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Computing quantum magic of state vectors

T0 review · 0 major / 2 minor · reviewed 2026-05-16 · grok-4.3

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Pith's one-line read Fast Hadamard transforms compute SRE and mana for qubit and qutrit state vectors at O(N d^{2N}) cost.

desk verdict This gives a workable exact method for SRE and mana via fast Hadamard transform with open code, improving scaling for state vector inputs. read the letter →

arxiv 2601.07824 v3 submitted 2026-01-12 quant-ph cond-mat.dis-nncond-mat.stat-mech

Piotr Sierant , Jofre Vallès-Muns , Artur Garcia-Saez This is my paper · ORCID
classification quant-phcond-mat.dis-nncond-mat.stat-mech
keywords quantummagicstabilizerRényientropymanafastHadamardtransformstatevectorsnon-stabilizernessmany-bodysystemscomputationalalgorithms
checked against Cost.FunctionalEquationalexander_duality_circle_linking
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper develops algorithms that use the fast Hadamard transform to calculate non-stabilizerness measures exactly for pure states supplied as vectors. For qubits the target is the stabilizer Rényi entropy; for qutrits it is the mana. The new scaling replaces the direct triple-exponential cost with a quadratic-exponential cost multiplied by a linear factor in the number of qudits. The same transform can be paired with Monte Carlo sampling to estimate SRE and can be extended to compute mana for mixed states. An open-source implementation supplies built-in support for multithreading, distributed parallelism and GPU acceleration.

What carries the argument

Fast Hadamard transform applied to the character sums that define SRE and mana.

What would settle it

For any small N where both methods fit in memory, the numerical value of SRE or mana produced by the fast-transform algorithm must agree with the value from the naive triple sum to machine precision.

Watch

Extended reading notes

Core claim

The fast Hadamard transform can be applied directly to the sums that define the stabilizer Rényi entropy for qubits and the mana for qutrits, yielding numerically exact values for any pure state vector at cost O(N d^{2N}).

Load-bearing premise

The quantum state must be supplied exactly as a full state vector and the defining sums must admit direct application of the fast Hadamard transform without approximation.

Editorial extensions

If this is right

  • SRE and mana become accessible for system sizes that were previously out of reach.
  • The algorithms admit straightforward parallelization across CPU cores and GPUs.
  • Monte Carlo sampling combined with the transform yields practical estimates of SRE for large vectors.
  • Mana can be evaluated for mixed states as well as pure states.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The reduced scaling opens the possibility of mapping how magic accumulates across quantum phase transitions.
  • The same transform technique may be adaptable to other resource measures that are expressed as sums over characters.
  • Integration with tensor-network or variational methods could push the reachable system sizes even farther.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The paper introduces numerically exact algorithms that use the fast Hadamard transform to evaluate the stabilizer Rényi entropy (SRE) for pure qubit states and the mana for pure qutrit states supplied as full state vectors. It claims an exact complexity of O(N d^{2N}) (versus the naive O(d^{3N})), with extensions to Monte Carlo sampling for SRE estimation and to mixed-state mana, all implemented in the open-source Julia package HadaMAG with multithreading, MPI, and GPU support.

Significance. If the exactness and complexity claims hold, the work removes a major computational bottleneck for quantifying non-stabilizerness, enabling routine calculations on systems an order of magnitude larger than before. The open-source, parallelized implementation is a concrete strength that directly supports reproducible large-scale studies of magic in many-body systems.

minor comments (2)
  1. [§3] The complexity analysis in the main text would benefit from an explicit step-by-step count of operations in the FHT application to the defining sums (currently summarized in the abstract).
  2. [Fig. 2] Figure captions should state the precise system sizes and hardware used for the reported timings to allow direct reproduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of our contributions, and recommendation to accept. No major comments were raised that require point-by-point response.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity in derivation chain

full rationale

The paper derives an O(N d^{2N}) algorithm for exact SRE and mana computation by applying the standard fast Hadamard transform directly to the sums that define these quantities for pure state vectors. This mapping is presented as a straightforward consequence of linear-algebra properties of the FHT and does not rely on self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations. The complexity improvement over the naive O(d^{3N}) scaling follows immediately from the known O(d^N log d^N) cost of the FHT without circular reduction. The open-source package provides an independent verification route, confirming the derivation is self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on standard mathematical properties of the fast Hadamard transform applied to sums over stabilizer states; no free parameters, ad-hoc axioms, or invented entities are introduced.

assumptions (1)
  • standard math The fast Hadamard transform efficiently computes the required sums defining SRE and mana for pure states given as vectors
    This follows from established properties of the Walsh-Hadamard transform in quantum information and signal processing.

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Cite this review

Pith. "Pith review of Computing quantum magic of state vectors." pith.science (2026). https://pith.science/paper/2601.07824

@misc{pith2026260107824,
  author       = {Pith},
  title        = {Pith review of: Computing quantum magic of state vectors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2601.07824}},
  note         = {Machine review of arXiv:2601.07824}
}
abstract

Non-stabilizerness, also known as ``magic,'' quantifies how far a quantum state departs from the stabilizer set. It is a central resource behind quantum advantage and a useful probe of the complexity of quantum many-body states. Yet standard magic quantifiers, such as the stabilizer R\'enyi entropy (SRE) for qubits and the mana for qutrits, are costly to evaluate numerically, with the computational complexity growing rapidly with the number $N$ of qudits. Here we introduce efficient, numerically exact algorithms that exploit the fast Hadamard transform to compute the SRE for qubits ($d=2$) and the mana for qutrits ($d=3$) for pure states given as state vectors. Our methods compute SRE and mana at cost $O(N d^{2N})$, providing an exponential improvement over the naive $O(d^{3N})$ scaling, with substantial parallelism and straightforward GPU acceleration. We further show how to combine the fast Hadamard transform with Monte Carlo sampling to estimate the SRE of state vectors, and we extend the approach to compute the mana of mixed states. All algorithms are implemented in the open-source Julia package HadaMAG ( https://github.com/bsc-quantic/HadaMAG.jl/ ), which provides a high-performance toolbox for computing SRE and mana with built-in support for multithreading, MPI-based distributed parallelism, and GPU acceleration. The package, together with the methods developed in this work, offers a practical route to large-scale numerical studies of magic in quantum many-body systems.

Figures

Figures reproduced from arXiv: 2601.07824 by the authors.

Figure 1
Figure 1. Left: Variance σf of the “energy” f(Xa) at β = 1, which controls the statistical uncertainty of the SRE in the sampling Algorithm 3, for states evolved under quantum circuits (46) of depth t, plotted as a function of system size N. Right: Absolute error of the SRE M2 at circuit depth t = 4, comparing the numerically exact result from Algorithm 2 with its sampling estimate Mˆ2, as a function of the number of samples … view at source ↗

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Forward citations

Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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    In U(1)-symmetric 1D random circuits the stabilizer Rényi entropy gap closes diffusively as 1/t, with the same scaling seen in an energy-conserving Ising chain.

  3. Non-Local Magic Resources for Fermionic Gaussian States

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    Non-local stabilizer entropies of fermionic Gaussian states admit a closed form from reduced Majorana covariance eigenvalues when minimized over local Gaussian unitaries.

  4. Exact stabilizer scars in two-dimensional $U(1)$ lattice gauge theory

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    Sublattice scars in the 2D Rokhsar–Kivelson model are exact stabilizer states: zero stabilizer Rényi entropy, quantized entanglement, and explicit Clifford-circuit preparation.

  5. Limits of Clifford Disentangling in Tensor Network States

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Pith tools

Reviewed May 16, 2026 · model on record in the stance chip above.