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Efficient implementation of single particle Hamiltonians in exponentially reduced qubit space

Quantum Journal

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Researchers Martin Plesch, Martin Friák, and Ijaz Ahamed Mohammad developed a logarithmic-qubit encoding that compresses N-site systems into just ⌈log₂N⌉ qubits, exponentially reducing hardware requirements for solid-state Hamiltonians. Their method pairs this encoding with a Gray-code-inspired measurement strategy, cutting measurement settings to logarithmic growth while preserving physical model fidelity in variational quantum algorithms. A new volumetric efficiency metric combines qubit count, circuit depth, and measurements into a single cost measure, revealing total resource reduction from N² to (log N)³ for hardware-efficient ansatz circuits. The approach enables parallel execution of multiple compressed circuits on single devices, maximizing near-term quantum hardware utilization for large-scale material simulations. This breakthrough extends variational quantum eigensolvers’ reach, allowing study of structured physical systems previously beyond current hardware limits.

Efficient implementation of single particle Hamiltonians in exponentially reduced qubit space

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AbstractCurrent and near-term quantum hardware is constrained by limited qubit counts, circuit depth, and the high cost of repeated measurements. We address these challenges for solid-state Hamiltonians by introducing a logarithmic-qubit encoding that maps a system with $N$ physical sites onto only $\lceil \log_2 N \rceil$ qubits while maintaining a clear correspondence with the underlying physical model. Within this reduced register, we construct a compatible variational circuit and a Gray-code-inspired measurement strategy whose number of global settings grows only logarithmically with system size. To quantify the overall hardware load, we introduce a volumetric efficiency metric that combines the number of qubits, circuit depth, and the number of measurement settings into a single measure, expressing the overall computation costs. Using this metric, we show that the total space--time sampling volume required in a variational loop can be reduced dramatically from $N^2$ to $(\log N)^3$ for a hardware-efficient ansatz, allowing an exponential reduction in time and size of the quantum hardware. These results demonstrate that large, structured solid-state Hamiltonians can be simulated on substantially smaller quantum registers with controlled sampling overhead and manageable circuit complexity, extending the reach of variational quantum algorithms on near-term devices.Featured image: Volumetric view of the proposed compression. Left: the original SES variational loop occupies a large cuboid spanned by qubit count (blue), circuit depth (green), and measurement settings (red); faded copies indicate repeated runs in different bases. Right (top): after the logarithmic encoding the same computation fits into a much thinner cuboid, exponentially smaller in qubits. Right (bottom): the small footprint allows many compressed circuits to be run in parallel on a single device.Popular summarySimulating quantum materials is one of the most promising applications of quantum computers, but present-day devices are still small and noisy. Every qubit, circuit operation, and measurement matters. Our work shows that, for an important class of single-particle Hamiltonians, the size of the quantum computer needed can be reduced exponentially. A system with N physical sites can be represented using only about log₂N qubits. We also show how to prepare suitable variational states and measure the required quantities efficiently, so that the savings in qubits are not lost elsewhere. To make this comparison precise, we introduce a resource-volume measure that combines qubit number, circuit depth, and measurement cost. Under this measure, the total cost of a variational quantum simulation can be reduced from N² to (log N)³ in the setting we study. These results suggest a practical route for using near-term quantum hardware to study larger structured physical systems than would otherwise fit on the available devices.► BibTeX data@article{Plesch2026efficient, doi = {10.22331/q-2026-05-08-2099}, url = {https://doi.org/10.22331/q-2026-05-08-2099}, title = {Efficient implementation of single particle {H}amiltonians in exponentially reduced qubit space}, author = {Plesch, Martin and Fri{\'{a}}k, Martin and Mohammad, Ijaz Ahamed}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2099}, month = may, year = {2026} }► References [1] Trygve Helgaker, Poul Jorgensen, and Jeppe Olsen. Molecular electronic-structure theory. John Wiley & Sons, 2013. [2] Attila Szabo and Neil S Ostlund. Modern quantum chemistry: introduction to advanced electronic structure theory. Courier Corporation, 2012. [3] Peter J Knowles and Nicholas C Handy. A new determinant-based full configuration interaction method.

Chemical Physics Letters, 111 (4-5): 315–321, 1984. 10.1016/0009-2614(84)85513-X. https://doi.org/10.1016/0009-2614(84)85513-X [4] Rodney J Bartlett and Monika Musiał. Coupled-cluster theory in quantum chemistry. Reviews of Modern Physics, 79 (1): 291–352, 2007. 10.1103/RevModPhys.79.291. https://doi.org/10.1103/RevModPhys.79.291 [5] Robert G Parr and Yang Weitao. Density-functional theory of atoms and molecules. 1995. [6] H Teng, T Fujiwara, T Hoshi, T Sogabe, S-L Zhang, and S Yamamoto. Efficient and accurate linear algebraic methods for large-scale electronic structure calculations with nonorthogonal atomic orbitals. Physical Review B, 83 (16): 165103, 2011. 10.1103/PhysRevB.83.165103. https://doi.org/10.1103/PhysRevB.83.165103 [7] Jakub Valdhans and Petr Klenovský. Combined tight-binding and configuration interaction study of unfolded electronic structure of the $g$ color center in si. Phys. Rev. B, 112: 165413, Oct 2025. 10.1103/8d5x-779h. URL https://doi.org/10.1103/8d5x-779h. https://doi.org/10.1103/8d5x-779h [8] Jun-Qiang Lu, HT Johnson, Vaishno Devi Dasika, and RS Goldman. Moments-based tight-binding calculations of local electronic structure in InAs/GaAs quantum dots for comparison to experimental measurements.

Applied Physics Letters, 88 (5), 2006. https://doi.org/10.1063/1.2171473. https://doi.org/10.1063/1.2171473 [9] Alexander Mittelstädt, Andrei Schliwa, and Petr Klenovskỳ. Modeling electronic and optical properties of III–V quantum dots—selected recent developments. Light: Science & Applications, 11 (1): 17, 2022. 10.1038/s41377-021-00700-9. https://doi.org/10.1038/s41377-021-00700-9 [10] Christopher C Paige. Error analysis of the Lanczos algorithm for tridiagonalizing a symmetric matrix. IMA Journal of Applied Mathematics, 18 (3): 341–349, 1976. 10.1093/imamat/18.3.341. https://doi.org/10.1093/imamat/18.3.341 [11] Beresford N Parlett and David S Scott. The Lanczos algorithm with selective orthogonalization. Mathematics of Computation, 33 (145): 217–238, 1979. 10.1090/S0025-5718-1979-0514820-3. https://doi.org/10.1090/S0025-5718-1979-0514820-3 [12] Taisuke Ozaki. $O(N)$ Krylov-subspace method for large-scale ab initio electronic structure calculations. Physical Review B, 74 (24): 245101, 2006. 10.1103/PhysRevB.74.245101. https://doi.org/10.1103/PhysRevB.74.245101 [13] Marco Cerezo, Andrew Arrasmith, Ryan Babbush, Simon 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. 10.1038/s42254-021-00348-9. https://doi.org/10.1038/s42254-021-00348-9 [14] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J Love, Alán Aspuru-Guzik, and Jeremy L O’brien. A variational eigenvalue solver on a photonic quantum processor. Nature Communications, 5 (1): 4213, 2014. 10.1038/ncomms5213. https://doi.org/10.1038/ncomms5213 [15] Dmitry A Fedorov, Bo Peng, Niranjan Govind, and Yuri Alexeev. VQE method: A short survey and recent developments. Materials Theory, 6 (1): 2, 2022. 10.1186/s41313-021-00032-6. https://doi.org/10.1186/s41313-021-00032-6 [16] 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. 10.1038/nature23879. https://doi.org/10.1038/nature23879 [17] Jules Tilly, Hongxiang Chen, Shuxiang Cao, Dario Picozzi, Kanav Setia, Ying Li, Edward Grant, Leonard Wossnig, Ivan Rungger, George H Booth, et al. The variational quantum eigensolver: a review of methods and best practices. Physics Reports, 986: 1–128, 2022. 10.1016/j.physrep.2022.08.003. https://doi.org/10.1016/j.physrep.2022.08.003 [18] Ijaz Ahamed Mohammad, Matej Pivoluska, and Martin Plesch. Meta-optimization of resources on quantum computers. Scientific Reports, 14 (1): 10312, 2024. 10.1038/s41598-024-59618-y. https://doi.org/10.1038/s41598-024-59618-y [19] Ivana Miháliková, Matej Pivoluska, Martin Plesch, Martin Friák, Daniel Nagaj, and Mojmír Šob. The cost of improving the precision of the variational quantum eigensolver for quantum chemistry. Nanomaterials, 12 (2): 243, 2022. 10.3390/nano12020243. https://doi.org/10.3390/nano12020243 [20] Oscar Higgott, Daochen Wang, and Stephen Brierley. Variational quantum computation of excited states. Quantum, 3: 156, 2019. 10.22331/q-2019-07-01-156. https://doi.org/10.22331/q-2019-07-01-156 [21] Jean-Marc Jancu, Reinhard Scholz, Fabio Beltram, and Franco Bassani. Empirical $spds^{*}$ tight-binding calculation for cubic semiconductors: General method and material parameters. Physical Review B, 57 (11): 6493, 1998. 10.1103/PhysRevB.57.6493. https://doi.org/10.1103/PhysRevB.57.6493 [22] Anh-Luan Phan, Alessandro Pecchia, Alessia Di Vito, and Matthias Auf der Maur. Empirical tight-binding method for large-supercell simulations of disordered semiconductor alloys. Physica Scripta, 99 (7): 075903, 2024. 10.1088/1402-4896/ad4f65. https://doi.org/10.1088/1402-4896/ad4f65 [23] P Vogl, Harold P Hjalmarson, and John D Dow. A semi-empirical tight-binding theory of the electronic structure of semiconductors. Journal of Physics and Chemistry of Solids, 44 (5): 365–378, 1983. 10.1016/0022-3697(83)90064-1. https://doi.org/10.1016/0022-3697(83)90064-1 [24] Yaohua Tan, Michael Povolotskyi, Tillmann Kubis, Yu He, Zhengping Jiang, Gerhard Klimeck, and Timothy B Boykin. Empirical tight binding parameters for GaAs and MgO with explicit basis through dft mapping. Journal of Computational Electronics, 12 (1): 56–60, 2013. 10.1007/s10825-013-0436-0. https://doi.org/10.1007/s10825-013-0436-0 [25] Kyle Sherbert, Frank Cerasoli, and Marco Buongiorno Nardelli. A systematic variational approach to band theory in a quantum computer. RSC Advances, 11 (62): 39438–39449, 2021. 10.1039/D1RA07451B. https://doi.org/10.1039/D1RA07451B [26] Michal Krejčí, Lucie Krejčí, Ijaz Ahamed Mohammad, Martin Plesch, and Martin Friák. Minimum measurements quantum protocol for band structure calculation. arXiv preprint arXiv:2511.04389, 2025. arXiv:2511.04389 [27] Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C Benjamin, and Xiao Yuan. Quantum computational chemistry. Reviews of Modern Physics, 92 (1): 015003, 2020. 10.1103/RevModPhys.92.015003. https://doi.org/10.1103/RevModPhys.92.015003 [28] Kyle Sherbert, Anooja Jayaraj, and Marco Buongiorno Nardelli. Quantum algorithm for electronic band structures with local tight-binding orbitals. Scientific Reports, 12 (1): 9867, 2022. 10.1038/s41598-022-13627-x. https://doi.org/10.1038/s41598-022-13627-x [29] Ijaz Ahamed Mohammad, Yury Chernyak, and Martin Plesch. HOPSO: A robust classical optimizer for VQE. arXiv preprint arXiv:2508.13651, 2025. arXiv:2508.13651 [30] A generic pure state on $n$ qubits lives in a an exponentially big Hilbert space and, up to normalization and global phase, requires $\mathcal{O}(2^n)$ real parameters to specify. By contrast, a hardware-efficient ansatz (HEA) with fixed layout and limited depth contains only $\mathcal{O}(p n)$ continuous parameters (for $p$ layers) and uses a restricted set of entangling operations. Starting from a simple product state such as $|0\rangle^{\otimes n}$ and applying a shallow, architecture-constrained circuit therefore explores only a tiny subset of the full state space. This induces an intrinsic bias toward low-complexity states and leaves most of Hilbert space effectively inaccessible at practical depths. [31] Tanuj Khattar and Craig Gidney. Rise of conditionally clean ancillae for efficient quantum circuit constructions. Quantum, 9: 1752, 2025. 10.22331/q-2025-05-21-1752. https://doi.org/10.22331/q-2025-05-21-1752 [32] Vivek V. Shende and Igor L. Markov. On the cnot-cost of toffoli gates. Quantum Info. Comput., 9 (5): 461–486, May 2009. ISSN 1533-7146. URL https://dl.acm.org/doi/10.5555/2011791.2011799. https://dl.acm.org/doi/10.5555/2011791.2011799 [33] Edgard N Gilbert. Gray codes and paths on the n-cube.

The Bell System Technical Journal, 37 (3): 815–826, 1958. 10.1002/j.1538-7305.1958.tb03887.x. https://doi.org/10.1002/j.1538-7305.1958.tb03887.x [34] Frank Gray. Pulse code communication, 1953. Filed 1947. [35] Carla D. Savage. A survey of combinatorial Gray codes. SIAM Review, 39 (4): 605–629, 1997. 10.1137/S0036144595295272. https://doi.org/10.1137/S0036144595295272 [36] Martin Plesch and Časlav Brukner. Quantum-state preparation with universal gate decompositions. Physical Review A, 83 (3): 032302, 2011. 10.1103/PhysRevA.83.032302. https://doi.org/10.1103/PhysRevA.83.032302Cited byCould not fetch Crossref cited-by data during last attempt 2026-05-08 08:40:42: Could not fetch cited-by data for 10.22331/q-2026-05-08-2099 from Crossref. This is normal if the DOI was registered recently. On SAO/NASA ADS no data on citing works was found (last attempt 2026-05-08 08:40:43).This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractCurrent and near-term quantum hardware is constrained by limited qubit counts, circuit depth, and the high cost of repeated measurements. We address these challenges for solid-state Hamiltonians by introducing a logarithmic-qubit encoding that maps a system with $N$ physical sites onto only $\lceil \log_2 N \rceil$ qubits while maintaining a clear correspondence with the underlying physical model. Within this reduced register, we construct a compatible variational circuit and a Gray-code-inspired measurement strategy whose number of global settings grows only logarithmically with system size. To quantify the overall hardware load, we introduce a volumetric efficiency metric that combines the number of qubits, circuit depth, and the number of measurement settings into a single measure, expressing the overall computation costs. Using this metric, we show that the total space--time sampling volume required in a variational loop can be reduced dramatically from $N^2$ to $(\log N)^3$ for a hardware-efficient ansatz, allowing an exponential reduction in time and size of the quantum hardware. These results demonstrate that large, structured solid-state Hamiltonians can be simulated on substantially smaller quantum registers with controlled sampling overhead and manageable circuit complexity, extending the reach of variational quantum algorithms on near-term devices.Featured image: Volumetric view of the proposed compression. Left: the original SES variational loop occupies a large cuboid spanned by qubit count (blue), circuit depth (green), and measurement settings (red); faded copies indicate repeated runs in different bases. Right (top): after the logarithmic encoding the same computation fits into a much thinner cuboid, exponentially smaller in qubits. Right (bottom): the small footprint allows many compressed circuits to be run in parallel on a single device.Popular summarySimulating quantum materials is one of the most promising applications of quantum computers, but present-day devices are still small and noisy. Every qubit, circuit operation, and measurement matters. Our work shows that, for an important class of single-particle Hamiltonians, the size of the quantum computer needed can be reduced exponentially. A system with N physical sites can be represented using only about log₂N qubits. We also show how to prepare suitable variational states and measure the required quantities efficiently, so that the savings in qubits are not lost elsewhere. To make this comparison precise, we introduce a resource-volume measure that combines qubit number, circuit depth, and measurement cost. Under this measure, the total cost of a variational quantum simulation can be reduced from N² to (log N)³ in the setting we study. These results suggest a practical route for using near-term quantum hardware to study larger structured physical systems than would otherwise fit on the available devices.► BibTeX data@article{Plesch2026efficient, doi = {10.22331/q-2026-05-08-2099}, url = {https://doi.org/10.22331/q-2026-05-08-2099}, title = {Efficient implementation of single particle {H}amiltonians in exponentially reduced qubit space}, author = {Plesch, Martin and Fri{\'{a}}k, Martin and Mohammad, Ijaz Ahamed}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2099}, month = may, year = {2026} }► References [1] Trygve Helgaker, Poul Jorgensen, and Jeppe Olsen. Molecular electronic-structure theory. John Wiley & Sons, 2013. [2] Attila Szabo and Neil S Ostlund. Modern quantum chemistry: introduction to advanced electronic structure theory. Courier Corporation, 2012. [3] Peter J Knowles and Nicholas C Handy. A new determinant-based full configuration interaction method.

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Efficient implementation of single particle Hamiltonians in exponentially reduced qubit space

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