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arxiv: 2601.00761 · v2 · submitted 2026-01-02 · 🪐 quant-ph · cond-mat.mes-hall· cond-mat.stat-mech

Exponentially Accelerated Sampling of Pauli Strings for Nonstabilizerness

Zhenyu Xiao , Shinsei Ryu This is my paper

Pith reviewed 2026-05-16 17:48 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hallcond-mat.stat-mech
keywords nonstabilizernessstabilizer Rényi entropyPauli string samplingquantum magicClifford circuitsMonte Carlo estimationT-doped circuitsscrambling ratio
0
0 comments X

The pith

A classical method samples Pauli strings for nonstabilizerness at linear cost in qubit number rather than exponential.

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

The paper introduces a classical framework that computes stabilizer Rényi entropies and stabilizer nullity for generic N-qubit wavefunctions. It pairs the fast Walsh-Hadamard transform with an exact partition of Pauli operators to cut the average cost per sampled Pauli string from exponential in N to linear in N. A Monte Carlo estimator that preconditions with Clifford circuits keeps the needed number of samples from growing visibly with N in the tested cases. When applied to T-doped random Clifford circuits, the work shows that the scrambling ratio of Clifford gates per T gate controls how nonstabilizerness accumulates, and that each T gate reaches nearly its full dilute-limit contribution with only modest scrambling.

Core claim

By combining the fast Walsh-Hadamard transform with an exact partition of the Pauli operators, the average computational cost to sample a Pauli string drops from O(2^N) to O(N), and a Clifford-preconditioned Monte Carlo estimator converges with a number of samples that shows no visible growth with N on T-doped Clifford circuits.

What carries the argument

Fast Walsh-Hadamard transform applied to an exact partition of Pauli operators, which generates the probability distribution over Pauli strings needed for the entropies at linear cost per sample.

If this is right

  • Stabilizer Rényi entropies become computable for qubit numbers previously out of reach for exact methods.
  • Magic growth in hybrid Clifford-T circuits can be tracked quantitatively through the single parameter η.
  • Each T gate contributes close to its maximum nonstabilizerness once modest Clifford scrambling is present.
  • Long-time nonequilibrium dynamics of nonstabilizerness can be followed in large systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The linear scaling may make quantitative magic studies feasible for systems with hundreds of qubits if sample count remains bounded.
  • Partitioning techniques of this kind could accelerate other computations that sum over Pauli operators.
  • If the N-independent sample count holds for typical entangled states, similar preconditioning may help sampling problems in other quantum measures.

Load-bearing premise

The Clifford-preconditioned Monte Carlo estimator stays unbiased and requires a number of samples independent of N for generic highly entangled states beyond the T-doped Clifford circuits examined.

What would settle it

For a random highly entangled state at N=20, run the estimator and observe that the variance of the entropy estimate or the number of samples required for convergence grows exponentially with N.

Figures

Figures reproduced from arXiv: 2601.00761 by Shinsei Ryu, Zhenyu Xiao.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Runtime for computing [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Quantum circuit begin with a [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Ensemble-averaged [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
read the original abstract

Quantum magic, quantified by nonstabilizerness, measures departures from stabilizer structure and underlies potential quantum speedups. We introduce an efficient classical framework for computing stabilizer R\'enyi entropies and stabilizer nullity of generic $N$-qubit wavefunctions. The method combines the fast Walsh-Hadamard transform with an exact partition of Pauli operators, reducing the average cost per sampled Pauli string from $\mathcal{O}(2^N)$ to $\mathcal{O}(N)$. We further develop a Monte Carlo estimator with Clifford preconditioning and find that the required number of samples shows no visible growth with $N$ in our benchmarks. Applying the method to $T$-doped random Clifford circuits, we identify the scrambling ratio $\eta$ (Clifford gates per $T$ gate) as the key parameter governing magic growth. Each $T$ gate approaches its dilute-limit nonstabilizerness power with only modest Clifford scrambling. Our approach enables quantitative studies of magic in highly entangled states and long-time nonequilibrium dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces an efficient classical algorithm for computing stabilizer Rényi entropies and stabilizer nullity of N-qubit states. It combines the fast Walsh-Hadamard transform with an exact partition of Pauli operators to reduce the average cost per sampled Pauli string from O(2^N) to O(N), and develops a Clifford-preconditioned Monte Carlo estimator. Benchmarks on T-doped random Clifford circuits show that the number of required samples exhibits no visible growth with N; the scrambling ratio η is identified as the parameter controlling magic growth, with each T gate approaching its dilute-limit nonstabilizerness contribution under modest Clifford scrambling.

Significance. If the Monte Carlo estimator's variance remains controlled beyond the tested family, the O(N) per-sample cost would enable quantitative studies of nonstabilizerness in large-scale entangled states and long-time dynamics that are inaccessible to exact methods. The empirical N-independence of sample count for T-doped Clifford circuits is a practical strength that could accelerate research on quantum magic.

major comments (2)
  1. [Benchmarks and Monte Carlo estimator description] The central claim that the Clifford-preconditioned Monte Carlo estimator requires an N-independent number of samples rests on benchmarks for T-doped random Clifford circuits only. No general variance bound or convergence proof is supplied for arbitrary states (e.g., Haar-random or Hamiltonian ground states), which is load-bearing for the assertion of applicability to generic highly entangled wavefunctions.
  2. [Algorithm and numerical results sections] Validation of the estimator against exact small-N results and explicit error bounds or variance estimates for the Monte Carlo sampling are not provided, leaving the reliability of the reported N-independence insufficiently quantified.
minor comments (2)
  1. [Abstract] The abstract states that 'the required number of samples shows no visible growth with N' without specifying the range of N tested or the number of circuit realizations used.
  2. [Introduction and method] Notation for the scrambling ratio η and the precise definition of the Pauli partition should be introduced earlier and used consistently throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive report and positive assessment of the method's potential. We address each major comment below, clarifying the scope of our claims and proposing targeted revisions.

read point-by-point responses

  1. Referee: The central claim that the Clifford-preconditioned Monte Carlo estimator requires an N-independent number of samples rests on benchmarks for T-doped random Clifford circuits only. No general variance bound or convergence proof is supplied for arbitrary states (e.g., Haar-random or Hamiltonian ground states), which is load-bearing for the assertion of applicability to generic highly entangled wavefunctions.

    Authors: We agree that the observed N-independence of the sample count is an empirical finding specific to the T-doped random Clifford circuit family studied in the benchmarks. The manuscript does not claim or provide a general variance bound or convergence proof that would hold for arbitrary states such as Haar-random states or Hamiltonian ground states. The algorithm and Clifford-preconditioned estimator are formulated for generic wavefunctions, but the sample complexity necessarily depends on the state's nonstabilizerness structure; we will revise the text in the abstract, introduction, and numerical results sections to explicitly state that the N-independence is demonstrated for the tested family and to qualify the broader applicability accordingly. revision: partial

  2. Referee: Validation of the estimator against exact small-N results and explicit error bounds or variance estimates for the Monte Carlo sampling are not provided, leaving the reliability of the reported N-independence insufficiently quantified.

    Authors: We will add a new subsection in the numerical results that validates the Monte Carlo estimator against exact computations for small N (up to N=8) across several circuit instances, including direct comparisons of the estimated stabilizer Rényi entropies and nullity. We will also include explicit variance estimates derived from the Monte Carlo samples and report standard errors for the N-independence plots to better quantify reliability. revision: yes

Circularity Check

0 steps flagged

No circularity: acceleration via standard transforms and empirical benchmarks

full rationale

The derivation relies on the fast Walsh-Hadamard transform and an exact Pauli partition to achieve the stated O(N) per-sample cost reduction; this follows directly from linear-algebraic properties of the Hadamard matrix and Pauli-string enumeration without any self-referential fitting or redefinition. The Monte Carlo estimator with Clifford preconditioning is introduced as a standard variance-reduction technique, with N-independent sample counts reported only as an empirical observation on the specific T-doped Clifford ensemble. No load-bearing self-citations, ansatz smuggling, or uniqueness theorems imported from prior author work appear in the chain; the central claims remain independent of the target quantities by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the correctness of the fast Walsh-Hadamard transform applied to Pauli strings and the convergence properties of the preconditioned Monte Carlo estimator, both drawn from standard quantum information theory without new free parameters or invented entities.

axioms (2)
  • standard math Pauli operators form a complete basis for N-qubit operators and admit exact partitioning compatible with the Walsh-Hadamard transform.
    Invoked implicitly in the cost-reduction claim; standard in quantum information.
  • domain assumption Clifford circuits can be used as preconditioners that do not bias the Monte Carlo estimator for nonstabilizerness.
    Central to the sample-count independence claim.

pith-pipeline@v0.9.0 · 5479 in / 1403 out tokens · 37229 ms · 2026-05-16T17:48:07.353797+00:00 · methodology

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Forward citations

Cited by 5 Pith papers

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  3. Computing quantum magic of state vectors

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    quant-ph 2026-04 unverdicted novelty 6.0

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Reference graph

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