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Simple ways of preparing qudit Dicke states

Noah B. Kerzner, Federico Galeazzi, Rafael I. Nepomechie

TL;DR

This work addresses efficient preparation of qudit Dicke states, including $SU(2)$ spin-$s$ Dicke states $|D^{(s)}_{n,k}\rangle$ and $SU(d)$ Dicke states $|D^{n}(\boldsymbol{k})\rangle$, on a qudit quantum computer. It develops two complementary strategies for each family: a deterministic sequential approach based on exact matrix product state representations and a probabilistic approach based on quantum phase estimation, with explicit circuit constructions. The results give concrete resource scalings: deterministic circuits with depth $\mathcal{O}(s k n)$ for spin-$s$ and $\mathcal{O}((n/d)^d)$ for $SU(d)$, and QPE-based circuits achieving log-depth $\mathcal{O}(\log(s n))$ or $\mathcal{O}(d\log n)$ as well as constant-depth variants at the cost of larger ancillas; worst-case and typical-case repetition counts are provided. Collectively, these methods broaden practical routes to high-dimensional entangled states, with explicit circuits and Cirq implementations facilitating deployment in metrology, error correction, and interferometric imaging contexts.

Abstract

Dicke states are permutation-invariant superpositions of qubit computational basis states, which play a prominent role in quantum information science. We consider here two higher-dimensional generalizations of these states: $SU(2)$ spin-$s$ Dicke states and $SU(d)$ Dicke states. We present various ways of preparing both types of qudit Dicke states on a qudit quantum computer, using two main approaches: a deterministic approach, based on exact canonical matrix product state representations; and a probabilistic approach, based on quantum phase estimation. The quantum circuits are explicit and straightforward, and are arguably simpler than those previously reported.

Simple ways of preparing qudit Dicke states

TL;DR

This work addresses efficient preparation of qudit Dicke states, including spin- Dicke states and Dicke states , on a qudit quantum computer. It develops two complementary strategies for each family: a deterministic sequential approach based on exact matrix product state representations and a probabilistic approach based on quantum phase estimation, with explicit circuit constructions. The results give concrete resource scalings: deterministic circuits with depth for spin- and for , and QPE-based circuits achieving log-depth or as well as constant-depth variants at the cost of larger ancillas; worst-case and typical-case repetition counts are provided. Collectively, these methods broaden practical routes to high-dimensional entangled states, with explicit circuits and Cirq implementations facilitating deployment in metrology, error correction, and interferometric imaging contexts.

Abstract

Dicke states are permutation-invariant superpositions of qubit computational basis states, which play a prominent role in quantum information science. We consider here two higher-dimensional generalizations of these states: spin- Dicke states and Dicke states. We present various ways of preparing both types of qudit Dicke states on a qudit quantum computer, using two main approaches: a deterministic approach, based on exact canonical matrix product state representations; and a probabilistic approach, based on quantum phase estimation. The quantum circuits are explicit and straightforward, and are arguably simpler than those previously reported.

Paper Structure

This paper contains 13 sections, 2 theorems, 84 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. $SU(2)$ spin-$s$ Dicke states $|D^{(s)}_{n,k}\rangle$
  3. Sequential preparation
  4. QPE preparation
  5. Log depth
  6. Constant depth
  7. $SU(d)$ Dicke states $|D^{n}(\vec{k})\rangle$
  8. Sequential preparation
  9. QPE preparation
  10. Log depth
  11. Constant depth
  12. Discussion
  13. Proof of Eq. \ref{['proved']}

Key Result

Lemma 1

If $\vec{y} \in \mathcal{A}^{i+1}(\vec{k})$ and $\vec{y}>_{\rm lex} \vec{a} + \hat{0}$ for some $\vec{a} \in \mathcal{A}^{i}(\vec{k})$, then $\vec{y} = \vec{x} + \hat{0}$ for some $\vec{x} \in \mathcal{A}^{i}(\vec{k})$.

Figures (9)

  • Figure 1: Circuit diagram for ${\rm I}^{(i)}_l$ defined in \ref{['eq:spinsIa']}, \ref{['eq:spinsIb']}.
  • Figure 2: Circuit diagram for preparing the state $|D^{(s)}_{n,k}\rangle$ sequentially \ref{['sequentialspins']} (a) $U_i=\overset{\curvearrowleft}{\prod}_l I^{(i)}_l$, with $x=\max(0,2s(i - n - 1) + k)$ and $y=\min(2si-1, k-1)$; (b) $\overset{\curvearrowleft}{\prod}_i U_i\, |\underline{0}\rangle|0\rangle^{\otimes n}$
  • Figure 3: Circuit diagram for preparing the state $|D^{(s)}_{n,k}\rangle$ in $\log$ depth using the standard QPE algorithm. All ancilla wires are qubits. The initial state of the bottom wire is \ref{['init_qpe']}, and $U$ is defined in \ref{['QPEUspins']}.
  • Figure 4: Circuit diagram for preparing the state $|D^{(s)}_{n,k}\rangle$ in constant depth using the Hadamard test. The top wire is a qudit of dimension $d=2sn+1$. The initial state of the bottom wire is \ref{['init_qpe']}, and $\mathcal{U}$ is defined in \ref{['calUdef']}.
  • Figure 5: Circuit diagram for preparing the state $|D^{(s)}_{n,k}\rangle$, which can be implemented in constant depth. The top $\ell$ wires are qubits, while all other wires are qudits of dimension $2s+1$. The state $|\psi(s,p)\rangle$ is given by \ref{['psip']}, and $U(x)$ is defined in \ref{['Vdef']}.
  • ...and 4 more figures

Theorems & Definitions (4)

  • Lemma
  • proof
  • Proposition
  • proof