Circuit complexity of quantum access models for encoding classical data

TL;DR

This paper investigates the quantum circuit complexity of data encoding models, providing near-optimal construction protocols and analyzing their efficiency for different matrix types in quantum algorithms.

Contribution

It introduces new quantum circuit constructions for data encoding models, including near-optimal protocols and analysis of their complexity for various matrix classes.

Findings

Sparse-access models require nearly linear circuit complexity.

Efficient construction when matrices are polynomial combinations of efficient unitaries.

Improved block encoding for Pauli string unitaries.

Abstract

Classical data encoding is usually treated as a black-box in the oracle-based quantum algorithms. On the other hand, their constructions are crucial for practical algorithm implementations. Here, we open the black-boxes of data encoding and study the Clifford$+T$ complexity of constructing some typical quantum access models. For general matrices, we show that both sparse-access input models and block-encoding require nearly linear circuit complexities relative to the matrix dimension, even if matrices are sparse. We also gives construction protocols achieving near-optimal gate complexities. On the other hand, the construction becomes efficient with respect to the data qubit when the matrix is the linear combination polynomial terms of efficient unitaries. As a typical example, we propose improved block encoding when these unitaries are Pauli strings. Our protocols are built upon improved quantum state preparation and a selective oracle for Pauli strings, which hold independent value. Our access model constructions offer considerable flexibility, allowing for tunable ancillary qubit number and offers corresponding space-time trade-offs.