Feasibility-Preserving Mixed Dicke Ansatz

• The paper introduces a feasibility-preserving mixed Dicke state ansatz that omits penalty terms by restricting the variational search to feasible Hamming-weight subspaces.
• It employs both pure and mixed state formulations—using density-matrix formalism—to address equality and inequality constraints in problems like multiclass portfolio optimization.
• Empirical results indicate improved performance in retaining feasibility and concentrating probability mass, though challenges in circuit depth and noise sensitivity remain.

Searching arXiv for the cited papers and closely related work on Dicke-state-based VQE and Dicke-state preparation. The feasibility-preserving mixed Dicke state ansatz is a variational construction for constrained combinatorial optimization in which the quantum state is restricted, by design, to the feasible Hamming-weight subspace, thereby eliminating penalty terms from the objective function. In the formulation introduced for variational quantum eigensolvers (VQE), pure Dicke states encode exact equality constraints, while mixed Dicke states extend the construction to inequality constraints through a density-matrix formalism and an ancilla-labeled superposition over feasible Hamming-weight sectors (Scursulim, 7 Jun 2026). In multiclass portfolio optimization, a related “mixed” usage denotes a tensor product of class-specific Dicke states, each enforcing a fixed class-wise selection count, so that every sampled portfolio automatically satisfies diversification constraints (Scursulim et al., 19 Aug 2025). The common principle is feasibility preservation: the ansatz explores only valid configurations rather than relying on Lagrange-multiplier penalties or post hoc rejection of invalid bitstrings.

1. Conceptual basis and problem setting

In constrained binary optimization, the decision vector $x \in \{0,1\}^n$ often satisfies Hamming-weight conditions such as

$$\sum_i x_i = b,\qquad \sum_i x_i \le b,\qquad \sum_i x_i \ge b.$$

Within portfolio optimization, these constraints determine the number of selected assets globally or within designated classes (Scursulim, 7 Jun 2026).

The central motivation for Dicke-state-based ansätze is that the feasible set may occupy only a small subspace of the full Hilbert space. A Dicke state is naturally supported on computational basis states with fixed Hamming weight, making it a canonical representation for equality-constrained subspaces (Scursulim, 7 Jun 2026). By restricting the variational family to feasible sectors from the outset, the method avoids two standard difficulties in constrained VQE: the inclusion of penalty terms in the Hamiltonian and the tuning of Lagrange multipliers needed to enforce feasibility (Scursulim, 7 Jun 2026).

In the multiclass portfolio setting, the same logic is applied to diversification constraints. A realistic portfolio must include assets from multiple classes, such as stocks, bonds, and commodities, and the ansatz is constructed so that each class automatically contains the required number of selected assets (Scursulim et al., 19 Aug 2025). This makes feasibility preservation literal: every basis state in the superposition already satisfies the class-wise allocation rules.

2. Pure Dicke state ansatz for equality constraints

For equality-constrained problems, the basic variational object is the parameterized Dicke state

$$\vert D^n_k (\vec{\theta})\rangle = \sum_i \mathcal{P}_i c_i(\vec{\theta}) \vert 0 \rangle^{\otimes(n-k)}\otimes\vert 1\rangle^k,$$

where $k$ is the exact Hamming weight, $\mathcal{P}_i$ are permutation operators over basis states of weight $k$, and $c_i(\vec{\theta})$ are variational amplitudes (Scursulim, 7 Jun 2026).

A closely related expression is used in multiclass portfolio optimization: $$\lvert D_k^n(\vec{\theta})\rangle = \sum_i \mathcal{P}_i \, a_i(\vec{\theta}) \lvert 0\rangle^{\otimes (n-k)}\otimes \lvert 1\rangle^{\otimes k},$$ with trainable amplitudes $a_i(\vec{\theta})$ (Scursulim et al., 19 Aug 2025). In both formulations, the variational degrees of freedom modulate amplitudes only within the fixed-weight manifold.

When the constraint is

$\sum_i x_i = b,$

a single Dicke state of weight $$\sum_i x_i = b,\qquad \sum_i x_i \le b,\qquad \sum_i x_i \ge b.$$0 suffices, and the reduced state is

$$\sum_i x_i = b,\qquad \sum_i x_i \le b,\qquad \sum_i x_i \ge b.$$1

This exactly preserves feasibility because every computational basis component has Hamming weight $$\sum_i x_i = b,\qquad \sum_i x_i \le b,\qquad \sum_i x_i \ge b.$$2 (Scursulim, 7 Jun 2026).

The search-space reduction is explicit. Instead of exploring all $$\sum_i x_i = b,\qquad \sum_i x_i \le b,\qquad \sum_i x_i \ge b.$$3 binary strings, the optimization is confined to the feasible combinations

$$\sum_i x_i = b,\qquad \sum_i x_i \le b,\qquad \sum_i x_i \ge b.$$4

which is stated as effectively reducing the search space from $$\sum_i x_i = b,\qquad \sum_i x_i \le b,\qquad \sum_i x_i \ge b.$$5 to $$\sum_i x_i = b,\qquad \sum_i x_i \le b,\qquad \sum_i x_i \ge b.$$6 in the constrained setting (Scursulim et al., 19 Aug 2025). This restricted exploration is the essential mechanism by which Dicke-state ansätze remove the need for penalties.

3. Mixed Dicke state formalism for inequality constraints

The main extension introduced in the 2026 formulation is a mixed Dicke state construction that handles inequality constraints directly through a density-matrix formalism (Scursulim, 7 Jun 2026). Rather than fixing a single Hamming weight, the ansatz forms a superposition over all feasible Dicke sectors: $$\sum_i x_i = b,\qquad \sum_i x_i \le b,\qquad \sum_i x_i \ge b.$$7 Here subsystem $$\sum_i x_i = b,\qquad \sum_i x_i \le b,\qquad \sum_i x_i \ge b.$$8 contains the Dicke state, subsystem $$\sum_i x_i = b,\qquad \sum_i x_i \le b,\qquad \sum_i x_i \ge b.$$9 is an ancilla register that labels the Hamming-weight branch, and $$\vert D^n_k (\vec{\theta})\rangle = \sum_i \mathcal{P}_i c_i(\vec{\theta}) \vert 0 \rangle^{\otimes(n-k)}\otimes\vert 1\rangle^k,$$0 is a learned probability distribution over feasible weights (Scursulim, 7 Jun 2026).

The corresponding density matrix is

$$\vert D^n_k (\vec{\theta})\rangle = \sum_i \mathcal{P}_i c_i(\vec{\theta}) \vert 0 \rangle^{\otimes(n-k)}\otimes\vert 1\rangle^k,$$1

Tracing out the ancilla yields the reduced state

$$\vert D^n_k (\vec{\theta})\rangle = \sum_i \mathcal{P}_i c_i(\vec{\theta}) \vert 0 \rangle^{\otimes(n-k)}\otimes\vert 1\rangle^k,$$2

which is the central mixed Dicke state ansatz (Scursulim, 7 Jun 2026).

This construction distinguishes equality and inequality constraints precisely. For $$\vert D^n_k (\vec{\theta})\rangle = \sum_i \mathcal{P}_i c_i(\vec{\theta}) \vert 0 \rangle^{\otimes(n-k)}\otimes\vert 1\rangle^k,$$3, the pure ansatz is recovered as a special case. For $$\vert D^n_k (\vec{\theta})\rangle = \sum_i \mathcal{P}_i c_i(\vec{\theta}) \vert 0 \rangle^{\otimes(n-k)}\otimes\vert 1\rangle^k,$$4, the mixture spans $$\vert D^n_k (\vec{\theta})\rangle = \sum_i \mathcal{P}_i c_i(\vec{\theta}) \vert 0 \rangle^{\otimes(n-k)}\otimes\vert 1\rangle^k,$$5; for $$\vert D^n_k (\vec{\theta})\rangle = \sum_i \mathcal{P}_i c_i(\vec{\theta}) \vert 0 \rangle^{\otimes(n-k)}\otimes\vert 1\rangle^k,$$6, it spans $$\vert D^n_k (\vec{\theta})\rangle = \sum_i \mathcal{P}_i c_i(\vec{\theta}) \vert 0 \rangle^{\otimes(n-k)}\otimes\vert 1\rangle^k,$$7 (Scursulim, 7 Jun 2026). The mixture is therefore a probability distribution over feasible Hamming weights rather than a single fixed-weight state.

For diagonal Ising Hamiltonians

$$\vert D^n_k (\vec{\theta})\rangle = \sum_i \mathcal{P}_i c_i(\vec{\theta}) \vert 0 \rangle^{\otimes(n-k)}\otimes\vert 1\rangle^k,$$8

the density-matrix VQE objective becomes

$$\vert D^n_k (\vec{\theta})\rangle = \sum_i \mathcal{P}_i c_i(\vec{\theta}) \vert 0 \rangle^{\otimes(n-k)}\otimes\vert 1\rangle^k,$$9

which reduces to a classical weighted sum over measurement probabilities (Scursulim, 7 Jun 2026). The appendix result

$k$0

makes explicit that the probability assigned to a basis state factorizes into a learned weight-sector probability and an intra-sector amplitude contribution (Scursulim, 7 Jun 2026).

A common misconception is to treat all uses of “mixed Dicke” as density-matrix mixtures. In fact, the 2025 multiclass portfolio work uses “mixed” in a different sense: a tensor product of multiple Dicke blocks, one per asset class, each with its own fixed Hamming weight (Scursulim et al., 19 Aug 2025). By contrast, the 2026 density-matrix formulation uses a genuine mixture over feasible Hamming-weight sectors to encode inequality constraints (Scursulim, 7 Jun 2026).

4. Multiclass and multi-group constructions

In multiclass portfolio optimization, if the portfolio contains multiple asset classes, each class is represented by its own Dicke state, and the full initialization is a tensor product

$k$1

Different classes may have different sizes $k$2 and required selected counts $k$3 (Scursulim et al., 19 Aug 2025). Feasibility follows immediately because each class block has fixed Hamming weight, so every sampled bitstring satisfies the class-wise diversification constraints.

The parameter count for this multiclass ansatz is

$k$4

which reduces in the single-block case to

$k$5

(Scursulim et al., 19 Aug 2025).

The 2026 framework generalizes the same principle to multiple independent constraint groups using tensor products of pure or mixed blocks: $k$6 This permits equality constraints in some groups and inequality constraints in others within one global ansatz (Scursulim, 7 Jun 2026).

A plausible implication is that the two works are structurally aligned despite using different notions of “mixed.” Both use blockwise composition to encode feasibility locally within groups of variables, and both treat the pure fixed-weight Dicke construction as the basic primitive. The distinction lies in whether “mixed” refers to multiple class-specific pure Dicke blocks (Scursulim et al., 19 Aug 2025) or to a reduced density matrix over multiple feasible Hamming weights (Scursulim, 7 Jun 2026).

5. Role in variational quantum eigensolver workflows

The VQE objective in the multiclass formulation is

$k$7

where $k$8 is derived from the portfolio optimization QUBO or Ising Hamiltonian (Scursulim et al., 19 Aug 2025). The classical problem begins from the mean-variance objective

$k$9

and, after QUBO conversion,

$\mathcal{P}_i$0

Because the Dicke ansatz enforces feasibility, the penalty can be removed by setting $\mathcal{P}_i$1 (Scursulim et al., 19 Aug 2025).

The same mean-variance objective is used in the 2026 study,

$\mathcal{P}_i$2

with Hamming-weight constraints on the number of selected assets (Scursulim, 7 Jun 2026). In both cases, the elimination of penalty terms is not a secondary convenience but a defining feature of the ansatz.

The hybrid loop follows the standard VQE pattern: prepare $\mathcal{P}_i$3, measure the cost Hamiltonian to estimate $\mathcal{P}_i$4, pass the estimate to a classical optimizer, update $\mathcal{P}_i$5, and repeat until convergence or an iteration limit (Scursulim et al., 19 Aug 2025). The 2025 work compares CMA-ES, COBYLA, Random Sampler, SPSA, and QNSPSA, using 100 random initializations, 4096 shots per circuit, up to 1000 iterations, and parameter initialization in $\mathcal{P}_i$6 (Scursulim et al., 19 Aug 2025). The 2026 work focuses on CMA-ES as primary optimizer, compares with COBYLA, and uses random search with replacement restricted to the feasible subspace as baseline; its sampling schedule is $\mathcal{P}_i$7 shots with $\mathcal{P}_i$8, $\mathcal{P}_i$9 iterations with $k$0, 100 independent runs per configuration, and random seed fixed to 42 (Scursulim, 7 Jun 2026).

6. Empirical behavior, performance measures, and scaling

The 2025 multiclass study evaluates the ansatz using three metrics: the approximation ratio,

$k$1

the frequency of finding the optimal solution across 100 trials, and the measurement probability $k$2 of the optimal state (Scursulim et al., 19 Aug 2025). The principal reported finding is that CMA-ES gives the best overall performance among the tested optimizers in terms of convergence rate, approximation ratio, and measurement probability (Scursulim et al., 19 Aug 2025). It has the highest frequency of obtaining the optimal solution, tends to concentrate probability mass near the ground state, and increasing the number of iterations improves $k$3; after roughly $k$4 in Scenario I, the average probability enters the region $k$5 (Scursulim et al., 19 Aug 2025). The same study notes that COBYLA is faster, QNSPSA can also perform well but is much slower, and Random Sampler is worst (Scursulim et al., 19 Aug 2025).

The 2026 study emphasizes a different comparison: performance against feasible random search as the feasible search space grows. It considers three portfolio scenarios of increasing complexity. Scenario I uses $k$6 assets with $k$7 and feasible search space

$k$8

Scenario II uses $k$9 with $c_i(\vec{\theta})$0 and

$c_i(\vec{\theta})$1

Scenario III uses four sector groups of five assets each, with constraints $c_i(\vec{\theta})$2, $c_i(\vec{\theta})$3, $c_i(\vec{\theta})$4, and $c_i(\vec{\theta})$5, producing

$c_i(\vec{\theta})$6

(Scursulim, 7 Jun 2026).

The reported trend is that the advantage over random feasible search is modest in small feasible spaces but increases as the feasible subspace grows from Scenario I to Scenario III, specifically in terms of the number of objective function calls needed to find the optimum (Scursulim, 7 Jun 2026). The paper interprets this as evidence that the ansatz concentrates probability mass effectively in large constrained spaces where exhaustive feasible sampling becomes impractical (Scursulim, 7 Jun 2026). It also notes that even when the optimizer does not fully collapse onto the optimal bitstring, the sampled distribution still yields good feasible portfolios, often near the classical efficient frontier (Scursulim, 7 Jun 2026).

7. Practical limitations, noise sensitivity, and related Dicke-state research

Both studies present the Dicke-state approach as promising but qualified by nontrivial implementation costs. The 2025 paper notes that Dicke-state circuits require a nontrivial number of parameters and entangling gates; for larger instances, circuit depth and hardware noise become concerns; and the method is explicitly sensitive to bit-flip, readout, and coherent errors (Scursulim et al., 19 Aug 2025). Because noise can destroy the fixed-Hamming-weight structure, optimization becomes harder. That work also states that the reported results are from noiseless simulation rather than hardware, and suggests error-mitigation methods including ZNE, PEC, PEA, and readout-error mitigation methods such as M3 or T-REX for noisy-device implementations (Scursulim et al., 19 Aug 2025).

The 2026 study includes hardware experiments on IBM processors ibm_kingston, ibm_fez, and ibm_marrakesh, using 4096 shots and Qiskit Sampler and Estimator primitives (Scursulim, 7 Jun 2026). Hardware and simulation distributions are compared using KL divergence,

$c_i(\vec{\theta})$7

and Hellinger fidelity,

$c_i(\vec{\theta})$8

(Scursulim, 7 Jun 2026). The reported hardware findings are unfavorable: sampler results show low fidelity and high KL divergence in most cases, and estimator results show relative errors around 50% across devices and scenarios (Scursulim, 7 Jun 2026). The paper attributes this primarily to two-qubit gate noise associated with conditional preparation blocks containing many entangling gates, and further notes sensitivity to bit-flip and readout errors because such errors can change Hamming weight and push states out of the feasible sector (Scursulim, 7 Jun 2026).

The same work models bit-flip noise as

$c_i(\vec{\theta})$9

and its two-qubit generalization as

$$\lvert D_k^n(\vec{\theta})\rangle = \sum_i \mathcal{P}_i \, a_i(\vec{\theta}) \lvert 0\rangle^{\otimes (n-k)}\otimes \lvert 1\rangle^{\otimes k},$$0

(Scursulim, 7 Jun 2026). It argues that the mixed Dicke ansatz is somewhat more robust than the pure one because only the boundary Hamming-weight sectors can be pushed outside feasibility by a single bit flip, whereas interior sectors require multiple flips (Scursulim, 7 Jun 2026). The paper explicitly identifies error mitigation and transpilation optimization to reduce two-qubit gate depth as open challenges for practical deployment (Scursulim, 7 Jun 2026).

A separate Dicke-state literature should be distinguished from the optimization ansatz literature. The state-preparation work on collective spin systems uses one-axis twisting, rapid adiabatic passage, and approximate counterdiabatic driving to prepare Dicke states such as $$\lvert D_k^n(\vec{\theta})\rangle = \sum_i \mathcal{P}_i \, a_i(\vec{\theta}) \lvert 0\rangle^{\otimes (n-k)}\otimes \lvert 1\rangle^{\otimes k},$$1 or, for odd $$\lvert D_k^n(\vec{\theta})\rangle = \sum_i \mathcal{P}_i \, a_i(\vec{\theta}) \lvert 0\rangle^{\otimes (n-k)}\otimes \lvert 1\rangle^{\otimes k},$$2, the symmetric superposition $$\lvert D_k^n(\vec{\theta})\rangle = \sum_i \mathcal{P}_i \, a_i(\vec{\theta}) \lvert 0\rangle^{\otimes (n-k)}\otimes \lvert 1\rangle^{\otimes k},$$3; it is explicitly described as a pure-state preparation protocol rather than a mixed-state ansatz in the density-matrix sense (Tang et al., 14 May 2026). Likewise, the optical study of hyperentangled mixed phased Dicke states concerns experimental generation and entanglement verification of noisy four-qubit phased Dicke states, not a variational feasibility-preserving ansatz for constrained optimization (Chiuri et al., 2010). These distinctions matter because the phrase “mixed Dicke state” appears in several research contexts with materially different meanings.

Taken together, the optimization-oriented literature defines the feasibility-preserving mixed Dicke state ansatz as a structured variational family that encodes Hamming-weight feasibility directly into the quantum state. In its pure form, it enforces equality constraints exactly; in its mixed density-matrix form, it extends that capability to inequality constraints; and in multiclass tensor-product form, it enforces diversification constraints across distinct variable groups (Scursulim, 7 Jun 2026, Scursulim et al., 19 Aug 2025). The principal technical trade-off is clear from the reported results: constraint handling is shifted from Hamiltonian penalization to state preparation, reducing the effective search space while introducing nontrivial circuit depth and noise-sensitivity challenges on current NISQ hardware.