Researchers at the Korea Institute of Science and Technology (KIST) have demonstrated a “subspace-search variational quantum eigensolver”, a method previously limited to finding ground states, now capable of calculating excited state energies. The team encoded qudits, quantum bits with more than two states, in the orbital angular momentum of single photons to realize the SSVQE on a photonic platform. They estimated excited state energies for a four-dimensional molecule and an eight-dimensional Heisenberg chain, demonstrating scalable excited state eigensolving in a high-dimensional Hilbert space and establishing photonic qudits as a resource-efficient pathway to quantum simulation.
Photonic Orbital Angular Momentum Qudits for SSVQE
By directly preparing mutually orthogonal qudit states using holographic patterns on a spatial light modulator, the researchers bypassed the need for complex gate sequences typically required for generating such inputs. This approach offers a pathway to scalability, a critical hurdle in quantum computation. The experimental setup uses the unique properties of photons to encode and manipulate quantum information, specifically exploiting the orbital angular momentum of each photon to define the qudit states.
Unlike many quantum platforms reliant on superconducting circuits or trapped ions, this photonic approach allows for the direct preparation of high-dimensional qudits, simplifying the process of creating the necessary orthogonal basis states for the subspace-search VQE (SSVQE) algorithm. The team then optimized a shared variational circuit, a series of quantum operations designed to find the lowest energy states of the simulated system, to estimate the excited state energies.
This capability represents a significant expansion of VQE methodology, which has historically been limited to calculating ground state energies. This achievement is particularly noteworthy given the increasing complexity of quantum simulations as the dimensionality of the system grows. The ability to accurately model higher-dimensional systems is essential for understanding complex materials and chemical reactions.
The authors write, “its experimental realization has remained elusive since SSVQE requires high-fidelity preparation of multiple mutually orthogonal input states and the reproducible application of a common parametrized unitary across them.” The photonic platform, with its ability to directly prepare orthogonal qudit states, addresses this key challenge.
Subspace-Search VQE for Excited State Energies
The ability to accurately calculate excited state energies has been experimentally demonstrated for systems up to eight dimensions using a photonic quantum computing platform, a feat previously limited by the demands of subspace-search variational quantum eigensolver (SSVQE) methodology. The team’s work extends the capabilities of variational quantum eigensolvers (VQEs), which have historically focused on finding the lowest energy, or ground, state of a quantum system.
By applying a shared variational circuit to multiple input states, and then processing the measurement outcomes, the researchers optimized the SSVQE cost function until convergence, allowing for the estimation of excited state energies. The iterative optimization process continued until the change in variational parameters fell below a defined tolerance, ensuring the accuracy of the estimated energies. The results establish photonic qudits as a resource-efficient pathway to excited state quantum simulation.
High-Dimensional Hilbert Space Implementation with Photons
The team’s approach uses the high-dimensional encoding capacity of single photons, offering a pathway to more compact quantum simulations. The core of this advance lies in the method of input state preparation, differing from approaches that rely on complex gate sequences.
Instead of building the necessary orthogonal states step-by-step, the team used holographic patterns displayed on the SLM to directly create a set of mutually orthogonal OAM qudit states, a technique advantageous for scaling to higher dimensions. The efficiency of this method extends to parameter optimization. The number of variational parameters can be reduced, leading to a more resource-efficient estimation process.
While the current architecture relies on single-qudit encoding, where encoding a n-qubit Hilbert space requires a qudit dimension of d = , the researchers acknowledge the trade-offs inherent in this approach. The present SLM-based implementation performs projective measurements sequentially, displaying one hologram at a time. Nevertheless, they envision extensions beyond single-particle encoding as important for truly scalable photonic quantum computing.
One promising route involves combining OAM qudits with additional photonic degrees of freedom, including path, polarization, time-bin and frequency, to further increase the accessible dimensionality. Another potential advancement lies in using correlated or entangled photon pairs generated via spontaneous parametric down-conversion, which could provide access to a -dimensional Hilbert space with two photons each encoded in a d-dimensional OAM space.
SSVQE Protocol and Variational Parameter Optimization
The number of variational parameters used in this photonic quantum simulation scales with the dimensionality of the system. This approach relies on iterative preparation and measurement of states on a photonic quantum processing unit, followed by classical processing to update variational parameters until convergence is achieved. The optimization process itself involves minimizing a cost function, tracking changes in the angle parameters with each iteration, and halting when the parameter updates fall below a defined tolerance.
This method was first benchmarked on a four-dimensional model Hamiltonian, validating the performance of the photonic implementation before scaling to more complex systems. To further demonstrate scalability, the researchers applied their subspace-search variational quantum eigensolver (SSVQE) to a three-qubit Heisenberg chain, a system with an eight-dimensional Hilbert space directly encodable within an orbital angular momentum qudit.
This represents a significant expansion beyond the four-dimensional molecular benchmark, demonstrating the potential of photonic qudits to handle increasingly complex quantum simulations. The current implementation uses a spatial light modulator (SLM) for projective measurements, displaying one hologram at a time to define the measurement basis.
Four- and Eight-Dimensional System Validation
This successful implementation extends the capabilities of SSVQE beyond ground state calculations, previously a significant limitation of noisy intermediate-scale quantum (NISQ) hardware. The team’s approach achieved convergence of energy levels toward theoretical values after an average of 81.6 ± 10.2 iterations for the four-dimensional molecular model, as shown in the lower panel of figure c, and 368 ± 57 iterations for the eight-dimensional Heisenberg chain.
This iterative optimization process, targeting specific energy levels, allows for a reduction in the number of variational parameters needed, enhancing resource efficiency. For instance, the eight-dimensional Heisenberg chain, typically requiring 63 parameters for a general unitary transformation, was solved using only 55 parameters by focusing on three targeted levels, a detail further elaborated in Supplementary Note III.
A key advantage of this method lies in its ability to perform subspace-selective eigensolving, enabling the estimation of only the desired energy levels without calculating the entire spectrum. This is particularly useful in complex systems where only specific excited states are of interest, streamlining the computational process. The experimental platform relies on hologram-based state preparation and measurement using spatial light modulators (SLMs), allowing for the creation of high-dimensional orthogonal states without the need for complex gate sequences.
The results, aligning closely with theoretical predictions, establish a pathway toward practical excited state eigensolving, an important step for interpreting phenomena like fluorescence and photochemical processes. As the researchers note, their scheme allows for simultaneous estimation of multiple excited states while scaling the number of variational parameters with the number of targeted energy levels, providing an experimentally practical route to resource-efficient excited state eigensolving.
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