spatially-coupled-quantum-codes

name	spatially-coupled-quantum-codes
description	Spatial coupling methodology for quantum LDPC/CSS codes. Proves that belief-propagation decoding on spatially coupled CSS codes (MN/HA-type) achieves the quantum erasure hashing bound. Uses coupled-vector potential method and density evolution analysis to show BP threshold equals MAP threshold.

Spatially Coupled Quantum Codes

Description

Spatial coupling is a technique from classical coding theory adapted to quantum error correction. By spatially coupling CSS codes (Calderbank-Shor-Steane), belief-propagation (BP) decoding achieves the maximum-a-posteriori (MAP) performance — specifically the quantum erasure hashing bound — despite BP being a suboptimal decoder. This methodology provides a DE-level proof for seeded BP decoding on finite-degree factor graphs.

Activation Keywords

• spatially coupled quantum code
• spatial coupling CSS
• quantum erasure hashing bound
• seeded belief propagation
• MN/HA CSS code
• coupled vector potential
• density evolution quantum
• 空间耦合量子码
• 量子删除信道哈希界

Core Concepts

Spatial Coupling

• A technique that couples multiple instances of a base code along a chain
• Enables BP decoding to achieve MAP threshold (threshold saturation)
• Originally discovered for classical LDPC codes; extends to quantum CSS codes

CSS Codes from MN/HA Ensembles

• MacKay-Neal/Hsu-Anastasopoulos punctured sparse ensembles
• Achieve capacity under MAP decoding
• Spatial coupling + seeded BP = achieves hashing bound under BP

Quantum Erasure Channel & Hashing Bound

• On the quantum erasure channel, erased qubits are replaced by maximally mixed states
• Hashing bound: R = 1 - 2p (for erasure probability p)
• Achieving the hashing bound means optimal performance for the given code rate

Density Evolution (DE) Analysis

• Tracks message distributions through BP iterations
• Five-message DE recursion for CSS erasure decoding
• Decomposes into Z-side and X-side constituent systems

Methodology

Step 1: CSS Ensemble Definition

Define the CSS code ensemble:

1. Specify sparse punctured parity-check matrices for X and Z checks
2. Define corresponding dense parity-check matrices
3. Set finite Z-side degree, X-side degree, and check degrees

Step 2: Erasure Model Setup

On an erased coordinate:

• Two binary Pauli components remain unresolved
• Erased qubit represented by four Pauli possibilities (I, X, Y, Z)
• Map to hard-erasure CSS decoding problem

Step 3: Density Evolution Derivation

Derive the DE recursion:

1. Write five-message DE recursion for the uncoupled system
2. Decompose into Z-side and X-side constituent systems
3. Define two constituent potentials (one per side)

Step 4: Coupled-Vector Potential Analysis

Apply the coupled-vector potential method:

1. Apply to Z-side constituent → Z-side BP threshold
2. Apply to X-side constituent → X-side BP threshold
3. Combined threshold = min(Z-side degree ratio, X-side complementary degree ratio)

Step 5: Hashing Bound Achievement

For X/Z equal-rate specialization:

• Z-side and X-side constituent design rates are equal
• BP threshold = hashing-bound channel parameter determined by design rate
• Proves seeded BP achieves hashing bound on finite-degree factor graphs

Workflow

Pattern 1: Spatially Coupled CSS Code Design

For constructing spatially coupled CSS codes:

1. Choose base MN/HA ensemble parameters (Z-degree, X-degree, check degree)
2. Define spatial coupling window and chain length
3. Design seeded BP decoder with finite-degree factor graphs
4. Verify DE threshold equals hashing bound for equal-rate case

Pattern 2: Erasure Channel Performance Analysis

For analyzing erasure channel performance:

1. Model erasure probability distribution
2. Run DE analysis for the coupled system
3. Compare BP threshold vs MAP threshold
4. Verify threshold saturation (BP → MAP)

Error Handling

Finite-Length Effects

DE analysis assumes infinite block length. For finite-length codes:

• Use BP concentration results for finite-block analysis
• Consider block-error convergence behavior
• Design finite-code realization of ideal DE seed

Unequal X/Z Rates

For unequal X/Z design rates:

• BP threshold = min(Z-side degree ratio, X-side complementary degree ratio)
• Not equal to hashing bound; requires different analysis

Resources

• Paper: arXiv:2606.32001
• Related: quantum-error-correction-methods (umbrella), loss-biased-qec (bias-tailored QEC), coset-based-qldpc-codes (CSS code construction)