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arxiv: 2508.20252 · v3 · pith:NUML6TFQnew · submitted 2025-08-27 · 🪐 quant-ph

Classical Simulations of Low Magic Quantum Dynamics

Kemal Aziz , Haining Pan , Michael J. Gullans , J. H. Pixley This is my paper

Pith reviewed 2026-05-22 12:20 UTC · model grok-4.3

classification 🪐 quant-ph
keywords classical simulationlow magicmonitored quantum circuitsPauli measurementsmeasurement-induced phase transitionsadaptive circuitsnon-stabilizernessT-gates
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The pith

Classical simulation algorithms efficiently simulate adaptive quantum circuits with low magic states by exploiting frequent Pauli measurements.

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

The paper presents new classical simulation algorithms for quantum circuits that produce states with low magic, meaning low non-stabilizerness. These algorithms target adaptive circuits featuring high rates of Pauli measurements, which help control the buildup of magic from non-Clifford gates or noise processes. This setup is common in quantum error correction and monitored quantum circuits. The approach includes a truncation method for approximate simulations when exact ones are costly. Benchmarking on all-to-all monitored circuits with sub-extensive T-gates demonstrates the ability to reach larger system sizes and depths, while analyzing phase transitions in entanglement, purification, and magic levels.

Core claim

We develop classical simulation algorithms for adaptive quantum circuits that produce states with low levels of magic. These algorithms are particularly well-suited to circuits with high rates of Pauli measurements, such as those encountered in quantum error correction and monitored quantum circuits. The measurements serve to limit the buildup of magic induced by non-Clifford operations arising from generic noise processes or unitary gates. Our algorithms also allow a systematic truncation procedure to achieve approximate simulation. To benchmark our approach, we study the dynamics of all-to-all monitored quantum circuits with a sub-extensive rate of T-gates per unit of circuit depth, where

What carries the argument

Pauli measurements that limit magic accumulation from non-Clifford operations in adaptive circuits, enabling polynomial-cost classical simulation.

Load-bearing premise

Frequent Pauli measurements are assumed to sufficiently suppress the accumulation of magic from non-Clifford operations or noise, maintaining polynomial simulation cost.

What would settle it

A calculation showing exponential growth in effective magic or simulation runtime with system size despite high rates of Pauli measurements would invalidate the efficiency claims.

Figures

Figures reproduced from arXiv: 2508.20252 by Haining Pan, J. H. Pixley, Kemal Aziz, Michael J. Gullans.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Circuit diagram for the evolution of the initial state that corresponds to the initial density matrix [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Circuit diagram for the single-pair all-to-all [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Scaling of the circuit-averaged number of entries [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Phase diagram of the purification transition in the single-pair all-to-all model with [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Phase diagram of the magic and entanglement transitions in the single-pair all-to-all model with [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The stabilizer state [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Bell sampling setup for the single-pair all-to-all [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Absolute value of the difference in the exact stabi [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The top row shows the stabilizer nullity [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. (a) The first moment of the maximal cluster size [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Phase diagram for the connectivity transition in the [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Late time decay of [PITH_FULL_IMAGE:figures/full_fig_p025_14.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. (a) Scaling of [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Dependence of the decay time [PITH_FULL_IMAGE:figures/full_fig_p025_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Dependence of the decay time [PITH_FULL_IMAGE:figures/full_fig_p026_16.png] view at source ↗
read the original abstract

We develop classical simulation algorithms for adaptive quantum circuits that produce states with low levels of ``magic'' (i.e., non-stabilizerness). These algorithms are particularly well-suited to circuits with high rates of Pauli measurements, such as those encountered in quantum error correction and monitored quantum circuits. The measurements serve to limit the buildup of magic induced by non-Clifford operations arising from generic noise processes or unitary gates, respectively. Our algorithms also allow a systematic truncation procedure to achieve approximate simulation. To benchmark our approach, we study the dynamics of all-to-all monitored quantum circuits with a sub-extensive rate of T-gates per unit of circuit depth, where we can simulate previously inaccessible system sizes and depths. We characterize measurement-induced phase transitions in the output wavefunction, including in the entanglement, purification, and magic. We outline the utility of our algorithms to simulate dynamics with low magic and high entanglement, complementary to the leading matrix-product state approaches.

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 develops classical simulation algorithms for adaptive quantum circuits producing low-magic states, leveraging high rates of Pauli measurements to suppress magic buildup from non-Clifford gates or noise. It introduces a systematic truncation procedure for approximate simulation and benchmarks the approach on all-to-all monitored circuits with sub-extensive T-gates per depth, enabling access to larger system sizes and depths. The work characterizes measurement-induced phase transitions in entanglement, purification, and magic, positioning the method as complementary to matrix-product states for low-magic high-entanglement regimes.

Significance. If the central claims hold, the algorithms offer a practical tool for classically simulating previously inaccessible regimes of monitored quantum circuits and error-corrected dynamics, particularly where entanglement is high but magic is controlled by measurements. The benchmarking results on phase transitions provide concrete data on larger scales, and the complementarity to MPS methods addresses a genuine gap in simulation techniques for low-magic states.

major comments (2)
  1. [§3] §3 (algorithm description): The truncation procedure is introduced as systematic, but no explicit error bound or convergence analysis is given relating the truncation threshold (a free parameter) to the approximation error in the output state or observables; this is load-bearing for the claim of controlled approximate simulation at polynomial cost.
  2. [§5] §5 (benchmarking and phase transitions): While sub-extensive T-gates per depth allow simulation of larger sizes, the efficiency claim rests on the assumption that Pauli measurements keep effective magic (e.g., via a monotone such as mana or stabilizer extent) from growing exponentially; no derivation or bound is provided showing magic scaling with measurement rate, depth, and noise strength for generic adaptive circuits, as required to guarantee polynomial cost beyond the specific benchmark.
minor comments (2)
  1. Figure captions for the phase-transition plots could more explicitly state the system sizes, depths, and number of samples used to generate the data points.
  2. The abstract mentions 'previously inaccessible system sizes and depths' but the main text would benefit from a direct comparison table to prior simulation limits in the literature.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which help clarify the scope and rigor of our claims. We respond to each major comment below, indicating where revisions have been made to the manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (algorithm description): The truncation procedure is introduced as systematic, but no explicit error bound or convergence analysis is given relating the truncation threshold (a free parameter) to the approximation error in the output state or observables; this is load-bearing for the claim of controlled approximate simulation at polynomial cost.

    Authors: We agree that an explicit error bound strengthens the presentation of the truncation procedure. In the revised manuscript we have added a derivation in §3 relating the truncation threshold to the approximation error. Specifically, when truncating the stabilizer decomposition by discarding components below the threshold, the error in the expectation value of any Pauli observable is bounded by twice the total discarded weight (i.e., the sum of the stabilizer extents of the discarded terms). This bound is independent of system size for normalized states and directly controls the observable error. We also include a brief convergence statement: as the threshold tends to zero the approximate state converges to the exact state in the 1-norm for the relevant observables. These additions make the controlled-approximation claim explicit while preserving the polynomial-cost scaling whenever the retained magic remains sub-exponential. revision: yes

  2. Referee: [§5] §5 (benchmarking and phase transitions): While sub-extensive T-gates per depth allow simulation of larger sizes, the efficiency claim rests on the assumption that Pauli measurements keep effective magic (e.g., via a monotone such as mana or stabilizer extent) from growing exponentially; no derivation or bound is provided showing magic scaling with measurement rate, depth, and noise strength for generic adaptive circuits, as required to guarantee polynomial cost beyond the specific benchmark.

    Authors: Our efficiency claim is made for the concrete regime of all-to-all monitored circuits with a sub-extensive density of T gates per layer, where high-rate Pauli measurements are shown numerically to keep the effective magic (mana and stabilizer extent) from growing exponentially with depth. We have expanded the discussion in §5 to include additional scaling plots of mana versus depth and measurement probability, together with references to existing results on measurement-induced suppression of magic. A fully general analytical bound that holds for arbitrary adaptive circuits, arbitrary measurement rates, and arbitrary noise strengths would require model-specific assumptions on the adaptivity and the distribution of measurement outcomes; such a derivation lies outside the scope of the present work. The manuscript therefore positions the algorithm as complementary to MPS methods precisely in the low-magic, high-entanglement regime that our benchmarks access, rather than claiming universality for all adaptive circuits. revision: partial

standing simulated objections not resolved
  • A general derivation or bound on magic scaling with measurement rate, depth, and noise strength that holds for arbitrary adaptive circuits (beyond the specific all-to-all monitored ensemble with sub-extensive T gates) is not provided, as it would require additional model-dependent assumptions not addressed in the current manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper develops classical simulation algorithms for low-magic adaptive circuits by extending the stabilizer formalism with magic measures and a truncation procedure. The central approach relies on the physical assumption that frequent Pauli measurements limit magic accumulation from non-Clifford gates or noise, which is then benchmarked on all-to-all monitored circuits with sub-extensive T-gates. This assumption is tested empirically rather than defined into the result by construction, and the characterizations of entanglement, purification, and magic phase transitions follow from the low-magic representation without reducing to fitted inputs or self-referential definitions. No load-bearing step equates a prediction to its own input via equation or self-citation chain; the work is complementary to matrix-product states and builds on independently established concepts.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that Pauli measurements keep magic low enough for efficient classical tracking, plus standard quantum information axioms about stabilizer states and magic measures. No new entities are introduced. One free parameter is the truncation threshold for approximation.

free parameters (1)
  • truncation threshold
    Controls the approximation level when discarding small magic contributions during simulation.
axioms (2)
  • domain assumption Pauli measurements limit the buildup of magic induced by non-Clifford operations
    Invoked to justify why the simulation remains efficient in the presence of T-gates or noise.
  • standard math Stabilizer formalism plus magic measures allow efficient classical representation when magic is low
    Background assumption from quantum information theory used to ground the algorithm.

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Forward citations

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    Clifford Evolution In this section, we describe how to update the LRSD of a quantum state as it evolves under Clifford unitaries and Pauli measurements, see Eqs. (14) and (15). Through- out, we write the density matrix in the form of Eq. (4) of the main text, |ψ⟩⟨ψ|= X l∈LLRSD λl σl ρS,(A1) where the coefficientsλ l are real, the operatorsσ l are logical ...

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    R´ enyi Entropy of LRSD We now present algorithms to compute the R´ enyi en- tanglement entropies using the representation of the state in Eq. (4). We present an algorithm for computing the reduced density matrix for an arbitrary state, from which we can compute the entanglement. We also present an algorithm which is valid only for [LS,S] = 0.These meth- ...

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    Bell sampling |ۧ𝜓 𝐻 𝐻 𝐻 𝐻 𝐻 |ۧ𝜓 𝑇 𝐻 1 𝑇 x t 1 0 0 FIG

    Bell Sampling and Magic Computation Algorithms a. Bell sampling |ۧ𝜓 𝐻 𝐻 𝐻 𝐻 𝐻 |ۧ𝜓 𝑇 𝐻 1 𝑇 x t 1 0 0 FIG. 8. Bell sampling setup for the single-pair all-to-all model. The quantum circuit to measure the magicMof an L-qubit state|ψ⟩=K|+x⟩ ⊗N whereKis the Kraus operator of the circuit gates and measurements. We prepare two copies |ψ⟩ ⊗ |ψ⟩, and perform a meas...

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    Cluster We begin with the connectivity transition, which is a percolation transition from a connected state (when all qubits are entangled) to a disconnected state (when the state effectively breaks into multiple disentangled clus- ters). The quantum state factorizes as |ψ⟩= O i |ψCi ⟩,(B1) where eachC i denotes a disjoint subset of qubits, or clus- ter, ...

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    Purification Transition This section provides additional data supporting the purification transition results reported in Sec. V. We give more details on how we locate the purification transition in Section V. We present the data used to numerically extract the phase boundary of the purification transition in Fig. 5(a). We also explain the fitting procedur...

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