arxiv: 2604.07909 · v1 · submitted 2026-04-09 · 🪐 quant-ph

Recognition: no theorem link

A Review of Variational Quantum Algorithms: Insights into Fault-Tolerant Quantum Computing

Zhirao Wang , Junxiang Huang , Runyu Ye , Qingyu Li , Qi-Ming Ding , Yiming Huang , Ting Zhang , Yumeng Zeng , Jianshuo Gao , Xiao Yuan , Yuan Yao

Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:08 UTC · model grok-4.3

classification 🪐 quant-ph

keywords variational quantum algorithmsfault-tolerant quantum computingnoisy intermediate-scale quantumbarren plateauserror mitigationparameterized quantum circuitsquantum error correctionquantum chemistry

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The pith

Variational quantum algorithms can be refined to operate in fault-tolerant regimes through targeted updates to ansatz design, optimization, and error handling.

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

This review examines how variational quantum algorithms, built on parameterized quantum circuits paired with classical optimizers, can evolve as hardware advances beyond noisy intermediate-scale devices. It breaks down the framework into ansatz choices, cost functions, gradient methods, and strategies for handling training difficulties such as barren plateaus. The discussion covers how error mitigation techniques can support performance during the early fault-tolerant transition and identifies specific obstacles that arise in fully error-corrected settings. Applications in physics, chemistry, machine learning, and optimization are surveyed to illustrate continued utility. The central contribution is a theoretical roadmap showing how variational methods can be updated systematically to stay viable rather than being superseded.

Core claim

The paper claims that deconstructing VQAs into ansatz design, classical optimization strategies including cost formulation and gradient computation, evaluation of barren plateaus with mitigation approaches, integration of quantum error mitigation for early fault-tolerant phases, and analysis of challenges in the full fault-tolerant phase together produce a theoretical roadmap for adapting these algorithms to future hardware generations so variational principles remain relevant and efficient within error-corrected environments.

What carries the argument

The theoretical roadmap for transitioning VQAs from NISQ to fault-tolerant regimes, which combines ansatz design, classical optimizers, barren plateau mitigations, and error mitigation to sustain algorithmic performance across hardware generations.

If this is right

VQAs can continue delivering value in many-body physics and quantum chemistry by incorporating partial error correction while retaining their hybrid structure.
Barren plateau issues can be managed through refined ansatzes to support scaling on more reliable devices.
Hybrid models will need adjustments to cost functions and gradient calculations to align with logical qubit operations in full fault-tolerance.
Applications in machine learning and mathematical optimization can extend directly to error-corrected hardware using the same variational core.
Systematic updates to the framework prevent VQAs from becoming obsolete as architectures mature.

Where Pith is reading between the lines

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

Error correction codes could be co-designed with specific variational ansatzes to improve compatibility and reduce overhead.
Classical optimization loops may require quantum-aware redesigns to handle operations on logical rather than physical qubits efficiently.
The roadmap implies that practical quantum advantage in chemistry could arrive earlier if mitigation techniques bridge the gap to partial correction.
Similar transition strategies might apply to other hybrid quantum-classical methods beyond the surveyed VQA variants.

Load-bearing premise

That current variational frameworks and their mitigation strategies can be extended and refined to work effectively once full error correction is available, without requiring entirely new non-variational paradigms.

What would settle it

A demonstration on early fault-tolerant hardware where VQAs, even after applying all reviewed ansatz improvements and error mitigation, show no sustained advantage over classical methods in a target application such as quantum chemistry simulation would undermine the roadmap.

Figures

Figures reproduced from arXiv: 2604.07909 by Jianshuo Gao, Junxiang Huang, Qi-Ming Ding, Qingyu Li, Runyu Ye, Ting Zhang, Xiao Yuan, Yiming Huang, Yuan Yao, Yumeng Zeng, Zhirao Wang.

**Figure 2.** Figure 2: FIG. 2. Applications of VQAs across major disciplines. [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

read the original abstract

Variational quantum algorithms (VQAs) have established themselves as a central computational paradigm in the Noisy Intermediate-Scale Quantum (NISQ) era. By coupling parameterized quantum circuits (PQCs) with classical optimization, they operate effectively under strict hardware limitations. However, as quantum architectures transition toward early fault-tolerant (EFT) and ultimate fault-tolerant (FT) regimes, the foundational principles and long-term viability of VQAs require systematic reassessment. This review offers an insightful analysis of VQAs and their progression toward the fault-tolerant regime. We deconstruct the core algorithmic framework by examining ansatz design and classical optimization strategies, including cost function formulation, gradient computation, and optimizer selection. Concurrently, we evaluate critical training bottlenecks, notably barren plateaus (BPs), alongside established mitigation strategies. The discussion then explores the EFT phase, detailing how the integration of quantum error mitigation and partial error correction can sustain algorithmic performance. Addressing the FT phase, we analyze the inherent challenges confronting current hybrid VQA models. Furthermore, we synthesize recent VQA applications across diverse domains, including many-body physics, quantum chemistry, machine learning, and mathematical optimization. Ultimately, this review outlines a theoretical roadmap for adapting quantum algorithms to future hardware generations, elucidating how variational principles can be systematically refined to maintain their relevance and efficiency within an error-corrected computational environment.

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

This is a standard literature review on VQAs that organizes existing ideas for the shift to fault tolerance but adds no new methods or results.

read the letter

This review compiles work on variational quantum algorithms and how they might hold up as hardware moves from NISQ toward early and full fault tolerance. It does not present new algorithms, proofs, or empirical tests, but it lays out the pieces in a straightforward sequence: ansatz design, cost functions, gradient methods, optimizer choices, barren plateaus with mitigation, error mitigation for the early fault-tolerant stage, and the harder problems that appear once full error correction is in place. The applications section covers many-body physics, quantum chemistry, machine learning, and optimization, and the closing roadmap tries to sketch how variational ideas could be adjusted for error-corrected machines. That structure is logical and covers the main topics people in the field already discuss. The soft spots are the usual ones for a review. The roadmap stays fairly general, and the value rests on whether the chosen papers are representative and up to date; nothing in the outline suggests selective omission or misreading, but a referee would still need to check the actual citations and depth. No internal contradictions or parameter-fitting issues appear, since there are no new derivations. This paper is mainly for researchers and students already working on VQAs who want a single place to see how the pieces connect to future hardware. It is not essential for experts who follow the literature closely. I would send it to peer review so referees can assess the completeness of the synthesis and the practicality of the roadmap.

Referee Report

0 major / 3 minor

Summary. The manuscript is a review of variational quantum algorithms (VQAs) that deconstructs their core framework by examining ansatz design, classical optimization strategies (including cost functions, gradients, and optimizers), evaluates training bottlenecks such as barren plateaus and associated mitigations, explores integration with error mitigation and partial correction in the early fault-tolerant (EFT) regime, analyzes challenges for hybrid models in the full fault-tolerant (FT) regime, synthesizes applications across many-body physics, quantum chemistry, machine learning, and optimization, and outlines a theoretical roadmap for refining variational principles in error-corrected environments.

Significance. If the literature synthesis is representative and the roadmap is substantiated, the review could provide significant value to the quantum computing community by consolidating current understanding of VQAs and offering guidance on their evolution beyond the NISQ era. The logical structure and coverage of key topics like barren plateaus and hardware transitions position it as a potentially useful reference for researchers adapting algorithms to improving quantum hardware.

minor comments (3)

[Abstract] Abstract: the self-referential claim that 'this review offers an insightful analysis' is subjective and should be revised to neutral language such as 'this review provides an analysis'.
[Roadmap discussion] The roadmap section would benefit from a summary table or bullet points listing concrete adaptation steps for key VQA components (e.g., ansatz modifications under error correction) to improve clarity and actionability.
[Throughout] Ensure consistent first-use definitions for acronyms such as PQC, BP, EFT, and FT throughout the manuscript.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and detailed summary of our manuscript, as well as for recommending minor revision. The assessment that the review consolidates understanding of VQAs and provides guidance for the transition beyond the NISQ era is appreciated. Since the report raises no specific major comments or points requiring clarification, we interpret the minor revision recommendation as an opportunity for general polishing.

Circularity Check

0 steps flagged

No significant circularity in this literature review

full rationale

This is a review paper that synthesizes existing literature on VQAs, ansatz design, barren plateaus, error mitigation, and adaptations to early and full fault-tolerant regimes without introducing original derivations, theorems, or quantitative models. The abstract and structure describe deconstruction of frameworks and outlining of a theoretical roadmap based on reviewed works, with no equations, fitted parameters, or self-referential definitions that reduce claims to inputs by construction. All load-bearing elements are external citations or standard field consensus, making the synthesis self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

As a review paper the central claims rest entirely on synthesis of prior literature; no new free parameters, axioms, or invented entities are introduced by the authors.

pith-pipeline@v0.9.0 · 5575 in / 1003 out tokens · 52246 ms · 2026-05-10T18:08:06.617870+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Recent Advances in Quantum Architecture Search
quant-ph 2026-04 unverdicted

A survey of core concepts, representative methodologies, applications, challenges, and future directions in Quantum Architecture Search for variational quantum algorithms.

Reference graph

Works this paper leans on

300 extracted references · 40 canonical work pages · cited by 1 Pith paper · 5 internal anchors

[1]

PCA In data science, PCA has also seen VQA advances: early quantum PCA algorithms [186] relied on quan- tum phase estimation and density matrix exponentia- tion, while subsequent variational approaches [165, 187] include a quantum state diagonalization algorithm (us- ing a cost function quantifying Hilbert-Schmidt distance between ˆρ(θ) =U(θ)ρU(θ) † and t...
[2]

[32], approxi- mates solutions to combinatorial optimization problems (e.g., Max-Cut, SAT) by discretizing adiabatic evolution into alternating phase and mixing layers

QAOA The QAOA, proposed by Farhi et al. [32], approxi- mates solutions to combinatorial optimization problems (e.g., Max-Cut, SAT) by discretizing adiabatic evolution into alternating phase and mixing layers. Variational pa- rameters are optimized classically to minimize the cost Hamiltonian expectation value, guiding the system to- ward the ground state ...
[3]

This section reviews VQA-based models across supervised, un- supervised, semi-supervised, and reinforcement learning frameworks

QML Quantum Machine Learning (QML) integrates quan- tum computing with machine learning, utilizing VQAs to potentially achieve computational advantages. This section reviews VQA-based models across supervised, un- supervised, semi-supervised, and reinforcement learning frameworks. a. Supervised Learning.Supervised VQC-based learning has evolved from found...
[4]

Quantum information theory VQAs have emerged as a unifying framework in quan- tum information theory (QIT) for characterizing di- verse quantum resources, particularly in the NISQ era. By employing shallow parameterized circuits and hy- brid quantum–classical optimization, these methods re- cast resource quantification, often involving extremal de- compos...
[5]

This is especially important in the NISQ setting, where errors typically accumulate with circuit depth and two-qubit gate count under realistic noise models [3, 61]

Quantum compilation In quantum compilation, the goal is to approximate a target unitaryVusing a native gate sequenceU(θ) with minimal or near-optimal circuit depth. This is especially important in the NISQ setting, where errors typically accumulate with circuit depth and two-qubit gate count under realistic noise models [3, 61]. Quantum compilation tasks ...
[6]

Quantum computing in the nisq era and beyond.Quantum, 2:79, 2018

John Preskill. Quantum computing in the nisq era and beyond.Quantum, 2:79, 2018

2018
[7]

Variational quantum algorithms.Nature Reviews Physics, 3(9):625–644, 2021

Marco Cerezo, Andrew Arrasmith, Ryan Babbush, Si- mon C Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, et al. Variational quantum algorithms.Nature Reviews Physics, 3(9):625–644, 2021

2021
[8]

Noisy intermediate-scale quantum algorithms.Reviews of Modern Physics, 94(1):015004, 2022

Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Al´ an Aspuru-Guzik. Noisy intermediate-scale quantum algorithms.Reviews of Modern Physics, 94(1):015004, 2022

2022
[9]

Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets.Nature, 549(7671):242–246, 2017

Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M Chow, and Jay M Gambetta. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets.Nature, 549(7671):242–246, 2017

2017
[10]

Early fault-tolerant quantum computing.PRX Quantum, 5(2):020101, 2024

Amara Katabarwa, Katerina Gratsea, Athena Caesura, and Peter D Johnson. Early fault-tolerant quantum computing.PRX Quantum, 5(2):020101, 2024. 19

2024
[11]

Peter W. Shor. Scheme for reducing decoherence in quantum computer memory.Physical Review A, 52(4):R2493–R2496, 1995

1995
[12]

Fault-tolerant quantum computation with constant error rate.SIAM Journal on Computing, 38(4):1207–1282, 2008

Dorit Aharonov and Michael Ben-Or. Fault-tolerant quantum computation with constant error rate.SIAM Journal on Computing, 38(4):1207–1282, 2008

2008
[13]

Suppressing quan- tum errors by scaling a surface code logical qubit.Na- ture, 614(7949):676–681, 2023

Rajeev Acharya, Igor Aleiner, et al. Suppressing quan- tum errors by scaling a surface code logical qubit.Na- ture, 614(7949):676–681, 2023

2023
[14]

A variational eigenvalue solver on a photonic quantum processor.Na- ture Communications, 5(1):4213, 2014

Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J Love, Al´ an Aspuru-Guzik, and Jeremy L O’brien. A variational eigenvalue solver on a photonic quantum processor.Na- ture Communications, 5(1):4213, 2014

2014
[15]

Hartree-fock on a superconducting qubit quantum computer.Science, 369(6507):1084–1089, 2020

Google AI Quantum, Collaborators*†, Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C Bardin, Rami Barends, Sergio Boixo, Michael Broughton, Bob B Buckley, et al. Hartree-fock on a superconducting qubit quantum computer.Science, 369(6507):1084–1089, 2020

2020
[16]

Yuwei Ma, Weiting Wang, Xianghao Mu, Weizhou Cai, Ziyue Hua, Xiaoxuan Pan, Dong-Ling Deng, Rebing Wu, Chang-Ling Zou, Lei Wang, et al. Experimental implementation of a qubit-efficient variational quantum eigensolver with analog error mitigation on a supercon- ducting quantum processor.Science China Physics, Me- chanics & Astronomy, 68(7):270311, 2025

2025
[17]

Demonstration of a cafqa-bootstrapped variational quantum eigensolver on a trapped-ion quantum com- puter.arXiv preprint arXiv:2408.06482, 2024

Qingfeng Wang, Liudmila Zhukas, Qiang Miao, Aniket S Dalvi, Peter J Love, Christopher Monroe, Frederic T Chong, and Gokul Subramanian Ravi. Demonstration of a cafqa-bootstrapped variational quantum eigensolver on a trapped-ion quantum com- puter.arXiv preprint arXiv:2408.06482, 2024

work page arXiv 2024
[18]

Kirmani, E

Ammar Kirmani, Elijah Pelofske, Andreas B¨ artschi, Stephan Eidenbenz, and Jian-Xin Zhu. Variational quantum simulations of a two-dimensional frustrated transverse-field ising model on a trapped-ion quantum computer.arXiv preprint arXiv:2505.22932, 2025

work page arXiv 2025
[19]

Photonic variational quantum eigensolver for nisq- compatible quantum technology.Nano Convergence, 12(1):60, 2025

Kang-Min Hu, Min Namkung, and Hyang-Tag Lim. Photonic variational quantum eigensolver for nisq- compatible quantum technology.Nano Convergence, 12(1):60, 2025

2025
[20]

Demonstration of Logical Qubits and Repeated Error Correction with Better-than- Physical Error Rates

A Paetznick, MP Da Silva, C Ryan-Anderson, JM Bello- Rivas, JP Campora III, A Chernoguzov, JM Dreil- ing, C Foltz, F Frachon, JP Gaebler, et al. Demon- stration of logical qubits and repeated error correction with better-than-physical error rates.arXiv preprint arXiv:2404.02280, 2024

work page arXiv 2024
[21]

A fault-tolerant neutral-atom archi- tecture for universal quantum computation.Nature, 649(8095):39–46, 2026

Dolev Bluvstein, Alexandra A Geim, Sophie H Li, Si- mon J Evered, J Pablo Bonilla Ataides, Gefen Baranes, Andi Gu, Tom Manovitz, Muqing Xu, Marcin Kali- nowski, et al. A fault-tolerant neutral-atom archi- tecture for universal quantum computation.Nature, 649(8095):39–46, 2026

2026
[22]

Scaling and networking a modular photonic quantum computer.Nature, 638(8052):912– 919, 2025

H Aghaee Rad, Thomas Ainsworth, Rafael N Alexan- der, Brandon Altieri, Mohsen F Askarani, R Baby, Leonardo Banchi, Ben Q Baragiola, J Eli Bourassa, RS Chadwick, et al. Scaling and networking a modular photonic quantum computer.Nature, 638(8052):912– 919, 2025

2025
[23]

Marciniak, Roman Stricker, Martin Ringbauer, Rainer Blatt, Philipp Schindler, Markus M¨ uller, and Thomas Monz

Lukas Postler, Sascha Heußen, Ivan Pogorelov, Manuel Rispler, Thomas Feldker, Michael Meth, Christian D. Marciniak, Roman Stricker, Martin Ringbauer, Rainer Blatt, Philipp Schindler, Markus M¨ uller, and Thomas Monz. Demonstration of fault-tolerant universal quan- tum gate operations.Nature, 605:675–680, 2022

2022
[24]

Beyond nisq: The megaquop machine, 2025

John Preskill. Beyond nisq: The megaquop machine, 2025

2025
[25]

Qvector: an al- gorithm for device-tailored quantum error correction

Peter D Johnson, Jonathan Romero, Jonathan Olson, Yudong Cao, and Al´ an Aspuru-Guzik. Qvector: an al- gorithm for device-tailored quantum error correction. arXiv preprint arXiv:1711.02249, 2017

work page arXiv 2017
[26]

Vari- ational circuit compiler for quantum error correction

Xiaosi Xu, Simon C Benjamin, and Xiao Yuan. Vari- ational circuit compiler for quantum error correction. Physical Review Applied, 15(3):034068, 2021

2021
[27]

A high performance compiler for very large scale surface code computations.Quantum, 8:1354, 2024

George Watkins, Hoang Minh Nguyen, Keelan Watkins, Steven Pearce, Hoi-Kwan Lau, and Alexandru Paler. A high performance compiler for very large scale surface code computations.Quantum, 8:1354, 2024

2024
[28]

Learning Encodings by Maximizing State Distinguishability: Variational Quantum Error Correction

Nico Meyer, Christopher Mutschler, Andreas Maier, and Daniel D Scherer. Learning encodings by maximiz- ing state distinguishability: Variational quantum error correction.arXiv preprint arXiv:2506.11552, 2025

work page internal anchor Pith review Pith/arXiv arXiv 2025
[29]

A fault-tolerant vari- ational quantum algorithm with limited t-depth.Quan- tum Science and Technology, 9(1):015015, 2023

Hasan Sayginel, Francois Jamet, Abhishek Agarwal, Dan E Browne, and Ivan Rungger. A fault-tolerant vari- ational quantum algorithm with limited t-depth.Quan- tum Science and Technology, 9(1):015015, 2023

2023
[30]

Barren plateaus in quantum neural network training landscapes.Nature Communications, 9(1):4812, 2018

Jarrod R McClean, Sergio Boixo, Vadim N Smelyanskiy, Ryan Babbush, and Hartmut Neven. Barren plateaus in quantum neural network training landscapes.Nature Communications, 9(1):4812, 2018

2018
[31]

Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum- classical algorithms.Advanced Quantum Technologies, 2(12):1900070, 2019

Sukin Sim, Peter D Johnson, and Al´ an Aspuru- Guzik. Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum- classical algorithms.Advanced Quantum Technologies, 2(12):1900070, 2019

2019
[32]

Variational quantum algorithms: fun- damental concepts, applications and challenges.Quan- tum Information Processing, 23(6):224, 2024

Han Qi, Sihui Xiao, Zhuo Liu, Changqing Gong, and Abdullah Gani. Variational quantum algorithms: fun- damental concepts, applications and challenges.Quan- tum Information Processing, 23(6):224, 2024

2024
[33]

Hybrid quantum-classical algorithms and quan- tum error mitigation.Journal of the Physical Society of Japan, 90(3):032001, 2021

Suguru Endo, Zhenyu Cai, Simon C Benjamin, and Xiao Yuan. Hybrid quantum-classical algorithms and quan- tum error mitigation.Journal of the Physical Society of Japan, 90(3):032001, 2021

2021
[34]

Truncated variational hamiltonian ansatz: ef- ficient quantum circuit design for quantum chemistry and material science.arXiv preprint arXiv:2505.19772, 2025

Clemens Possel, Walter Hahn, Reza Shirazi, Marina Walt, Peter Pinski, Frank K Wilhelm, and Dmitry Bagrets. Truncated variational hamiltonian ansatz: ef- ficient quantum circuit design for quantum chemistry and material science.arXiv preprint arXiv:2505.19772, 2025

work page arXiv 2025
[35]

Quantum information driven ansatz (qida): shallow-depth empirical quantum circuits from quantum chemistry.The Journal of Physical Chemistry A, 128(39):8533–8543, 2024

Davide Materia, Leonardo Ratini, Celestino Angeli, and Leonardo Guidoni. Quantum information driven ansatz (qida): shallow-depth empirical quantum circuits from quantum chemistry.The Journal of Physical Chemistry A, 128(39):8533–8543, 2024

2024
[36]

Mario Motta, Kevin J Sung, K Birgitta Whaley, Martin Head-Gordon, and James Shee. Bridging physical intu- ition and hardware efficiency for correlated electronic states: the local unitary cluster jastrow ansatz for elec- tronic structure.Chemical Science, 14(40):11213–11227, 2023

2023
[37]

A Quantum Approximate Optimization Algorithm

Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum approximate optimization algorithm.arXiv preprint arXiv:1411.4028, 2014

work page internal anchor Pith review Pith/arXiv arXiv 2014
[38]

Progress towards practical quantum variational algorithms.Physical Review A, 92(4):042303, 2015

Dave Wecker, Matthew B Hastings, and Matthias Troyer. Progress towards practical quantum variational algorithms.Physical Review A, 92(4):042303, 2015. 20

2015
[39]

Pansatz: Pulse- based ansatz for variational quantum algorithms.Fron- tiers in Quantum Science and Technology, 2:1273581, 2023

Dekel Meirom and Steven H Frankel. Pansatz: Pulse- based ansatz for variational quantum algorithms.Fron- tiers in Quantum Science and Technology, 2:1273581, 2023

2023
[40]

Chong, Song Han, Yiyu Shi, and Xuehai Qian

Zhiding Liang, Jinglei Cheng, Hang Ren, Hanrui Wang, Fei Hua, Yongshan Ding, Frederic T. Chong, Song Han, Yiyu Shi, and Xuehai Qian. PAN: Pulse ansatz on NISQ machines.arXiv preprint arXiv:2208.01215, 2022

work page arXiv 2022
[41]

Zhiding Liang, Jinglei Cheng, Hang Ren, Hanrui Wang, Fei Hua, Zhixin Song, Yongshan Ding, Fred- eric T Chong, Song Han, Xuehai Qian, et al. Napa: intermediate-level variational native-pulse ansatz for variational quantum algorithms.IEEE Transactions on Computer-Aided Design of Integrated Circuits and Sys- tems, 43(6):1834–1847, 2024

2024
[42]

Parametriza- tion and optimizability of pulse-level variational quantum eigensolvers.Physical Review Applied, 23(2):024036, 2025

Kyle M Sherbert, Hisham Amer, Sophia E Economou, Edwin Barnes, and Nicholas J Mayhall. Parametriza- tion and optimizability of pulse-level variational quantum eigensolvers.Physical Review Applied, 23(2):024036, 2025

2025
[43]

Pulse variational quan- tum eigensolver on cross-resonance-based hardware

Daniel J Egger, Chiara Capecci, Bibek Pokharel, Pana- giotis Kl Barkoutsos, Laurin E Fischer, Leonardo Guidoni, and Ivano Tavernelli. Pulse variational quan- tum eigensolver on cross-resonance-based hardware. Physical Review Research, 5(3):033159, 2023

2023
[44]

Pulse-based optimization of quantum many-body states with ryd- berg atoms in optical tweezer arrays.arXiv preprint arXiv:2507.19153, 2025

Kazuma Nagao, Sergi Juli` a-Farr´ e, Joseph Vovrosh, Alexandre Dauphin, and Seiji Yunoki. Pulse-based optimization of quantum many-body states with ryd- berg atoms in optical tweezer arrays.arXiv preprint arXiv:2507.19153, 2025

work page arXiv 2025
[45]

Efficient variational quantum algorithms via circuit knitting and architecture search.arXiv preprint arXiv:2508.03376, 2025

Jun Wu, Jiaqi Yang, Jicun Li, Wei Xie, and Xiang- Yang Li. Efficient variational quantum algorithms via circuit knitting and architecture search.arXiv preprint arXiv:2508.03376, 2025

work page arXiv 2025
[46]

Doubling the size of quantum simulators by entanglement forging.PRX Quantum, 3(1):010309, 2022

Andrew Eddins, Mario Motta, Tanvi P Gujarati, Sergey Bravyi, Antonio Mezzacapo, Charles Hadfield, and Sarah Sheldon. Doubling the size of quantum simulators by entanglement forging.PRX Quantum, 3(1):010309, 2022

2022
[47]

Tensor-network-assisted variational quantum algorithm.Physical Review A, 108(5):052407, 2023

Junxiang Huang, Wenhao He, Yukun Zhang, Yusen Wu, Bujiao Wu, and Xiao Yuan. Tensor-network-assisted variational quantum algorithm.Physical Review A, 108(5):052407, 2023

2023
[48]

Synergistic pretraining of parametrized quantum cir- cuits via tensor networks.Nature Communications, 14(1):8367, 2023

Manuel S Rudolph, Jacob Miller, Danial Motlagh, Jing Chen, Atithi Acharya, and Alejandro Perdomo-Ortiz. Synergistic pretraining of parametrized quantum cir- cuits via tensor networks.Nature Communications, 14(1):8367, 2023

2023
[49]

Quan- tum circuit design search.Quantum Machine Intelli- gence, 3(2):25, 2021

Mohammad Pirhooshyaran and Tam´ as Terlaky. Quan- tum circuit design search.Quantum Machine Intelli- gence, 3(2):25, 2021

2021
[50]

Differentiable quantum architecture search

Shi-Xin Zhang, Chang-Yu Hsieh, Shengyu Zhang, and Hong Yao. Differentiable quantum architecture search. Quantum Science and Technology, 7(4):045023, 2022

2022
[51]

Quantum circuit architecture search for variational quantum algorithms.npj Quantum In- formation, 8(1):62, 2022

Yuxuan Du, Tao Huang, Shan You, Min-Hsiu Hsieh, and Dacheng Tao. Quantum circuit architecture search for variational quantum algorithms.npj Quantum In- formation, 8(1):62, 2022

2022
[52]

Aqea-qas: An adaptive quantum evolu- tionary algorithm for quantum architecture search.En- tropy, 27(7):733, 2025

Yaochong Li, Jing Zhang, Rigui Zhou, Yi Qu, and Ruiqing Xu. Aqea-qas: An adaptive quantum evolu- tionary algorithm for quantum architecture search.En- tropy, 27(7):733, 2025

2025
[53]

Model-free deep recurrent q- network reinforcement learning for quantum circuit ar- chitectures design.Quantum Reports, 4(4):380–389, 2022

Tomah Sogabe, Tomoaki Kimura, Chih-Chieh Chen, Kodai Shiba, Nobuhiro Kasahara, Masaru Sogabe, and Katsuyoshi Sakamoto. Model-free deep recurrent q- network reinforcement learning for quantum circuit ar- chitectures design.Quantum Reports, 4(4):380–389, 2022

2022
[54]

Kuo, Y.-L

En-Jui Kuo, Yao-Lung L Fang, and Samuel Yen-Chi Chen. Quantum architecture search via deep reinforce- ment learning.arXiv preprint arXiv:2104.07715, 2021

work page arXiv 2021
[55]

Reinforcement learning-based architecture search for quantum machine learning.Machine Learn- ing: Science and Technology, 6(1):015041, 2025

Frederic Rapp, David A Kreplin, Marco F Huber, and Marco Roth. Reinforcement learning-based architecture search for quantum machine learning.Machine Learn- ing: Science and Technology, 6(1):015041, 2025

2025
[56]

Quantumdarts: differentiable quantum ar- chitecture search for variational quantum algorithms

Wenjie Wu, Ge Yan, Xudong Lu, Kaisen Pan, and Junchi Yan. Quantumdarts: differentiable quantum ar- chitecture search for variational quantum algorithms. InInternational conference on machine learning, pages 37745–37764. PMLR, 2023

2023
[57]

Gradient-based optimization for quantum architecture search.Neural Networks, 179:106508, 2024

Zhimin He, Jiachun Wei, Chuangtao Chen, Zhiming Huang, Haozhen Situ, and Lvzhou Li. Gradient-based optimization for quantum architecture search.Neural Networks, 179:106508, 2024

2024
[58]

McClean, Jonathan Romero, Ryan Babbush, and Al´ an Aspuru-Guzik

Jarrod R. McClean, Jonathan Romero, Ryan Babbush, and Al´ an Aspuru-Guzik. The theory of variational hybrid quantum-classical algorithms.New Journal of Physics, 18(2):023023, 2016

2016
[59]

Cost function dependent bar- ren plateaus in shallow parametrized quantum circuits

Marco Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cin- cio, and Patrick J Coles. Cost function dependent bar- ren plateaus in shallow parametrized quantum circuits. Nature Communications, 12(1):1791, 2021

2021
[60]

Quantum fingerprinting.Physical Re- view Letters, 87(16):167902, 2001

Harry Buhrman, Richard Cleve, John Watrous, and Ronald De Wolf. Quantum fingerprinting.Physical Re- view Letters, 87(16):167902, 2001

2001
[61]

The swap test and the hong-ou-mandel effect are equivalent.arXiv preprint arXiv:1303.6814, 2013

Juan Carlos Garcia-Escartin and Pedro Chamorro- Posada. The swap test and the hong-ou-mandel effect are equivalent.arXiv preprint arXiv:1303.6814, 2013

work page arXiv 2013
[62]

Izmaylov

Vladyslav Verteletskyi, Tzu-Ching Yen, and Artur F. Izmaylov. Measurement optimization in the variational quantum eigensolver using a minimum clique cover.The Journal of Chemical Physics, 152(12):124114, 2020

2020
[63]

Predicting many properties of a quantum system from very few measurements.Nature Physics, 16(10):1050– 1057, 2020

Hsin-Yuan Huang, Richard Kueng, and John Preskill. Predicting many properties of a quantum system from very few measurements.Nature Physics, 16(10):1050– 1057, 2020

2020
[64]

Temme, S

Kristan Temme, Sergey Bravyi, and Jay M Gam- betta. Error mitigation for short-depth quantum cir- cuits.arXiv preprint arXiv:1612.02058, 2016

work page arXiv 2016
[65]

Error mitigation extends the computational reach of a noisy quantum processor.Nature, 567(7749):491– 495, 2019

Abhinav Kandala, Kristan Temme, Antonio D C´ orcoles, Antonio Mezzacapo, Jerry M Chow, and Jay M Gam- betta. Error mitigation extends the computational reach of a noisy quantum processor.Nature, 567(7749):491– 495, 2019

2019
[66]

Benjamin, Sug- uru Endo, William J

Zhenyu Cai, Ryan Babbush, Simon C. Benjamin, Sug- uru Endo, William J. Huggins, Ying Li, Jarrod R. Mc- Clean, and Thomas E. O’Brien. Quantum error mitiga- tion.Rev. Mod. Phys., 95:045005, Dec 2023

2023
[67]

Quantum error mitigation as a uni- versal error reduction technique: Applications from the nisq to the fault-tolerant quantum computing eras.PRX Quantum, 3(1):010345, 2022

Yasunari Suzuki, Suguru Endo, Keisuke Fujii, and Yuuki Tokunaga. Quantum error mitigation as a uni- versal error reduction technique: Applications from the nisq to the fault-tolerant quantum computing eras.PRX Quantum, 3(1):010345, 2022

2022
[68]

Hardware-efficient entangled measurements for variational quantum algorithms.Physical Review Applied, 20(3):034044, 2023

Francisco Escudero, David Fern´ andez-Fern´ andez, Gabriel Jaum` a, Guillermo F Pe˜ nas, and Luciano Pereira. Hardware-efficient entangled measurements for variational quantum algorithms.Physical Review Applied, 20(3):034044, 2023. 21

2023
[69]

Estimating the gradient and higher-order derivatives on quantum hardware.Physical Review A, 103(1):012405, 2021

Andrea Mari, Thomas R Bromley, and Nathan Killoran. Estimating the gradient and higher-order derivatives on quantum hardware.Physical Review A, 103(1):012405, 2021

2021
[70]

Natural gradient works efficiently in learning.Neural computation, 10(2):251–276, 1998

Shun-Ichi Amari. Natural gradient works efficiently in learning.Neural computation, 10(2):251–276, 1998

1998
[71]

Quantum natural gradient.Quan- tum, 4:269, 2020

James Stokes, Josh Izaac, Nathan Killoran, and Giuseppe Carleo. Quantum natural gradient.Quan- tum, 4:269, 2020

2020
[72]

Wolf, and Fil- ippo Miatto

Yuan Yao, Pierre Cussenot, Richard A. Wolf, and Fil- ippo Miatto. Complex natural gradient optimization for optical quantum circuit design.Physical Review A, 105(5):052402, 2022

2022
[73]

Evaluating analytic gra- dients on quantum hardware.Physical Review A, 99(3):032331, 2019

Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran. Evaluating analytic gra- dients on quantum hardware.Physical Review A, 99(3):032331, 2019

2019
[74]

Adam: A Method for Stochastic Optimization

Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization.arXiv preprint arXiv:1412.6980, 2014

work page internal anchor Pith review Pith/arXiv arXiv 2014
[75]

On the limited memory bfgs method for large scale optimization.Mathematical Programming, 45(1-3):503–528, 1989

Dong C Liu and Jorge Nocedal. On the limited memory bfgs method for large scale optimization.Mathematical Programming, 45(1-3):503–528, 1989

1989
[76]

Rie- mannian optimization of photonic quantum circuits in phase and Fock space.SciPost Phys., 17:082, 2024

Yuan Yao, Filippo Miatto, and Nicol´ as Quesada. Rie- mannian optimization of photonic quantum circuits in phase and Fock space.SciPost Phys., 17:082, 2024

2024
[77]

Benchmarking a wide range of optimisers for solving the fermi–hubbard model using the variational quantum eigensolver.Quantum Science and Technology, 10(4):045032, 2025

Benjamin DM Jones, Lana Mineh, and Ashley Mon- tanaro. Benchmarking a wide range of optimisers for solving the fermi–hubbard model using the variational quantum eigensolver.Quantum Science and Technology, 10(4):045032, 2025

2025
[78]

James C. Spall. Multivariate stochastic approxima- tion using a simultaneous perturbation gradient approx- imation.IEEE Transactions on Automatic Control, 37(3):332–341, 1992

1992
[79]

An empirical com- parison of optimizers for quantum machine learning with spsa-based gradients

Marco Wiedmann, Marc H¨ olle, Maniraman Periyasamy, Nico Meyer, Christian Ufrecht, Daniel D Scherer, Axel Plinge, and Christopher Mutschler. An empirical com- parison of optimizers for quantum machine learning with spsa-based gradients. In2023 IEEE International Conference on Quantum Computing and Engineering (QCE), volume 1, pages 450–456. IEEE, 2023

2023
[80]

Michael J. D. Powell. Direct search algorithms for opti- mization calculations.Acta Numerica, 7:287–336, 1998

1998

Showing first 80 references.