HadaMAG.jl

High-performance Stabilizer Rényi Entropy and Mana computations in Julia

HadaMAG.jl is an optimized Julia library for computing the Stabilizer Rényi Entropy (SRE) and Mana of pure quantum states, and Mana of mixed quantum states. It is designed for both research and high‐performance computing environments, with support for multi-threading, MPI, GPU acceleration (using CUDA), and multi-GPU systems (using MPI + CUDA).

Paper: https://arxiv.org/abs/2601.07824

Key features:

• Exact SRE – Computes the SRE exactly, which applies a sequence of Fast Hadamard Transforms (FHT) to reduce the naive $O(8^N)$ cost down to $O(N 4^N)$ for $N$ qubits.
• Monte Carlo SRE – Estimates the SRE using stochastic sampling.
• Mana for qutrits – Numerically exact mana computation for statevectors of qutrit systems, reducing the naive $O(27^N)$ cost down to $O(N 9^N)$ for $N$ qutrits.
• Mana for mixed states of qutrits – Numerically exact mana computation mana for mixed states of qutrits, with complexity $O(N 9^N)$.
• Multiple backends – Choose between different execution backends (:serial, :threads, :mpi_threads, :cuda, :mpi_cuda) for optimal performance on your hardware.

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Installation

To install HadaMAG.jl, use the Julia package manager:

julia> using Pkg
julia> Pkg.add(url="https://github.com/bsc-quantic/HadaMAG.jl/")

Stabilizer Rényi Entropy (SRE)

The Stabilizer Rényi Entropy (SRE) of order $q$ is a measure of the non-stabilizerness of a pure quantum state $| ψ ⟩$. HadaMAG.jl provides both exact and Monte Carlo methods to compute the SRE efficiently:

julia> using HadaMAG

# Haar-random 16-qubit state
julia> ψ = rand_haar(16; depth=2)
StateVec{ComplexF64,2}(n=16, dim=65536, mem=1.0 MiB)

# Compute the q=2 Stabilizer Rényi Entropy
julia> (sre2, lost_norm) = SRE(ψ, 2)
[==================================================] 100.0%  (65536/65536)
(8.213760134602566, 3.9968028886505635e-15)

# Estimate the q=2 SRE using Monte Carlo with 10000 samples
julia> sre2_mc = MC_SRE(ψ, 2; Nsamples=10000)
[==================================================] 100.0%  (10000/10000)
8.218601276704227

Mana Computation

HadaMAG.jl provides functionality to compute the mana of pure states for qutrit systems:

julia> using HadaMAG

# Haar-random 10-qutrit state
julia> ψ = rand_haar(10; depth=3, q=3)
StateVec{ComplexF64,3}(n=10, dim=59049, mem=923.0 KiB)

# Compute the mana
julia> mana = Mana(ψ)
4.720097873481441

Additionally, you can compute the mana of mixed quantum states of qutrit systems, represented as density matrices using the DensityMatrix{T,q} struct:

julia> using HadaMAG

# Haar-random 10-qutrit state
julia> ψ = rand_haar(10; depth=3, q=3)
StateVec{ComplexF64,3}(n=10, dim=59049, mem=923.0 KiB)

# Reduced density matrix by tracing out part of the system
julia> ρA = reduced_density_matrix(ψ, 6)
DensityMatrix{ComplexF64,3}(n=6, size=(729, 729), mem=8.11 MiB)

# Compute the mana of the mixed state
julia> mana_mixed = Mana(ρA)
3.5551656874727082

Backends

HadaMAG.jl supports multiple execution backends:

• :serial – single-threaded CPU execution.
• :threads – multi-threaded CPU execution.
• :mpi_threads – MPI-based execution with multi-threading (requires MPI.jl).
• :cuda – GPU execution using CUDA (requires CUDA.jl).
• :mpi_cuda – hybrid MPI + GPU execution for multi-GPU systems (requires both MPI.jl and CUDA.jl).

By default, backend = :auto selects the fastest available backend. For more details on configuring and using backends, see the Backend Configuration guide.

Documentation

Full documentation is available at HadaMAG.jl Documentation.

Acknowledgements

HadaMAG.jl uses the FFHT C library for efficient bit‐reversed Fast Hadamard Transforms.