Quantum Logical Spectroscopy

---
name: quantum-logical-spectroscopy
category: quantum-computing
description: Logical spectroscopy methodology for constructing addressable conjugate bases in Abelian lifted-product quantum LDPC codes using CRT decomposition of group algebras.
trigger_words: ["logical spectroscopy", "lifted-product", "qLDPC", "CRT decomposition", "group algebra", "logical operators", "addressable bases", "hypergraph-product"]
---

# Quantum Logical Spectroscopy for qLDPC Codes

Methodology from arXiv:2607.05386 (Lee, Jul 2026).

## Problem

Quantum LDPC memories encode many logical qubits, but dimension alone doesn't make them usable. Applications need explicit conjugate logical operators with structured labels and physical representatives. For hypergraph-product (HGP) codes this structure is transparent (binary matrices, row-reduction over F2), but Abelian lifted-product codes are subtler — seed entries are shifts in a group-algebra ring, not a field, so pivot blocks may not be invertible and global row reduction can fail.

## Solution: Logical Spectroscopy

A spectral construction that replaces global row reduction with finite-field computations in Frobenius character packets of the Abelian lift group.

### Core Steps

1. **CRT Decomposition**: Decompose the group algebra into Frobenius character packets using the Chinese Remainder Theorem
2. **Packet Computation**: In each packet, compute kernels, quotients, and product-complex homology over finite fields
3. **CRT Lifting**: Lift resulting representatives back using CRT idempotents
4. **Logical Pairing**: Pair X and Z logicals through reciprocal trace-dual packets
5. **Design Diagnostics**: Use packet ranks to show how logical sectors split, certify basis width bounds

### Key Properties

- Gives complete addressable conjugate logical bases for finite Abelian lifted products LP(A,B)
- Preserves layout freedom of group-algebra lifts while gaining HGP-like transparency
- Packet ranks serve as working coordinates: label logical sectors, certify basis width, attribute structured erasure failures
- Under bounded seed-shape and group-basis-support assumptions, construction gives qLDPC families with HGP-like features

## Application

Use when designing or analyzing quantum LDPC error-correcting codes, especially:
- Constructing logical operator bases for Abelian lifted-product codes
- Certifying logical qubit layouts and erasure failure attribution
- Designing qLDPC families with addressable logical operators

## Implementation Pattern

```
Group Algebra → CRT Decomposition → Character Packets → 
  [per-packet: kernel, quotient, homology] → 
  CRT Lift → Conjugate Pairing → Addressable Basis
```