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Constant-Depth Unitary Preparation of Dicke States

Francisca Vasconcelos, Malvika Raj Joshi

TL;DR

Constant-Depth Unitary Preparation of Dicke States demonstrates unitary, constant-depth circuits for exact Dicke-state preparation by exploiting global interactions in QAC$^0$ and QAC$^0_f$ architectures. The work establishes a Dicke-to-EXACT reduction, enabling exact constant-weight Dicke states in QAC$^0$ and arbitrary-weight Dicke states in QAC$^0_f$ through parallel amplification and FAN-OUT, with explicit constructions and fidelity analyses. It provides exact $EXACT_k$ and THRESHOLD$_k$ implementations in QAC$^0$ (polynomial ancillae) and constant-ancilla approximate $EXACT_1$, enabling constant-depth synthesis of $|D^n_k\rangle$ and, with FAN-OUT, arbitrary-weight Dicke states. These results illuminate how global interactions unlock constant-depth state preparation, offer a potential separation between neutral-atom and trapped-ion hardware, and push forward the understanding of quantum circuit depth in relation to highly entangled states.

Abstract

Dicke states serve as a critical resource in quantum metrology, communication, and computation. However, unitary preparation of Dicke states is limited to logarithmic depth in standard circuit models and existing constant-depth protocols require measurement and feed-forward. In this work, we present the first unitary, constant-depth protocols for exact Dicke state preparation. We overcome the logarithmic-depth barrier by moving beyond the standard circuit model and leveraging global interactions (native to architectures such as neutral atoms and trapped ions). Specifically, utilizing unbounded CZ gates (i.e. within the QAC$^0$ circuit class), we offer circuits for exact computation of constant-weight Dicke states, using polynomial ancillae, and approximation of weight-1 Dicke states (i.e. $W$ states), using only constant ancillae. Granted additional access to the quantum FAN-OUT operation (i.e. upgrading to the QAC$_f^0$ circuit class), we also achieve exact preparation of arbitrary-weight Dicke states, with polynomial ancillae. These protocols distinguish the constant-depth capabilities of quantum architectures based on connectivity and offer a novel path toward resolving a long-standing quantum complexity conjecture.

Constant-Depth Unitary Preparation of Dicke States

TL;DR

Constant-Depth Unitary Preparation of Dicke States demonstrates unitary, constant-depth circuits for exact Dicke-state preparation by exploiting global interactions in QAC and QAC architectures. The work establishes a Dicke-to-EXACT reduction, enabling exact constant-weight Dicke states in QAC and arbitrary-weight Dicke states in QAC through parallel amplification and FAN-OUT, with explicit constructions and fidelity analyses. It provides exact and THRESHOLD implementations in QAC (polynomial ancillae) and constant-ancilla approximate , enabling constant-depth synthesis of and, with FAN-OUT, arbitrary-weight Dicke states. These results illuminate how global interactions unlock constant-depth state preparation, offer a potential separation between neutral-atom and trapped-ion hardware, and push forward the understanding of quantum circuit depth in relation to highly entangled states.

Abstract

Dicke states serve as a critical resource in quantum metrology, communication, and computation. However, unitary preparation of Dicke states is limited to logarithmic depth in standard circuit models and existing constant-depth protocols require measurement and feed-forward. In this work, we present the first unitary, constant-depth protocols for exact Dicke state preparation. We overcome the logarithmic-depth barrier by moving beyond the standard circuit model and leveraging global interactions (native to architectures such as neutral atoms and trapped ions). Specifically, utilizing unbounded CZ gates (i.e. within the QAC circuit class), we offer circuits for exact computation of constant-weight Dicke states, using polynomial ancillae, and approximation of weight-1 Dicke states (i.e. states), using only constant ancillae. Granted additional access to the quantum FAN-OUT operation (i.e. upgrading to the QAC circuit class), we also achieve exact preparation of arbitrary-weight Dicke states, with polynomial ancillae. These protocols distinguish the constant-depth capabilities of quantum architectures based on connectivity and offer a novel path toward resolving a long-standing quantum complexity conjecture.
Paper Structure (42 sections, 12 theorems, 74 equations, 1 table)

This paper contains 42 sections, 12 theorems, 74 equations, 1 table.

Table of Contents

  1. Introduction
  2. Paper Overview.
  3. Preliminaries
  4. The QAC$^0$ Circuit Class.
  5. The QAC$^0_f$ Circuit Class.
  6. Progress on QAC$^0$ vs QAC$^0_f$.
  7. Constant-Weight Dicke States via Global $\texttt{CZ}$ Gates
  8. A Lg Reduction from Lg to EXACTLg
  9. Implementing Constant-Weight EXACT in Lg
  10. Exact EXACT$_k$.
  11. Approximate EXACT$_1$.
  12. Challenges in Extending Beyond Constant-Weight
  13. Arbitrary-Weight Dickes State via Global FAN-OUT
  14. 1) Initalization.
  15. 2) LSB Selection.
  16. ...and 27 more sections

Key Result

Theorem 1

For any $k=\mathcal{O}(1)$, there exists a QAC$^0$ circuit for exact preparation of the weight-$k$ Dicke state, using $\mathcal{O}(n^{k+1})$ ancillae.

Theorems & Definitions (21)

  • Theorem 1: Exact Constant-Weight Dicke in
  • Theorem 2: Approximate $W$ State in
  • Theorem 3: Exact Arbitrary-Weight Dicke in
  • Lemma 4: Constant-Weight $\texttt{THRESHOLD}\xspace_k$ in QAC$^0$
  • proof : Proof of \ref{['thm:threshk_qac0']}
  • Corollary 5: Exact EXACT$_k$ in QAC$^0$
  • proof : Proof of \ref{['thm:exactk_qac0']}
  • Lemma 6: Approximate EXACT$_1$ in QAC$^0$
  • proof : Proof of \ref{['thm:approx_exact_imp']}
  • Lemma 7: Reducing $\ket{D_k^n}$ to EXACT$_k$
  • ...and 11 more