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.
This paper deals with separable and entangled qudits | ψ d ⟩ (quantum states in dimension d) constructed from Dicke states made of N = d − 1 qubits. Such qudits present the property to be totally symmetric under the interchange of the N qubits. We discuss the notion of perma-concurrence P d for the qudit | ψ d ⟩ , introduced by the authors (Entropy 2018, 20, 292), as a parameter for characterizing the entanglement degree of | ψ d ⟩ . For d = 3 , the perma-concurrence P 3 constitutes an alternative to the concurrence C for symmetric two-qubit states. We give several expressions of P d (in terms of matrix permanent and in terms of unit vectors of R 3 pointing on the Bloch sphere) and precise the range of variation of P d (going from separable to maximally entangled states). Numerous examples are presented for P d . Special attention is devoted to states of W type and to maximally entangled states of Bell and Greenberger–Horne–Zeilinger type.
An
n-qubit Dicke state of weight
k, is the uniform superposition over all
n-bit strings of Hamming weight
k. Dicke states are central to quantum algorithms exhibiting speedups, such as Decoded Quantum Interferometry (Jordan et al., \emph{Nature}, 2025). In the NISQ era, quantum hardware is constrained by both depth and locality, motivating the question of which global operations suffice to prepare such states. QAC
0, the quantum analogue of AC
0, minimally extends local
O(1)-depth quantum circuits by allowing arbitrary-width Toffoli (reversible AND) gates.
We show that Dicke states of
polylog(n) weight can be prepared in QAC
0. This gives the first QAC
0 construction of any super-constant-weight
n-qubit Dicke state, since previous constructions relied on the much more powerful FANOUT
n gate. In general, we show that any weight-
k Dicke state can be constructed using FANOUT
min(k,n−k) gates. Combined with recent hardness results, this yields a tight characterization: for
k≤n/2, a
n-qubit weight-
k Dicke state can be prepared in QAC
0 if and only if FANOUT
k ∈ QAC
0. We develop a limited-fanout state-synthesis toolkit for QAC
0 that yields further constant-depth, poly
(n)-ancilla constructions:
1. Every
n-qubit symmetric state supported on Hamming weight
≤k can be prepared using FANOUT
k gates.
2. Every
O(logn)-qubit state can be prepared using quantum random-access memory (QRAM
n), which refers to a coherent indexing gate. QRAM
n is a potentially weaker resource than FANOUT
n and can be implemented in QAC
f0.
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.