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Recursive algorithm for constructing antisymmetric fermionic states in first quantization mapping

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Researchers developed a deterministic quantum algorithm for constructing antisymmetric fermionic states in first quantization, improving upon sorting-based methods by initializing particles independently rather than requiring ordered inputs. The algorithm achieves O(η²√N) T-gate complexity for η particles and N single-particle states, outperforming alternatives when η≲√N, while using O(√N) dirty ancilla qubits for intermediate calculations. A recursive approach enables simultaneous state preparation and antisymmetrization, leveraging single-particle orthogonality to detect exchanges, with efficiency gains when prior knowledge of states is available. A measurement-based variant cuts gate costs by roughly half, demonstrated through two- and three-particle circuit examples, with noise impact analyzed in a three-particle Clifford+T decomposition. The method excels for simple single-particle states (e.g., lattice sites), offering advantages for systems with ≤40 particles, while factorizing preparation costs for complex states.

Recursive algorithm for constructing antisymmetric fermionic states in first quantization mapping

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AbstractWe devise a deterministic quantum algorithm to produce antisymmetric states of single-particle orbitals in the first quantization mapping. Unlike sorting-based antisymmetrization algorithms, which require ordered input states and high Clifford-gate overhead, our approach initializes the state of each particle independently. For a system of $\eta$ particles and $N$ single-particle states, our algorithm prepares antisymmetrized states of non-trivial localized (e.g., Hartree-Fock) orbitals using $O(\eta^2\sqrt{N})$ $T$-gates, outperforming alternative algorithms when $\eta\lesssim \sqrt{N}$. To achieve such scaling, we require $O(\sqrt{N})$ dirty ancilla qubits for intermediate calculations. Knowledge of the single-particle states to be antisymmetrized can be leveraged to further improve the efficiency of the circuit, and a measurement-based variant reduces gate cost by roughly a factor of two. We show example circuits for two- and three-particle systems and discuss the generalization to an arbitrary number of particles. For a specific three-particle example, we decompose the circuit into Clifford$+T$ gates and study the impact of noise on the prepared state.Featured image: Illustration of our recursive algorithm used to prepare a totally antisymmetric state on four particles. At each step, a new particle is added and the resulting state is made to include all possible exchanges between the added particle and those within the existing antisymmetric state.Popular summaryTo describe an interacting system of identical fermions on a quantum computer requires a choice of encoding scheme. When the number of accessible single-particle states is large compared to the particle number, an efficient choice is the first quantization mapping, where the state of each particle is encoded and the number of required qubits grows logarithmically with the number of states. In first quantization, however, many-body fermionic states are not antisymmetric by default. In this work, we introduce a novel recursive algorithm for constructing antisymmetric fermionic states in the first quantization mapping. Existing antisymmetrization algorithms are predominantly based on sorting lists of integers. Our algorithm is qualitatively different, using the fact that the single-particle states are orthogonal to detect where particle exchanges have occurred. In our approach, state preparation and antisymmetrization occur simultaneously. From a resource standpoint, it is useful to consider two limiting cases: First, when the single-particle states are very simple (for example, each particle occupies a particular site in a lattice), then state preparation is trivial and the leading ($T$-gate) costs arise from antisymmetrization. In this regime, our approach can outperform alternatives for systems with $\lesssim$ 40 particles. At the other extreme, if the single-particle states are complicated mixtures of the underlying basis states, then state preparation dominates the ($T$-gate) cost. Due to the recursive nature of our algorithm, state preparation factorizes across each single-particle Hilbert space, avoiding costly operations on the entire many-body Hilbert space.► BibTeX data@article{Rule2026recursivealgorithm, doi = {10.22331/q-2026-04-08-2056}, url = {https://doi.org/10.22331/q-2026-04-08-2056}, title = {Recursive algorithm for constructing antisymmetric fermionic states in first quantization mapping}, author = {Rule, E. and Chernyshev, I. A. and Stetcu, I. and Carlson, J. and Weiss, R.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2056}, month = apr, year = {2026} }► References [1] Kenneth W Ford and John A Wheeler. ``Semiclassical description of scattering''. Annals of Physics 7, 259–286 (1959). https://doi.org/10.1016/0003-4916(59)90026-0 [2] M. Brack and R. Bhaduri. ``Semiclassical physics''. Frontiers in Physics. Avalon Publishing. (2003). url: https://doi.org/10.1201/9780429502828. https://doi.org/10.1201/9780429502828 [3] D.M. Brink. ``Semi-classical methods for nucleus-nucleus scattering''.

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Advanced Quantum Technologies 2, 1900015 (2019). https://doi.org/10.1002/qute.201900015 [51] Matthew Amy, Dmitri Maslov, Michele Mosca, and Martin Roetteler. ``A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits''. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 32, 818–830 (2013). https://doi.org/10.1109/TCAD.2013.2244643Cited byCould not fetch Crossref cited-by data during last attempt 2026-04-08 12:41:31: Could not fetch cited-by data for 10.22331/q-2026-04-08-2056 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-04-08 12:41:32: No response from ADS or unable to decode the received json data when getting the list of citing works.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. AbstractWe devise a deterministic quantum algorithm to produce antisymmetric states of single-particle orbitals in the first quantization mapping. Unlike sorting-based antisymmetrization algorithms, which require ordered input states and high Clifford-gate overhead, our approach initializes the state of each particle independently. For a system of $\eta$ particles and $N$ single-particle states, our algorithm prepares antisymmetrized states of non-trivial localized (e.g., Hartree-Fock) orbitals using $O(\eta^2\sqrt{N})$ $T$-gates, outperforming alternative algorithms when $\eta\lesssim \sqrt{N}$. To achieve such scaling, we require $O(\sqrt{N})$ dirty ancilla qubits for intermediate calculations. Knowledge of the single-particle states to be antisymmetrized can be leveraged to further improve the efficiency of the circuit, and a measurement-based variant reduces gate cost by roughly a factor of two. We show example circuits for two- and three-particle systems and discuss the generalization to an arbitrary number of particles. For a specific three-particle example, we decompose the circuit into Clifford$+T$ gates and study the impact of noise on the prepared state.Featured image: Illustration of our recursive algorithm used to prepare a totally antisymmetric state on four particles. At each step, a new particle is added and the resulting state is made to include all possible exchanges between the added particle and those within the existing antisymmetric state.Popular summaryTo describe an interacting system of identical fermions on a quantum computer requires a choice of encoding scheme. When the number of accessible single-particle states is large compared to the particle number, an efficient choice is the first quantization mapping, where the state of each particle is encoded and the number of required qubits grows logarithmically with the number of states. In first quantization, however, many-body fermionic states are not antisymmetric by default. In this work, we introduce a novel recursive algorithm for constructing antisymmetric fermionic states in the first quantization mapping. Existing antisymmetrization algorithms are predominantly based on sorting lists of integers. Our algorithm is qualitatively different, using the fact that the single-particle states are orthogonal to detect where particle exchanges have occurred. In our approach, state preparation and antisymmetrization occur simultaneously. From a resource standpoint, it is useful to consider two limiting cases: First, when the single-particle states are very simple (for example, each particle occupies a particular site in a lattice), then state preparation is trivial and the leading ($T$-gate) costs arise from antisymmetrization. In this regime, our approach can outperform alternatives for systems with $\lesssim$ 40 particles. At the other extreme, if the single-particle states are complicated mixtures of the underlying basis states, then state preparation dominates the ($T$-gate) cost. Due to the recursive nature of our algorithm, state preparation factorizes across each single-particle Hilbert space, avoiding costly operations on the entire many-body Hilbert space.► BibTeX data@article{Rule2026recursivealgorithm, doi = {10.22331/q-2026-04-08-2056}, url = {https://doi.org/10.22331/q-2026-04-08-2056}, title = {Recursive algorithm for constructing antisymmetric fermionic states in first quantization mapping}, author = {Rule, E. and Chernyshev, I. A. and Stetcu, I. and Carlson, J. and Weiss, R.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2056}, month = apr, year = {2026} }► References [1] Kenneth W Ford and John A Wheeler. ``Semiclassical description of scattering''. Annals of Physics 7, 259–286 (1959). https://doi.org/10.1016/0003-4916(59)90026-0 [2] M. Brack and R. Bhaduri. ``Semiclassical physics''. Frontiers in Physics. Avalon Publishing. (2003). url: https://doi.org/10.1201/9780429502828. https://doi.org/10.1201/9780429502828 [3] D.M. Brink. ``Semi-classical methods for nucleus-nucleus scattering''.

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Recursive algorithm for constructing antisymmetric fermionic states in first quantization mapping

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