Researchers at the Iranian Center for Quantum Technologies have calculated the ground and excited-state energies of the neutral helium atom using a photonic quantum circuit. The work encodes helium’s complex electronic structure within a specific “CAS(2,2) subspace” and represents it using a continuous-variable quantum circuit initialized in the Fock state |1100⟩.
In ideal simulations, the team observed “survival probability of the ground and the excited states” reaching unity, a result attributed to the preservation of total photon number and spin sector balance. This approach demonstrates a method for determining electronic energies while maintaining orthogonality with negligible errors.
Photonic Circuit Encoding of Helium’s CAS(2,2) Space
The electronic structure of helium, specifically within a CAS(2,2) subspace, is now being modeled using a four-mode photonic quantum circuit constructed from single-mode phase shifts, beam splitters and cross-Kerr interactions. This approach encodes the helium atom’s complex electronic structure into a bosonic Fock-space representation, then optimizes it with a continuous-variable quantum circuit, offering a distinct pathway for simulating molecular electronic behavior.
Researchers benchmarked the circuit’s performance against exact diagonalization of the active-space configuration-interaction (CASCI) Hamiltonian, a standard method for calculating electronic energies. The work extends beyond the helium ground state to optimize calculations for up to three excited states, employing the variational quantum deflation (VQD) method within the CAS(2,2) space.
The team’s design emphasizes the preservation of both total photon number and photon number within the two spin sectors, alpha and beta, which are critical for maintaining the integrity of the quantum simulation. “By naturally encoding quantum information into optical spatial or temporal modes, photonics provides room-temperature operability, ultra-low decoherence, and most crucially exact intrinsic conservation of particle number and spin projection through unitary, beam-splitter-mediated linear optical transformations,” the study notes.
In the ideal noiseless simulation setting, the survival probability of the ground and the excited states within the target CAS subspace is unity. Analysis of the complete N=2, M=4 bosonic Fock-space population distribution for the optimized states reveals complete confinement within the physical CAS(2,2) subspace, with a population probability of 100%. This indicates the circuit effectively isolates the relevant electronic configurations, preventing leakage into unwanted states and ensuring accurate energy level determination.
The researchers demonstrated the feasibility of determining the singlet ground state of neutral helium, with an energy of approximately -2. 8703 Ha, on a four-mode integrated photonic processor in a preceding investigation. Optimization trajectories for the sequential deflation protocol, using Powell’s direction-set algorithm, show the circuit converging on accurate energy levels. The study reports that the final converged energies and energy deviations are consistent with exact full configuration interaction (FCI) calculations within the CAS(2,2) active space.
Further refinements, the researchers suggest, could involve more expressive circuit designs, the incorporation of additional photonic modes and nonlinear interactions, and the development of more effective optimization strategies. These findings establish a foundation for further exploration of continuous-variable photonic approaches to variational quantum chemistry, potentially offering a scalable and precise method for simulating complex molecular systems.
Variational Quantum Deflation for Helium Excited States
The photonic circuit used in this work maintains complete confinement of the helium atom’s electronic state within the target computational space, achieving 100% population within the CAS(2,2) subspace, according to results published in a recent study. This complete confinement demonstrates the circuit’s ability to isolate relevant electronic configurations, which is an important step in accurately simulating excited states and preventing unwanted leakage into unmodeled regions of the electronic structure.
The circuit’s initial state, the Fock state |1100⟩, served as the starting point for optimizing the helium atom’s energy levels using the variational quantum deflation (VQD) algorithm. This specific initialization, combined with the circuit architecture, allowed for the sequential determination of excited states, extending the method from the ground state to the first three excited states within the CAS(2,2) subspace. This comparison confirms the viability of the photonic implementation as a competitive alternative for calculating electronic energies.
A key finding of the study centers on the preservation of fundamental quantum properties during the simulation process. This preservation isn’t merely a byproduct of the simulation, but a direct consequence of the circuit’s design and its ability to accurately represent the underlying physics of the helium atom. Maintaining these conserved quantities is essential for ensuring the fidelity of the calculated energy levels and preventing spurious results.
The VQD algorithm employed in this work relies on sequentially resolving excited electronic states while enforcing mutual orthogonality. The study also demonstrates the ability to determine both ground and excited-state energies within the CAS(2,2) electronic Hilbert space while satisfying orthogonality conditions with negligible errors. The work, led by F. Mirmasoudi and B., demonstrates that the proposed approach enables the determination of the ground and excited-state energies within the CAS(2,2) electronic Hilbert space using a continuous-variable quantum circuit and the variational quantum deflation (VQD) algorithm, while satisfying the orthogonality conditions with negligible errors.
Continuous-Variable Circuit Architecture & Parameters
This circuit, operating across a four-mode spatial-rail optical architecture, is initialized in the Fock state |1100⟩, a specific configuration chosen to represent the initial electronic state of the helium atom. This expansion relied on the variational quantum deflation (VQD) method, a technique designed to systematically refine the energy spectrum. The photonic circuit’s architecture imposes constraints on the dynamics within the CAS(2,2) subspace, effectively preventing leakage into non-physical sectors of the broader electronic configuration space.
Analysis of the physical CAS survival probability, defined as the sum of squared amplitudes of the target states, confirmed 100% projection fidelity for each of the four states examined. This level of accuracy is further supported by the observed distributions of states, which align with theoretical predictions for electronic configurations, specifically showing the filling of the lowest molecular orbital by two electrons with opposite spins in the ground state.
The study also highlights the sensitivity of excited-state calculations to the expressibility of the variational ansatz and the optimization landscape associated with the deflation terms. While the overall structure of the low-lying spectrum was successfully recovered, the accuracy of these excited states proved more vulnerable to these factors. These findings illustrate both the potential and current limitations of symmetry-preserving continuous-variable circuits for variational quantum chemistry.
Number Conservation & State Fidelity in Simulation
The fidelity of quantum simulations hinges on maintaining the integrity of the simulated state, and this work demonstrates a survival probability reaching unity for both ground and excited states within a defined electronic structure. This complete preservation of state is achieved because the total photon number, alongside the photon count in the α and β spin sectors, remains constant throughout the calculation. This inherent conservation stems from the use of beam-splitter-mediated linear optical transformations, a key feature of the photonic circuit architecture employed in this study.
The researchers mapped the four fermionic states defining the CAS(2,2) subspace of helium, representing specific electronic configurations, into a four-mode bosonic Fock-space representation. This translation isn’t merely a mathematical convenience; it “naturally guarantees that the total particle number and spin projections remain invariant during the optical unitary evolution,” ensuring the simulation stays within physically plausible boundaries.
Specifically, the states |1α 1β⟩, |1α 2β⟩, |1β 2α⟩, and |2α 2β⟩ are used to represent the electronic structure, with each state corresponding to a unique occupation-number Fock state. This explicit encoding is critical for preventing unwanted leakage into non-physical states outside the defined CAS(2,2) subspace. Beyond simply preventing errors, the approach demonstrably achieves high accuracy in energy calculations.
Using a continuous-variable quantum circuit and the variational quantum deflation (VQD) algorithm, the team determined the ground-state energy to be -2. 87015497 Ha, differing from the exact FCI benchmark of -2. 87016200 Ha by only 7. 03× Ha. This result falls well below the standard chemical accuracy threshold of 1. 59× Ha, indicating a remarkably precise simulation. The circuit architecture under study is constructed of single-mode phase shifts, beam splitters and cross-Kerr interactions.
Benchmarking Against CASCI: Ground & Excited Energies
This comparison revealed a post-selection survival probability of unity, meaning no quantum information leaked into unphysical states during the calculation, a critical indicator of a well-behaved simulation. Scaling the penalty to 64.0 Hartree was necessary to prevent spectral collapse when targeting the second excited state, demonstrating the sensitivity of the optimization process. Analysis of the resulting energy levels revealed distinct configurations corresponding to each excited state; the first and second excited states converged to energies of approximately -1.399 and -0.949 Hartree, respectively, as shown by the convergence of projected energies.
The third, high-energy state resided in the |0011⟩ configuration, signifying a doubly-excited state where both electrons were promoted from the lowest to the highest energy spatial orbital. This configuration-specific mapping demonstrates the circuit’s ability to accurately represent the complex interplay of electron promotion and orbital occupancy.
Initializing the circuit in the Fock state |1100⟩ proved to be an effective starting point for the variational procedure, allowing the algorithm to efficiently explore the relevant portion of the Hilbert space. The results suggest a pathway toward more complex simulations of atomic and molecular systems using photonic quantum computing, potentially offering new insights into chemical processes and material properties.
👉 More information
🗞 Helium States in a CAS(2,2) Subspace Inspired by Quantum Photonics
✍️ Forouzan Mirmasoudi and Badie Ghavami
🧠 ArXiv: http://arxiv.org/abs/2611.06448v1
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