Efficient mutual magic and magic capacity with matrix product states

Missing some relevant literature.

This paper introduces efficient algorithms to compute the mutual von Neumann stabilizer Rényi entropy (SRE) and a new quantity called the magic capacity, with applications to matrix product states (MPS) and statevector simulations.
The numerical results are clear and relevant: mutual SREs behave as robust probes of criticality (also under basis rotation), and the magic capacity connects naturally to the anti-flatness of the Pauli spectrum, offering an operational handle on sampling complexity.

The paper is technically solid and well written. The methods build on prior developments—especially those on stabilizer Rényi entropies and Pauli sampling—but refine them meaningfully and apply them to both ground-state transitions and random circuit ensembles. The Monte Carlo improvements and hybrid sampling approaches are nicely done and practically useful.

To better situate the work in the current literature, the authors should cite: - arXiv: 2501.18679 - arXiv: 2503.07468 - arXiv:2412.10229 - arXiv:2408.16047 - arXiv:2502.20455 - arXiv: 2305.11797

These papers are relevant for the discussion of Pauli statistics, randomness in Clifford+T circuits, and resource theory of magic in many-body systems.

Publish (easily meets expectations and criteria for this Journal; among top 50%)

The manuscript by P. S. Tarabunga and T. Haug provides a comprehensive study to different magic diagnostic for the quantum many-body physics. The pros are that they explain in details each definition of the diagnostic, but the cons are that these different quantities are not well organized in a coherent logical flow, and the demonstration of the methods needs to be improved. Overall, their computation methods could benefit the community, but the current presented version of the manuscript is not yet ready for publication in SciPost. My detailed comments / suggestions are as follows:

1. Parallel definitions of the distribution functions and entropies: In Sec III, their Eq.6-8 are alternative definitions of the SRE based on an alternative distribution function q=tr(rho P rho P)/2^N, in comparison with p=tr(rho P)^2/tr(rho^2)/2^N in Eq.3-4 of Sec II. They should both be placed in Sec II, before introducing the finer quantities of mutual magic and magic capacity. After all, the probability distribution is the basic of everything that follows.

2. Motivation for the alternative distribution: They should explain earlier the motivation why they need to introduce a different distribution q, otherwise it is very confusing – in principle one can define a lot of different distribution functions from a quantum state. I didnot understand their motivation until reading their paragraph following Eq.17-18 in Sec IV – is it simply because such definition saves the computation of mutual magic by roughly a factor 3 (since one does not need to evaluate A and B separately)?

3. I’m a bit confused what purpose Fig.1 serves? It is embedded in Sec IV explaining methods. But Fig.1a (with its caption and texts in paragraphs) does not tell us the efficiency and accuracy neither the physical implication of the anisotropic Heisenberg ground state. Fig.1b and its texts only tell us the convergence of small bond dimension MPS methods, and Monte Carlo based on finite bond dimension as well.

4. Sec. V-A, Clifford + T circuit physics: in their Fig.2b, the data is noisy, and there is too limited number of data points near the suspicious “crossing point”, which is far from convincing that the observable scales large for z<zc and scales to 0 for z>zc. Similarly for the data at z>zc’ in Fig.2c. It is hard to claim “critical point(s)” based on the existing data presentation in Fig.2bc. For Fig.2a, what “(entanglement/magic/information) order parameter” can distinguish the two phases if this is a “transition”?

5. Sec. V-B, in Fig.3 anisotropic Heisenberg chain ground state: this is a paradigmatic integrable model with very rich analytic and numerical understanding in the literature, concerning its universal features in long-wave-length limit. The magic calculation is new – can the authors elucidate what known or unknown universal scaling exponents can be extracted from the magic quantities they calculate? If they go to stronger anisotropic coupling to gap out the gapless quantum chain, what do they observe in the magic quantities? Can they tell the phase transition?

6. Fig.4-5 for TFIM – can they show the data collapse? Without the prior knowledge of the exact solution hc=1, what numerical hc can they get? Which scaling operators are responsible for the scaling of 1-SRE and 2-SRE here?

Ask for major revision