Efficient Equivalent of Shallow Quantum Hashing

TL;DR

This paper establishes a connection between shallow quantum hashing and single-qubit quantum hashing for the amplitude form and proposes a circuit of depth 1 that achieves the same collision resistance.

Abstract

Quantum hashing is a widely used technique in quantum computation that allows us to design space-efficient algorithms and protocols. Recently, Vasiliev has shown that the phase form of shallow quantum hashing can be implemented by a circuit of depth 2. In this paper, we establish a connection between shallow quantum hashing and single-qubit quantum hashing for the amplitude form. For a shallow circuit, we propose a circuit of depth 1 that achieves the same collision resistance.

Figures (6)

• Figure 1: Controlled rotation gates.
• Figure 2: A circuit for amplitude form of quantum hashing.
• Figure 3: Uniformly controlled rotation.
• Figure 4: Shallow uniformly controlled rotation decomposition.
• Figure 5: A circuit for shallow quantum hashing.

Theorems & Definitions (11)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof