# Linear and logarithmic time compositions of quantum many-body operators

## TL;DR

This paper introduces new methods for constructing many-body quantum operators efficiently in linear or logarithmic time, significantly improving over existing strategies by using superposition, dynamical decoupling, and ancilla-based parallelization.

## Contribution

It presents a generalized framework for fast quantum operator construction, including exact decompositions and protocols that reduce time complexity from linear to logarithmic using ancilla qubits.

## Key findings

- Linear time protocol using superposition and dynamical decoupling
- Logarithmic time protocol with ancilla registers and parallel chaining
- Significant reductions in time and space complexity for many-body operators

## Abstract

We develop a generalized framework for constructing many-body-interaction operations either in linear time, or in logarithmic time with a linear number of ancilla qubits. Exact gate decompositions are given in particular for Pauli strings, many-control Toffoli gates, number-~and parity-conserving interactions, Unitary Coupled Cluster operations, and sparse matrix generators. We provide a linear time protocol that works by creating a superposition of exponentially many different possible operator strings and then uses dynamical decoupling methodology to undo all the unwanted terms. A logarithmic time protocol overcomes the speed limit of the first by using ancilla registers to condition evolution to the support of the desired many-body interaction before using parallel chaining operations to expand the string length. The two techniques improve substantially on current strategies (reductions in time and space can range from linear to exponential), are applicable to different physical interaction mechanisms such as CNOT, $XX$, and $XX+YY$, and generalize to a wide range of many-body operators.