplitude transfer step and uses a specifically imbalanced database with the zero-index state acting as a probability reservoir. Both protocols are described in detail in Section \ref{['sec:extend_database']}.
• Figure 3: Circuit diagram for creating the balanced quantum superposition state with $k=22$ elements (where $\log_2(k) \not \in \mathbb{N}$). The presented quantum circuit acts on the index register of the database, and we apply the identity otherwise. The resulting state here is given by $\tfrac{1}{\sqrt{22}} \sum_{j=0}^{21} \ket{j}$.
• Figure 4: Circuit diagram for creating the quantum superposition state with $k=14$ elements (where $\log_2(k) \not \in \mathbb{N}$). The presented quantum circuit acts on the index register of the database, and we apply the identity otherwise. The resulting state here is given by $\sqrt{\frac{1+l}{14+l}} \ket{0} + \sqrt{\frac{1}{14+l}} \sum_{j=1}^{14} \ket{j}$ .
• Figure 5: Binary tree that visualizes the step-by-step creation of new indices for a database with a probability reservoir. In this case, $14$ elements are initialized (up to binary element $1101$) and visualized in a binary tree structure. Each branching point visualizes the application of a (conditional) $Y(p)$ gate that leads to the creation of a new state per branching, starting from the root towards the tree's leaves.

Theorems & Definitions (5)

• Lemma 1
• proof
• Theorem 1: Definition mor1999disentanglingbandyopadhyay1999disentanglement
• Lemma 2
• proof