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Variational quantum algorithms (VQAs) have consolidated as a tool in quantum information in noisy, intermediate-scale quantum hardware, trading circuit depth for classical-quantum feedback. This Collection, curated by our Associate Editor Keisuke Fujii, showcases some of the relevant papers published in this area in our journal. From redesigning the ansatz itself to pushing quantum approximate optimization toward large, noisy hardware; from adapting variational time-evolution to simulate gauge theories, batteries, and spin chains, to squeezing more signal from every measurement via joint readouts, ensemble learning, and machine-learned noise models.

Adaptive Circuit Construction & Trainability

A key to variational quantum algorithms is the ansatz it trains and how it matches with the Hilbert space without depleting resources. These papers show different approaches to the problem that highlight the importance of treating circuit structure itself as a design space.

TETRIS-ADAPT-VQE: An adaptive algorithm that yields shallower, denser circuit Ansätze
Panagiotis G. Anastasiou, Yanzhu Chen, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou
Phys. Rev. Research 6, 013254 (2024)

Scaling adaptive quantum simulation algorithms via operator pool tiling
John S. Van Dyke, Karunya Shirali, George S. Barron, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou
Phys. Rev. Research 6, L012030 (2024)

Training variational quantum algorithms with random gate activation
Shuo Liu, Shi-Xin Zhang, Shao-Kai Jian, and Hong Yao
Phys. Rev. Research 5, L032040 (2023)

Benchmarking variational quantum eigensolvers for the square-octagon-lattice Kitaev model
Andy C. Y. Li, M. Sohaib Alam, Thomas Iadecola, Ammar Jahin, Joshua Job, Doga Murat Kurkcuoglu, Richard Li, Peter P. Orth, A. Barış Özgüler, Gabriel N. Perdue, and Norm M. Tubman
Phys. Rev. Research 5, 033071 (2023)

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Quantum Approximate Optimization Algorithm for Combinatorial Optimization

The Quantum Approximate Optimization Algorithm may very well be one of the most tested algorithms on real quantum processors. These papers represent a mapping on how QAOA is evolving from a promising heuristic toward an algorithm whose behavior, limits, and hardware requirements are increasingly well understood.

Large-scale quantum approximate optimization on nonplanar graphs with machine learning noise mitigation
Stefan H. Sack and Daniel J. Egger
Phys. Rev. Research 6, 013223 (2024)

Quantifying the impact of precision errors on quantum approximate optimization algorithms
Gregory Quiroz, Paraj Titum, Phillip Lotshaw, Pavel Lougovski, Kevin Schultz, Eugene Dumitrescu, and Itay Hen
Phys. Rev. Research 7, 023240 (2025)

Universal resources for quantum approximate optimization algorithm and quantum annealing
Pablo Díez-Valle, Fernando J. Gómez-Ruiz, Diego Porras, and Juan José García-Ripoll
Phys. Rev. Research 8, 013211 (2026)

Quantum dropout: On and over the hardness of quantum approximate optimization algorithm
Zhenduo Wang (王朕铎), Pei-Lin Zheng, Biao Wu (吴飙), and Yi Zhang
Phys. Rev. Research 5, 023171 (2023)

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Variational Simulation of Quantum Dynamics

Variational Algorithms are increasingly used to simulate the evolution of quantum systems in time. These papers present complementary approaches to tackle the ever-present dilemma of computational resources and show how a reformulation of the problem alleviates it.

Variational quantum simulation of U(1) lattice gauge theories with qudit systems
Pavel P. Popov, Michael Meth, Maciej Lewestein, Philipp Hauke, Martin Ringbauer, Erez Zohar, and Valentin Kasper
Phys. Rev. Research 6, 013202 (2024)

Subspace variational quantum simulator
Kentaro Heya, Ken M. Nakanishi, Kosuke Mitarai, Zhiguang Yan, Kun Zuo, Yasunari Suzuki, Takanori Sugiyama, Shuhei Tamate, Yutaka Tabuchi, Keisuke Fujii, and Yasunobu Nakamura
Phys. Rev. Research 5, 023078 (2023)

Variational quantum time evolution without the quantum geometric tensor
Julien Gacon, Jannes Nys, Riccardo Rossi, Stefan Woerner, and Giuseppe Carleo
Phys. Rev. Research 6, 013143 (2024)

Variational quantum algorithm for ergotropy estimation in quantum many-body batteries
Duc Tuan Hoang, Friederike Metz, Andreas Thomasen, Tran Duong Anh-Tai, Thomas Busch, and Thomás Fogarty
Phys. Rev. Research 6, 013038 (2024)

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Measurement Efficiency & Quantum-Classical Learning

A common challenge in Variational Algorithms is the measurement budget and noise floor of the hardware reading it out. Measurement Theory and Machine Learning are surfacing as key resources to engineer more efficient methods to extract information.

Accelerated variational quantum eigensolver with joint Bell measurement
Chenfeng Cao, Hiroshi Yano, and Yuya O. Nakagawa
Phys. Rev. Research 6, 013205 (2024)

Ensemble-learning error mitigation for variational quantum shallow-circuit classifiers
Qingyu Li, Yuhan Huang, Xiaokai Hou, Ying Li, Xiaoting Wang, and Abolfazl Bayat
Phys. Rev. Research 6, 013027 (2024)

VQE-generated quantum circuit dataset for machine learning
Akimoto Nakayama, Kosuke Mitarai, Leonardo Placidi, Takanori Sugimoto, and Keisuke Fujii
Phys. Rev. Research 7, 033048 (2025)

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