⚛️ quantum physics

Quantum approximate optimization of finite-state bosonic systems

This paper proposes a Hamiltonian-based Quantum Approximate Optimization Algorithm (QAOA) that utilizes specific mixing Hamiltonians to exclude infeasible subspaces when solving finite-state bosonic problems on qubit hardware, demonstrating that binary, symmetric, and unary mapping techniques offer varying implementation costs and successfully applying the framework to find the ground state of the repulsive Bose-Hubbard model.

Original authors: Shakib Daryanoosh

Published 2026-02-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Shakib Daryanoosh

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a complex puzzle, but you only have a box of square pegs (qubits), while your puzzle pieces are round, triangular, and hexagonal (qudits or multi-level systems). This is the fundamental challenge when trying to simulate complex physical systems, like groups of interacting atoms, on today's quantum computers.

This paper, by Shakib Daryanoosh, proposes a clever way to fit those round puzzle pieces into the square box without wasting time or energy on pieces that don't fit.

Here is the breakdown using simple analogies:

1. The Problem: The "Forbidden Room"

In nature, many systems (like bosons in a Bose-Hubbard model) have a specific number of states. Let's say a system has 3 states (like a light switch that can be Off, Low, or High).

The Old Way: To put this on a quantum computer, we usually use binary code (0s and 1s). Two switches can make 4 combinations (00, 01, 10, 11).
The Issue: We only need 3 of those combinations. The 4th one (11) is a "forbidden room" or an infeasible state. It doesn't exist in our real-world physics problem.
The Old Fix: Previously, scientists would tell the computer, "If you go into the forbidden room, I'll give you a huge penalty." But this is like trying to find a needle in a haystack by telling the computer to avoid the whole barn. As the problem gets bigger, the "forbidden rooms" explode in number, making the search incredibly slow and inefficient.

2. The Solution: Building a "One-Way Door"

Instead of penalizing the computer for going to the wrong place, the author suggests building a mixer (a specific type of quantum operation) that physically prevents the system from ever entering the forbidden room in the first place.

Think of it like a maze.

Old Method: Let the maze runner wander everywhere, but if they hit a wall (the forbidden state), you shout "Stop!" and make them start over.
New Method: Build the maze with walls that only allow movement through the valid paths. The runner never even sees the dead ends.

3. The Three Maps (Encoding Schemes)

To translate the physics problem into the quantum computer's language, the author tests three different "maps" or encoding schemes:

Binary Encoding (The Compact Map): This is like using a standard phone keypad. It uses the fewest buttons (qubits) but creates a lot of "forbidden rooms" that need to be avoided. It's efficient in space but messy to navigate.
Unary Encoding (The One-Hot Map): This is like having a row of light switches where only one can be on at a time. It uses a lot of switches (qubits) but is very easy to understand. However, the "mixer" (the mechanism that moves between states) becomes very heavy and expensive to build.
Symmetric Encoding (The Balanced Map): This is the paper's star player. It arranges the switches so that the "forbidden rooms" are naturally excluded by the laws of symmetry.
- The Analogy: Imagine a dance floor where dancers must always hold hands in pairs. The "mixer" is a dance move that swaps partners. If you design the dance move correctly, you can never end up with a dancer standing alone (the forbidden state).

4. The Big Discovery: Why Symmetry Wins

The author calculated the "cost" of these methods. In quantum computing, the most expensive and error-prone operations are entangling gates (specifically CNOT gates), which are like the "glue" that connects qubits.

The Result: The Symmetric Encoding is the clear winner.
- For the other methods, as you add more layers to your algorithm (making the simulation deeper), the number of "glue" operations needed grows exponentially. It's like trying to build a skyscraper where every new floor requires double the amount of cement.
- For Symmetric Encoding, the standard "mixer" (a simple flip of switches) works perfectly. It requires zero extra glue (entangling gates) to keep the system in the valid zone. It's like a magic trick where the rules of the game naturally keep you on the right path.

5. Real-World Applications

The author tested this idea on two specific physics problems:

Quantum Thermalization (Heating things up): Simulating how a system reaches a comfortable temperature (like a cup of coffee cooling down). The Symmetric method reached the "correct" temperature state much faster and with higher accuracy than the Binary method.
The Bose-Hubbard Model (Superfluids vs. Insulators): This models how atoms behave in a grid.
- In a strong interaction regime (atoms hate each other and stay put), the Symmetric method found the solution almost instantly with very few resources.
- In a weak interaction regime (atoms flow freely like a superfluid), the solution is more complex and "entangled." Here, the Symmetric method still performed well, though it required a deeper "circuit" (more steps) to capture the complexity, just as a more complex dance requires more steps.

The Takeaway

This paper is a guidebook for building better quantum simulations. It tells us: "Don't just try to punish the computer for making mistakes. Instead, design the rules of the game so that mistakes are impossible to make."

By using Symmetric Encoding, we can simulate complex bosonic systems (like atoms and light) much more efficiently, using fewer quantum resources and avoiding the "forbidden rooms" that slow down current methods. It's a shift from brute-force searching to elegant, rule-based navigation.

1. Problem Statement

Many physical systems, particularly in quantum mechanics, chemistry, and material science, are naturally described by finite-dimensional $D$ -level states (qudits), such as bosonic modes with a truncated Fock space. To simulate these on current qubit-based quantum hardware, one must map the $D$ -dimensional Hilbert space to a multi-qubit space of dimension $2^K$ (where $K = \lceil \log_2 D \rceil$ ).

The Core Challenge:

Infeasible Subspace: Unless $D = 2^K$ , the mapping is not one-to-one. This creates a large "infeasible" or "illegitimate" subspace within the $2^K$ qubit space that does not correspond to valid physical states.
Inefficiency of Penalties: A common approach to handle this is adding penalty terms to the cost Hamiltonian to push the system out of the infeasible subspace. However, as $D$ grows, the infeasible subspace becomes exponentially larger than the physical subspace, making the search for the optimal solution inefficient.
Resource Cost: Standard Quantum Approximate Optimization Algorithm (QAOA) mixers (typically sums of Pauli-X operators) do not preserve the feasible subspace, requiring complex constraints or penalties.

2. Methodology

The author proposes a Hamiltonian-based QAOA (specifically the Quantum Alternating Operator Ansatz) approach where the mixing Hamiltonian ( $\hat{H}_M$ ) is explicitly designed to preserve the feasible subspace, thereby ruling out infeasible configurations without penalty terms.

Key Components:

Encoding Schemes: The paper investigates three distinct mapping strategies from qudits to qubits:
- Binary Encoding: Maps $|d\rangle$ to the binary representation of $d$ . Efficient in qubit count but creates a large infeasible subspace for $D < 2^K$ .
- Symmetric Encoding: Maps $|d\rangle$ to a superposition of all $K$ -qubit states with Hamming weight $d$ (where $K=D-1$ ). This utilizes the full symmetric subspace.
- Unary Encoding: Maps $|d\rangle$ to a state with a single excitation at position $d$ (requiring $K=D$ qubits).
Mixing Hamiltonian Design:
- Instead of the standard mixer $\hat{H}_M = \sum X_k$ , the paper derives specific mixers for each encoding that commute with the constraints of the feasible subspace.
- Cost Metric: The primary metric for evaluation is the CNOT gate count (entangling operations), as these are the most error-prone and costly operations in near-term quantum devices.
Applications:
- Quantum Approximate Thermalization: Generating thermal (Gibbs) states for coupled harmonic oscillators.
- Bose-Hubbard Model: Finding the ground state of repulsive bosons in a lattice, covering both strong (interaction-dominated) and weak (kinetic-dominated) regimes.

3. Key Contributions

A. Optimal Mixing for Symmetric Encoding

The paper proves that for Symmetric Encoding, the standard mixer (sum of Pauli-X operators, $\sum X_k$ ) is the optimal choice.

Theorem: The standard mixer commutes with all permutation operators of the qubits, thereby preserving the symmetric subspace (states with fixed Hamming weight).
Resource Efficiency: Unlike Binary and Unary encodings, the standard mixer for Symmetric encoding requires zero CNOT gates (it decomposes entirely into single-qubit $X$ rotations).
Scalability: For Binary and Unary encodings, the required CNOT count for the mixer scales linearly with the number of QAOA layers ( $p$ ) and grows exponentially with the system dimension $D$ . In contrast, Symmetric encoding maintains a constant, minimal CNOT cost regardless of $D$ .

B. Comprehensive Resource Analysis

The author provides a detailed breakdown of implementation costs, including:

Initial State Preparation: Symmetric encoding allows preparation via simple Hadamard gates (no entanglement), whereas other encodings or specific mixers may require entangled initial states (e.g., Bell states).
Measurement: While Symmetric encoding requires entangling measurements (Bell measurements) to read out in the computational basis, the paper argues that the savings in the mixer and circuit depth outweigh this cost, especially for high $D$ .

C. Numerical Validation

The framework was tested on two bosonic systems:

Coupled Harmonic Oscillators (Thermalization):
- Symmetric encoding achieved higher state fidelity ( $F \approx 0.93$ ) and lower quantum relative entropy compared to Binary encoding ( $F \approx 0.89$ ).
- It demonstrated robustness against temperature changes and depolarizing noise.
Bose-Hubbard Model (Ground State):
- Strong Interaction Regime: The ground state is a product state (vacuum). Both encodings converged quickly, but Symmetric required fewer resources.
- Weak Interaction Regime: The ground state is highly entangled. The paper notes that while Symmetric encoding is more efficient in gate count, the complex entangled nature of the ground state in this regime requires deeper circuits ( $p=50$ ) to achieve high fidelity, highlighting a trade-off between ansatz expressibility and gate cost.

4. Results

Gate Count Reduction: For a $p$ -layer QAOA, Binary and Unary encodings exhibit a $p$ -fold increase in CNOT gates compared to the optimal Symmetric encoding. For high-dimensional systems, the CNOT count for Binary/Unary grows exponentially, while Symmetric remains constant (effectively zero for the mixer).
Performance: Symmetric encoding consistently outperformed Binary encoding in terms of solution quality (fidelity) and resource efficiency for the thermalization task.
Noise Resilience: The Symmetric approach showed slightly better resilience to depolarizing noise on CNOT gates due to the reduced number of entangling operations.

5. Significance

Paradigm Shift: The paper challenges the reliance on penalty-based methods for handling constraints in QAOA, demonstrating that constraint-preserving mixers are superior for finite-state bosonic systems.
Hardware Efficiency: By identifying that Symmetric encoding allows the use of the standard, hardware-efficient mixer without entangling gates, the work provides a practical pathway for running complex bosonic simulations on noisy intermediate-scale quantum (NISQ) devices.
Scalability: The findings suggest that for problems with large $D$ , Symmetric encoding is the superior strategy, as the exponential growth of CNOT requirements in other encodings makes them infeasible for deep circuits.
Future Directions: The work opens avenues for exploring qudit-based architectures and developing error mitigation strategies that leverage the large infeasible subspaces available in symmetric and unary mappings.

In summary, this paper establishes Symmetric Encoding combined with the Standard Mixer as the optimal strategy for quantum approximate optimization of finite-state bosonic systems, offering a significant reduction in entangling gate overhead and improved solution fidelity compared to traditional Binary or Unary mappings.