add hyperspace sampling

report/references.bib

@@ -1,6 +1,5 @@
 @article{quantum-advantage-bounds,
-    title = {Information-Theoretic Bounds on Quantum Advantage in Machine
-             Learning },
+  title         = {Information-Theoretic Bounds on Quantum Advantage in Machine Learning},
   author        = {Huang, Hsin-Yuan and Kueng, Richard and Preskill, John},
   journal       = {Phys. Rev. Lett.},
   volume        = {126},
@@ -11,103 +10,38 @@
   month         = {May},
   publisher     = {American Physical Society},
   doi           = {10.1103/PhysRevLett.126.190505},
-    url = {https://link.aps.org/doi/10.1103/PhysRevLett.126.190505},
+  url           = {https://link.aps.org/doi/10.1103/PhysRevLett.126.190505}
 }
-
 @article{quantum-advantage-learning,
-    author = {Hsin-Yuan Huang and Michael Broughton and Jordan Cotler and Sitan
-              Chen and Jerry Li and Masoud Mohseni and Hartmut Neven and Ryan
-              Babbush and Richard Kueng and John Preskill and Jarrod R. McClean },
+  author        = {Hsin-Yuan Huang and Michael Broughton and Jordan Cotler and Sitan Chen and Jerry Li and Masoud Mohseni and Hartmut Neven and Ryan Babbush and Richard Kueng and John Preskill and Jarrod R. McClean},
   title         = {Quantum advantage in learning from experiments},
   journal       = {Science},
   volume        = {376},
   number        = {6598},
-    pages = {1182-1186},
+  pages         = {1182--1186},
   year          = {2022},
   doi           = {10.1126/science.abn7293},
-    URL = {https://www.science.org/doi/abs/10.1126/science.abn7293},
+  url           = {https://www.science.org/doi/abs/10.1126/science.abn7293},
   eprint        = {https://www.science.org/doi/pdf/10.1126/science.abn7293},
-    abstract = {Quantum technology promises to revolutionize how we learn about
-                the physical world. An experiment that processes quantum data
-                with a quantum computer could have substantial advantages over
-                conventional experiments in which quantum states are measured and
-                outcomes are processed with a classical computer. We proved that
-                quantum machines could learn from exponentially fewer experiments
-                than the number required by conventional experiments. This
-                exponential advantage is shown for predicting properties of
-                physical systems, performing quantum principal component analysis
-                , and learning about physical dynamics. Furthermore, the quantum
-                resources needed for achieving an exponential advantage are quite
-                modest in some cases. Conducting experiments with 40
-                superconducting qubits and 1300 quantum gates, we demonstrated
-                that a substantial quantum advantage is possible with today’s
-                quantum processors. There is considerable interest in extending
-                the recent success of quantum computers in outperforming their
-                conventional classical counterparts (quantum advantage) from some
-                model mathematical problems to more meaningful tasks. Huang et
-                al. show how manipulating multiple quantum states can provide an
-                exponential advantage over classical processing of measurements
-                of single-quantum states for certain learning tasks. These
-                include predicting properties of physical systems, performing
-                quantum principal component analysis on noisy states, and
-                learning approximate models of physical dynamics (see the
-                Perspective by Dunjko). In their proof-of-principle experiments
-                using up to 40 qubits on a Google Sycamore quantum processor, the
-                authors achieved almost four orders of magnitude of reduction in
-                the required number of experiments over the best-known classical
-                lower bounds. —YS Quantum-enhanced strategies can provide a
-                dramatic performance boost in learning useful information from
-                quantum experiments.},
+  abstract      = {Quantum technology promises to revolutionize how we learn about the physical world. An experiment that processes quantum data with a quantum computer could have substantial advantages over conventional experiments in which quantum states are measured and outcomes are processed with a classical computer. We proved that quantum machines could learn from exponentially fewer experiments than the number required by conventional experiments. This exponential advantage is shown for predicting properties of physical systems, performing quantum principal component analysis , and learning about physical dynamics. Furthermore, the quantum resources needed for achieving an exponential advantage are quite modest in some cases. Conducting experiments with 40 superconducting qubits and 1300 quantum gates, we demonstrated that a substantial quantum advantage is possible with today's quantum processors. There is considerable interest in extending the recent success of quantum computers in outperforming their conventional classical counterparts (quantum advantage) from some model mathematical problems to more meaningful tasks. Huang et al. show how manipulating multiple quantum states can provide an exponential advantage over classical processing of measurements of single-quantum states for certain learning tasks. These include predicting properties of physical systems, performing quantum principal component analysis on noisy states, and learning approximate models of physical dynamics (see the Perspective by Dunjko). In their proof-of-principle experiments using up to 40 qubits on a Google Sycamore quantum processor, the authors achieved almost four orders of magnitude of reduction in the required number of experiments over the best-known classical lower bounds. --YS Quantum-enhanced strategies can provide a dramatic performance boost in learning useful information from quantum experiments.}
 }
-
 @article{expressibility-and-entanglement,
-    author = {Sim, Sukin and Johnson, Peter D. and Aspuru-Guzik, Alán},
-    title = {Expressibility and Entangling Capability of Parameterized Quantum
-             Circuits for Hybrid Quantum-Classical Algorithms},
+  author        = {Sim, Sukin and Johnson, Peter D. and Aspuru-Guzik, Al\'{a}n},
+  title         = {Expressibility and Entangling Capability of Parameterized Quantum Circuits for Hybrid Quantum-Classical Algorithms},
   journal       = {Advanced Quantum Technologies},
   volume        = {2},
   number        = {12},
   pages         = {1900070},
   keywords      = {quantum algorithms, quantum circuits, quantum computation},
   doi           = {https://doi.org/10.1002/qute.201900070},
-    url = {
-           https://advanced.onlinelibrary.wiley.com/doi/abs/10.1002/qute.201900070
-           },
-    eprint = {
-              https://advanced.onlinelibrary.wiley.com/doi/pdf/10.1002/qute.201900070
-              },
-    abstract = {Abstract Parameterized quantum circuits (PQCs) play an essential
-                role in the performance of many variational quantum algorithms.
-                One challenge in implementing such algorithms is choosing an
-                effective circuit that well represents the solution space while
-                maintaining a low circuit depth and parameter count. To
-                characterize and identify expressible, yet compact, circuits,
-                several descriptors are proposed, including expressibility and
-                entangling capability, that are statistically estimated from
-                classical simulations. These descriptors are computed for
-                different circuit structures, varying the qubit connectivity and
-                selection of gates. From these simulations, circuit fragments
-                that perform well with respect to the descriptors are identified.
-                In particular, a substantial improvement in performance of
-                two-qubit gates in a ring or all-to-all connected arrangement,
-                compared to that of those on a line, is observed. Furthermore,
-                improvement in both descriptors is achieved by sequences of
-                controlled X-rotation gates compared to sequences of controlled
-                Z-rotation gates. In addition, it is investigated how
-                expressibility “saturates” with increased circuit depth, finding
-                that the rate and saturated value appear to be distinguishing
-                features of a PQC. While the correlation between each descriptor
-                and algorithm performance remains to be investigated, methods and
-                results from this study can be useful for algorithm development
-                and design of experiments.},
-    year = {2019},
+  url           = {https://advanced.onlinelibrary.wiley.com/doi/abs/10.1002/qute.201900070},
+  eprint        = {https://advanced.onlinelibrary.wiley.com/doi/pdf/10.1002/qute.201900070},
+  abstract      = {Abstract Parameterized quantum circuits (PQCs) play an essential role in the performance of many variational quantum algorithms. One challenge in implementing such algorithms is choosing an effective circuit that well represents the solution space while maintaining a low circuit depth and parameter count. To characterize and identify expressible, yet compact, circuits, several descriptors are proposed, including expressibility and entangling capability, that are statistically estimated from classical simulations. These descriptors are computed for different circuit structures, varying the qubit connectivity and selection of gates. From these simulations, circuit fragments that perform well with respect to the descriptors are identified. In particular, a substantial improvement in performance of two-qubit gates in a ring or all-to-all connected arrangement, compared to that of those on a line, is observed. Furthermore, improvement in both descriptors is achieved by sequences of controlled X-rotation gates compared to sequences of controlled Z-rotation gates. In addition, it is investigated how expressibility ``saturates'' with increased circuit depth, finding that the rate and saturated value appear to be distinguishing features of a PQC. While the correlation between each descriptor and algorithm performance remains to be investigated, methods and results from this study can be useful for algorithm development and design of experiments.},
+  year          = {2019}
 }
-
 @article{quantum-dynamics-physical-resource,
   title         = {Quantum dynamics as a physical resource},
-    author = {Nielsen, Michael A. and Dawson, Christopher M. and Dodd, Jennifer
-              L. and Gilchrist, Alexei and Mortimer, Duncan and Osborne, Tobias
-              J. and Bremner, Michael J. and Harrow, Aram W. and Hines, Andrew},
+  author        = {Nielsen, Michael A. and Dawson, Christopher M. and Dodd, Jennifer L. and Gilchrist, Alexei and Mortimer, Duncan and Osborne, Tobias J. and Bremner, Michael J. and Harrow, Aram W. and Hines, Andrew},
   journal       = {Phys. Rev. A},
   volume        = {67},
   issue         = {5},
@@ -117,42 +51,33 @@
   month         = {May},
   publisher     = {American Physical Society},
   doi           = {10.1103/PhysRevA.67.052301},
-    url = {https://link.aps.org/doi/10.1103/PhysRevA.67.052301},
+  url           = {https://link.aps.org/doi/10.1103/PhysRevA.67.052301}
 }
-
 @article{scaling-variational-circuit-depth,
   doi           = {10.22331/q-2020-05-28-272},
   url           = {https://doi.org/10.22331/q-2020-05-28-272},
-    title = {Scaling of variational quantum circuit depth for condensed matter
-             systems},
-    author = {Bravo-Prieto, Carlos and Lumbreras-Zarapico, Josep and Tagliacozzo
-              , Luca and Latorre, Jos{\'{e}} I.},
+  title         = {Scaling of variational quantum circuit depth for condensed matter systems},
+  author        = {Bravo-Prieto, Carlos and Lumbreras-Zarapico, Josep and Tagliacozzo , Luca and Latorre, Jos{\'{e}} I.},
   journal       = {{Quantum}},
   issn          = {2521-327X},
-    publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den
-                 Quantenwissenschaften}},
+  publisher     = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
   volume        = {4},
   pages         = {272},
   month         = may,
-    year = {2020},
+  year          = {2020}
 }
-
 @misc{akash,
-    title = {Reinforcement learning-assisted quantum architecture search for
-             variational quantum algorithms},
+  title         = {Reinforcement learning-assisted quantum architecture search for variational quantum algorithms},
   author        = {Akash Kundu},
   year          = {2024},
   eprint        = {2402.13754},
-    archivePrefix = {arXiv},
-    primaryClass = {quant-ph},
-    url = {https://arxiv.org/abs/2402.13754},
+  archiveprefix = {arXiv},
+  primaryclass  = {quant-ph},
+  url           = {https://arxiv.org/abs/2402.13754}
 }
-
 @article{supernet-qas,
-    author = {Du, Yuxuan and Huang, Tao and You, Shan and Hsieh, Min-Hsiu and
-              Tao, Dacheng},
-    title = {Quantum circuit architecture search for variational quantum
-             algorithms},
+  author        = {Du, Yuxuan and Huang, Tao and You, Shan and Hsieh, Min-Hsiu and Tao, Dacheng},
+  title         = {Quantum circuit architecture search for variational quantum algorithms},
   journal       = {npj Quantum Information},
   year          = {2022},
   month         = {May},
@@ -160,102 +85,47 @@
   volume        = {8},
   number        = {1},
   pages         = {62},
-    abstract = {Variational quantum algorithms (VQAs) are expected to be a path
-                to quantum advantages on noisy intermediate-scale quantum
-                devices. However, both empirical and theoretical results exhibit
-                that the deployed ansatz heavily affects the performance of VQAs
-                such that an ansatz with a larger number of quantum gates enables
-                a stronger expressivity, while the accumulated noise may render a
-                poor trainability. To maximally improve the robustness and
-                trainability of VQAs, here we devise a resource and runtime
-                efficient scheme termed quantum architecture search (QAS). In
-                particular, given a learning task, QAS automatically seeks a
-                near-optimal ansatz (i.e., circuit architecture) to balance
-                benefits and side-effects brought by adding more noisy quantum
-                gates to achieve a good performance. We implement QAS on both the
-                numerical simulator and real quantum hardware, via the IBM cloud,
-                to accomplish data classification and quantum chemistry tasks. In
-                the problems studied, numerical and experimental results show
-                that QAS cannot only alleviate the influence of quantum noise and
-                barren plateaus but also outperforms VQAs with pre-selected
-                ansatze.},
+  abstract      = {Variational quantum algorithms (VQAs) are expected to be a path to quantum advantages on noisy intermediate-scale quantum devices. However, both empirical and theoretical results exhibit that the deployed ansatz heavily affects the performance of VQAs such that an ansatz with a larger number of quantum gates enables a stronger expressivity, while the accumulated noise may render a poor trainability. To maximally improve the robustness and trainability of VQAs, here we devise a resource and runtime efficient scheme termed quantum architecture search (QAS). In particular, given a learning task, QAS automatically seeks a near-optimal ansatz (i.e., circuit architecture) to balance benefits and side-effects brought by adding more noisy quantum gates to achieve a good performance. We implement QAS on both the numerical simulator and real quantum hardware, via the IBM cloud, to accomplish data classification and quantum chemistry tasks. In the problems studied, numerical and experimental results show that QAS cannot only alleviate the influence of quantum noise and barren plateaus but also outperforms VQAs with pre-selected ansatze.},
   issn          = {2056-6387},
   doi           = {10.1038/s41534-022-00570-y},
-    url = {https://doi.org/10.1038/s41534-022-00570-y},
+  url           = {https://doi.org/10.1038/s41534-022-00570-y}
 }
-
-
 @article{evolutionary-architecture-search,
-    AUTHOR = {Ding, Li and Spector, Lee},
-    TITLE = {Multi-Objective Evolutionary Architecture Search for Parameterized
-             Quantum Circuits},
-    JOURNAL = {Entropy},
-    VOLUME = {25},
-    YEAR = {2023},
-    NUMBER = {1},
-    ARTICLE-NUMBER = {93},
-    URL = {https://www.mdpi.com/1099-4300/25/1/93},
-    PubMedID = {36673234},
-    ISSN = {1099-4300},
-    ABSTRACT = {Recent work on hybrid quantum-classical machine learning systems
-                has demonstrated success in utilizing parameterized quantum
-                circuits (PQCs) to solve the challenging reinforcement learning
-                (RL) tasks, with provable learning advantages over classical
-                systems, e.g., deep neural networks. While existing work
-                demonstrates and exploits the strength of PQC-based models, the
-                design choices of PQC architectures and the interactions between
-                different quantum circuits on learning tasks are generally
-                underexplored. In this work, we introduce a Multi-objective
-                Evolutionary Architecture Search framework for parameterized
-                quantum circuits (MEAS-PQC), which uses a multi-objective genetic
-                algorithm with quantum-specific configurations to perform
-                efficient searching of optimal PQC architectures. Experimental
-                results show that our method can find architectures that have
-                superior learning performance on three benchmark RL tasks, and
-                are also optimized for additional objectives including reductions
-                in quantum noise and model size. Further analysis of patterns and
-                probability distributions of quantum operations helps identify
-                performance-critical design choices of hybrid quantum-classical
-                learning systems.},
-    DOI = {10.3390/e25010093},
+  author        = {Ding, Li and Spector, Lee},
+  title         = {Multi-Objective Evolutionary Architecture Search for Parameterized Quantum Circuits},
+  journal       = {Entropy},
+  volume        = {25},
+  year          = {2023},
+  number        = {1},
+  article-number = {93},
+  url           = {https://www.mdpi.com/1099-4300/25/1/93},
+  pubmedid      = {36673234},
+  issn          = {1099-4300},
+  abstract      = {Recent work on hybrid quantum-classical machine learning systems has demonstrated success in utilizing parameterized quantum circuits (PQCs) to solve the challenging reinforcement learning (RL) tasks, with provable learning advantages over classical systems, e.g., deep neural networks. While existing work demonstrates and exploits the strength of PQC-based models, the design choices of PQC architectures and the interactions between different quantum circuits on learning tasks are generally underexplored. In this work, we introduce a Multi-objective Evolutionary Architecture Search framework for parameterized quantum circuits (MEAS-PQC), which uses a multi-objective genetic algorithm with quantum-specific configurations to perform efficient searching of optimal PQC architectures. Experimental results show that our method can find architectures that have superior learning performance on three benchmark RL tasks, and are also optimized for additional objectives including reductions in quantum noise and model size. Further analysis of patterns and probability distributions of quantum operations helps identify performance-critical design choices of hybrid quantum-classical learning systems.},
+  doi           = {10.3390/e25010093}
 }
-
 @misc{generative-quantum-eigensolver,
-    title = {The generative quantum eigensolver (GQE) and its application for
-             ground state search},
-    author = {Kouhei Nakaji and Lasse Bjørn Kristensen and Ryota Kemmoku and
-              Jorge A. Campos-Gonzalez-Angulo and Mohammad Ghazi Vakili and
-              Haozhe Huang and Mohsen Bagherimehrab and Christoph Gorgulla and
-              FuTe Wong and Alex McCaskey and Jin-Sung Kim and Thien Nguyen and
-              Pooja Rao and Qi Gao and Michihiko Sugawara and Naoki Yamamoto and
-              Alán Aspuru-Guzik},
+  title         = {The generative quantum eigensolver (GQE) and its application for ground state search},
+  author        = {Kouhei Nakaji and Lasse Bj\o{}rn Kristensen and Ryota Kemmoku and Jorge A. Campos-Gonzalez-Angulo and Mohammad Ghazi Vakili and Haozhe Huang and Mohsen Bagherimehrab and Christoph Gorgulla and FuTe Wong and Alex McCaskey and Jin-Sung Kim and Thien Nguyen and Pooja Rao and Qi Gao and Michihiko Sugawara and Naoki Yamamoto and Al\'{a}n Aspuru-Guzik},
   year          = {2025},
   eprint        = {2401.09253},
-    archivePrefix = {arXiv},
-    primaryClass = {quant-ph},
-    url = {https://arxiv.org/abs/2401.09253},
+  archiveprefix = {arXiv},
+  primaryclass  = {quant-ph},
+  url           = {https://arxiv.org/abs/2401.09253}
 }
-
 @inproceedings{calibration-aware-transpilation,
   author        = {Ji, Yanjun and Brandhofer, Sebastian and Polian, Ilia},
-    booktitle = {2022 IEEE International Conference on Quantum Computing and
-                 Engineering (QCE)},
-    title = {Calibration-Aware Transpilation for Variational Quantum
-             Optimization},
+  booktitle     = {2022 IEEE International Conference on Quantum Computing and Engineering (QCE)},
+  title         = {Calibration-Aware Transpilation for Variational Quantum Optimization},
   year          = {2022},
   volume        = {},
   number        = {},
-    pages = {204-214},
-    keywords = {Computers;Quantum computing;Quantum algorithm;Program
-                processors;Error analysis;Logic
-                gates;Calibration;Calibration-Aware;Transpilation;NISQ;QAOA;Benchmarking;Quantum
-                Computing},
-    doi = {10.1109/QCE53715.2022.00040},
+  pages         = {204--214},
+  keywords      = {Computers;Quantum computing;Quantum algorithm;Program processors;Error analysis;Logic gates;Calibration;Calibration-Aware;Transpilation;NISQ;QAOA;Benchmarking;Quantum Computing},
+  doi           = {10.1109/QCE53715.2022.00040}
 }
-
 @article{topology-driven-search,
-    author = {Su, Junjian and Fan, Jiacheng and Wu, Shengyao and Li, Guanghui
-              and Qin, Sujuan and Gao, Fei},
+  author        = {Su, Junjian and Fan, Jiacheng and Wu, Shengyao and Li, Guanghui and Qin, Sujuan and Gao, Fei},
   title         = {Topology-driven quantum architecture search framework},
   journal       = {Science China Information Sciences},
   year          = {2025},
@@ -264,110 +134,46 @@
   volume        = {68},
   number        = {8},
   pages         = {180507},
-    abstract = {The limitations of noisy intermediate-scale quantum (NISQ)
-                devices have motivated the development of variational quantum
-                algorithms (VQAs), which are designed to potentially achieve
-                quantum advantage for specific tasks. Quantum architecture search
-                (QAS) algorithms play a critical role in automating the design of
-                high-performance parameterized quantum circuits (PQCs) for VQAs.
-                However, existing QAS approaches struggle with large search
-                spaces, leading to substantial computational overhead when
-                optimizing large-scale quantum circuits. Extensive empirical
-                analysis reveals that circuit topology has a greater impact on
-                quantum circuit performance than gate types. Based on this
-                insight, we propose the topology-driven quantum architecture
-                search (TD-QAS) framework, which first identifies optimal circuit
-                topologies and then fine-tunes the gate types. In the fine-tuning
-                phase, the QAS inherits parameters from the topology search phase
-                , eliminating the need for training from scratch. By decoupling
-                the large search space into separate topology and gate-type
-                components, TD-QAS avoids exploring gate configurations within
-                low-performance topologies, thereby significantly reducing
-                computational complexity. Numerical simulations across various
-                tasks, under both noiseless and noisy conditions, validate the
-                effectiveness of the TD-QAS framework. This framework advances
-                standard QAS algorithms by enabling the identification of
-                high-performance quantum circuits while minimizing computational
-                demands. These findings indicate that TD-QAS deepens our
-                understanding of VQAs and offers broad potential for the
-                development of future QAS algorithms.},
+  abstract      = {The limitations of noisy intermediate-scale quantum (NISQ) devices have motivated the development of variational quantum algorithms (VQAs), which are designed to potentially achieve quantum advantage for specific tasks. Quantum architecture search (QAS) algorithms play a critical role in automating the design of high-performance parameterized quantum circuits (PQCs) for VQAs. However, existing QAS approaches struggle with large search spaces, leading to substantial computational overhead when optimizing large-scale quantum circuits. Extensive empirical analysis reveals that circuit topology has a greater impact on quantum circuit performance than gate types. Based on this insight, we propose the topology-driven quantum architecture search (TD-QAS) framework, which first identifies optimal circuit topologies and then fine-tunes the gate types. In the fine-tuning phase, the QAS inherits parameters from the topology search phase , eliminating the need for training from scratch. By decoupling the large search space into separate topology and gate-type components, TD-QAS avoids exploring gate configurations within low-performance topologies, thereby significantly reducing computational complexity. Numerical simulations across various tasks, under both noiseless and noisy conditions, validate the effectiveness of the TD-QAS framework. This framework advances standard QAS algorithms by enabling the identification of high-performance quantum circuits while minimizing computational demands. These findings indicate that TD-QAS deepens our understanding of VQAs and offers broad potential for the development of future QAS algorithms.},
   issn          = {1869-1919},
   doi           = {10.1007/s11432-024-4486-x},
-    url = {https://doi.org/10.1007/s11432-024-4486-x},
+  url           = {https://doi.org/10.1007/s11432-024-4486-x}
 }
-
 @article{Hirsbrunner2024beyondmp,
   doi           = {10.22331/q-2024-11-26-1538},
   url           = {https://doi.org/10.22331/q-2024-11-26-1538},
-    title = {Beyond {MP}2 initialization for unitary coupled cluster quantum
-             circuits},
-    author = {Hirsbrunner, Mark R. and Chamaki, Diana and Mullinax, J. Wayne and
-              Tubman, Norm M.},
+  title         = {Beyond {MP}2 initialization for unitary coupled cluster quantum circuits},
+  author        = {Hirsbrunner, Mark R. and Chamaki, Diana and Mullinax, J. Wayne and Tubman, Norm M.},
   journal       = {{Quantum}},
   issn          = {2521-327X},
-    publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den
-                 Quantenwissenschaften}},
+  publisher     = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
   volume        = {8},
   pages         = {1538},
   month         = nov,
-    year = {2024},
+  year          = {2024}
 }
-
 @misc{liu2025haqgnnhardwareawarequantumkernel,
-    title = {HaQGNN: Hardware-Aware Quantum Kernel Design Based on Graph Neural
-             Networks},
-    author = {Yuxiang Liu and Fanxu Meng and Lu Wang and Yi Hu and Sixuan Li and
-              Xutao Yu and Zaichen Zhang},
+  title         = {HaQGNN: Hardware-Aware Quantum Kernel Design Based on Graph Neural Networks},
+  author        = {Yuxiang Liu and Fanxu Meng and Lu Wang and Yi Hu and Sixuan Li and Xutao Yu and Zaichen Zhang},
   year          = {2025},
   eprint        = {2506.21161},
-    archivePrefix = {arXiv},
-    primaryClass = {quant-ph},
-    url = {https://arxiv.org/abs/2506.21161},
+  archiveprefix = {arXiv},
+  primaryclass  = {quant-ph},
+  url           = {https://arxiv.org/abs/2506.21161}
 }
-
 @article{training-free,
   title         = {Training-Free Quantum Architecture Search},
   volume        = {38},
   url           = {https://ojs.aaai.org/index.php/AAAI/article/view/29135},
-    DOI = {10.1609/aaai.v38i11.29135},
-    abstractNote = {Variational quantum algorithm (VQA) derives advantages from
-                    its error resilience and high flexibility in quantum resource
-                    requirements, rendering it broadly applicable in the noisy
-                    intermediate-scale quantum era. As the performance of VQA
-                    highly relies on the structure of the parameterized quantum
-                    circuit, it is worthwhile to propose quantum architecture
-                    search (QAS) algorithms to automatically search for
-                    high-performance circuits. Nevertheless, existing QAS methods
-                    are time-consuming, requiring circuit training to assess
-                    circuit performance. This study pioneers training-free QAS by
-                    utilizing two training-free proxies to rank quantum circuits,
-                    in place of the expensive circuit training employed in
-                    conventional QAS. Taking into account the precision and
-                    computational overhead of the path-based and
-                    expressibility-based proxies, we devise a two-stage
-                    progressive training-free QAS (TF-QAS). Initially, directed
-                    acyclic graphs (DAGs) are employed for circuit representation
-                    , and a zero-cost proxy based on the number of paths in the
-                    DAG is designed to filter out a substantial portion of
-                    unpromising circuits. Subsequently, an expressibility-based
-                    proxy, finely reflecting circuit performance, is employed to
-                    identify high-performance circuits from the remaining
-                    candidates. These proxies evaluate circuit performance
-                    without circuit training, resulting in a remarkable reduction
-                    in computational cost compared to current training-based QAS
-                    methods. Simulations on three VQE tasks demonstrate that
-                    TF-QAS achieves a substantial enhancement of sampling
-                    efficiency ranging from 5 to 57 times compared to
-                    state-of-the-art QAS, while also being 6 to 17 times faster.},
+  doi           = {10.1609/aaai.v38i11.29135},
+  abstractnote  = {Variational quantum algorithm (VQA) derives advantages from its error resilience and high flexibility in quantum resource requirements, rendering it broadly applicable in the noisy intermediate-scale quantum era. As the performance of VQA highly relies on the structure of the parameterized quantum circuit, it is worthwhile to propose quantum architecture search (QAS) algorithms to automatically search for high-performance circuits. Nevertheless, existing QAS methods are time-consuming, requiring circuit training to assess circuit performance. This study pioneers training-free QAS by utilizing two training-free proxies to rank quantum circuits, in place of the expensive circuit training employed in conventional QAS. Taking into account the precision and computational overhead of the path-based and expressibility-based proxies, we devise a two-stage progressive training-free QAS (TF-QAS). Initially, directed acyclic graphs (DAGs) are employed for circuit representation , and a zero-cost proxy based on the number of paths in the DAG is designed to filter out a substantial portion of unpromising circuits. Subsequently, an expressibility-based proxy, finely reflecting circuit performance, is employed to identify high-performance circuits from the remaining candidates. These proxies evaluate circuit performance without circuit training, resulting in a remarkable reduction in computational cost compared to current training-based QAS methods. Simulations on three VQE tasks demonstrate that TF-QAS achieves a substantial enhancement of sampling efficiency ranging from 5 to 57 times compared to state-of-the-art QAS, while also being 6 to 17 times faster.},
   number        = {11},
   journal       = {Proceedings of the AAAI Conference on Artificial Intelligence},
-    author = {He, Zhimin and Deng, Maijie and Zheng, Shenggen and Li, Lvzhou and
-              Situ, Haozhen},
+  author        = {He, Zhimin and Deng, Maijie and Zheng, Shenggen and Li, Lvzhou and Situ, Haozhen},
   year          = {2024},
   month         = {Mar.},
-    pages = {12430-12438},
+  pages         = {12430--12438}
 }
-
 @article{npqas,
   doi           = {10.1088/2632-2153/ac28dd},
   url           = {https://doi.org/10.1088/2632-2153/ac28dd},
@@ -377,86 +183,48 @@
   volume        = {2},
   number        = {4},
   pages         = {045027},
-    author = {Zhang, Shi-Xin and Hsieh, Chang-Yu and Zhang, Shengyu and Yao,
-              Hong},
+  author        = {Zhang, Shi-Xin and Hsieh, Chang-Yu and Zhang, Shengyu and Yao, Hong},
   title         = {Neural predictor based quantum architecture search},
   journal       = {Machine Learning: Science and Technology},
-    abstract = {Variational quantum algorithms (VQAs) are widely speculated to
-                deliver quantum advantages for practical problems under the
-                quantum–classical hybrid computational paradigm in the near term.
-                Both theoretical and practical developments of VQAs share many
-                similarities with those of deep learning. For instance, a key
-                component of VQAs is the design of task-dependent parameterized
-                quantum circuits (PQCs) as in the case of designing a good neural
-                architecture in deep learning. Partly inspired by the recent
-                success of AutoML and neural architecture search (NAS), quantum
-                architecture search (QAS) is a collection of methods devised to
-                engineer an optimal task-specific PQC. It has been proven that
-                QAS-designed VQAs can outperform expert-crafted VQAs in various
-                scenarios. In this work, we propose to use a neural network based
-                predictor as the evaluation policy for QAS. We demonstrate a
-                neural predictor guided QAS can discover powerful quantum circuit
-                ansatz, yielding state-of-the-art results for various examples
-                from quantum simulation and quantum machine learning. Notably,
-                neural predictor guided QAS provides a better solution than that
-                by the random-search baseline while using an order of magnitude
-                less of circuit evaluations. Moreover, the predictor for QAS as
-                well as the optimal ansatz found by QAS can both be transferred
-                and generalized to address similar problems.},
+  abstract      = {Variational quantum algorithms (VQAs) are widely speculated to deliver quantum advantages for practical problems under the quantum–classical hybrid computational paradigm in the near term. Both theoretical and practical developments of VQAs share many similarities with those of deep learning. For instance, a key component of VQAs is the design of task-dependent parameterized quantum circuits (PQCs) as in the case of designing a good neural architecture in deep learning. Partly inspired by the recent success of AutoML and neural architecture search (NAS), quantum architecture search (QAS) is a collection of methods devised to engineer an optimal task-specific PQC. It has been proven that QAS-designed VQAs can outperform expert-crafted VQAs in various scenarios. In this work, we propose to use a neural network based predictor as the evaluation policy for QAS. We demonstrate a neural predictor guided QAS can discover powerful quantum circuit ansatz, yielding state-of-the-art results for various examples from quantum simulation and quantum machine learning. Notably, neural predictor guided QAS provides a better solution than that by the random-search baseline while using an order of magnitude less of circuit evaluations. Moreover, the predictor for QAS as well as the optimal ansatz found by QAS can both be transferred and generalized to address similar problems.}
 }
-
 @article{hea-kandala,
-    author = {Kandala, Abhinav and Mezzacapo, Antonio and Temme, Kristan and
-              Takita, Maika and Brink, Markus and Chow, Jerry M. and Gambetta,
-              Jay M.},
-    title = {Hardware-efficient variational quantum eigensolver for small
-             molecules and quantum magnets},
+  author        = {Kandala, Abhinav and Mezzacapo, Antonio and Temme, Kristan and Takita, Maika and Brink, Markus and Chow, Jerry M. and Gambetta, Jay M.},
+  title         = {Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets},
   journal       = {Nature},
   year          = {2017},
   month         = {Sep},
   day           = {01},
   volume        = {549},
   number        = {7671},
-    pages = {242-246},
-    abstract = {The ground-state energy of small molecules is determined
-                efficiently using six qubits of a superconducting quantum
-                processor.},
+  pages         = {242--246},
+  abstract      = {The ground-state energy of small molecules is determined efficiently using six qubits of a superconducting quantum processor.},
   issn          = {1476-4687},
   doi           = {10.1038/nature23879},
-    url = {https://doi.org/10.1038/nature23879},
+  url           = {https://doi.org/10.1038/nature23879}
 }
-
 @article{tensorcircuit,
   doi           = {10.22331/q-2023-02-02-912},
   url           = {https://doi.org/10.22331/q-2023-02-02-912},
-    title = {Tensor{C}ircuit: a {Q}uantum {S}oftware {F}ramework for the {NISQ}
-             {E}ra},
-    author = {Zhang, Shi-Xin and Allcock, Jonathan and Wan, Zhou-Quan and Liu,
-              Shuo and Sun, Jiace and Yu, Hao and Yang, Xing-Han and Qiu,
-              Jiezhong and Ye, Zhaofeng and Chen, Yu-Qin and Lee, Chee-Kong and
-              Zheng, Yi-Cong and Jian, Shao-Kai and Yao, Hong and Hsieh, Chang-Yu
-              and Zhang, Shengyu},
+  title         = {Tensor{C}ircuit: a {Q}uantum {S}oftware {F}ramework for the {NISQ} {E}ra},
+  author        = {Zhang, Shi-Xin and Allcock, Jonathan and Wan, Zhou-Quan and Liu, Shuo and Sun, Jiace and Yu, Hao and Yang, Xing-Han and Qiu, Jiezhong and Ye, Zhaofeng and Chen, Yu-Qin and Lee, Chee-Kong and Zheng, Yi-Cong and Jian, Shao-Kai and Yao, Hong and Hsieh, Chang-Yu and Zhang, Shengyu},
   journal       = {{Quantum}},
   issn          = {2521-327X},
-    publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den
-                 Quantenwissenschaften}},
+  publisher     = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
   volume        = {7},
   pages         = {912},
   month         = feb,
-    year = {2023},
+  year          = {2023}
 }
-
 @misc{genetic-expressibility,
-    title = {Genetic optimization of ansatz expressibility for enhanced
-             variational quantum algorithm performance},
+  title         = {Genetic optimization of ansatz expressibility for enhanced variational quantum algorithm performance},
   author        = {Manish Mallapur and Ronit Raj and Ankur Raina},
   year          = {2025},
   eprint        = {2509.05804},
-    archivePrefix = {arXiv},
-    primaryClass = {quant-ph},
-    url = {https://arxiv.org/abs/2509.05804},
+  archiveprefix = {arXiv},
+  primaryclass  = {quant-ph},
+  url           = {https://arxiv.org/abs/2509.05804}
 }
-
 @article{Zhang_2022,
   doi           = {10.1088/2058-9565/ac87cd},
   url           = {https://doi.org/10.1088/2058-9565/ac87cd},
@@ -466,36 +234,13 @@
   volume        = {7},
   number        = {4},
   pages         = {045023},
-    author = {Zhang, Shi-Xin and Hsieh, Chang-Yu and Zhang, Shengyu and Yao,
-              Hong},
+  author        = {Zhang, Shi-Xin and Hsieh, Chang-Yu and Zhang, Shengyu and Yao, Hong},
   title         = {Differentiable quantum architecture search},
   journal       = {Quantum Science and Technology},
-    abstract = {Quantum architecture search (QAS) is the process of automating
-                architecture engineering of quantum circuits. It has been desired
-                to construct a powerful and general QAS platform which can
-                significantly accelerate current efforts to identify quantum
-                advantages of error-prone and depth-limited quantum circuits in
-                the NISQ era. Hereby, we propose a general framework of
-                differentiable quantum architecture search (DQAS), which enables
-                automated designs of quantum circuits in an end-to-end
-                differentiable fashion. We present several examples of circuit
-                design problems to demonstrate the power of DQAS. For instance,
-                unitary operations are decomposed into quantum gates, noisy
-                circuits are re-designed to improve accuracy, and circuit layouts
-                for quantum approximation optimization algorithm are
-                automatically discovered and upgraded for combinatorial
-                optimization problems. These results not only manifest the vast
-                potential of DQAS being an essential tool for the NISQ
-                application developments, but also present an interesting
-                research topic from the theoretical perspective as it draws
-                inspirations from the newly emerging interdisciplinary paradigms
-                of differentiable programming, probabilistic programming, and
-                quantum programming.},
+  abstract      = {Quantum architecture search (QAS) is the process of automating architecture engineering of quantum circuits. It has been desired to construct a powerful and general QAS platform which can significantly accelerate current efforts to identify quantum advantages of error-prone and depth-limited quantum circuits in the NISQ era. Hereby, we propose a general framework of differentiable quantum architecture search (DQAS), which enables automated designs of quantum circuits in an end-to-end differentiable fashion. We present several examples of circuit design problems to demonstrate the power of DQAS. For instance, unitary operations are decomposed into quantum gates, noisy circuits are re-designed to improve accuracy, and circuit layouts for quantum approximation optimization algorithm are automatically discovered and upgraded for combinatorial optimization problems. These results not only manifest the vast potential of DQAS being an essential tool for the NISQ application developments, but also present an interesting research topic from the theoretical perspective as it draws inspirations from the newly emerging interdisciplinary paradigms of differentiable programming, probabilistic programming, and quantum programming.}
 }
-
 @article{qdea,
-    title = {A survey on Quality-Diversity optimization: Approaches,
-             applications, and challenges},
+  title         = {A survey on Quality-Diversity optimization: Approaches, applications, and challenges},
   journal       = {Swarm and Evolutionary Computation},
   volume        = {100},
   pages         = {102240},
@@ -503,33 +248,20 @@
   issn          = {2210-6502},
   doi           = {https://doi.org/10.1016/j.swevo.2025.102240},
   url           = {https://www.sciencedirect.com/science/article/pii/S2210650225003979},
-    author = {Haoxiang Qin and Yi Xiang and Hainan Zhang and Yuyan Han and
-              Yuting Wang and Xinrui Tao and Yiping Liu},
-    keywords = {Quality-Diversity optimization, Evolutionary computation,
-                Feature space, MAP-Elites},
-    abstract = {Quality-Diversity (QD) optimization is a paradigm of
-                evolutionary computation (EC) that extends the classic approaches
-                , aiming to generate a collection of solutions that are both
-                diverse and high-performing. Unlike traditional evolutionary
-                algorithms (EAs), QD methods emphasize the illumination (or
-                coverage) of a user-defined feature space, while simultaneously
-                aiming for local optimization within each discovered region of
-                the feature space. Over the past decade, QD has rapidly developed
-                and proven effective in areas such as evolutionary robotics and
-                video games. However, a systematic review of this growing field
-                remains lacking. To date, the most recent review article on QD
-                was published in 2021. Therefore, to offer a more comprehensive
-                overview of the latest QD research, this paper provides a
-                thorough survey of QD optimization, covering its foundational
-                principles and representative algorithmic frameworks such as
-                Novelty Search with Local Competition (NSLC), MAP-Elites, the
-                unified modular QD framework, and RIBS. In addition, we divide
-                the algorithm improvement part into three modules for discussion:
-                containers, selection, and mutation. Then, the evaluation metrics
-                widely used in QD optimization are listed for researchers. We
-                further explore its diverse applications across domains such as
-                evolutionary robotics, video games, scheduling, software testing,
-                and engineering design. Finally, we discuss the current
-                challenges in the field and outline promising directions for
-                future research.},
+  author        = {Haoxiang Qin and Yi Xiang and Hainan Zhang and Yuyan Han and Yuting Wang and Xinrui Tao and Yiping Liu},
+  keywords      = {Quality-Diversity optimization, Evolutionary computation, Feature space, MAP-Elites},
+  abstract      = {Quality-Diversity (QD) optimization is a paradigm of evolutionary computation (EC) that extends the classic approaches , aiming to generate a collection of solutions that are both diverse and high-performing. Unlike traditional evolutionary algorithms (EAs), QD methods emphasize the illumination (or coverage) of a user-defined feature space, while simultaneously aiming for local optimization within each discovered region of the feature space. Over the past decade, QD has rapidly developed and proven effective in areas such as evolutionary robotics and video games. However, a systematic review of this growing field remains lacking. To date, the most recent review article on QD was published in 2021. Therefore, to offer a more comprehensive overview of the latest QD research, this paper provides a thorough survey of QD optimization, covering its foundational principles and representative algorithmic frameworks such as Novelty Search with Local Competition (NSLC), MAP-Elites, the unified modular QD framework, and RIBS. In addition, we divide the algorithm improvement part into three modules for discussion: containers, selection, and mutation. Then, the evaluation metrics widely used in QD optimization are listed for researchers. We further explore its diverse applications across domains such as evolutionary robotics, video games, scheduling, software testing, and engineering design. Finally, we discuss the current challenges in the field and outline promising directions for future research.}
+}
+@article{Eppstein_Har-Peled_Sidiropoulos_2020,
+  title         = {Approximate greedy clustering and distance selection for graph metrics},
+  volume        = {11},
+  url           = {https://jocg.org/index.php/jocg/article/view/3115},
+  doi           = {10.20382/jocg.v11i1a25},
+  abstractnote  = {\<p\>In this paper, we consider two important problems defined on finite metric spaces, and provide efficient new algorithms and approximation schemes for these problems on inputs given as graph shortest path metrics or high-dimensional Euclidean metrics. The first of these problems is the greedy permutation (or farthest-first traversal) of a finite metric space: a permutation of the points of the space in which each point is as far as possible from all previous points. We describe randomized algorithms to compute $(1+\varepsilon)$-approximate greedy permutations of any graph with $n$ vertices and $m$ edges in expected time $O\bigl(\varepsilon^{-1}(m+n)\log n\log(n/\varepsilon) \bigr)$. We also present an algorithm that computes a $(1+\varepsilon)$-approximate greedy permutations of points in high-dimensional Euclidean spaces in expected time $O(\varepsilon^{-2} n^{1+1/(1+\varepsilon)^2 + o(1)})$. Additionally we describe a deterministic algorithm to find exact greedy permutations of any graph with $n$ vertices and treewidth $O(1)$ in worst-case time $O(n^{3/2}\log^{O(1)} n)$.\</p\>\<p\>The second problem we consider is distance selection: given $k \le \binom{n}{2}$, we are interested in computing the $k$th smallest distance in the given metric space. We show that for planar graph metrics one can approximate this distance, up to a constant factor, in near linear time.\</p\>},
+  number        = {1},
+  journal       = {Journal of Computational Geometry},
+  author        = {Eppstein, David and Har-Peled, Sariel and Sidiropoulos, Anastasios},
+  year          = {2020},
+  month         = {Dec.},
+  pages         = {629–652}
 }

src/ga_qas.py

@@ -28,6 +28,54 @@ gate_set: list[GateType] = [
 ]
 
 
+def sample_hyperspace(
+    *args: tuple[int, int] | tuple[float, float], seed: int = 2010392991
+):
+    minimums: tuple[float | int, ...] = tuple(arg[0] for arg in args)
+    maximums: tuple[float | int, ...] = tuple(arg[1] for arg in args)
+    diffs: tuple[float | int, ...] = tuple(
+        ma - mi for mi, ma in zip(minimums, maximums)
+    )
+    idiffs: tuple[float, ...] = tuple(1.0 / diff for diff in diffs)
+    rng: random.Random = random.Random(seed)
+
+    previous_points: set[tuple[float | int, ...]] = set()
+
+    def dist(point: tuple[float | int, ...], other: tuple[float | int, ...]) -> float:
+        return sum(((p - o) * idiff) ** 2 for p, o, idiff in zip(point, other, idiffs))
+
+    while True:
+        sampled_points: list[tuple[float | int, ...]] = [
+            tuple(
+                min
+                + (
+                    rng.randint(min, min + diff)
+                    if isinstance(min, int) and isinstance(diff, int)
+                    else rng.uniform(min, min + diff)
+                )
+                for min, diff in zip(minimums, diffs)
+            )
+            for _ in range(10)
+        ]
+
+        if len(previous_points) == 0:
+            previous_points.add(sampled_points[0])
+            yield sampled_points[0]
+
+        min_distances: list[float] = [
+            min((dist(point, other) for other in previous_points))
+            for point in sampled_points
+        ]
+
+        mdist: float = max(min_distances)
+        point: tuple[float | int, ...] = sampled_points[
+            [i for i, j in enumerate(min_distances) if j == mdist][0]
+        ]
+
+        previous_points.add(point)
+        yield point
+
+
 def plot_best_circuits(best_circuits: list[QuantumCircuit]) -> None:
     fig, ax = plt.subplots()