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

arXiv:2610.11026 · quant-ph · Submitted 2026-10-08 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Demonstration of subspace-search variational quantum eigensolver using photonic orbital angular momentum qudits".

Mira: The gist:

Kai: First, who's behind it and why it matters.

Title and authors: Kai: So we're talking about this paper now called "Demonstration of subspace-search variational quantum eigensolver using photonic orbital angular momentum qudits." It’s about taking a method that usually only finds the lowest energy state and making it work for excited states in a high-dimensional setting.

Mira: Exactly, and they’re doing this using light—specifically, orbital angular momentum of single photons—to encode those high-dimensional states. That sounds like it could be really interesting because standard VQEs are stuck on the ground state, which is what we usually care about in things like molecular vibrations or chemistry.

Lev: From a hardware standpoint, if this works as described, it means we could be doing something beyond just finding the lowest energy level on NISQ devices. I’m interested in how robust these orthogonal states are when you try to run the actual optimization loop.

Kai: Well, what they're showing is that they can use this subspace-search variational quantum eigensolver protocol to estimate excited state energies for a four-dimensional HeH+ molecule and an eight-dimensional Heisenberg chain. They’re validating the SSVQE method on a photonic platform, showing it works in these high-dimensional Hilbert spaces.

Mira: That's a big deal because they are specifically avoiding those ancilla-based overlap measurements that usually kill excited state approaches, which makes it much more suitable for current noisy hardware. It lets you target only the energies you want to know about without having to map out the entire spectrum first.

Lev: And they give some numbers on convergence, which is always important when we talk about running this on real hardware. For that three-qubit Heisenberg chain in its eight-dimensional space, they saw an average convergence time of three hundred sixty-eight plus or minus fifty-seven iterations to estimate three energy levels simultaneously.

Kai: That’s a solid result, and it shows the experimental setup is actually functional enough for this kind of simulation. The core idea here is using the orbital angular momentum states directly, rather than needing tons of deep gate sequences to generate those inputs.

Mira: And they’re doing that by using holographic patterns displayed on a Spatial Light Modulator, which lets them prepare the set of mutually orthogonal input states without relying on those complex gate sequences you usually have to worry about. It’s a much more direct way to get those specific states you need for the SSVQE cost function.

Title and authors: Lev: I think that direct preparation is key for scalability, because if preparing each state requires too many sequential operations, it just becomes a slow classical bottleneck before the quantum part even starts.

Kai: The paper also talks about how they reduce the number of variational parameters needed by using a column-wise synthesis strategy inspired by multiport interferometer architectures, which helps keep the classical optimization manageable. They aren't parameterizing every possible transformation; they’re just synthesizing the specific columns that define their targeted subspace.

Mira: That’s smart because if you have to optimize thousands of parameters for a full unitary, you're in trouble on NISQ hardware anyway. By only synthesizing the necessary components, they make the classical optimization step much less demanding.

Lev: So we’re looking at a method that manages parameter count while still achieving this simultaneous estimation of excited states, which is really pushing the boundaries of what we can do right now with these types of variational approaches.

Kai: It’s about making sure that when you run this on a real chip, the classical optimization part isn't so huge that it drowns out the quantum advantage you're trying to find. This whole setup is designed to be resource-efficient for excited state eigensolving in these high-dimensional spaces.

Mira: And I want to stress that the encoding itself—using OAM qudits—is what gives them access to that large, potentially unbounded photonic Hilbert space, which is a huge advantage over trying to fit everything into just a few qubits. It’s about compact routing for high-dimensional simulations.

Lev: But we have to be careful with the trade-off they mention: encoding an n-qubit Hilbert space requires a qudit dimension of d equals two n, which means if you want to simulate something bigger, you need more photons or a higher dimension qudit.

Kai: Exactly. And that’s where the future work comes in—they point out that for truly scalable photonic quantum computing, we probably won't just stick to single-qudit encoding anymore. We’ll likely need to combine those OAM qudits with other degrees of freedom like polarization or path information.

Mira: It sounds like they’re suggesting a multi-mode approach is the next logical step for scaling up this type of excited state simulation, moving beyond just one degree of freedom in the photon.

Title and authors: Lev: If we think about running this on a real quantum computer, the fidelity issues they noted—like non-unity purity and state fidelity due to imperfect optical alignment—that’s something we have to seriously address when building the actual experimental apparatus.

Kai: So, to wrap up this part of "Demonstration of subspace-search variational quantum eigensolver using photonic orbital angular momentum qudits," they show that this platform is a scalable way to do excited state eigensolving efficiently on NISQ hardware by avoiding ancilla measurements and using OAM states directly.

Mira: They proved the protocol works for both the four-dimensional HeH+ molecule and the eight-dimensional Heisenberg chain, showing good agreement with theoretical values across different physical conditions like interatomic distance or spin ratios.

Lev: It’s a solid demonstration of the SSVQE protocol in action, even if we have to work on reducing those experimental deviations from perfect fidelity.

Kai: We see this as a pathway for using light sources to tackle excited state problems where current VQEs fall short. It’s a concrete piece of experimental realization that moves the concept closer to practical use.

Mira: The implication is that we can start designing simulations for systems with more complexity, like larger molecules or more intricate condensed matter models, using these photonic architectures.

Lev: For us in error correction research, this is interesting because it shows a concrete way to structure the problem so that the required measurements are simpler than what we usually need for excited states.

Kai: So if you’re driving and thinking about what this means for quantum simulation, it’s a proof of concept showing how photonic OAM qudits can be resource-efficient tools for finding excited state energies in high-dimensional systems.

Mira: It really puts the focus on encoding rather than just brute force computation when dealing with complex Hamiltonians.

Lev: The numbers they gave, like three hundred sixty-eight plus or minus fifty-seven iterations, give us a baseline for how much classical optimization time we might expect before we hit a wall.

Kai: Before we move on to the next paper, this one was all about proving that subspace-search variational quantum eigensolver using photonic orbital angular momentum qudits is actually feasible to build and test experimentally.

Mira: It’s a good step toward making these types of simulations more accessible than they are right now.

Lev: We'll keep an eye on how the error correction researchers see this platform fitting into larger quantum error correcting architectures down the line.

The paper's summary: Kai: So, this paper is showing how you can use light—specifically orbital angular momentum of single photons—to tackle excited state problems in high-dimensional systems using a technique called subspace search VQE.

Mira: Yeah, they’re using these qudits to encode states, and they're doing it on a photonic platform, which means we're talking about light instead of just superconducting circuits for this simulation.

Kai: Exactly. The main point is that they’ve built a protocol that lets you estimate excited state energies without needing those extra overlap measurements that usually bog down these kinds of methods.

Mira: That’s the big practical thing right there, because it makes the method much better suited for the kind of noisy hardware we have today, like NISQ devices. It lets you zero in on only the energy levels you care about instead of trying to map out every single possible state first.

Kai: They tested this on two things—a four-dimensional molecule and an eight-dimensional chain—and they showed that their results actually converged toward the correct theoretical values as they ran more iterations.

Mira: The numbers are interesting because for the chain, they needed about three hundred sixty-eight iterations on average to find three different energy levels simultaneously, which gives us a real benchmark for how long these classical optimization steps might take.

Kai: And they also showed that by being smart about how they build the unitary operator—using a column-wise synthesis approach—they only need to optimize a much smaller set of parameters than if you had to parameterize the whole thing from scratch.

Mira: That makes sense, because reducing those variational parameters helps keep the classical optimization step manageable on hardware with limited memory and time. It’s about being more resource-efficient with your quantum resources.

Kai: But there are some caveats they mention, like how preparing those orthogonal OAM states needs specific holographic patterns displayed on a spatial light modulator to get them right in the first place.

Mira: Right, so the success depends heavily on getting that initial state preparation perfect using the SLM setup and making sure those optical alignments stay consistent throughout the measurement process.

Kai: So what this means for us is that we can start thinking about using photonic systems not just for ground states, but for simulating more complex chemical or magnetic systems where excited states are just as important.

Mira: It opens up a new avenue where you might be able to simulate things that are too big or too complex to handle with standard qubit architectures alone, provided you can scale up the light encoding properly.

Kai: We need to keep an eye on how they plan to extend this beyond just one type of photon encoding, because they’re suggesting we combine OAM with other features like polarization or time-bin for even bigger Hilbert spaces.

The paper's improvements: Tom: So, this paper isn't just showing how to run the VQE protocol; they’re also suggesting ways to make it even more efficient for real use on hardware.

Kai: Right, they talk about simplifying the construction of those high-dimensional unitary operators by using a column-wise synthesis strategy instead of trying to parameterize every possible rotation angle in one go.

Mira: That’s smart because if you have to optimize thousands of parameters for a full transformation, it becomes impossible on current computers; synthesizing just the specific columns needed keeps the classical optimization much less demanding.

Kai: It also mentioned adding a simple diagonal phase gate to handle some of those relative phase degrees of freedom while still cutting down on the number of parameters you have to tune.

Mira: That’s a necessary trade-off, because you lose some generality, but it keeps the parameter count low enough for NISQ devices to actually attempt the optimization without crashing. It’s about finding a sweet spot between accuracy and feasibility.

Kai: They also highlighted that they can target only a subset of energy levels by choosing specific input states and using a weighted sum cost function, which means you don't have to calculate the whole spectrum just to find one specific excited state.

Mira: That’s a practical improvement because it lets you focus your computational power where it matters most, making the simulation much more targeted when dealing with complex molecules or materials. It moves us away from brute-force spectral mapping towards selective discovery.

Kai: So, instead of trying to find everything at once, we can now design the search to specifically look for certain energy ranges that we already suspect might hold an excited state.

Mira: Exactly, it shifts the focus from a general solver to a targeted search tool that is much more efficient for finding specific states in large Hilbert spaces. It’s about making the classical optimization phase smarter, not just bigger.

Kai: The authors also pointed out that if we want to scale this up further, we have to think beyond just one type of photon and start combining OAM with other degrees of freedom like path or polarization.

Mira: That makes sense because a single-mode encoding has its limits, so the next step in making these simulations powerful will likely involve using multiple properties of the photon to encode even richer, higher-dimensional states.

Kai: So the immediate improvement is about making the existing method more efficient with fewer parameters, and the long-term direction is about building bigger light sources to handle even larger problems.

Conclusion: Kai: So we’re wrapping up on "Demonstration of subspace-search variational quantum eigensolver using photonic orbital angular momentum qudits." This work basically shows how you can use light to find excited state energies in high-dimensional systems without needing those complicated overlap measurements.

Mira: Yeah, it proves that the SSVQE protocol works on a photonic platform, and it uses orbital angular momentum states as the way to encode those high-dimensional inputs.

Kai: It’s really cool because they validated it on a four-dimensional molecule and an eight-dimensional chain, showing that the energy estimates actually converged toward the correct theoretical values as they ran more iterations.

Mira: And that convergence time for the chain, around three hundred sixty-eight iterations on average to find three levels, gives us a concrete number we can use to judge how long this classical optimization part will take on real hardware.

Lev: From my side, those iteration counts are actually quite impressive for a variational approach; it shows the protocol is feasible enough to even attempt on current noisy systems, though we’ll need to work hard on that state fidelity issue they mentioned.

Kai: Right, so the takeaway here is that this approach bypasses the ancilla-assisted overlap estimation which is usually a huge hurdle for excited states in these types of quantum algorithms.

Mira: Exactly; it lets you target only the energy levels you care about without having to map out the entire spectrum first, which is a big deal for efficiency.

Lev: For error correction, this structure would be helpful because if we could implement this on a fault-tolerant machine, the measurement requirements seem simpler than what we usually face when trying to characterize excited states.

Kai: So we’ve seen how they use holographic patterns on spatial light modulators to prepare those orthogonal OAM states directly instead of relying on lengthy gate sequences.

Mira: That direct preparation is a key part of their method, it avoids the deep gate sequences that usually complicate things and makes the setup much more scalable in terms of input state generation.

Lev: It’s still a big experimental hurdle, though; the paper flagged that some of those experimental deviations came down to imperfect optical alignment and state purity issues with the light modes themselves.

Kai: True, so they did get close to the theoretical values across different physical conditions like interatomic distance or spin ratios for the molecules they tested.

Mira: It’s a solid demonstration of how this encoding can provide access to a large photonic Hilbert space that would be hard to reach otherwise with just a few qubits.

Lev: We need to see how error correction researchers can integrate this kind of state preparation into larger architectures, because that’s where we see the real potential for using these light-based methods in quantum computing.

Kai: It’s a good step toward making these kinds of simulations more accessible by showing a concrete experimental realization of an excited state method on a photonic chip.

Mira: Indeed, so this paper proves that subspace-search variational quantum eigensolver using photonic orbital angular momentum qudits is actually a viable tool for high-dimensional excited state eigensolving.

Lev: We’ll need to keep watching how they extend this beyond single-photon encoding because the future of scalable quantum computing will probably require combining OAM with other degrees of freedom like polarization or time-bin information.

Kang-Min Hu, Myung-Hyun Sohn, Yong-Su Kim, Kwang Jo Lee, Hyun Woo Kim, Hyang-Tag Lim

Center for Quantum Technology, Korea Institute of Science and Technology (KIST) · Korea University of Science and Technology · Department of Applied Physics, Kyung Hee University · Department of Chemistry, Gwangju Institute of Science and Technology

quant-ph

Submitted: 2026-10-08

Updated: 2026-10-08

Comments: 8 pages, 4 figures, Supplementary Note

Journal ref: npj Quantum Information (2026)

DOI: 10.1038/s41534-026-01380-2

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 85/100

The gist: The gist: This work demonstrates an experimental implementation of subspace-search variational quantum eigensolver (SSVQE) on a photonic platform by encoding orthogonal qudit states in the orbital

Key concepts

Subspace-Search VQE (SSVQE)
This is an extension of the standard VQE method designed to find multiple excited states simultaneously. It works by using a weighted cost function that targets specific energy levels within a defined subspace by exploiting orthogonal input states and a shared variational unitary operator.
Orbital Angular Momentum (OAM) Qudits
OAM is used to encode quantum information into single photons. The researchers use holographic patterns on spatial light modulators to prepare a set of mutually orthogonal OAM states, which act as the high-dimensional input 'qudits' for the quantum computation.
Variational Unitary Construction
To keep the optimization manageable, instead of defining a full unitary matrix with many parameters, this method synthesizes only the necessary orthonormal columns. These columns are built from simple two-mode rotations to reduce the classical optimization problem while maintaining high dimensionality.

Terminology

Summary

The gist: This work demonstrates an experimental implementation of subspace-search variational quantum eigensolver (SSVQE) on a photonic platform by encoding orthogonal qudit states in the orbital angular momentum (OAM) of single photons, establishing photonic OAM qudits as a scalable and resource-efficient architecture for excited state eigensolving in high-dimensional Hilbert spaces.

SSVQE Protocol

The SSVQE protocol extends the standard VQE framework to simultaneously target excited states by exploiting orthogonal input states and a shared variational unitary For each input state, the corresponding energy estimate is calculated as ⟨Ej (⃗α)⟩ = ⟨ϕj U†(⃗α)HU(⃗α)ϕj⟩ The SSVQE cost function is defined as a weighted sum L(⃗α) = ∑wj Ej (⃗α), where w0 > w1 > · · · > wN−1 Minimizing L(⃗α) drives ⟨Ej (⃗α)⟩ toward the jth eigenenergy Ej within the target subspace.

Photonic Implementation and Scalability

The experimental realization exploits high-dimensional qudit encoding in the orbital angular momentum (OAM) of single photons Rather than generating required orthogonal inputs through deep gate sequences, the researchers directly prepare a set of mutually orthogonal OAM qudit states using holographic patterns on a spatial light modulator (SLM), which is advantageous for high-dimensional scalability. The procedure involves preparing N mutually orthogonal input states ψj (⃗αn)⟩ N−1 j=0 by displaying the corresponding holograms on SLM Each state is then measured on the photonic QPU using a set of Pauli-string observables implemented with a second SLM, from which the Hamiltonian expectation values are estimated.

Experimental Results and Validation

The SSVQE protocol was validated by estimating excited state energies for a four-dimensional HeH+ molecule and an eight-dimensional Heisenberg chain For the four-dimensional HeH+ model, the estimated energies converged toward theoretical values with increasing iteration number n. Similarly, for the three-qubit Heisenberg chain in its eight-dimensional Hilbert space, they simultaneously estimated three energy levels with an average convergence of 368 ± 57 iterations. The results showed good agreement with theoretical values across the full range of interatomic distance R for HeH+ and for the ratio J/B in the Heisenberg chain.

Advantages and Future Directions

A key advantage of SSVQE is that it eliminates the need for ancilla-assisted overlap estimation, which is commonly required in deflation-based or penalty-based methods. The OAM-based qudit encoding provides access to a large (in principle unbounded) photonic Hilbert space, offering a compact route to high-dimensional quantum simulations with a limited number of photons. However, the single-qudit encoding has an intrinsic tradeoff where encoding an n-qubit Hilbert space requires a qudit dimension of d = 2n. Future scalable photonic quantum computing will likely require extensions beyond a single-particle encoding, such as combining OAM qudits with additional photonic degrees of freedom like path, polarization, time-bin, and frequency.

Unitary Construction

To reduce the number of parameters required for constructing high-dimensional unitary operators, the authors adopted a column-wise synthesis strategy inspired by multiport interferometer architectures Instead of parameterizing a full SU(d) transformation with d squared − 1 parameters, they synthesize only the set of N orthonormal columns that define the targeted subspace. Each column is constructed from a product of two-mode rotations T(s)m (θs, ϕs), which allows for substantial reuse of rotation angles across columns while reducing the dimensionality of the classical optimization. To retain generality under this reduction, a diagonal phase gate D⃗ω = diag1, eiω1,..., eiωd−1 is prepended to account for relative phase degrees of freedom.

Experimental Setup Details

The experimental setup utilizes a continuous-wave 403 nm laser pumping a PPKTP nonlinear crystal to generate heralded single photons via type-II spontaneous parametric down-conversion (SPDC) A horizontally polarized single-photon is directed onto SLM1 for state preparation, where N mutually orthogonal input states are prepared sequentially by displaying the corresponding holograms encoded in an OAM qudit basis. For projective measurements, SLM2 displays holograms that implement the measurement bases for the Pauli-string observables The measured outcomes are processed on the CPU to reconstruct Hamiltonian expectation values and compute the SSVQE cost function L, after which an optimizer updates the variational parameters. Detailed information on their experimental setup is provided in Supplementary Fig. 1. The use of both azimuthal index l and radial index p of Laguerre–Gaussian (LG) modes was employed to ensure the orthogonality of the basis while equalizing their coupling efficiencies.

Parameterization for Subspace Selection

For the four-dimensional case, they use 15 variational parameters corresponding to d squared − 1 for d = 4, and these are grouped into vector components such as⃗θ, ϕ⃗, and ⃗ω to conveniently scale the parametrization when targeting a reduced subspace. This parameterization scheme allows for the estimation of only the ground and first excited states by preparing N = 2 orthogonal ansatz states in a four-level system. The weight coefficients used for the energy estimates are provided in Supplementary Table III. The optimization terminates after an average of 81.6±10.2 iterations for the HeH+ model.

Heisenberg Chain Application

To demonstrate scalability, the SSVQE was applied to a three-qubit Heisenberg chain, whose Hilbert space is eight-dimensional and can be encoded directly in an OAM qudit The Hamiltonian for this system is given by H = J X ⟨i,j⟩ (XiXj + YiYj + ZiZj) + B X 3 i=1 Zi, where J is the nearest-neighbor spin–spin interaction strength and B is the external magnetic field. The researchers simultaneously estimated three energy levels using this eight-dimensional OAM qudit, with an average convergence of 368 ± 57 iterations. The remaining experimental deviations were mainly attributed to non-unity purity and state fidelity of the prepared OAM modes, primarily due to imperfect optical alignment.

Data Availability

The datasets generated and/or analyzed during the current study are not publicly available in accordance with the data management policies of the Korea Institute of Science and Technology (KIST), but are available from the corresponding author on reasonable request. The code used to generate figures within this paper and other findings of this study is available from the corresponding author upon request. The authors declared no competing interests in their work.

References

The paper references numerous works covering molecular Hamiltonians, VQE methods, and photonic quantum computing platforms Key references include those on fluorescence [3], equation of motion approaches [25], and the foundational work on variational quantum eigensolvers [8, 9]. The paper also cites specific literature regarding OAM qudit encoding in VQE [15] and experimental realization using single-photon platforms. The references span theoretical foundations like Jordan-Wigner transformation [38] and experimental techniques such as spontaneous parametric down-conversion [39].

Improvements for AI systems

  1. A photonic subspace-search VQE (SSVQE) system can be developed to estimate excited state energies for high-dimensional molecular Hamiltonians like a four-dimensional HeH+ molecule or an eight-dimensional Heisenberg chain. This allows for scalable excited state eigensolving in a high-dimensional Hilbert space using OAM qudits as the encoding mechanism.

  2. The SSVQE protocol can be implemented to avoid ancilla-based overlap measurements, which is a key limitation of many excited state approaches, making it more suitable for NISQ hardware. This enables subspace-selective eigensolving: only the energy levels of interest need to be targeted, without determining the full spectrum.

  3. The system can utilize holographic patterns on a Spatial Light Modulator (SLM) to directly prepare mutually orthogonal OAM qudit states, bypassing the need for deep gate sequences for input state generation. This is achieved by generating holograms from the columns of the variational unitary, synthesizing relevant output states directly.

  4. The experimental setup can be adapted to estimate multiple excited states simultaneously by using a cost function defined as a weighted sum of energy estimates: "L(⃗α) = N Σ wj Ej (⃗α)⟩, w0 > w1 > · · · > wN−1. This allows for the simultaneous estimation of ground and excited state energies" within a chosen subspace.

  5. The system can scale the number of variational parameters by targeting specific energy levels, allowing for a more resource-efficient estimation when estimating only a subset of states, as demonstrated by reducing parameters to 55 for three targeted levels in an eight-dimensional Heisenberg chain.

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 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.

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