Decoded Quantum Interferometry Benchmark

---
name: decoded-quantum-interferometry-benchmark
description: Complexity-theoretic benchmarking methodology for decoded quantum interferometry (DQI) and bounded-degree constraint satisfaction problems. Analyzes quantum advantage limits, classical vs quantum decoder performance, and approximation hardness over finite fields.
category: quantum-information
version: "1.0"
created: "2026-06-14"
source: "arXiv:2606.13570"
authors: "Maximilian J. Kramer, Carsten Schubert, Jens Eisert"
tags:
  - quantum-computing
  - information-theory
  - constraint-satisfaction
  - decoded-quantum-interferometry
  - approximation-algorithms
  - complexity-theory
  - finite-fields
  - quantum-decoding
  - bounded-degree
---

# Decoded Quantum Interferometry Benchmark

## Overview

Methodology for analyzing the complexity-theoretic limits of quantum algorithms on bounded-degree constraint satisfaction problems (CSPs), specifically decoded quantum interferometry (DQI), QAOA, and classical heuristics.

## Core Theory

### Bounded-Degree CSP Complexity Landscape

For max-k-XORSAT with bounded degree D (each variable appears in at most D constraints):

1. **Unbounded vs Bounded**: Unbounded max-k-XORSAT (k≥3) has no polynomial-time algorithm better than random guessing (NP-hard). Bounded-degree instances allow beating random by O(1/√D).

2. **Finite Field Extension**: Extended hardness to max-Ek-LINSAT(q,r) over arbitrary finite fields Fq: NP-hard to exceed r/q + O_{q,r}(1/√D).

3. **Quantum Advantage Limit**: Any quantum advantage on bounded-degree instances is confined to the constant prefactor only - not asymptotic scaling.

### Decoder Performance Barriers

| Decoder Type | Scaling Barrier | Can Match Hardness? |
|-------------|-----------------|---------------------|
| Classical DQI | O(1/√(D log D)) | ❌ No |
| Quantum DQI | O(1/√D) | ✅ Yes |

## Implementation Patterns

### Pattern 1: Complexity-Theoretic Benchmark

```python
def compute_approximation_bound(degree_D, field_q, r_value=1):
    """
    Compute the approximation hardness bound for max-Ek-LINSAT(q,r)
    
    Args:
        degree_D: Maximum degree (constraints per variable)
        field_q: Finite field size
        r_value: Target value parameter
    
    Returns:
        approximation_threshold: r/q + O(1/sqrt(D))
    """
    import math
    baseline = r_value / field_q
    hard_additive = 1 / math.sqrt(degree_D)
    return baseline + hard_additive
```

### Pattern 2: Decoder Performance Comparison

```python
def decoder_comparison(degree_D):
    """
    Compare classical vs quantum decoder performance scaling
    
    Args:
        degree_D: Maximum degree
    
    Returns:
        dict with classical and quantum decoder barriers
    """
    import math
    return {
        'classical_barrier': 1 / math.sqrt(degree_D * math.log(degree_D)),
        'quantum_barrier': 1 / math.sqrt(degree_D),
        'quantum_advantage_factor': math.sqrt(math.log(degree_D))
    }
```

### Pattern 3: Quantum Advantage Assessment

```python
def assess_quantum_advantage(problem_type, degree_D, algorithm='DQI'):
    """
    Assess whether quantum advantage is possible for bounded-degree CSPs
    
    Returns:
        dict with advantage assessment and limitations
    """
    results = {
        'advantage_type': 'constant_prefactor_only',
        'asymptotic_scaling': 'same_as_classical',
        'decoder_critical': True,
        'quantum_decoder_required': algorithm == 'DQI',
        'recommendation': 'Focus on improving constant prefactor, not scaling'
    }
    return results
```

## Key Findings

1. **Information-Theoretic Barrier**: Classical decoders face 1/√(D log D) barrier
2. **Quantum Decoding Essential**: Quantum decoders can match 1/√D complexity-theoretic scaling
3. **Constant Prefactor Focus**: Quantum advantage is limited to constant factor improvement
4. **Finite Field Generality**: Results hold for arbitrary finite fields Fq
5. **Benchmark for Algorithms**: Provides complexity-theoretic benchmark for DQI, QAOA, and classical heuristics

## When to Use

- Analyzing quantum algorithms for constraint satisfaction problems
- Benchmarking DQI, QAOA, or classical heuristics on bounded-degree instances
- Assessing fundamental limits of quantum advantage
- Comparing classical vs quantum decoder performance
- Designing quantum algorithms with realistic performance expectations

## References

- arXiv:2606.13570 - "Approximability limits for bounded-degree max-LINSAT and implications for decoded quantum interferometry"
- Trevisan (hardness result for Boolean instances)
- Barak et al. (algorithmic guarantee for Boolean instances)