Pure and mixed Dicke state ansatz for equality and inequality constraints in variational quantum eigensolver

This paper introduces a feasibility-preserving mixed Dicke state ansatz for the Variational Quantum Eigensolver that structurally encodes both equality and inequality Hamming weight constraints to eliminate the need for penalty terms, demonstrating superior performance over random search in combinatorial portfolio optimization while highlighting remaining challenges for NISQ hardware deployment.

The Gist

The Big Picture: Finding the Best Team in a Sea of Options

Imagine you are a manager trying to build the perfect team of employees from a pool of 100 candidates. You have two main goals:

1. Maximize performance (get the best results).
2. Follow strict rules (e.g., "You must pick exactly 5 people," or "You must pick between 3 and 7 people").

In the world of finance, this is called Portfolio Optimization. Instead of employees, you are picking stocks. Instead of performance, you are looking for high returns with low risk.

The problem is that as the number of candidates grows, the number of possible teams explodes. Checking every single combination one by one (like a brute-force search) takes forever. This is where Quantum Computing comes in. It promises to explore these massive possibilities much faster than a regular computer.

The Problem: The "Penalty" Trap

In the past, when scientists tried to solve this with quantum computers, they used a method called the Variational Quantum Eigensolver (VQE). Think of VQE as a student trying to solve a math problem.

To make sure the student follows the rules (like "pick exactly 5 stocks"), the teacher usually adds a penalty.

• Teacher: "If you pick 6 stocks, you get a huge red mark on your paper."
• Student: "Okay, I'll try to avoid the red mark."

The problem is that the teacher has to guess how big that red mark should be. If the penalty is too small, the student ignores the rules. If it's too big, the student gets confused and can't find the best solution. Tuning this "penalty" is a huge headache and often leads to bad results.

The Solution: Building the Rules into the Blueprint

This paper introduces a new way to build the quantum computer's "student" (called an Ansatz). Instead of adding penalties after the fact, the authors build the rules directly into the student's DNA.

They use something called Dicke States.

• The Analogy: Imagine a magical box that only spits out teams of exactly 5 people. You can't ask the box to give you 4 or 6. It is physically impossible for the box to break the rule.
• Pure Dicke State: This is the box that only spits out teams of exactly 5. This solves the "Equality Constraint" (must be exactly 5).
• Mixed Dicke State: This is the paper's big innovation. Imagine a box that can spit out teams of 3, 4, 5, 6, or 7 people, but never 2 or 8. It is a "mixture" of different valid team sizes. This solves the "Inequality Constraint" (must be between 3 and 7).

By using Density Matrices (a fancy math way of describing a mix of possibilities), the authors created a quantum circuit that only ever explores valid solutions.

• No Penalties Needed: Since the machine physically cannot generate an invalid team, you don't need to add red marks or penalties.
• No Tuning: You don't need to guess how strict the rules should be; the rules are hard-wired into the machine.

How They Tested It

The authors tested this idea using a "Combinatorial Portfolio Optimization" problem (picking the best mix of stocks). They created three scenarios, like climbing a mountain with increasing difficulty:

1. Scenario 1 (Small Hill): Pick up to 4 stocks from 11 options.
2. Scenario 2 (Medium Hill): Pick between 3 and 6 stocks from 11 options.
3. Scenario 3 (Big Mountain): A complex mix where different groups of stocks have different rules (e.g., "Pick exactly 3 from Energy," "Pick 1 or 2 from Finance").

They compared their new "Rule-Built-In" quantum method against a Random Search (just guessing valid teams randomly).

The Results:

• As the number of possible valid teams got bigger (from Scenario 1 to 3), their method got much better than random guessing.
• Random guessing is like throwing darts blindfolded; eventually, you might hit the bullseye, but it takes a long time. Their method is like a guided missile that only flies toward the valid targets.
• They found high-quality solutions (portfolios on the "efficient frontier," which is the best possible balance of risk and reward) much faster than random search.

The Catch: Real-World Noise

The paper also tested this on real quantum computers (IBM's noisy machines).

• The Issue: Real quantum computers are like delicate instruments; they get "noisy." A tiny bit of interference can flip a bit (change a 0 to a 1).
• The Risk: If a bit flips, a valid team of 5 might accidentally become a team of 6, breaking the rule.
• The Finding: The authors found that their "Mixed" method (the box that allows 3, 4, 5, 6, or 7) is actually more robust against these errors than the strict "Pure" method. If a single error happens, the "Mixed" box is more likely to stay within the valid range than the strict box.
• The Reality Check: Despite this advantage, the real hardware is still very noisy. The results on real machines had about a 50% error rate compared to simulations. The paper concludes that while the idea is brilliant, we need better "noise cancellation" technology before this can be used for real money management.

Summary

This paper proposes a clever trick for quantum computers: Stop punishing bad answers; instead, build a machine that can't even make them. By structurally encoding the rules (like "pick 3 to 7 stocks") directly into the quantum circuit using "Mixed Dicke States," they eliminated the need for tricky penalty tuning. Their experiments showed this method finds the best solutions much faster than random guessing, especially for complex problems, though real-world hardware noise remains a hurdle to overcome.

Technical Summary

Technical Summary: Pure and Mixed Dicke State Ansatz for Equality and Inequality Constraints in Variational Quantum Eigensolver

Problem Statement
The paper addresses a central challenge in applying Variational Quantum Algorithms (VQAs), specifically the Variational Quantum Eigensolver (VQE), to combinatorial optimization: designing an ansatz expressive enough to explore the feasible subspace of the Hilbert space while respecting problem constraints. Traditional approaches often rely on penalty terms in the objective function to enforce constraints (such as Hamming weight limits), requiring the difficult tuning of Lagrange multipliers. Furthermore, as the number of variables and constraints scales, the computational cost of finding optimal solutions grows exponentially, rendering exact classical methods intractable and forcing reliance on suboptimal approximations. The specific problem domain highlighted is combinatorial portfolio optimization, where the goal is to select a subset of assets to maximize return while minimizing risk, subject to constraints on the number of assets selected (Hamming weight).

Methodology
The authors propose a feasibility-preserving framework that structurally encodes both equality and inequality constraints directly into the quantum circuit ansatz, eliminating the need for penalty terms.

1. Pure and Mixed Dicke States:

   • Equality Constraints: For constraints requiring an exact number of selected assets ($k=b$), the authors utilize a parameterized pure Dicke state ansatz. This state is a superposition of computational basis states with a fixed Hamming weight $k$, naturally restricting the search to the feasible subspace.
   • Inequality Constraints: For constraints involving upper or lower bounds ($k \leq b$ or $k \geq b$), the authors extend the formalism to mixed Dicke states. By employing the density matrix formalism and open quantum systems theory, they construct an ansatz that represents an ensemble (mixture) of Dicke states with varying Hamming weights within the feasible range.
   • Implementation: The mixed state is generated using ancilla qubits to control the conditional preparation of different Dicke states. The reduced density matrix of the system (obtained by tracing out the ancilla) describes the mixture. The VQE objective function is generalized to minimize the trace of the product of this density matrix and the problem Hamiltonian: $\min \text{tr}(\sigma H)$.

2. Tensor Product Structure:
   The framework generalizes to problems with multiple constraint groups (e.g., different sectors in a portfolio) by constructing the global ansatz as a tensor product of individual pure or mixed Dicke state density matrices. This allows for the simultaneous satisfaction of diverse constraint types (equality and inequality) across different subsets of variables.

3. Experimental Setup:
   The approach was validated using combinatorial portfolio optimization with three scenarios of increasing complexity:

   • Scenario I: 11 assets with an upper bound constraint ($k \leq 4$).
   • Scenario II: 11 assets with a range constraint ($3 \leq k \leq 6$).
   • Scenario III: A multi-sector portfolio (4 sectors, 5 assets each) with a mix of equality and inequality constraints, utilizing a tensor product ansatz.
     The optimization was performed using the CMA-ES (Covariance Matrix Adaptation Evolution Strategy) optimizer, a gradient-free method, and compared against random search restricted to the feasible subspace. Simulations were conducted on classical hardware (CPU/GPU) and validated on IBM NISQ processors.

Key Results

• Search Efficiency: As the size of the feasible search space increased (from 561 in Scenario I to 45,000 in Scenario III), the proposed VQE-Dicke framework demonstrated a clear advantage over random search. It required significantly fewer objective function calls to identify high-quality solutions and the optimal solution.
• Solution Quality: In all scenarios, the bitstrings sampled from the optimized distribution mapped to portfolios lying on or near the classical efficient frontier. Even when the optimizer did not fully concentrate the density matrix on the single optimal bitstring, the sampled solutions remained high-quality feasible candidates.
• Hardware Performance: Experiments on IBM NISQ processors confirmed the theoretical feasibility but highlighted practical limitations. The ansatz showed relative errors of approximately 50% on quantum processing units (QPUs). The authors note that while the mixed state ansatz offers greater theoretical robustness to single bit-flip errors (which might displace a pure state out of the feasible set), the cumulative noise from two-qubit gates remains a significant challenge.

Significance and Claims
The paper claims to introduce the first feasibility-preserving mixed Dicke state ansatz capable of handling both equality and inequality constraints in VQE without penalty terms. The primary significance lies in:

1. Structural Constraint Encoding: By embedding constraints directly into the ansatz via the density matrix formalism, the method removes the need for Lagrange multiplier tuning and reduces the optimization landscape complexity.
2. Scalability: The approach is particularly effective in regimes where the feasible subspace is large enough to make exhaustive random sampling impractical, yet small enough to be a fraction of the total Hilbert space ($O(n^k)$ vs $O(2^n)$).
3. Generality: While validated on portfolio optimization, the framework is asserted to be directly applicable to other combinatorial problems with Hamming weight constraints, such as feature selection and ensemble selection.

The authors maintain a modest stance regarding practical deployment, acknowledging that noise mitigation and circuit transpilation optimizations remain open challenges for NISQ-era hardware. They emphasize that the current work serves as a theoretical and experimental validation of the ansatz's potential, rather than a fully realized commercial solution.