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arXiv:2606.24659 · Journal of Physics A Mathematical and Theoretical

Preparing multi-qudit states in a definite-weight subspace

Nabi Zare Harofteh, Rafael I. Nepomechie

quant-phSubmitted Jun 23, 2026• Announced Jun 24, 2026• 0 citations• Revised Oct 14, 2026 (v3)

v1 · Jun 23, 2026 v2 · Jun 27, 2026 v3 · Oct 14, 2026

Abstract

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.

Journal reference: J. Phys. A: Math. Theor. 59, 415302 (2026)
DOI: 10.1088/1751-8121/aeaf22
Comments: 21 pages, 6 figures, implementations of circuits in cirq are available on GitHub; v2: references added; v3: new appendix on noise and error detection, and more references added

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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.

Spin-

s

U(1)

-eigenstate preparation

We formulate a deterministic algorithm for preparing a general

U(1)

-eigenstate of a spin-

s

chain of length

n

. These states consist of linear combinations of computational basis states

|\vec{m}\rangle

of

n

qudits, each with

(2s+1)

levels and

s= 1/2, 1, 3/2, \ldots

, whose ditstrings

\vec{m}

have a fixed digit sum. Exploiting a Gray code for bounded integer compositions, whose consecutive ditstrings obey the Gray property, the quantum state is prepared by applying corresponding ``Gray gates.'' We use this algorithm to prepare exact eigenstates of integrable spin-

s

XXX Hamiltonians. We also consider the preparation of AKLT states and spin-

s

Dicke states.

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.

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.

Universal quantum algorithms that prepare arbitrary n-qubit quantum states require

{O\left(2^{n}\right)}

gate complexity. The complexity can be reduced by considering specific families of quantum states depending on the task at hand. In particular, multipartite quantum states that are invariant under permutations, e.g. Dicke states, have intriguing properties. In this paper, we consider states invariant under cyclic permutations, which we call cyclic states. We present a quantum algorithm that deterministically prepares cyclic states with gate complexity

{O\left(n\right)}

without requiring any ancillary qubit. Through both analytical and numerical analyses, we show that our algorithm is more efficient than existing ones.

The

n

-qubit

k

-weight Dicke states

|D^n_k\rangle

, defined as the uniform superposition of all computational basis states with exactly

k

qubits in state

|1\rangle

, form a basis of the symmetric subspace and represent an important class of entangled quantum states with broad applications in quantum computing. We propose deterministic quantum circuits for Dicke state preparation under two commonly seen qubit connectivity constraints: 1. All-to-all qubit connectivity: our circuit has depth

O(\log(k)\log(n/k)+k)

, which improves the previous best bound of

O(k\log(n/k))

. 2. Grid qubit connectivity (

(n_1\times n_2)

-grid,

n_1\le n_2

): (a) For

k\ge n_2/n_1

, we design a circuit with depth

O(k\log(n/k)+n_2)

, surpassing the prior

O(\sqrt{nk})

bound. (b) For

k< n_2/n_1

, we design an optimal-depth circuit with depth

O(n_2)

. Furthermore, we establish the depth lower bounds of

Ω(\log(n))

for all-to-all qubit connectivity and

Ω(n_2)

for

(n_1\times n_2)

-grid connectivity constraints, demonstrating the near-optimality of our constructions.