Circuit Design Informed Adaptive Variational Quantum Algorithms

Dimitris G. Angelakis; Muhammad Umer

REVIEW 2 major objections 5 minor 37 references

Circuit Design Informed Adaptive Variational Quantum Algorithms

T0 review · 2 major / 5 minor · reviewed 2026-07-11 · grok-4.5

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Folding Hadamard-test circuit rules into adaptive gate pools cuts measurement overhead by 25–55% while still building expressive low-depth ansätze for the nonlinear Schrödinger ground state.

desk verdict Useful HT-constrained adaptive-pool idea with real NLSE demos, but the headline 25–55% measurement cut is estimated under random pool growth, not the adaptive policy that produced the circuits. read the letter →

arxiv 2607.04110 v1 pith:IGRREXUX submitted 2026-07-05 quant-ph

Muhammad Umer , Dimitris G. Angelakis This is my paper · ORCID

classification quant-ph

keywords variationalquantumalgorithmsadaptiveansätzeHadamardtestmeasurementoverheadnonlinearSchrödingerequationNISQgatepoolpruninghardware-awarecircuits

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Adaptive variational quantum algorithms build their circuits by repeatedly testing a large pool of candidate gates, which creates heavy measurement costs on NISQ hardware. This paper shows that the structural rules of the Hadamard-test circuit layout—together with hardware connectivity and a simple non-redundancy rule—can be built directly into that candidate pool so that many gates are never even considered. The pruned pool grows far more slowly: its size scales as constant or linear rather than linear or quadratic at the first iteration, and stays below 75% of its unconstrained maximum thereafter. On the ground-state problem of the nonlinear Schrödinger equation the resulting adaptive circuits still reach 95–99% fidelity with shallow depth. The practical claim is therefore that circuit-design constraints are not merely compilation details but a systematic way to make adaptive algorithms resource-friendly without sacrificing expressivity.

What carries the argument

The dynamically constrained gate pool O_pool: at each adaptive iteration only controlled rotations whose control qubit has already appeared as a target (or single-qubit gate) are admitted, subject also to nearest-neighbour connectivity and temporary exclusion of recently used gates; this single object simultaneously enforces HT-compatible circuit structure and shrinks the measurement budget.

What would settle it

Re-run the identical adaptive procedure on the same NLSE instances while logging the exact size of O_pool at every iteration under the true (cost- or gradient-based) selection rule; if the observed pool sizes stay within a few percent of the random-selection curves in Fig. 8, the resource claim stands; otherwise it does not transfer.

Watch

Extended reading notes

Core claim

Incorporating Hadamard-test design constraints (controls may act only on qubits already touched by earlier gates), hardware-aware connectivity, and non-redundancy into the candidate gate pool of adaptive variational algorithms reduces the measurement resources needed to screen that pool by at least 25% and up to 50–55%, while the same constrained pool still yields low-depth, high-fidelity ansätze for the nonlinear Schrödinger ground state.

Load-bearing premise

The claimed 25–55% measurement savings rests on randomly picking three gates per iteration and averaging pool size over 100 trials; if the real adaptive selection rule expands the admissible set differently, the percentage reduction may not hold for the algorithm as actually run.

Editorial extensions

If this is right

Adaptive ansätze for any algorithm that already uses a Hadamard-test layout can inherit the same pool pruning and the associated measurement reduction.
Nearest-neighbour hardware with ring connectivity becomes markedly more practical for adaptive VQAs once HT constraints keep the first-iteration pool size constant rather than linear in n.
Layered HT circuits that simply repeat a fixed pattern can be replaced by the adaptive sequences generated here, which reach higher fidelity at comparable or lower depth.
The same three constraints can be applied to other two-qubit gate sets (CR_x, CR_z, etc.) without changing the scaling argument.
Measurement overhead, not just gate count, becomes a first-class design metric when circuit architecture is folded into the adaptive loop.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same control-qubit ancestry rule could prune pools in other adaptive schemes that are not built around the Hadamard test, provided the target circuit family has an analogous directed dependency among qubits.
Because the pool remains under 75% of maximum even after many iterations, the method may remain advantageous on devices whose connectivity is denser than nearest-neighbour, provided non-redundancy is kept.
A natural next experiment is to measure wall-clock shot counts on real hardware for the constrained versus unconstrained pools on the same NLSE instances; any discrepancy would quantify the gap between random and adaptive pool growth.
If the relaxed acceptance criterion used for the min-cost rule is replaced by a true rollback, the pool-size trajectory might change, offering a clean test of how selection policy interacts with the design constraints.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The one thing worth knowing: they fold Hadamard-test control rules (plus connectivity and non-redundancy) into an adaptive gate pool, show that this shrinks first-iteration scaling from linear/quadratic to constant/linear, and then build shallow adaptive ansätze that hit high fidelity on NLSE ground states where fixed layered HT layouts stall around 50–90%.

What is actually new is the dynamical pool under HT eligibility—control qubits only after they have been targets—together with a quantified pool-size fraction over iterations and the NLSE adaptive circuits. Prior ADAPT-style work and their own HT layered paper are the base; the combination and the resource accounting are the extension. The contrast in the paper is clean: layered HT (Fig. 4) is under-expressive for these g values; min-cost and max-gradient adaptive runs with the constrained pool (Figs. 5–7) reach ~95–99% fidelity at modest gate counts, and the example circuits in Fig. 6 look compact. First-iteration scaling arguments (Fig. 2) are straightforward and solid. Circularity is low—the fidelities and pool pruning are numerical outcomes, not forced by definitions.

Soft spots, in proportion. The central resource claim (abstract, §III C, Fig. 8) averages |O_pool|/S0 under random selection of three gates per step, not under the actual min-cost/max-gradient trajectories used for the NLSE results. Because eligibility is path-dependent, those fractions need not match the algorithm as run; they never report pool size or cumulative evaluations along Figs. 5–7. That weakens the 25–55% number as stated, though the qualitative pruning story and the ≤75% bound under the constraints still stand. Systems are n=3–5, no error bars, f' and m are hand-chosen, and data/code are closed. Self-citation of their HT and SGEO papers is heavy but supplies tools rather than manufacturing the main claim.

This is for people already building adaptive VQAs or HT-style PDE solvers on NISQ hardware. It is a methods paper with a clear, limited-scope result—not a new technology class. Math and numerics are competent; citation pattern is normal for a follow-on. I would send it to peer review. Engage if you care about pool overhead or HT ansätze; skip if you need large-n or open artifacts. Worth a careful read, not a rewrite of your roadmap.

Referee Report

2 major / 5 minor

Summary. The manuscript proposes adaptive variational quantum algorithms whose candidate gate pool is pruned by three constraints: hardware qubit connectivity, Hadamard-test (HT) circuit-design rules that admit a control qubit only after it has already been a target, and a non-redundancy rule that temporarily removes a gate after selection. The authors argue that these constraints reduce the pool size (and therefore the measurement overhead of repeated pool evaluations) from linear/quadratic to constant/linear scaling in the first iteration and keep the pool below 75% of its unconstrained maximum thereafter, yielding claimed savings of at least 25% and up to 50–55%. Expressivity is demonstrated on the ground-state problem of the nonlinear Schrödinger equation (NLSE) for n=3–5 qubits and several nonlinearity strengths g, using both minimum-cost and maximum-gradient selection criteria; the resulting adaptive HT ansätze reach high state fidelities while remaining shallow. Layered HT ansätze of fixed layout are shown to be less expressive on the same instances.

Significance. If the resource-reduction claim holds for the algorithm as actually run, the work supplies a concrete, hardware- and circuit-aware design principle that can be ported to other adaptive VQAs and that directly addresses a known measurement bottleneck of ADAPT-style methods. The numerical demonstration that constrained adaptive circuits outperform fixed-layer HT ansätze on the NLSE is useful for the growing literature on variational solvers for nonlinear PDEs. The paper is transparent about its free parameters (m, f′, ε_threshold) and builds on the authors’ prior HT and cost-estimation machinery without reducing the new claims to tautologies. The main quantitative claim, however, rests on a proxy experiment whose fidelity to the real adaptive trajectories is not established; that gap limits the strength of the resource-efficiency conclusion as currently written.

major comments (2)

§III C and Fig. 8: The central quantitative claim (abstract; §II B; §III C) of ≥25% (often 50–55%) measurement-resource reduction is obtained by randomly selecting three gates per iteration, updating O_pool under the three constraints, and averaging over 100 realizations. Because the admissible set is path-dependent (a control becomes eligible only after it has been a target), the growth of |O_pool| under random selection need not match the growth under the actual min-cost / max-gradient policy used for the NLSE circuits of Figs. 5–7. The manuscript never reports |O_pool| or cumulative circuit evaluations along those real adaptive trajectories. Without that comparison (or an argument that the fraction relative to S0 is policy-independent), the 25–55% savings figure does not transfer to the algorithm as run and should be either recomputed on the actual trajectories or clearly labeled as a
§II C / Algorithm 1 and §III B: The relaxed acceptance criterion (C′_k ≤ f′ C_{k−1} with f′=1.35, and forced acceptance after r>2) is introduced without systematic justification or sensitivity analysis. Because the minimum-cost results of Figs. 5a,c and 7 rely on this rule, and because the rule can permanently enlarge the ansatz even when the cost increases, the paper should either (i) report how the final fidelities and gate counts change under a range of f′ (or under pure rollback), or (ii) restrict the main claims to the maximum-gradient criterion, for which C′_k ≤ C_{k−1} always holds and the extra machinery is unnecessary.

minor comments (5)

Fig. 4 caption states results for g=250 in (a,b) and g=750 in (c,d), but the body text of §III A refers only to panels (a–d) without clarifying that (c,d) exist; the figure itself appears to show only two panels in the provided text. Align caption, body, and figure.
Fig. 5 caption likewise refers to panels (a–d) while the surrounding text and the figure block as rendered mention only (a,b); ensure consistent labeling.
§II B: The non-redundancy rule is described only informally (“reinstate it only after another gate acts on at least one of the control−target qubits”). A precise statement of when a gate re-enters O_pool would aid reproducibility.
The manuscript cites the authors’ own HT design [23] and SGEO/cost-estimation work [32] heavily; a short paragraph situating the present pool-pruning idea relative to other adaptive pool constructions (e.g., qubit-ADAPT, TETRIS-ADAPT, hardware-native pools) would help readers assess novelty.
Typographical: “Schr¨ odinger” and similar spacing artifacts appear throughout; “ans¨ atze” should be consistently “ansätze” or “ansatze”.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor non-load-bearing self-citation of the authors' prior HT design [23] and cost/SGEO routines [32]; pool-size reduction and NLSE fidelities are independent numerical outcomes, not identities forced by those citations.

self citation load bearing [§II A (Hadamard Test Constraints) and Algorithm 1 / §II C]
"Recently, design constraints that leverage the logical composition of HT circuits were investigated in Ref. [23] and shown to facilitate low-depth circuit constructions. ... we evaluate cost function C_k(λ_{k-1},λ_p) by utilizing the procedure discussed in Ref. [32]."

The HT-compatible control restriction that prunes O_pool and the cost-evaluation routine that drives gate selection are justified solely by citations whose author lists overlap with the present paper. The citations are not machine-checked uniqueness theorems and are not re-derived here; they simply import the design rule and the optimizer. The subsequent numerical claims remain independent of those imports, so the circularity is minor and non-load-bearing for the 25–55% savings or the fidelity results.

full rationale

The paper's central quantitative claims (pool scaling reduced from linear/quadratic to constant/linear in the first iteration; |O_pool| stays ≤75% of S0 thereafter, yielding ≥25% and often 50–55% measurement savings; adaptive min-cost/max-gradient ansätze reach >95–99% fidelity on NLSE ground states) are obtained by direct enumeration of the constrained candidate set and by explicit variational optimization against an independent classical imaginary-time benchmark. These are not algebraic identities or fitted predictions. The only self-citations supply reusable tools: the HT control-target ordering rule restated from Ref. [23] and the cost-evaluation/SGEO procedure of Ref. [32]. Neither citation is invoked as a uniqueness theorem that forbids alternatives, nor does any equation reduce by construction to a parameter fitted on the same data. The random-growth protocol used for Fig. 8 is a methodological limitation on transferability, not a circularity. Hence the circularity burden is limited to ordinary building-on-prior-work self-citation and scores 2.

Assumptions & free parameters 5 free parameters · 5 assumptions · 1 invented entities

The central claim rests on standard VQA/ADAPT machinery, the authors’ prior HT circuit design, a finite-difference NLSE encoding, hand-chosen algorithmic knobs (m, f', thresholds), and the modeling choice that random pool growth proxies measurement cost. No new physical entities are postulated; the ‘constrained gate pool’ is an algorithmic construction.

free parameters (5)

f' (relaxed acceptance factor) = 1.35
Set to 1.35 for the minimum-cost criterion so gates can be kept when C'_k ≤ f' C_{k-1}; directly controls whether the ansatz grows and thus the reported circuits and fidelities.
m (gates appended per iteration) = 2 (n=3); 3 (n=4,5)
Fixed by hand: m=2 for n=3, m=3 for n=4,5; shapes ansatz depth and pool evolution.
ε_threshold (convergence indicator cutoff)
Terminates adaptive growth when ε_k is small; value not numerically specified in the text but is a free algorithmic stop.
V0 (quadratic potential depth) = 1000
Set to 10^3 for all NLSE instances; defines the target ground states being approximated.
g (nonlinearity strengths studied) = 250, 500, 750, 1000
Chosen problem instances (250, 500, 750, 1000); not fitted, but select which regimes support the expressivity claim.

assumptions (5)

domain assumption Hadamard-test compatible ansätze may use only intra-register controlled rotations whose control qubits have previously been targets (plus one ancilla-controlled X), as in Ref. [23].
§II A; this is the design rule that prunes the pool and is taken as the circuit architecture of interest.
domain assumption Nearest-neighbor ring connectivity defines the hardware-admissible two-qubit pairs for the main resource analysis.
§II B, §III B–C; pool scaling and Fig. 8 are for this connectivity model.
domain assumption NLSE ground state on n qubits is faithfully encoded by finite-difference discretization with cost ⟨⟨E⟩⟩ = EI + EK + EP and 3n+1 qubits total.
§III and Refs. [5,32]; all fidelity claims are relative to this classical discretization benchmark.
domain assumption Cost- or gradient-based selection among candidates in O_pool yields a sufficiently expressive adaptive ansatz when the pool is HT-pruned.
Algorithm 1 and §II C; standard ADAPT-style premise adapted to the constrained pool.
standard math Standard linear algebra / quantum circuit composition for controlled rotations and expectation estimation.
Throughout; no nonstandard math axioms.

invented entities (1)

HT-constrained dynamical adaptive gate pool O_pool
purpose: Restrict candidate gates by connectivity, HT control history, and non-redundancy so measurement overhead falls while adaptive growth remains possible.
Algorithmic construction, not a physical object; independent_evidence is false because its value is demonstrated only inside this paper’s numerics.

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Cite this review

Pith. "Pith review of Circuit Design Informed Adaptive Variational Quantum Algorithms." pith.science (2026). https://pith.science/paper/IGRREXUX

@misc{pith2026260704110,
  author       = {Pith},
  title        = {Pith review of: Circuit Design Informed Adaptive Variational Quantum Algorithms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IGRREXUX}},
  note         = {Machine review of arXiv:2607.04110}
}

read the original abstract

Resource-efficient computation is of central importance in the noisy intermediate-scale quantum (NISQ) era, where decoherence, gate errors, and restricted qubit connectivity severely limit the reliable execution of quantum algorithms. In this work, we demonstrate that incorporating circuit design considerations is crucial for developing resource-efficient variational quantum algorithms. By focusing on the Hadamard test circuit architecture, hardware-aware qubit connectivity, and problem-specific adaptive framework, we analyze how circuit design constraints can systematically reduce the measurement overhead associated with repeated evaluations of the candidate gate pool in adaptive algorithms. Specifically, we demonstrate reductions in the required measurement resources ranging from at least 25% to as high as 50% - 55%. To assess the effectiveness of our approach, we investigate the ground state problem of the nonlinear Schr\"{o}dinger equation. Overall, our work contributes to resource-friendly strategies for quantum computation and underscores that algorithmic frameworks should systematically integrate circuit design constraints with hardware-aware and problem-specific structures to enhance the practical feasibility of quantum devices in the NISQ era.

Figures

Figures reproduced from arXiv: 2607.04110 by the authors.

**Figure 3.** FIG. 3. Constraints embedded in the gate pool [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** FIG. 4. Behavior of state infidelity as a function of number [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Figures from the paper (2 more)

**Figure 6.** Figure 6: FIG. 6. Adaptive HT ans¨atze. Panels are organized accord [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 7.** Figure 7: FIG. 7. State infidelity as a function of the number of gates in [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

37 extracted references · 2 canonical work pages

[1]

Preskill, Quantum computing in the nisq era and be- yond, Quantum2, 79 (2018)

J. Preskill, Quantum computing in the nisq era and be- yond, Quantum2, 79 (2018)

2018
[2]

Peruzzo, J

A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’brien, A variational eigenvalue solver on a photonic quantum processor, Nature communications5, 4213 (2014)

2014
[3]

Kandala, A

A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, Hardware- efficient variational quantum eigensolver for small molecules and quantum magnets, Nature549, 242 (2017)

2017
[4]

Cerezo, A

M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio,et al., Variational quantum algorithms, Nature Reviews Physics3, 625 (2021)

2021
[5]

Lubasch, J

M. Lubasch, J. Joo, P. Moinier, M. Kiffner, and D. Jaksch, Variational quantum algorithms for nonlinear problems, Physical Review A101, 010301 (2020)

2020
[6]

Jaksch, P

D. Jaksch, P. Givi, A. J. Daley, and T. Rung, Variational quantum algorithms for computational fluid dynamics, AIAA journal61, 1885 (2023)

2023
[7]

Cirstoiu, Z

C. Cirstoiu, Z. Holmes, J. Iosue, L. Cincio, P. J. Coles, and A. Sornborger, Variational fast forwarding for quan- tum simulation beyond the coherence time, npj Quantum Information6, 82 (2020)

2020
[8]

S.-H. Lin, R. Dilip, A. G. Green, A. Smith, and F. Poll- mann, Real- and imaginary-time evolution with com- pressed quantum circuits, PRX Quantum2, 010342 (2021)

2021

Show all 37 references

[9]

Linteau, S

D. Linteau, S. Barison, N. H. Lindner, and G. Carleo, Adaptive projected variational quantum dynamics, Phys. Rev. Res.6, 023130 (2024)

2024
[10]

Farhi, J

E. Farhi, J. Goldstone, and S. Gutmann, A quantum approximate optimization al- gorithm, arXiv preprint arXiv:1411.4028 https://doi.org/10.48550/arXiv.1411.4028 (2014), https://arxiv.org/abs/1411.4028

2014 doi arXiv
[11]

Tan, M.-A

B. Tan, M.-A. Lemonde, S. Thanasilp, J. Tangpanitanon, and D. G. Angelakis, Qubit-efficient encoding schemes for binary optimisation problems, Quantum5, 454 (2021)

2021
[12]

Bengoechea, P

S. Bengoechea, P. Over, D. Jaksch, and T. Rung, Toward variational quantum algorithms for generalized linear and nonlinear transport phenomena, AIAA Journal64, 585 (2026), https://doi.org/10.2514/1.J065582

2026 doi
[13]

A. J. Pool, A. D. Somoza, C. Mc Keever, M. Lubasch, and B. Horstmann, Nonlinear dynamics as a ground-state so- lution on quantum computers, Phys. Rev. Res.6, 033257 (2024)

2024
[14]

Siegl, G

P. Siegl, G. S. Reese, T. Hashizume, N.-L. van H¨ ulst, and D. Jaksch, Tensor-programmable quantum circuits for solving differential equations, Phys. Rev. Res.8, 013052 (2026)

2026
[15]

Setty, R

A. Setty, R. Abdusalamov, and F. Motzoi, Self-adaptive physics-informed quantum machine learning for solving differential equations, Machine Learning: Science and Technology6, 015002 (2025)

2025
[16]

E. X. Huber, B. Y. Tan, P. R. Griffin, and D. G. An- gelakis, Exponential qubit reduction in optimization for financial transaction settlement, EPJ Quantum Technol- ogy11, 52 (2024)

2024
[17]

Sarma, T

A. Sarma, T. W. Watts, M. Moosa, Y. Liu, and P. L. McMahon, Quantum variational solving of nonlinear and multidimensional partial differential equations, Phys. Rev. A109, 062616 (2024)

2024
[18]

H. R. Grimsley, S. E. Economou, E. Barnes, and N. J. Mayhall, An adaptive variational algorithm for exact molecular simulations on a quantum computer, Nature communications10, 3007 (2019)

2019
[19]

S. Sim, J. Romero, J. F. Gonthier, and A. A. Kunitsa, Adaptive pruning-based optimization of parameterized quantum circuits, Quantum Science and Technology6, 025019 (2021)

2021
[20]

Bennakhi, G

A. Bennakhi, G. T. Byrd, and P. Franzon, Analyzing quantum circuit depth reduction with ancilla qubits in mcx gates, in2024 IEEE International Conference on Quantum Computing and Engineering (QCE), Vol. 2 (IEEE, 2024) pp. 510–511

2024
[21]

A. Chan, Z. Shi, L. Dellantonio, W. D¨ ur, and C. A. Muschik, Measurement-based infused circuits for vari- ational quantum eigensolvers, Phys. Rev. Lett.132, 240601 (2024). 9

2024
[22]

Tserkis, M

S. Tserkis, M. Umer, and D. G. Ange- lakis, Depth optimization of cnot ladder circuits, arXiv preprint arXiv:2511.13256 https://doi.org/10.48550/arXiv.2511.13256 (2025)

2025 doi
[23]

Mastorakis, M

E. Mastorakis, M. Umer, M. Guevara-Bertsch, J. Ulma- nis, F. Rohde, and D. G. Angelakis, Resource-efficient hadamard test tailored variational framework for nonlin- ear dynamics on quantum computers, Quantum Science and Technology11, 015061 (2026)

2026
[24]

H. L. Tang, V. Shkolnikov, G. S. Barron, H. R. Grim- sley, N. J. Mayhall, E. Barnes, and S. E. Economou, Qubit-adapt-vqe: An adaptive algorithm for construct- ing hardware-efficient ans¨ atze on a quantum processor, PRX Quantum2, 020310 (2021)

2021
[25]

Zhang, N

F. Zhang, N. Gomes, Y. Yao, P. P. Orth, and T. Iadecola, Adaptive variational quantum eigensolvers for highly ex- cited states, Phys. Rev. B104, 075159 (2021)

2021
[26]

Y. S. Yordanov, V. Armaos, C. H. Barnes, and D. R. Arvidsson-Shukur, Qubit-excitation-based adaptive vari- ational quantum eigensolver, Communications Physics4, 228 (2021)

2021
[27]

L. Zhu, H. L. Tang, G. S. Barron, F. A. Calderon-Vargas, N. J. Mayhall, E. Barnes, and S. E. Economou, Adaptive quantum approximate optimization algorithm for solving combinatorial problems on a quantum computer, Phys. Rev. Res.4, 033029 (2022)

2022
[28]

P. G. Anastasiou, Y. Chen, N. J. Mayhall, E. Barnes, and S. E. Economou, Tetris-adapt-vqe: An adaptive algo- rithm that yields shallower, denser circuit ans¨ atze, Phys. Rev. Res.6, 013254 (2024)

2024
[29]

Feniou, M

C. Feniou, M. Hassan, B. Claudon, A. Courtat, O. Ad- joua, Y. Maday, and J.-P. Piquemal, Greedy gradient-free adaptive variational quantum algorithms on a noisy in- termediate scale quantum computer, Scientific Reports 15, 18689 (2025)

2025
[30]

Wu, Y.-Q

S.-Y. Wu, Y.-Q. Song, R.-Z. Li, S.-J. Qin, Q.-Y. Wen, and F. Gao, Resource-efficient adaptive variational quan- tum algorithm for combinatorial optimization problems, Advanced Quantum Technologies , 2400484 (2025)

2025
[31]

S. Herbert, On the depth overhead incurred when running quantum algorithms on near- term quantum computers with limited qubit connectivity, arXiv preprint arXiv:1805.12570 https://doi.org/10.48550/arXiv.1805.12570 (2018)

2018 doi arXiv
[32]

M. Umer, E. Mastorakis, and D. G. Angelakis, Efficient estimation and sequential optimization of cost functions in variational quantum algorithms, Quantum Science and Technology10, 035022 (2025)

2025
[33]

E. P. Gross, Structure of a quantized vortex in boson systems, Il Nuovo Cimento (1955-1965)20, 454 (1961); L. P. Pitaevskii, Vortex lines in an imperfect bose gas, Sov. Phys. JETP13, 451 (1961)

1955
[34]

Dalfovo, S

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of bose-einstein condensation in trapped gases, Reviews of modern physics71, 463 (1999)

1999
[35]

G. P. Agrawal, Nonlinear fiber optics, inNonlinear Sci- ence at the Dawn of the 21st Century, edited by P. L. Christiansen, M. P. Sørensen, and A. C. Scott (Springer Berlin Heidelberg, Berlin, Heidelberg, 2000) pp. 195–211

2000
[36]

Edwards and K

M. Edwards and K. Burnett, Numerical solution of the nonlinear schr¨ odinger equation for small samples of trapped neutral atoms, Physical Review A51, 1382 (1995)

1995
[37]

M. Umer, E. Mastorakis, S. Evangelou, and D. G. An- gelakis, Probing the limits of variational quantum algo- rithms for nonlinear ground states on real quantum hard- ware: The effects of noise, Phys. Rev. A111, 012626 (2025)

2025

Pith tools

Reviewed July 11, 2026 · model on record in the stance chip above.

[1] [1]

Preskill, Quantum computing in the nisq era and be- yond, Quantum2, 79 (2018)

J. Preskill, Quantum computing in the nisq era and be- yond, Quantum2, 79 (2018)

2018

[2] [2]

Peruzzo, J

A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’brien, A variational eigenvalue solver on a photonic quantum processor, Nature communications5, 4213 (2014)

2014

[3] [3]

Kandala, A

A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, Hardware- efficient variational quantum eigensolver for small molecules and quantum magnets, Nature549, 242 (2017)

2017

[4] [4]

Cerezo, A

M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio,et al., Variational quantum algorithms, Nature Reviews Physics3, 625 (2021)

2021

[5] [5]

Lubasch, J

M. Lubasch, J. Joo, P. Moinier, M. Kiffner, and D. Jaksch, Variational quantum algorithms for nonlinear problems, Physical Review A101, 010301 (2020)

2020

[6] [6]

Jaksch, P

D. Jaksch, P. Givi, A. J. Daley, and T. Rung, Variational quantum algorithms for computational fluid dynamics, AIAA journal61, 1885 (2023)

2023

[7] [7]

Cirstoiu, Z

C. Cirstoiu, Z. Holmes, J. Iosue, L. Cincio, P. J. Coles, and A. Sornborger, Variational fast forwarding for quan- tum simulation beyond the coherence time, npj Quantum Information6, 82 (2020)

2020

[8] [8]

S.-H. Lin, R. Dilip, A. G. Green, A. Smith, and F. Poll- mann, Real- and imaginary-time evolution with com- pressed quantum circuits, PRX Quantum2, 010342 (2021)

2021