# Deterministic Preparation of Dicke States

## TL;DR

This paper introduces a deterministic quantum algorithm for efficiently preparing Dicke states with linear depth and no ancillas, advancing quantum state preparation and compression techniques for symmetric states.

## Contribution

The paper presents a novel, deterministic quantum circuit for Dicke state preparation with linear depth and no ancillas, applicable to linear nearest neighbor architectures.

## Key findings

- Uses O(kn) gates and O(n) depth for state preparation
- Enables linear-depth preparation of symmetric pure states
- Provides a quasilinear-depth circuit for quantum information compression

## Abstract

The Dicke state $|D_k^n\rangle$ is an equal-weight superposition of all $n$-qubit states with Hamming Weight $k$ (i.e. all strings of length $n$ with exactly $k$ ones over a binary alphabet). Dicke states are an important class of entangled quantum states that among other things serve as starting states for combinatorial optimization quantum algorithms. We present a deterministic quantum algorithm for the preparation of Dicke states. Implemented as a quantum circuit, our scheme uses $O(kn)$ gates, has depth $O(n)$ and needs no ancilla qubits. The inductive nature of our approach allows for linear-depth preparation of arbitrary symmetric pure states and -- used in reverse -- yields a quasilinear-depth circuit for efficient compression of quantum information in the form of symmetric pure states, improving on existing work requiring quadratic depth. All of these properties even hold for Linear Nearest Neighbor architectures.

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Source: https://tomesphere.com/paper/1904.07358