Can photonic orbital angular momentum qudits demonstrate subspace-search VQE for excited states?

Excited states need a different variational strategy

Standard variational quantum eigensolvers minimize the energy expectation value of a trial state, and that bound only pushes the state toward the ground state. Excited-state energies matter for interpreting fluorescence and photochemical processes, but reaching them on noisy intermediate-scale quantum hardware has largely required extra machinery: ancilla-assisted overlap measurements, controlled operations, penalty terms, or classical diagonalization of large linear systems.

Subspace-search VQE (SSVQE) avoids that overhead. It prepares several mutually orthogonal input states and applies one shared parametrized unitary to all of them. Because orthogonality is fixed at the input and preserved by the shared unitary, the method needs no ancilla-based overlap estimation and no measurement settings beyond those of standard VQE. The catch, which the paper states directly, is that SSVQE had not been experimentally demonstrated because it demands high-fidelity preparation of multiple orthogonal states and reproducible application of a common unitary across them.

Orthogonal OAM modes carry the target subspace

The experiment encodes each state in the orbital angular momentum (OAM) of a single photon, using both the azimuthal index l and the radial index p of Laguerre–Gaussian modes. Combining l and p keeps the basis orthogonal while equalizing coupling efficiencies. For the four-dimensional Hamiltonian the modes are (l, p) = (−3, 4), (−1, 4), (1, 4), (3, 4); for the eight-dimensional case the paper lists eight (l, p) pairs spanning l from −7 to 7.

Rather than building N orthogonal inputs and then physically applying a full multi-mode unitary to each, the implementation prepares the N orthogonal output states directly as OAM qudit states. A classical processing unit computes the target output coefficients from the variational parameters, converts them into the OAM basis, and generates phase-amplitude holograms. The hologram for the j-th state comes from the j-th column of the effective unitary at that iteration and is displayed on a spatial light modulator (SLM). In other words, the hardware synthesizes only the columns of the unitary that the cost function actually needs.

Column-wise synthesis shrinks the parameter space

A general SU(d) transformation would need d² − 1 parameters. The authors instead synthesize only the N orthonormal columns defining the targeted subspace and complete the remaining columns by enforcing orthonormality, so the matrix stays unitary. Each column is built from a product of two-mode rotations, and the block structure lets the optimizer extract only the rotation parameters relevant to the chosen columns. A diagonal phase gate prepended to the product accounts for relative phases that would otherwise be absorbed into the unspecified columns.

This is what makes subspace selectivity concrete rather than rhetorical. In the four-dimensional case the full construction uses 15 variational parameters, matching d² − 1 for d = 4. For the eight-dimensional Heisenberg chain, a general unitary would require 63 parameters, but targeting only three levels reduces the count to 55 by excluding parameters tied to unused subspaces. The number of variational parameters therefore scales with the number of targeted energy levels, not with the full dimension.

HeH+ yields four levels in one optimization run

The first benchmark is a four-dimensional HeH+ model Hamiltonian generated with the OpenFermion–Psi4 library. In a single optimization run the experiment estimates the full four-level spectrum simultaneously. At the equilibrium bond length R = 0.90 Å, the 15 variational parameters are initialized arbitrarily and the estimated energies converge toward the theoretical values as iterations proceed; error bars are one standard deviation from 1000 Monte Carlo resamplings.

Across interatomic distances R, the experimental markers track the theoretical curves, and the lower panel of the corresponding figure plots energy deviations on a logarithmic scale to show all four levels converging. Using the stated convergence criterion, optimization terminates after an average of 81.6 ± 10.2 iterations. The paper also notes that SSVQE can be configured to estimate only the ground and first excited states by preparing N = 2 orthogonal ansatz states, which reduces the parameter count accordingly.

An eight-dimensional chain tests scalability

The second benchmark is a three-qubit Heisenberg chain with nearest-neighbor spin–spin coupling J and an external field B along Z. Its Hilbert space is eight-dimensional and is encoded directly in one OAM qudit. The experiment targets the lowest three energy levels of the eight-level spectrum, with an average convergence of 368 ± 57 iterations.

The ratio J/B organizes the physics: for J/B ≤ 1 the spectrum is dominated by Zeeman splitting with small interaction-induced shifts, while for J/B > 1 the interaction dominates, producing strongly correlated eigenstates and enhanced level repulsion. At J/B = 1, where the two contributions are comparable, the estimated energies remain in good agreement with theory. The authors attribute remaining deviations mainly to non-unity purity and state fidelity of the prepared OAM modes, primarily from imperfect optical alignment, and describe these as technical rather than fundamental limits that improved mode preparation and alignment could mitigate.

What the single-qudit encoding trades away

The compactness has a price the paper states explicitly. Encoding an n-qubit Hilbert space requires a qudit dimension of d = 2ⁿ, and the present SLM-based implementation performs projective measurements sequentially, displaying one hologram at a time. The architecture is therefore suited to near-term demonstrations of high-dimensional variational eigensolvers, but scalable photonic quantum computing would require extensions beyond single-particle encoding.

The authors point to two directions: combining OAM with other photonic degrees of freedom such as path, polarization, time-bin, and frequency to raise accessible dimensionality, and using correlated or entangled photon pairs so that two photons each in a d-dimensional OAM space reach a d²-dimensional Hilbert space. Multi-qudit or multi-particle architectures would open larger Hilbert spaces but introduce new challenges in photon generation, control, and detection.

Where the photonic qudit route fits

The paper's own comparison is between SSVQE and other excited-state approaches: it avoids ancilla-assisted overlap estimation, and OAM encoding offers a large, in principle unbounded photonic Hilbert space, giving a compact route to high-dimensional simulation with a limited number of photons. The demonstrated evidence covers a four-dimensional molecular model and an eight-dimensional spin chain, with three of eight levels targeted in the latter.

The practical priority that follows is subspace selectivity under a fixed photon budget. When only a few low-lying levels are needed, the parameter count tracks the number of targeted levels — 55 instead of 63 in the eight-dimensional case — and the shared unitary removes the ancilla overhead. When the target is a full high-dimensional spectrum or a multi-particle problem, the d = 2ⁿ dimension requirement and sequential SLM measurements become the binding constraints, and the paper's suggested extensions to additional degrees of freedom or entangled pairs are the relevant next step rather than the demonstrated single-qudit setup.

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