Demonstration of subspace-search variational quantum eigensolver using photonic orbital angular momentum qudits

AI summary

Overview

• Research area: Quantum computing — variational quantum algorithms and photonic quantum simulation, specifically excited-state eigensolving using high-dimensional orbital angular momentum (OAM) qudits.
• Technical level: Intermediate. The core idea of subspace-search VQE is explained self-containedly, but familiarity with VQE, Pauli-string decompositions, and photonic state preparation helps.
• Scope (1 sentence): The paper reports the first experimental realization of subspace-search variational quantum eigensolver (SSVQE) on a photonic platform, using single-photon OAM qudits to simultaneously estimate ground and excited state energies of a four-dimensional HeH⁺ model and an eight-dimensional three-qubit Heisenberg chain.

What This Paper Is About

Variational quantum eigensolvers (VQEs) work well for ground states but have largely been confined to them, even though excited states govern fluorescence and photochemical processes. SSVQE offers a NISQ-friendly route to excited states that avoids ancilla-assisted overlap measurements, but it had never been demonstrated experimentally because it requires preparing multiple mutually orthogonal input states and applying a shared variational unitary to all of them. This paper solves that practical bottleneck by encoding the orthogonal states directly in the orbital angular momentum of single photons and optimizing a shared parametrized circuit.

Key Contributions

1. First experimental demonstration of SSVQE. The authors state that SSVQE "has lacked an experimental demonstration" since its proposal, and this work realizes it on a photonic platform.
2. Hologram-based orthogonal state preparation in OAM qudits. Rather than generating orthogonal inputs through deep gate sequences, they prepare the required mutually orthogonal states directly with phase–amplitude holograms on a spatial light modulator (SLM), using both the azimuthal index l and radial index p of Laguerre–Gaussian modes to guarantee orthogonality while equalizing coupling efficiencies.
3. State synthesis instead of full-unitary synthesis. The implementation does not physically realize the whole multi-mode unitary U(α⃗); instead the CPU computes the relevant columns of U and the SLM synthesizes the corresponding output states |ψⱼ(α⃗ₙ)⟩, which is equivalent for evaluating the SSVQE cost function.
4. Column-wise unitary parametrization for subspace-selective eigensolving. A triangular-mesh-inspired construction (plus a diagonal phase gate D_ω⃗) synthesizes only the N orthonormal columns defining the target subspace, reducing the parameter count when only a subset of levels is requested.

Main Findings

• Full four-level spectrum of HeH⁺ from one optimization run: Using 15 variational parameters (equal to d² − 1 for d = 4), the experiment simultaneously estimates ⟨E₀⟩, ⟨E₁⟩, ⟨E₂⟩, and ⟨E₃⟩ at the equilibrium bond length R = 0.90 Å, and the energies converge toward the theoretical values with increasing iteration number.
• Convergence speed for the molecular benchmark: With the paper's convergence criterion, optimization terminates after an average of 81.6 ± 10.2 iterations. Error bars are one standard deviation from 1,000 Monte Carlo resamplings.
• Energy curves versus interatomic distance: Estimated ground and excited state energies of HeH⁺ track theory across the full range of R, with deviations plotted on a logarithmic scale in the lower panel of Fig. 3c. The main text does not report explicit numerical deviation values.
• Eight-dimensional demonstration: For a three-qubit Heisenberg chain, the three lowest energy levels are estimated simultaneously using an eight-dimensional OAM qudit, with an average convergence of 368 ± 57 iterations, and remain in good agreement with theory as a function of J/B.
• Parameter reduction in practice: A general eight-dimensional unitary requires 63 parameters, but targeting only three levels reduces the count to 55 by excluding parameters associated with unused subspaces.
• Subspace selectivity works: In the four-level system, SSVQE can be configured with N = 2 orthogonal ansatz states to estimate only the ground and first excited states, using fewer variational parameters.
• Source of residual error: Deviations are attributed mainly to non-unity purity and state fidelity of the prepared OAM modes, primarily from imperfect optical alignment — described as technical rather than fundamental limitations.
• No advantage in measurement cost: SSVQE requires no measurement settings beyond standard VQE, so existing measurement-grouping and error-mitigation strategies apply directly.

Methodology in Plain English

A continuous-wave 403 nm laser pumps a 20-mm-long periodically poled potassium titanyl phosphate (PPKTP) crystal, producing heralded single photons by type-II spontaneous parametric down-conversion; one photon is detected to herald the other, which enters the quantum processing unit. The signal photon is shaped by a first SLM, which displays a hologram imprinting the desired superposition of OAM modes — these are the variational ansatz states, prepared one at a time and in sequence. A second SLM displays holograms that rotate the state into the measurement basis for each Pauli string, and the light is Fourier transformed and coupled into a single-mode fiber so that photon counting yields the required expectation values. A classical CPU combines these counts into the Hamiltonian expectation values, evaluates the weighted SSVQE cost function, and a COBYLA optimizer updates the parameters; the prepare–measure–update loop repeats until the parameter change falls below a tolerance ϵ. The molecule's Hamiltonian was generated with the OpenFermion–Psi4 library.

Why This Matters

• Research impact: It closes the gap between the theory of SSVQE and hardware, showing that a high-dimensional single-qudit encoding can access excited states without ancilla-assisted overlap estimation, controlled operations, or penalty hyperparameters that complicate competing methods.
• Fluorescence spectroscopy: Excited-state energies are the quantities that interpret fluorescence, which the paper cites as a primary motivation.
• Photochemistry and catalysis: The paper motivates excited states via chemical reactions and hydrogen bonding, and lists catalysis among target applications.
• Battery research and drug discovery: Both are explicitly named among applications where excited-state properties are central.
• Industry relevance: The work targets NISQ-era hardware, argues that OAM offers an in-principle unbounded photonic Hilbert space with few photons, and points toward compact, resource-efficient quantum simulators — relevant to photonic quantum computing developers and quantum-chemistry software and hardware groups.

Future Directions

• Combine OAM with other photonic degrees of freedom: Path, polarization, time-bin, and frequency are proposed to further increase accessible dimensionality.
• Use entangled photon pairs: Two photons each encoded in a d-dimensional OAM space would give access to a d²-dimensional Hilbert space, moving beyond single-particle encoding.
• Build multi-qudit or multi-particle architectures: The authors note these would access larger Hilbert spaces but introduce new challenges in photon generation, control, and detection; they also flag that the present SLM-based scheme measures sequentially, one hologram at a time.
• Improve mode preparation and alignment: Since residual deviations trace to non-unity purity and state fidelity, better optical alignment and mode preparation are the immediate route to higher accuracy.

Target Audience

Quantum-information experimentalists working on photonic platforms; researchers in quantum chemistry and materials simulation interested in excited-state algorithms for near-term devices; and graduate students or algorithm developers who want a concrete, hardware-realistic picture of how SSVQE behaves when implemented with high-dimensional qudit encoding.

Authors’ abstract

Accurate excited state energies are essential for interpreting fluorescence and photochemical processes, yet variational quantum eigensolvers (VQEs) on noisy intermediate-scale quantum (NISQ) hardware have largely been confined to ground states. Subspace-search VQE (SSVQE) offers a NISQ-friendly route to excited states without ancilla-based overlap measurements, however it has lacked an experimental demonstration since it requires the coherent preparation and manipulation of multiple orthogonal basis states under a shared variational unitary. Here we realize SSVQE on a photonic platform by encoding orthogonal qudit states in the orbital angular momentum of single photons and optimizing a shared variational circuit. We estimate excited state energies for a four-dimensional $\rm{HeH^+}$ molecule and an eight-dimensional Heisenberg chain, thereby demonstrating scalable excited state eigensolving in a high-dimensional Hilbert space. Our results establish photonic qudits as a resource-efficient pathway to excited state quantum simulation.