pith. sign in

arxiv: 2605.18985 · v1 · pith:OMVJZFGOnew · submitted 2026-05-18 · 🪐 quant-ph

Efficient Fourier-Based Linear Combination of Unitaries and Applications in Quantum Optimization

Almudena Carrera Vazquez , Daniel J. Egger , Stefan Woerner This is my paper

Pith reviewed 2026-05-20 10:38 UTC · model grok-4.3

classification 🪐 quant-ph
keywords linear combination of unitariesFourier decompositionquantum approximate optimization algorithmconstraint penaltiesLagrangian relaxationquantum algorithmssuperconducting qubits
0
0 comments X

The pith

Fourier-based LCU decomposes complex unitaries into simpler gates for quantum optimization at polynomial sampling cost.

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

The paper demonstrates that ancilla-free linear combinations of unitaries built from Fourier series can approximate a wide range of diagonal and non-diagonal operators that appear in quantum optimization routines. This replacement turns highly connected multi-qubit interactions into sequences of single-qubit gates, at the expense of a polynomial number of additional circuit runs. Because many optimization tasks only require sampling good candidate solutions that are then checked classically, the extra samples remain affordable. The same framework also supplies a direct link between quantum penalty terms and classical Lagrangian relaxation, giving a single picture for how constraints are enforced. Large-scale simulations and hardware runs on superconducting processors confirm that solution quality is preserved while circuit depth and connectivity demands drop.

Core claim

Fourier-based LCU constructions efficiently decompose broad classes of diagonal and non-diagonal unitaries, replacing highly connected qubit interactions with single-qubit gate layers or significantly simpler structures at the cost of a polynomial sampling overhead. Applied to algorithms such as QAOA, this yields efficient, hardware-friendly decompositions of cardinality-constraint penalties and the fully connected XY-mixer, while maintaining rigorous performance guarantees compared to fully coherent implementations. The work also establishes a formal connection between Fourier-based quantum penalties and Lagrangian relaxation.

What carries the argument

ancilla-free Fourier-based linear combination of unitaries (LCU), which expresses target operators through a Fourier expansion and realizes the combination by sampling terms from that expansion.

If this is right

  • QAOA circuits for constrained problems can be realized with only single-qubit layers plus classical sampling.
  • Cardinality penalties and fully connected XY-mixers acquire hardware-efficient implementations that still obey the original performance bounds.
  • Constraint handling in quantum optimization receives a unified description that matches classical Lagrangian relaxation.
  • Circuit complexity can be traded for sampling overhead while preserving rigorous approximation guarantees.

Where Pith is reading between the lines

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

  • The same Fourier-LCU technique may simplify other variational quantum algorithms that currently demand dense qubit connectivity.
  • Classical post-processing of the sampled bitstrings could further amortize the polynomial overhead in practice.
  • Different choices of Fourier basis functions might reduce the sampling factor for particular families of objective functions.

Load-bearing premise

The optimization task succeeds when high-quality bitstrings are sampled and then evaluated classically, without any need to reproduce the full quantum output distribution.

What would settle it

A head-to-head comparison on a fixed optimization instance in which the Fourier-LCU sampler, after accounting for its polynomial overhead, produces solution quality statistically worse than the corresponding fully coherent circuit.

Figures

Figures reproduced from arXiv: 2605.18985 by Almudena Carrera Vazquez, Daniel J. Egger, Stefan Woerner.

Figure 1
Figure 1. Figure 1: Quantum circuit for implementing Φ+ jk(ρ)−Φ − jk(ρ), where P denotes a phase gate applying ϕjk. Consequently, expectation values of observables can be estimated directly from samples of this randomized pro￾cedure. For an observable O, we obtain tr(OU(ρ)) = Xm j=0 |cj | 2 tr(OVjρV † j ) + X k<j 2|cj ||ck|Ea [(−1)a tr(Oρa)] , where ρa denotes the post-measurement state condi￾tioned on obtaining outcome a. Th… view at source ↗
Figure 3
Figure 3. Figure 3: Penalty-based LCU basis circuit with a warm [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: The considered 3-regular graph and corresponding [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Results for the 12-node densest k-subgraph simulations with a penalty term. (Left) Distribution of objective values obtained from samples of the different experiments, restricted to the range [−10, 4] for clarity. (Right) Corresponding distribu￾tions restricted to feasible samples (unnormalized). Both panels also show “Exp. 1/Γ2”, representing the coherent distribution scaled by the sampling overhead from … view at source ↗
Figure 5
Figure 5. Figure 5: XY-mixer-based LCU basis circuit with a warm [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Results for the 12-node densest k-subgraph simulations with XY-mixer. (Left) Distribution of objective values ob￾tained from samples of the different experiments, restricted to the range [−10, 4] for clarity. (Right) Corresponding distributions restricted to feasible samples (unnormalized). Both panels also show “Exp. 1/Γ2”, representing the distribution of Experiment 1 scaled by the sampling overhead of E… view at source ↗
Figure 7
Figure 7. Figure 7: Cumulative distribution functions of the CVaR val [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Problem graph with 106 nodes resulting from a [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Problem graph with 106 nodes obtained from a 5 [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
read the original abstract

We investigate ancilla-free linear combination of unitaries (LCU) as a framework for approximating complex quantum circuits. This is particularly effective for quantum optimization algorithms, where candidate solutions can be evaluated classically and the task is to sample high-quality bitstrings rather than reproduce the full output distribution. We show that Fourier-based LCU constructions efficiently decompose broad classes of diagonal and non-diagonal unitaries, replacing highly connected qubit interactions with single-qubit gate layers or significantly simpler structures at the cost of a polynomial sampling overhead. Applied to algorithms such as QAOA, this yields efficient, hardware-friendly decompositions of, for instance, cardinality-constraint penalties and the fully connected XY-mixer, while maintaining rigorous performance guarantees compared to fully coherent implementations. Furthermore, we establish a formal connection between Fourier-based quantum penalties and Lagrangian relaxation, offering a unified perspective on constraint handling. We validate our approach using exact statevector simulations of 12-qubit circuits and large-scale experiments on 106 superconducting qubits. Our results illustrate how approximate sampling via an LCU systematically trades circuit complexity for sampling overhead, extending the practical reach of near-term quantum optimization.

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 ancilla-free Fourier-based linear combination of unitaries (LCU) framework for approximating complex quantum circuits, with a focus on quantum optimization algorithms such as QAOA. It claims that Fourier analysis enables efficient decompositions of broad classes of diagonal and non-diagonal unitaries (e.g., cardinality penalties and fully connected XY-mixers) into simpler single-qubit gate layers, at the cost of polynomial sampling overhead rather than deep entangling circuits. The approach is asserted to maintain rigorous performance guarantees relative to fully coherent implementations, while establishing a formal connection to Lagrangian relaxation for constraint handling. Validation is provided via exact statevector simulations on 12-qubit circuits and experiments on 106 superconducting qubits.

Significance. If the error bounds and guarantee preservation hold under the bitstring-sampling regime, the work could meaningfully extend the reach of near-term quantum optimization by reducing hardware connectivity demands. The unification with Lagrangian relaxation offers a useful theoretical perspective, and the 106-qubit experiment provides a concrete demonstration of scale. These elements would strengthen the case for approximate LCU as a practical tool when full coherence is not required.

major comments (2)
  1. [§4] §4 (QAOA applications and XY-mixer decomposition): the central claim that the Fourier-based LCU approximation 'maintains rigorous performance guarantees compared to fully coherent implementations' is load-bearing, yet the manuscript does not supply an explicit bound on how finite Fourier truncation plus ancilla-free probabilistic sampling distorts the variational landscape or effective expectation values. A concrete error propagation argument (or numerical evidence isolating the approximation ratio degradation) is required to substantiate transfer of guarantees when only high-quality bitstrings are sampled.
  2. [§3.2] §3.2 (Fourier LCU construction): the efficiency statement that the method replaces 'highly connected qubit interactions with single-qubit gate layers' at polynomial sampling cost lacks a precise statement of the polynomial degree and the truncation threshold needed to keep the approximation ratio within stated bounds. This directly affects the claimed hardware-friendliness.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'polynomial sampling overhead' should be accompanied by the leading-order degree (e.g., O(n^k)) to allow immediate assessment of practicality.
  2. [Notation] Notation: ensure the Fourier coefficients and truncation parameter are defined consistently between the general LCU section and the specific penalty/mixer applications.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report on our manuscript. We address each major comment below and have incorporated revisions to strengthen the presentation of error bounds and efficiency claims.

read point-by-point responses

  1. Referee: [§4] §4 (QAOA applications and XY-mixer decomposition): the central claim that the Fourier-based LCU approximation 'maintains rigorous performance guarantees compared to fully coherent implementations' is load-bearing, yet the manuscript does not supply an explicit bound on how finite Fourier truncation plus ancilla-free probabilistic sampling distorts the variational landscape or effective expectation values. A concrete error propagation argument (or numerical evidence isolating the approximation ratio degradation) is required to substantiate transfer of guarantees when only high-quality bitstrings are sampled.

    Authors: We thank the referee for this observation. The manuscript establishes that performance guarantees transfer because the algorithm only requires sampling high-quality bitstrings (rather than reproducing the full coherent output distribution), and the Fourier truncation error is controlled in operator norm while the LCU sampling introduces only a multiplicative overhead that does not alter the relative ordering of bitstring probabilities above a threshold. To make this explicit, the revised manuscript will add a dedicated error-propagation paragraph in Section 4 deriving that the deviation in the sampled expectation value is bounded by the Fourier tail plus an additive O(1/√M) term from M samples, together with additional numerical results from the 12-qubit statevector simulations that isolate the approximation-ratio degradation for the truncation levels employed. revision: yes

  2. Referee: [§3.2] §3.2 (Fourier LCU construction): the efficiency statement that the method replaces 'highly connected qubit interactions with single-qubit gate layers' at polynomial sampling cost lacks a precise statement of the polynomial degree and the truncation threshold needed to keep the approximation ratio within stated bounds. This directly affects the claimed hardware-friendliness.

    Authors: We agree that an explicit characterization of the polynomial degree and truncation threshold improves clarity. The original text states only that the overhead is polynomial; the revised Section 3.2 will now include a precise statement that, for the diagonal and XY-mixer families considered, the number of samples scales as O(1/ε²) to achieve an additive error ε in the approximation ratio, with the Fourier truncation order chosen so that the series tail is smaller than ε in operator norm. For the smooth penalty and mixer functions treated, this truncation order is O(log(1/ε)) or better, yielding an overall polynomial dependence on system size and 1/ε. A supporting lemma will be added. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from Fourier analysis

full rationale

The paper presents Fourier-based LCU constructions as derived from standard Fourier analysis applied to classes of unitaries, yielding explicit decompositions that replace complex interactions with simpler gate layers plus sampling overhead. The abstract and described approach frame this as a direct methodological advance for QAOA and constraint handling, with a claimed formal link to Lagrangian relaxation presented as unification rather than redefinition. Validation via statevector simulations and hardware experiments provides independent empirical content. No load-bearing step reduces by construction to a fitted parameter, self-referential definition, or unverified self-citation chain; the central claims remain externally falsifiable through the stated approximation error bounds and sampling overhead. This is the expected outcome for a methods paper whose core contribution is an algorithmic construction rather than a closed tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that Fourier series provide efficient decompositions for the relevant unitaries and that sampling overhead is acceptable for the target use case; no explicit free parameters or invented entities are stated in the abstract.

axioms (1)
  • domain assumption Fourier series can be used to decompose the target diagonal and non-diagonal unitaries efficiently
    Core mechanism described in the abstract for replacing complex interactions with simpler gates.

pith-pipeline@v0.9.0 · 5731 in / 1159 out tokens · 55519 ms · 2026-05-20T10:38:21.076389+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

77 extracted references · 77 canonical work pages · 2 internal anchors

  1. [1]

    Parameter optimization is performed

    Optimize the fully coherent QAOA circuit param- eters to maximize⟨H⟩, yielding optimal parame- ters (β ∗, γ∗). Parameter optimization is performed ... q1 RY(θinit) e−iγH1 RZ(θ) RY(−θinit) RZ(β) RY(θinit) qn RY(θinit) RZ(θ) RY(−θinit) RZ(β) RY(θinit) Figure 3. Penalty-based LCU basis circuit with a warm- started initial state and mixer [37], the cost opera...

    work page

  2. [2]

    Evaluate the corresponding Fourier-based LCU ap- proximation at (β ∗, γ∗) and compute the resulting sampling overhead Γ

    work page

  3. [3]

    Re-optimize (β, γ) to maximize the CVaR 1/Γ us- ing weighted LCU samples collected from alln+ 1 circuits, noting that Γ depends onγ

    work page

  4. [4]

    For comparability, we fix Γ to the value obtained in Experiment 3

    Optimize a single LCU basis circuit, jointly max- imizing over (β, γ, θ) using CVaR 1/Γ as objective. For comparability, we fix Γ to the value obtained in Experiment 3

    work page

  5. [5]

    Exp. 1/Γ 2

    Re-optimize the fully coherent QAOA circuit using CVaR1/Γ as the objective, again fixing Γ to the value obtained in Experiment 3, to put the LCU- based results into context. For all experiments, we report the resulting expecta- tion values, the sampling overhead Γ, the corresponding CVaR values where applicable, the probability of sam- pling a feasible so...

    work page

  6. [6]

    As before, we first perform a grid search followed by fine-tuning with COBYLA

    Optimize the fully coherent XY-QAOA circuit parameters to maximize CVaR p0 feasible using the penalty-based QUBO objective function, yielding optimal parameters (β ∗, γ∗). As before, we first perform a grid search followed by fine-tuning with COBYLA

    work page

  7. [7]

    Evaluate the sampled LCU circuits at (β ∗, γ∗) and estimate the resulting sampling overhead Γ

    work page

  8. [8]

    Note that Γ depends on the current value ofβ

    Re-optimize (β, γ) to maximize CVaR p0 feasible/Γ us- ing sampled solutions collected across all sampled LCU circuits. Note that Γ depends on the current value ofβ

    work page

  9. [9]

    Here,αis irrelevant since the correspondingR Z rotation appears im- mediately before measurement, andβis absent as the mixer Hamiltonian is realized via the LCU de- composition

    Optimize a single parametrized LCU basis circuit over (γ, ϑ, χ), again using CVaR p0 feasible/Γ as objec- tive and fixing Γ to the value obtained in Exper- iment 3 for comparability. Here,αis irrelevant since the correspondingR Z rotation appears im- mediately before measurement, andβis absent as the mixer Hamiltonian is realized via the LCU de- composition

    work page

  10. [10]

    Additionally, we run the following experiments to com- pare penalty-based and XY-mixer-based constraint han- dling:

    Re-optimize the coherent QAOA circuit using CVaRp0 feasible/Γ with the same Γ as in Experiment 3 for comparison. Additionally, we run the following experiments to com- pare penalty-based and XY-mixer-based constraint han- dling:

    work page

  11. [11]

    Optimize the XY-QAOA circuit parameters to maximize CVaR 1/Γp 3 , where Γ p 3 denotes the sam- pling overhead from the penalty-based Experi- ment 3 in Sec. VI A. Apart from the choice of ob- jective, this setup matches Experiment 1

    work page

  12. [12]

    Optimize a single parametrized LCU basis circuit over (β, γ, α, ϑ, χ) using CVaR1/Γp 3 as the objective, mirroring Experiment 4 with the objective of Ex- periment 6. For all experiments, we report expectation values, sam- pling overhead Γ, risk levelsηand corresponding CVaR values, the probability of sampling a feasible solution, the probability of sampli...

    work page

  13. [13]

    The considered circuits follow the structures shown in Figs

    together with the corresponding analytic expec- tation value⟨H⟩= 23.5452 as a reference. The considered circuits follow the structures shown in Figs. 3 and 5. They use 106 qubits and 886 CZ gates to implementH 1 and the SWAP gates, with a two-qubit- gate depth of 25. This corresponds to approximately 35 CZ gates in parallel, i.e., on average acting on abo...

    work page

  14. [14]

    Evaluate all 107 Fourier-based LCU circuits for the penalty-based approach, aggregate the sam- pled probabilities using the LCU weights, and com- pute the CVaR1/Γ for comparison with⟨H⟩, where Γ = 104.1328

    work page

  15. [15]

    The resulting angles are denoted (β∗ 2 , γ∗ 2 , θ∗ 2)

    Optimize a single penalty-based LCU basis circuit by jointly maximizing CVaR1/Γ over (β, γ, θ), with the same Γ as in Experiment 1, using COBYLA and initializing at (β ∗ 1 , γ∗ 1 , θj∗), whereθ j∗ denotes the LCU basis angle achieving the best CVaR value in Experiment 1. The resulting angles are denoted (β∗ 2 , γ∗ 2 , θ∗ 2)

    work page

  16. [16]

    Exp. 1/Γ2

    Optimize a single XY-mixer-based LCU basis circuit by maximizing CVaR 1/Γ over (γ, ϑ, χ), 11 # Experiment⟨H⟩ΓηCVaR η P-feasibleP-optimal⟨H⟩ feasible 1 Coherent XY-QAOA (CVaR) -7.8079 – 0.2384 2.6525 0.2384 0.0220 2.6350 2 Fourier-based LCU -42.5502 342.1372 0.0007 4.0000 0.1159 0.0059 1.7146 3 Fourier-based LCU re-optimized -27.2421 336.9378 0.0007 4.0000...

    work page

  17. [17]

    This ensures that the optimization starts from the best solution found in Experiment 2 and exploits the additional degrees of freedom to further improve the results

    The initial parameters are chosen as (γ∗ 2 , ϑβ∗ 2 ,θ∗ 2 ,θinit , χβ∗ 1 ,θ∗ 2 ,θinit), such that they repro- duce the optimal solution from Experiment 2 up to a global phase (see Appendix G). This ensures that the optimization starts from the best solution found in Experiment 2 and exploits the additional degrees of freedom to further improve the results....

    work page

  18. [18]

    An Erd˝ os-R´ enyi graph on nnodes is constructed by including each edge (i, j) with probabilityp∈[0,1]

    Graph problems on Erd˝ os-R´ enyi graphs with high edge probability We first consider graph optimization problems such as MAXCUT or Maximum Independent Set (MIS) on un- weighted Erd˝ os-R´ enyi graphs. An Erd˝ os-R´ enyi graph on nnodes is constructed by including each edge (i, j) with probabilityp∈[0,1]. Whenp >1/2, it is often advantageous to begin with...

    work page

  19. [19]

    Skewed Sherrington-Kirkpatrick model The Sherrington-Kirkpatrick (SK) model corresponds to MAXCUT on a complete graph with random edge 15 weights, commonly drawn fromU({−1,+1}). In the skewedSK model, we assign edge weights as follows: with probabilityp, the weight is set to +1, and with probabil- ity 1−p, it is set to−p/(1−p), ensuring zero mean edge wei...

    work page

  20. [20]

    , n}and edge setE ⊆ V × V

    Maximum Independent Set with indicator-function penalties LetG= (V,E) be a graph with vertex setV= {1, . . . , n}and edge setE ⊆ V × V. The MIS problem can be formulated as max x∈{0,1}n nX i=1 xi subject to:x ixj = 0∀(i, j)∈ E. In standard QAOA formulations, violations are penalized using the quadratic function g(x) = X (i,j)∈E xixj. An alternative approa...

    work page

  21. [21]

    The optimization problem is given by min x∈{0,1}n (x−1) T Σ(x−1) subject to:ℓ≤x T Σx≤u

    Index tracking with risk constraints We finally consider index tracking with risk constraints for a portfolio ofnassets, characterized by a covariance matrix Σ of returns. The optimization problem is given by min x∈{0,1}n (x−1) T Σ(x−1) subject to:ℓ≤x T Σx≤u. Here, the objective minimizes the tracking error relative to the full index, while the constraint...

    work page

  22. [22]

    Abbas, A

    A. Abbas, A. Ambainis, B. Augustino, A. B¨ artschi, H. Buhrman, C. Coffrin, G. Cortiana, V. Dunjko, D. J. Egger, B. G. Elmegreen, N. Franco, F. Fratini, B. Fuller, J. Gacon, C. Gonciulea, S. Gribling, S. Gupta, S. Had- field, R. Heese, G. Kircher, T. Kleinert, T. Koch, G. Korpas, S. Lenk, J. Mareˇ cek, V. Markov, G. Maz- zola, S. Mensa, N. Mohseni, G. Nan...

    work page 2024

  23. [23]

    Quantum optimization benchmarking library – the in- tractable decathlon,

    T. Koch, D. E. Bernal Neira, Y. Chen, G. Cortiana, D. J. Egger, R. Heese, N. N. Hegade, A. Gomez Ca- david, R. Huang, T. Itoko, T. Kleinert, P. M. Xavier, N. Mohseni, J. A. Monta˜ nez-Barrera, K. Nakano, G. Nannicini, C. O’Meara, J. Pauckert, M. Proissl, A. Ramesh, M. Schicker, N. Shimada, M. Takeori, V. Valls, D. Van Bulck, S. Woerner, and C. Zoufal, “Qu...

    work page arXiv 2025

  24. [24]

    A Quantum Approximate Optimization Algorithm

    E. Farhi, J. Goldstone, and S. Gutmann, “A quan- tum approximate optimization algorithm,” (2014), arXiv:1411.4028 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  25. [25]

    Hadfield, Z

    S. Hadfield, Z. Wang, B. O’Gorman, E. G. Rieffel, D. Venturelli, and R. Biswas, Algorithms12, 34 (2019)

    work page 2019

  26. [26]

    Z. Wang, N. C. Rubin, J. M. Dominy, and E. G. Rieffel, Physical Review A101, 012320 (2020)

    work page 2020

  27. [27]

    Weidenfeller, L

    J. Weidenfeller, L. C. Valor, J. Gacon, C. Tornow, L. Bello, S. Woerner, and D. J. Egger, Quantum6, 870 (2022)

    work page 2022

  28. [28]

    Dr˘ agoi, A

    S. Dr˘ agoi, A. Baiardi, and D. J. Egger, Physical Review Research8, 023159 (2026)

    work page 2026

  29. [29]

    T. Peng, A. W. Harrow, M. Ozols, and X. Wu, Physical Review Letters125, 150504 (2020)

    work page 2020

  30. [30]

    Mitarai and K

    K. Mitarai and K. Fujii, New Journal of Physics23, 023021 (2021)

    work page 2021

  31. [31]

    M. A. Perlin, Z. H. Saleem, M. Suchara, and J. C. Os- born, npj Quantum Information7, 64 (2021)

    work page 2021

  32. [32]

    W. Tang, T. Tomesh, M. Suchara, J. Larson, and M. Martonosi, inProceedings of the 26th ACM Inter- national Conference on Architectural Support for Pro- gramming Languages and Operating Systems, ASPLOS ’21 (Association for Computing Machinery, New York, NY, USA, 2021) pp. 473–486

    work page 2021

  33. [33]

    Piveteau and D

    C. Piveteau and D. Sutter, IEEE Transactions on Infor- mation Theory70, 2734 (2024)

    work page 2024

  34. [34]

    A. Lowe, M. Medvidovi´ c, A. Hayes, L. J. O’Riordan, T. R. Bromley, J. M. Arrazola, and N. Killoran, Quan- tum7, 934 (2023)

    work page 2023

  35. [35]

    Brenner, C

    L. Brenner, C. Piveteau, and D. Sutter, IEEE Transac- tions on Information Theory71, 7742 (2025)

    work page 2025

  36. [36]

    Schmitt, C

    L. Schmitt, C. Piveteau, and D. Sutter, Quantum9, 1634 (2025)

    work page 2025

  37. [37]

    A. W. Harrow and A. Lowe, PRX Quantum6, 010316 (2025)

    work page 2025

  38. [38]

    Carrera Vazquez, R

    A. Carrera Vazquez, R. Hiptmair, and S. Woerner, ACM Transactions on Quantum Computing3, 1 (2022)

    work page 2022

  39. [39]

    Carrera Vazquez, D

    A. Carrera Vazquez, D. J. Egger, D. Ochsner, and S. Wo- erner, Quantum7, 1067 (2023)

    work page 2023

  40. [40]

    Temme, S

    K. Temme, S. Bravyi, and J. M. Gambetta, Physical Review Letters119, 180509 (2017)

    work page 2017

  41. [41]

    S. Endo, S. C. Benjamin, and Y. Li, Physical Review X 8, 031027 (2018)

    work page 2018

  42. [42]

    Piveteau, D

    C. Piveteau, D. Sutter, and S. Woerner, npj Quantum Information8, 12 (2022)

    work page 2022

  43. [43]

    van den Berg, Z

    E. van den Berg, Z. K. Minev, A. Kandala, and K. Temme, Nature Physics19, 1116 (2023)

    work page 2023

  44. [44]

    Robledo-Moreno, M

    J. Robledo-Moreno, M. Motta, H. Haas, A. Javadi- Abhari, P. Jurcevic, W. Kirby, S. Martiel, K. Sharma, S. Sharma, T. Shirakawa, I. Sitdikov, R.-Y. Sun, K. J. Sung, M. Takita, M. C. Tran, S. Yunoki, and A. Mezza- capo, Science Advances11, eadu9991 (2025)

    work page 2025

  45. [45]

    Quantum- centric algorithm for sample-based Krylov diagonaliza- tion,

    J. Yu, J. Robledo Moreno, J. T. Iosue, L. Bertels, D. Claudino, B. Fuller, P. Groszkowski, T. S. Hum- ble, P. Jurcevic, W. Kirby, T. A. Maier, M. Motta, 17 # Experiment Best solution CVaR 1/Γ P-feasible 1 Fourier-based LCU 55 39.979368 0.027335 55 40.014231 0.027151 56 39.971362 0.027403 57 39.967406 0.027162 57 40.003091 0.027470 58 39.944893 0.027174 58...

    work page arXiv 2025

  46. [46]

    Piccinelli, A

    S. Piccinelli, A. Baiardi, S. Barison, M. Rossmannek, A. Carrera Vazquez, F. Tacchino, S. Mensa, E. Alta- mura, A. Alavi, M. Motta, J. Robledo-Moreno, W. Kirby, K. Sharma, A. Mezzacapo, and I. Tavernelli, “Quan- tum chemistry with provable convergence via randomized sample-based Krylov quantum diagonalization,” (2026), arXiv:2508.02578 [quant-ph]

    work page arXiv 2026

  47. [47]

    Feige, G

    U. Feige, G. Kortsarz, and D. Peleg, Algorithmica29, 410 (2001)

    work page 2001

  48. [48]

    A. M. Childs and N. Wiebe, Quantum Information and Computation12, 901 (2012)

    work page 2012

  49. [49]

    D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, Physical Review Letters114, 090502 (2015)

    work page 2015

  50. [50]

    P. K. Faehrmann, M. Steudtner, R. Kueng, M. Kieferov´ a, and J. Eisert, Quantum6, 806 (2022)

    work page 2022

  51. [51]

    Chakraborty, Quantum8, 1496 (2024)

    S. Chakraborty, Quantum8, 1496 (2024)

    work page 2024

  52. [52]

    S. V. Barron, D. J. Egger, E. Pelofske, A. B¨ artschi, S. Eidenbenz, M. Lehmkuehler, and S. Woerner, Nature Computational Science4, 865 (2024)

    work page 2024

  53. [53]

    Welch, D

    J. Welch, D. Greenbaum, S. Mostame, and A. Aspuru- Guzik, New Journal of Physics16, 033040 (2014)

    work page 2014

  54. [54]

    A. V. Oppenheim, A. S. Willsky, and S. H. Nawab,Sig- nals and Systems, 2nd ed. (Prentice Hall, Upper Saddle River, NJ, 1999)

    work page 1999

  55. [55]

    B¨ aumer and S

    E. B¨ aumer and S. Woerner, Physical Review Research7, 023120 (2025)

    work page 2025

  56. [56]

    Goodman and N

    R. Goodman and N. R. Wallach,Symmetry, Representa- tions, and Invariants, Graduate Texts in Mathematics, Vol. 255 (Springer, 2009)

    work page 2009

  57. [57]

    Leaser,Fourier Analysis on SU(2), Master’s thesis, East Carolina University (2012)

    T. Leaser,Fourier Analysis on SU(2), Master’s thesis, East Carolina University (2012)

    work page 2012

  58. [58]

    D. J. Egger, J. Mareˇ cek, and S. Woerner, Quantum5, 479 (2021)

    work page 2021

  59. [59]

    Streif and M

    M. Streif and M. Leib, Quantum Science and Technology 5, 034008 (2020). 18

    work page 2020

  60. [60]

    Pauli propagation: A computational framework for simulating quantum systems,

    M. S. Rudolph, T. Jones, Y. Teng, A. Angrisani, and Z. Holmes, “Pauli propagation: A computational framework for simulating quantum systems,” (2025), arXiv:2505.21606 [quant-ph]

    work page arXiv 2025

  61. [61]

    For Fixed Control Parameters the Quantum Approximate Optimization Algorithm's Objective Function Value Concentrates for Typical Instances

    F. G. S. L. Brand˜ ao, M. Broughton, E. Farhi, S. Gut- mann, and H. Neven, “For fixed control parameters the quantum approximate optimization algorithm’s ob- jective function value concentrates for typical instances,” (2018), arXiv:1812.04170 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  62. [62]

    Akshay, D

    V. Akshay, D. Rabinovich, E. Campos, and J. Biamonte, Physical Review A104, L010401 (2021)

    work page 2021

  63. [63]

    Galda, E

    A. Galda, E. Gupta, J. Falla, X. Liu, D. Lykov, Y. Alex- eev, and I. Safro, Frontiers in Quantum Science and Technology2, 1200975 (2023)

    work page 2023

  64. [64]

    P. K. Barkoutsos, G. Nannicini, A. Robert, I. Tavernelli, and S. Woerner, Quantum4, 256 (2020)

    work page 2020

  65. [65]

    Gabbassov, G

    E. Gabbassov, G. Rosenberg, and A. Scherer, Physical Review Research7, 023305 (2025)

    work page 2025

  66. [66]

    T. V. Le and V. Kekatos, Physical Review A110, 022430 (2024)

    work page 2024

  67. [67]

    Cutting slack: Quantum optimization with slack-free methods for combinatorial benchmarks,

    M. Sharma and H. C. Lau, “Cutting slack: Quantum optimization with slack-free methods for combinatorial benchmarks,” (2025), arXiv:2507.12159 [quant-ph]

    work page arXiv 2025

  68. [68]

    Kotil, E

    A. Kotil, E. Pelofske, S. Riedm¨ uller, D. J. Egger, S. Ei- denbenz, T. Koch, and S. Woerner, Nature Computa- tional Science5, 1168 (2025)

    work page 2025

  69. [69]

    IBM,IBM ILOG CPLEX Optimization Studio, IBM (2026), accessed 2026-05-18

    work page 2026

  70. [70]

    M. J. D. Powell, inAdvances in Optimization and Nu- merical Analysis, Mathematics and Its Applications, Vol. 275, edited by S. G´ omez and J.-P. Hennart (Springer, Dordrecht, 1994) pp. 51–67

    work page 1994

  71. [71]

    Z. He, R. Shaydulin, S. Chakrabarti, D. Herman, C. Li, Y. Sun, and M. Pistoia, npj Quantum Information9, 121 (2023)

    work page 2023

  72. [72]

    IBM Quantum,IBM Quantum Platform, IBM (2026), accessed 2026-05-18

    work page 2026

  73. [73]

    Bacon, I

    D. Bacon, I. L. Chuang, and A. W. Harrow, inProceed- ings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’07 (Society for Indus- trial and Applied Mathematics, Philadelphia, PA, 2007) pp. 1235–1244

    work page 2007

  74. [74]

    Matrix integrals over unitary groups: An application of Schur–Weyl duality,

    L. Zhang, “Matrix integrals over unitary groups: An application of Schur–Weyl duality,” (2014), arXiv:1408.3782 [quant-ph]

    work page arXiv 2014

  75. [75]

    J. S. Frame, G. d. B. Robinson, and R. M. Thrall, Cana- dian Journal of Mathematics6, 316 (1954)

    work page 1954

  76. [76]

    J. J. Sakurai,Modern Quantum Mechanics(Addison- Wesley, Reading, MA, 1994)

    work page 1994

  77. [77]

    Bucher, J

    D. Bucher, J. Stein, S. Feld, and C. Linnhoff-Popien, Physical Review A112, 062605 (2025)

    work page 2025