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

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

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
    """
     math
     {
        :  / math.sqrt(degree_D * math.log(degree_D)),
        :  / math.sqrt(degree_D),
        : math.sqrt(math.log(degree_D))
    }