Improving quantum communication rates with permutation-invariant codes

This repository contains MATLAB code used to compute the coherent information of certain permutation-invariant quantum codes using representation theory as described in the paper

Sujeet Bhalerao, Felix Leditzky: "Improving quantum communication rates with permutation-invariant codes", arXiv:2508.09978.

Requirements

• Parallel Computing Toolbox
• MATLAB's particleswarm function, available in the "Global Optimization Toolbox"
• fastexpm function for computing matrix exponentials of sparse matrices (available at MATLAB File Exchange here)

Usage

To optimize over permutation-invariant codes:

1. Run the optimization script pauli_opt.m with the desired parameters. This is currently set up to optimize over permutation-invariant codes for the $2$-Pauli channel with $9$ copies at a noise level of $0.2271$, which is slightly beyond the hashing bound for this channel.

2. The results will be saved in the optimization_results directory.

To compute coherent information for a specific permutation-invariant code:

Run the script single_code_ci.m with the desired parameters and input density matrices. The current parameters are set for $9$ copies of a $2$-Pauli channel using the code listed in Table 4 of the paper.

• Using purifications: For qubit channels, this can compute the coherent information for a single code for up to $100$ channel qubits. For channels with qutrit output (e.g. the dephrasure channel), this can compute the coherent information for a single code for up to $30$ channel qubits.

• Using complementary channels: For channels with a $3$-dimensional environment (e.g. the $2$-Pauli channel), this can compute the coherent information for a single code for upto $40$ channel qubits. For channels with a $4$-dimensional environment (e.g. the BB84 channel), this can compute the coherent information for a single code for upto $15$ channel qubits.

Files

Optimization

• pauli_opt.m: Main optimization script that sets up the problem and calls the particle swarm optimization function.

• compute_ci_symmetries_opt.m: Objective function for the particle swarm optimization, which computes the coherent information for a given set of input states and probabilities using the complementary channel.

• compute_entropy_d2.m, compute_entropy_d3.m, compute_entropy_d4.m: Functions to compute the entropy of states in dimensions $2$, $3$, and $4$ respectively (e.g. as outputs of qubit channels with environments of dimensions $3$ and $4$) using representation theory.

Evaluating specific codes

• eval_codes.m: Script to verify the coherent information for the specific codes listed in Table 4 of the paper. Each case in the script sets the required channel parameters and code states, then calls the appropriate computation function. This script can be used as a template to evaluate other codes by modifying the parameters within a 'case' block.

• compute_ci_symmetries.m: Function to compute coherent information for a given set of input states and probabilities using our representation-theoretic approach - makes use of the complementary channel.

• compute_ci_purification.m: Function to compute coherent information using purifications for a given set of pure input states and probabilities using our representation-theoretic approach.

• compute_ci_purification_qutrit.m: Similar to the function above but for channels with qutrit output.

Auxiliary Files

• compute_symm_rep_formula.m: computes matrix elements for the GL(2) irrep on the symmetric subspace (see Appendix B in the paper).

• compute_S_RA_GL2.m, compute_S_RA_log.m, compute_S_RA_purification_qutrit.m: Functions to compute entropy at the environment for the coherent information. compute_S_RA_GL2 uses explicit GL(2) irreps, compute_S_RA_log uses a log/exp approach, and compute_S_RA_purification_qutrit is for qutrit channels.

• get_states_bloch.m and get_states_measure.m: two parametrization schemes for qubit input states, one using Bloch vectors and the other using measurement outcomes. See Appendix D of the paper for details.

• comp_kraus.m: Function to compute the Kraus operators of the complementary channel for a given set of operators and output dimensions.

• q_logm.m: Function to compute logarithm of the positive part of a matrix.

• The folders E_matrices and E_matrices_sparse contain precomputed matrices for the representation of the Lie algebra $\mathfrak{gl}_d(\mathbb{C})$ as described in the paper.

• The folders dimWlambda and gt_patterns contain scripts for computing the dimensions of Specht modules and generating Gelfand-Tsetlin patterns for a given partition, respectively.

SageMath Scripts

To install SageMath, follow the instructions at SageMath Installation.

• dimWlambda/spechtmodule_dim.py: Script to compute the dimension of the Specht module for a given partition.
• gt_patterns/gen_gt_patterns.py: Script to generate Gelfand-Tsetlin patterns for a given partition and number of rows.