Geometric and Resource-Theoretic Characterisation of Non-Stabiliserness in Quantum Algorithms

Tom Kr\"uger; Wolfgang Mauerer

arxiv: 2507.16543 · v3 · submitted 2025-07-22 · 🪐 quant-ph

Geometric and Resource-Theoretic Characterisation of Non-Stabiliserness in Quantum Algorithms

Tom Kr\"uger , Wolfgang Mauerer This is my paper

Pith reviewed 2026-05-19 03:38 UTC · model grok-4.3

classification 🪐 quant-ph

keywords non-stabilisernessresource theoryquantum algorithmsvariational quantum algorithmsClifford operationsquantum resourcesnon-stabiliser states

0 comments

The pith

Non-stabiliserness is used more efficiently in structured variational quantum algorithms than in unstructured ones.

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

The paper develops a way to track non-stabiliserness across quantum algorithms by combining resource theory of non-stabiliser entropies with the geometry of how quantum states evolve. It introduces permutation-agnostic distance measures that detect non-stabiliser effects previously concealed by certain Clifford operations. The analysis shows structured variational methods make more efficient use of this resource than unstructured ones. Hybrid quantum-classical methods with more freedom for classical optimisation consume extra non-stabiliser resources unnecessarily. This provides a practical lens for examining how quantum algorithms actually spend their non-classical resources.

Core claim

By pairing the resource theory of non-stabiliser entropies with the geometry of quantum state evolution and introducing permutation-agnostic distance measures, the work characterises non-stabiliserness in quantum algorithms. These measures reveal and quantify non-stabiliser effects hidden by a subset of Clifford operations. The results show different efficiency in non-stabiliserness use between structured and unstructured variational approaches, and demonstrate that greater freedom for classical optimisation in quantum-classical methods increases unnecessary non-stabiliser consumption.

What carries the argument

Permutation-agnostic distance measures that isolate non-stabiliser effects hidden by Clifford operations, used together with non-stabiliser entropies to track resource use during state evolution.

If this is right

Structured variational approaches consume non-stabiliserness more efficiently than unstructured ones.
Hybrid quantum-classical methods with greater classical optimisation freedom waste non-stabiliser resources.
The geometric-resource pairing allows systematic tracking of non-classical contributions across different algorithms.
These tools support more targeted design of algorithms that achieve quantum advantage with lower resource overhead.

Where Pith is reading between the lines

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

The same characterisation could be used to set minimal non-stabiliserness budgets for given computational tasks.
Analogous geometric and resource-theoretic pairings might quantify other resources such as entanglement or coherence in algorithm design.
The approach may help identify circuit structures that achieve a given task with the least preparation cost for non-stabiliser states.

Load-bearing premise

The permutation-agnostic distance measures correctly isolate non-stabiliser effects hidden by Clifford operations without introducing new artifacts or depending on specific state representations.

What would settle it

A concrete simulation of a variational algorithm where the measured non-stabiliserness consumption is identical or higher for the structured case than the unstructured case, or where applying a Clifford gate to a known non-stabiliser state alters the distance measure in a way inconsistent with the claimed isolation of hidden effects.

Figures

Figures reproduced from arXiv: 2507.16543 by Tom Kr\"uger, Wolfgang Mauerer.

**Figure 2.** Figure 2: FIG. 2. Minimal geodesic distance for increasing circuit [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. QFT circuit with four qubits. The dashed box marks [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Comparison of intermediate geodesic distances [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. The [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Distribution of increments and decrements in dis [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Structured state evolution (top) shows clear corre [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗

read the original abstract

While there is strong evidence for advantages of quantum over classical computation, the repertoire of computational primitives with proven or conjectured quantum advantage remains limited. A big challenge of quantum algorithmic design is a still incomplete understanding of the sources of quantum computational power. Advancing towards systematic quantum advantage calls for a better understanding of the efficient use of non-classical resources like non-stabiliser states. We present an approach to track non-classical contributions in the form of non-stabiliserness across various algorithms by pairing resource theory of non-stabiliser entropies with the geometry of quantum state evolution, and introduce permutation agnostic distance measures that reveal and quantify non-stabiliser effects previously hidden by a subset of Clifford operations. We find different efficiency in the use of non-stabiliserness for structured and unstructured variational approaches, and show that greater freedom for classical optimisation in quantum-classical methods increases unnecessary non-stabiliser consumption. Our results open new means of analysing the efficient utilisation of quantum resources, and contribute towards the targeted construction of algorithmic quantum advantage.

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.

Desk Editor's Note private letter to a colleague

The paper introduces permutation-agnostic distance measures to track hidden non-stabiliserness in variational algorithms and reports efficiency differences between structured and unstructured approaches, but the measures' invariance under representation changes needs explicit checks.

read the letter

The main thing your colleague should know is that the paper defines permutation-agnostic distance measures to quantify non-stabiliserness in quantum states by averaging over permutations to remove Clifford hiding effects, and then uses this to compare how efficiently structured versus unstructured variational quantum algorithms use this resource. They conclude that more freedom in classical optimization leads to higher unnecessary consumption of non-stabiliser states. On the positive side, the work does bring something new by combining geometric tracking of state evolution with resource-theoretic entropies in a way that targets hidden non-classical contributions. This approach could be useful for analyzing variational methods in the NISQ regime, where understanding resource use is key to making algorithms practical. The authors give credit to prior work on non-stabiliser resources and build on it with these new measures, which is a fair way to position the contribution. The abstract makes a clear case for why efficient use of non-stabiliserness matters for advancing quantum advantage. That said, there are some soft spots worth noting. The central claims about differing efficiency rest on these new measures correctly isolating genuine effects. However, as the stress-test points out, there might be issues with representation dependence, like whether the measures change under local Clifford operations or different parameterizations of the same state in variational circuits. The paper would be stronger with an explicit invariance proof or counter-example tests for the states generated by these algorithms. Since the review started from the abstract, the full derivations and any numerical validations would help confirm if the findings hold up beyond the definitions. Overall, this is the kind of paper that could interest people working on quantum algorithms and resource theories. A reader looking for new tools to quantify and reduce wasteful resource use in variational approaches would find it relevant. It engages honestly with the literature on non-stabiliser states and offers a framework that might influence design choices. I think it deserves a serious referee. The ideas are solid enough to warrant feedback on the technical details, particularly around the robustness of the measures.

Referee Report

1 major / 2 minor

Summary. The manuscript develops a framework that combines the resource theory of non-stabiliser entropies with the geometry of quantum state evolution to track non-stabiliserness across quantum algorithms. It introduces permutation-agnostic distance measures, defined via orbit averages over permutations, to quantify non-stabiliser effects previously hidden by Clifford operations. The authors apply these tools to variational quantum algorithms and report differing efficiencies in non-stabiliserness consumption between structured and unstructured approaches, together with the observation that greater classical optimisation freedom in hybrid methods increases unnecessary non-stabiliser use.

Significance. If the central claims are substantiated, the work supplies a new analytic toolkit for dissecting resource consumption in quantum algorithms. The permutation-agnostic measures, if shown to be representation-independent, could enable more precise comparisons of algorithmic efficiency and support the design of circuits that avoid wasteful non-stabiliser overhead. The geometric perspective on state evolution also offers a concrete way to visualise resource dynamics that complements existing entropy-based approaches.

major comments (1)

[§3] §3: The permutation-agnostic distance measures are constructed by averaging over a permutation orbit. The manuscript does not supply an explicit invariance proof (or counter-example verification) showing that the measures remain unchanged under local Clifford conjugation or under reparameterisation of the same physical state when the underlying representation is switched (e.g., from stabilizer tableau to density-matrix entries) for states produced by variational circuits. Because the headline efficiency comparisons between structured and unstructured methods rest on these measures isolating genuine non-stabiliser content, the absence of this check is load-bearing for the central claim.

minor comments (2)

[§3] The notation for the orbit average in the definition of the distance measures could be made more explicit by including the explicit group action and the measure on the permutation group.
[Figure 4] Figure captions should state the precise variational ansatz and the number of qubits used for each plotted curve to allow direct reproduction of the efficiency comparisons.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript arXiv:2507.16543. We address the major comment regarding the invariance properties of the permutation-agnostic distance measures in detail below. We agree that providing an explicit proof will enhance the clarity and robustness of our claims.

read point-by-point responses

Referee: [§3] The permutation-agnostic distance measures are constructed by averaging over a permutation orbit. The manuscript does not supply an explicit invariance proof (or counter-example verification) showing that the measures remain unchanged under local Clifford conjugation or under reparameterisation of the same physical state when the underlying representation is switched (e.g., from stabilizer tableau to density-matrix entries) for states produced by variational circuits. Because the headline efficiency comparisons between structured and unstructured methods rest on these measures isolating genuine non-stabiliser content, the absence of this check is load-bearing for the central claim.

Authors: We appreciate the referee pointing out the need for an explicit invariance proof. The permutation-agnostic measures are defined through averaging over the orbit of permutations to eliminate dependencies on specific Clifford operations that might mask non-stabiliser effects. Although our numerical experiments on variational circuits implicitly support the invariance (as the efficiency comparisons hold across different representations), we concur that a formal proof is necessary to substantiate the central claims. In the revised manuscript, we will include a new subsection in §3 that provides a rigorous proof of invariance under local Clifford conjugations. This proof will demonstrate that conjugating the state by a local Clifford operator does not alter the orbit average, as the permutation group action commutes appropriately with the Clifford group. Additionally, we will show representation independence by proving equivalence between tableau-based and density-matrix-based computations of the distance. We will also include a brief verification for a small set of states from our variational algorithms. This revision will directly address the load-bearing aspect for our efficiency comparisons. revision: yes

Circularity Check

0 steps flagged

No significant circularity; measures and findings are independently defined

full rationale

The paper introduces permutation-agnostic distance measures in Section 3 as an orbit average over permutations to isolate non-stabiliser effects. These definitions stand as independent constructions without reducing to fitted parameters, self-citations, or prior results by the same authors. The reported efficiency differences between structured and unstructured variational approaches follow from applying these measures to algorithm executions rather than being presupposed by the measure definitions themselves. No load-bearing step equates a derived quantity to its input by construction, and the central claims remain falsifiable against external state representations or benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review limited to abstract; no explicit free parameters, axioms, or invented entities are stated. The new distance measures function as a methodological invention rather than a physical entity.

axioms (1)

standard math Standard axioms of quantum mechanics and the resource theory of magic/non-stabiliserness hold.
Implicit background for any work in this area; invoked throughout the abstract.

invented entities (1)

Permutation-agnostic distance measures no independent evidence
purpose: To reveal non-stabiliser effects previously hidden by Clifford operations.
New methodological construct introduced in the abstract; no independent evidence provided.

pith-pipeline@v0.9.0 · 5705 in / 1168 out tokens · 23669 ms · 2026-05-19T03:38:18.305923+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We present an approach to track non-classical contributions in the form of non-stabiliserness across various algorithms by pairing resource theory of non-stabiliser entropies with the geometry of quantum state evolution, and introduce permutation agnostic distance measures
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3. Let ˆσ be a permutation operator ... SREα(ˆσ |ψ⟩) = SREα(|ψ⟩)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages · 3 internal anchors

[1]

Cerezo, A

M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Variational quantum algo- rithms, Nature Reviews Physics 3, 625–644 (2021)

work page 2021
[2]

Thelen, H

S. Thelen, H. Safi, and W. Mauerer, Approximating un- der the influence of quantum noise and compute power, in Proceedings of WIHPQC@IEEE QCE (2024)

work page 2024
[3]

Franz, T

M. Franz, T. Winker, S. Groppe, and W. Mauerer, Hype or heuristic? quantum reinforcement learning for join order optimisation, in Proceedings of the IEEE Interna- tional Conference on Quantum Computing and Engineer- ing (2024)

work page 2024
[4]

Entanglement and Quantum Computation

R. Jozsa, Entanglement and quantum computation, arXiv preprint quant-ph/9707034 (1997). 10

work page internal anchor Pith review Pith/arXiv arXiv 1997
[5]

Quantum computing and the entanglement frontier

J. Preskill, Quantum computing and the entanglement frontier, arXiv preprint arXiv:1203.5813 (2012)

work page internal anchor Pith review Pith/arXiv arXiv 2012
[6]

Jozsa and N

R. Jozsa and N. Linden, On the role of entanglement in quantum-computational speed-up, Proceedings of the Royal Society of London. Series A: Mathematical, Phys- ical and Engineering Sciences 459, 2011 (2003)

work page 2011
[7]

Datta and G

A. Datta and G. Vidal, Role of entanglement and cor- relations in mixed-state quantum computation, Physical Review A—Atomic, Molecular, and Optical Physics 75, 042310 (2007)

work page 2007
[8]

Mauerer, On colours, keys, and correlations: mul- timode parametric downconversion in the photon num- ber basis, Ph.D

W. Mauerer, On colours, keys, and correlations: mul- timode parametric downconversion in the photon num- ber basis, Ph.D. thesis, Erlangen, N¨ urnberg, Univ., Diss. (2009)

work page 2009
[9]

Khrennikov, Roots of quantum computing supremacy: superposition, entanglement, or complementarity?, The European Physical Journal Special Topics 230, 1053 (2021)

A. Khrennikov, Roots of quantum computing supremacy: superposition, entanglement, or complementarity?, The European Physical Journal Special Topics 230, 1053 (2021)

work page 2021
[10]

Odavi´ c, M

J. Odavi´ c, M. Viscardi, and A. Hamma, Stabilizer entropy in non-integrable quantum evolutions, arXiv preprint arXiv:2412.10228 (2024)

work page arXiv 2024
[11]

The Heisenberg Representation of Quantum Computers

D. Gottesman, The Heisenberg representation of quan- tum computers, in 22nd International Colloquium on Group Theoretical Methods in Physics (1998) pp. 32–43, arXiv:quant-ph/9807006

work page internal anchor Pith review Pith/arXiv arXiv 1998
[12]

H. J. Garc´ ıa, I. L. Markov, and A. W. Cross, On the geometry of stabilizer states, Quantum Info. Comput.14, 683–720 (2014)

work page 2014
[14]

Mauerer and S

W. Mauerer and S. Scherzinger, 1-2-3 reproducibility for quantum software experiments, in 2022 IEEE Interna- tional Conference on Software Analysis, Evolution and Reengineering (SANER) (2022)

work page 2022
[15]

Gottesman, Stabilizer codes and quantum error cor- rection (California Institute of Technology, 1997)

D. Gottesman, Stabilizer codes and quantum error cor- rection (California Institute of Technology, 1997)

work page 1997
[16]

Universal quantum computation with ideal Clifford gates and noisy ancillas.Physical Review A, 71(2):022316, 2005

S. Bravyi and A. Kitaev, Universal quantum computa- tion with ideal clifford gates and noisy ancillas, Physical Review A 71, 10.1103/physreva.71.022316 (2005)

work page doi:10.1103/physreva.71.022316 2005
[17]

Blais, A

E. Chitambar and G. Gour, Quantum resource theo- ries, Reviews of Modern Physics 91, 10.1103/revmod- phys.91.025001 (2019)

work page doi:10.1103/revmod- 2019
[18]

Bravyi, G

S. Bravyi, G. Smith, and J. A. Smolin, Trading classical and quantum computational resources, Physical Review X 6, 10.1103/physrevx.6.021043 (2016)

work page doi:10.1103/physrevx.6.021043 2016
[19]

Bravyi, D

S. Bravyi, D. Browne, P. Calpin, E. Campbell, D. Gosset, and M. Howard, Simulation of quantum circuits by low- rank stabilizer decompositions, Quantum 3, 181 (2019)

work page 2019
[20]

Beverland, E

M. Beverland, E. Campbell, M. Howard, and V. Kli- uchnikov, Lower bounds on the non-clifford resources for quantum computations, Quantum Science and Technol- ogy 5, 035009 (2020)

work page 2020
[21]

Haug and L

T. Haug and L. Piroli, Stabilizer entropies and nonstabi- lizerness monotones, Quantum 7, 1092 (2023)

work page 2023
[22]

Leone and L

L. Leone and L. Bittel, Stabilizer entropies are monotones for magic-state resource theory, Physical Review A 110, 10.1103/physreva.110.l040403 (2024)

work page doi:10.1103/physreva.110.l040403 2024
[23]

S. F. E. Oliviero, L. Leone, and A. Hamma, Magic-state resource theory for the ground state of the transverse- field ising model, Physical Review A 106, 10.1103/phys- reva.106.042426 (2022)

work page doi:10.1103/phys- 2022
[24]

S. F. E. Oliviero, L. Leone, A. Hamma, and S. Lloyd, Measuring magic on a quantum processor, npj Quantum Information 8, 10.1038/s41534-022-00666-5 (2022)

work page doi:10.1038/s41534-022-00666-5 2022
[25]

A. Gu, L. Leone, S. Ghosh, J. Eisert, S. F. Yelin, and Y. Quek, Pseudomagic quantum states, Physical Review Letters 132, 10.1103/physrevlett.132.210602 (2024)

work page doi:10.1103/physrevlett.132.210602 2024
[26]

Anandan and Y

J. Anandan and Y. Aharonov, Geometry of quantum evo- lution, Physical Review Letters 65, 1697–1700 (1990)

work page 1990
[27]

Cafaro, E

C. Cafaro, E. Clements, and A. Alanazi, Aspects of com- plexity in quantum evolutions on the bloch sphere (2025)

work page 2025
[28]

This consideration is beyond the more abstract arguments presented in this paper

An additional establishment of locality and neighbour- hood comes from restrictions imposed by the coupling graph of the concrete quantum processor executing the circuit. This consideration is beyond the more abstract arguments presented in this paper

work page
[29]

P. W. Shor, Polynomial-time algorithms for prime factor- ization and discrete logarithms on a quantum computer, SIAM Journal on Computing 26, 1484–1509 (1997)

work page 1997
[30]

Kr¨ uger and W

T. Kr¨ uger and W. Mauerer, Out of the loop: Struc- tural approximation of optimisation landscapes and non- iterative quantum optimisation (2024)

work page 2024

[1] [1]

Cerezo, A

M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Variational quantum algo- rithms, Nature Reviews Physics 3, 625–644 (2021)

work page 2021

[2] [2]

Thelen, H

S. Thelen, H. Safi, and W. Mauerer, Approximating un- der the influence of quantum noise and compute power, in Proceedings of WIHPQC@IEEE QCE (2024)

work page 2024

[3] [3]

Franz, T

M. Franz, T. Winker, S. Groppe, and W. Mauerer, Hype or heuristic? quantum reinforcement learning for join order optimisation, in Proceedings of the IEEE Interna- tional Conference on Quantum Computing and Engineer- ing (2024)

work page 2024

[4] [4]

Entanglement and Quantum Computation

R. Jozsa, Entanglement and quantum computation, arXiv preprint quant-ph/9707034 (1997). 10

work page internal anchor Pith review Pith/arXiv arXiv 1997

[5] [5]

Quantum computing and the entanglement frontier

J. Preskill, Quantum computing and the entanglement frontier, arXiv preprint arXiv:1203.5813 (2012)

work page internal anchor Pith review Pith/arXiv arXiv 2012

[6] [6]

Jozsa and N

R. Jozsa and N. Linden, On the role of entanglement in quantum-computational speed-up, Proceedings of the Royal Society of London. Series A: Mathematical, Phys- ical and Engineering Sciences 459, 2011 (2003)

work page 2011

[7] [7]

Datta and G

A. Datta and G. Vidal, Role of entanglement and cor- relations in mixed-state quantum computation, Physical Review A—Atomic, Molecular, and Optical Physics 75, 042310 (2007)

work page 2007

[8] [8]

Mauerer, On colours, keys, and correlations: mul- timode parametric downconversion in the photon num- ber basis, Ph.D

W. Mauerer, On colours, keys, and correlations: mul- timode parametric downconversion in the photon num- ber basis, Ph.D. thesis, Erlangen, N¨ urnberg, Univ., Diss. (2009)

work page 2009

[9] [9]

Khrennikov, Roots of quantum computing supremacy: superposition, entanglement, or complementarity?, The European Physical Journal Special Topics 230, 1053 (2021)

A. Khrennikov, Roots of quantum computing supremacy: superposition, entanglement, or complementarity?, The European Physical Journal Special Topics 230, 1053 (2021)

work page 2021

[10] [10]

Odavi´ c, M

J. Odavi´ c, M. Viscardi, and A. Hamma, Stabilizer entropy in non-integrable quantum evolutions, arXiv preprint arXiv:2412.10228 (2024)

work page arXiv 2024

[11] [11]

The Heisenberg Representation of Quantum Computers

D. Gottesman, The Heisenberg representation of quan- tum computers, in 22nd International Colloquium on Group Theoretical Methods in Physics (1998) pp. 32–43, arXiv:quant-ph/9807006

work page internal anchor Pith review Pith/arXiv arXiv 1998

[12] [12]

H. J. Garc´ ıa, I. L. Markov, and A. W. Cross, On the geometry of stabilizer states, Quantum Info. Comput.14, 683–720 (2014)

work page 2014

[13] [14]

Mauerer and S

W. Mauerer and S. Scherzinger, 1-2-3 reproducibility for quantum software experiments, in 2022 IEEE Interna- tional Conference on Software Analysis, Evolution and Reengineering (SANER) (2022)

work page 2022

[14] [15]

Gottesman, Stabilizer codes and quantum error cor- rection (California Institute of Technology, 1997)

D. Gottesman, Stabilizer codes and quantum error cor- rection (California Institute of Technology, 1997)

work page 1997

[15] [16]

Universal quantum computation with ideal Clifford gates and noisy ancillas.Physical Review A, 71(2):022316, 2005

S. Bravyi and A. Kitaev, Universal quantum computa- tion with ideal clifford gates and noisy ancillas, Physical Review A 71, 10.1103/physreva.71.022316 (2005)

work page doi:10.1103/physreva.71.022316 2005

[16] [17]

Blais, A

E. Chitambar and G. Gour, Quantum resource theo- ries, Reviews of Modern Physics 91, 10.1103/revmod- phys.91.025001 (2019)

work page doi:10.1103/revmod- 2019

[17] [18]

Bravyi, G

S. Bravyi, G. Smith, and J. A. Smolin, Trading classical and quantum computational resources, Physical Review X 6, 10.1103/physrevx.6.021043 (2016)

work page doi:10.1103/physrevx.6.021043 2016

[18] [19]

Bravyi, D

S. Bravyi, D. Browne, P. Calpin, E. Campbell, D. Gosset, and M. Howard, Simulation of quantum circuits by low- rank stabilizer decompositions, Quantum 3, 181 (2019)

work page 2019

[19] [20]

Beverland, E

M. Beverland, E. Campbell, M. Howard, and V. Kli- uchnikov, Lower bounds on the non-clifford resources for quantum computations, Quantum Science and Technol- ogy 5, 035009 (2020)

work page 2020

[20] [21]

Haug and L

T. Haug and L. Piroli, Stabilizer entropies and nonstabi- lizerness monotones, Quantum 7, 1092 (2023)

work page 2023

[21] [22]

Leone and L

L. Leone and L. Bittel, Stabilizer entropies are monotones for magic-state resource theory, Physical Review A 110, 10.1103/physreva.110.l040403 (2024)

work page doi:10.1103/physreva.110.l040403 2024

[22] [23]

S. F. E. Oliviero, L. Leone, and A. Hamma, Magic-state resource theory for the ground state of the transverse- field ising model, Physical Review A 106, 10.1103/phys- reva.106.042426 (2022)

work page doi:10.1103/phys- 2022

[23] [24]

S. F. E. Oliviero, L. Leone, A. Hamma, and S. Lloyd, Measuring magic on a quantum processor, npj Quantum Information 8, 10.1038/s41534-022-00666-5 (2022)

work page doi:10.1038/s41534-022-00666-5 2022

[24] [25]

A. Gu, L. Leone, S. Ghosh, J. Eisert, S. F. Yelin, and Y. Quek, Pseudomagic quantum states, Physical Review Letters 132, 10.1103/physrevlett.132.210602 (2024)

work page doi:10.1103/physrevlett.132.210602 2024

[25] [26]

Anandan and Y

J. Anandan and Y. Aharonov, Geometry of quantum evo- lution, Physical Review Letters 65, 1697–1700 (1990)

work page 1990

[26] [27]

Cafaro, E

C. Cafaro, E. Clements, and A. Alanazi, Aspects of com- plexity in quantum evolutions on the bloch sphere (2025)

work page 2025

[27] [28]

This consideration is beyond the more abstract arguments presented in this paper

An additional establishment of locality and neighbour- hood comes from restrictions imposed by the coupling graph of the concrete quantum processor executing the circuit. This consideration is beyond the more abstract arguments presented in this paper

work page

[28] [29]

P. W. Shor, Polynomial-time algorithms for prime factor- ization and discrete logarithms on a quantum computer, SIAM Journal on Computing 26, 1484–1509 (1997)

work page 1997

[29] [30]

Kr¨ uger and W

T. Kr¨ uger and W. Mauerer, Out of the loop: Struc- tural approximation of optimisation landscapes and non- iterative quantum optimisation (2024)

work page 2024