Practical block encodings of matrix polynomials that can also be trivially controlled

TL;DR

This paper introduces practical quantum circuit implementations for block encoding matrix polynomials, significantly reducing depth overhead and enabling controlled operations with minimal additional cost, thus improving quantum algorithm efficiency.

Contribution

It presents explicit, efficient block encoding circuits for matrix polynomials using FOQCS-LCU, reducing depth overhead and allowing controlled operations with negligible extra cost.

Findings

Reduced circuit depth scales linearly with polynomial degree d

Controlled implementations have negligible overhead

Explicit circuits provided for spin models

Abstract

Quantum circuits naturally implement unitary operations on input quantum states. However, non-unitary operations can also be implemented through block encodings, where additional ancilla qubits are introduced and later measured. While block encoding has a number of well-established theoretical applications, its practical implementation has been prohibitively expensive for current quantum hardware. In this paper, we present practical and explicit block encoding circuits implementing matrix polynomial transformations of a target matrix. With standard approaches, block-encoding a degree-$d$ matrix polynomial requires a circuit depth scaling as $d$ times the depth for block-encoding the original matrix alone. By leveraging the recently introduced Fast One-Qubit Controlled Select LCU (FOQCS-LCU) framework, we show that the additional circuit-depth overhead required for encoding matrix polynomials can be reduced to scale linearly in $d$ with no dependence on system size or the cost of block encoding the original matrix. Moreover, we demonstrate that the FOQCS-LCU circuits and their associated matrix polynomial transformations can be controlled with negligible overhead, enabling efficient applications such as Hadamard tests. Finally, we provide explicit circuits for representative spin models, together with detailed non-asymptotic gate counts and circuit depths.