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arXiv:2507.13308 · Quantum Information and Computation

Simple Ways of Preparing Qudit Dicke States

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

Published Dec 1, 2025• 1 citation
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Abstract

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 ( d ) spin- s Dicke states and SU ( 2 ) 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.

DOI
10.2478/qic-2025-0036

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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 ( d ) spin- s Dicke states and SU ( 2 ) 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.
We derive an exact canonical matrix product state (MPS) representation for Dicke states $|{D}_{k}^{n}\ensuremath{\rangle}$ with a minimal bond dimension $\ensuremath{\chi}=k+1$, for general values of nnn and kkk, for which the WWW state is the simplest case k=1k=1k=1. We use this MPS to formulate a quantum circuit for sequentially preparing Dicke states deterministically, relating it to the recursive algorithm of B\"artschi and Eidenbenz. We also find exact canonical MPS representations with a minimal bond dimension for higher-spin and qudit Dicke states.
Abstract The notion of spin‐ Dicke states is introduced, which are higher‐spin generalizations of usual (spin‐1/2) Dicke states. These multi‐qudit states can be expressed as superpositions of qudit Dicke states. They satisfy a recursion formula, which is used to formulate an efficient quantum circuit for their preparation, whose size scales as , where is the number of qudits and is the number of times the total spin‐lowering operator is applied to the highest‐weight state. The algorithm is deterministic and does not require ancillary qudits.
We formulate a deterministic algorithm for preparing arbitrary multi-qudit states in a definite-weight subspace. By ordering the corresponding computational basis states according to a Gray code for multiset permutations, the state-preparation task is reduced to performing a sequence of controlled 2-qudit Gray rotations. We use this algorithm to prepare exact eigenstates of the SU(3)-invariant Heisenberg Hamiltonian, whose Bethe ansatz is nested. In particular, we describe the preparation of the Bethe states, which are SU(3) highest-weight states, as well as their lower-weight descendants. We also consider the preparation of SU(d) Dicke states and their q-deformations.
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Spin-sss U(1)U(1)U(1)-eigenstate preparation
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