Towards reconstructing quantum structured light on a quantum computer

TL;DR

This work tackles quantum state tomography in high-dimensional structured light by recasting the reconstruction as an Ising energy minimization solved with a variational quantum eigensolver on near-term hardware. The method maps a least-squares tomography cost to an Ising Hamiltonian using a one-to-one encoding of density-matrix components, enabling a hybrid quantum–classical optimization workflow. Demonstrations on SPDC-generated OAM entangled photons show high fidelities both in simulation ($F\approx 0.995$–$0.999$) and on IBM devices ($F\approx 0.995$–$0.996$), with shallow circuits delivering the best performance under realistic noise. This establishes a flexible platform for scalable encodings and noise-mitigated quantum tomography in high-dimensional structured light, potentially enabling efficient characterization beyond classical bottlenecks.

Abstract

We introduce a variational quantum computing approach for reconstructing quantum states from measurement data. By mapping the reconstruction cost function onto an Ising model, the problem can be solved using a variational eigensolver on present-day quantum hardware. As a proof of concept, we demonstrate the method on quantum structured light, in particular, entangled photons carrying orbital angular momentum and show that the reconstruction procedure can yield reliable performance even on noisy devices. Our results highlight the potential of variational algorithms for efficient quantum state tomography, particularly for high-dimensional structured light, where classical approaches can face bottlenecks.

Figures (7)

• Figure 1: (a) We highlight the conceptual layout of the tomography problem. Similar to reconstructing an image of a 3D object from its projections, in QST we aim to reconstruct the density matrix $\rho$ (our version of the 3D object) from a set of observables. (b) The underlying state we wish to uncover is that of a two-photon state described within the OAM degree of freedom. The two photons here are entangled. Assuming that each photon is defined using a two two-level system of OAM states, each photon is measured using a projection $M^{\pm}_{u,v}$, collapsing each photon on the Bloch sphere. (c) Representation of a two-level system of OAM states on the Bloch sphere, which is equivalent to that for standard qubit states. The states selected on the sphere (and shown in the inset) constitute an overcomplete set of measurements obtained from the eigenvalues of the Pauli operators. (d) Using the joint measurement outcomes $\langle P_k \rangle \equiv \langle M^{\pm}_u \otimes M^{\pm}_{v} \rangle$ for photons A and B, an optimisation routine is implemented and executed on the quantum computer to find the underlying density matrix.
• Figure 2: VQE Algorithm Workflow: (a) The input variables are transformed into spin variables, converting the least-squares problem into an Ising spin model. (b) The cost function is expressed as a Hamiltonian in terms of spin observables. (c) A quantum computer evaluates the Hamiltonian's expectation value based on ansatz circuit parameters, which are iteratively optimized by a classical computer. (d) Once the expectation value is minimized, the optimized circuit parameters are used to sample the bitstring distribution through measurements in the computational basis. The density matrix corresponding to bitstrings with the highest counts is used to obtain the density matrix representation of the best solution.
• Figure 3: (a) Setup for extracting measurement (m) data using non-degenerate SPDC. A continuous-wave pump laser passes through a Nonlinear Crystal (NC) to generate signal and idler photons at different wavelengths. A Dichroic Mirror (DM) separates the two wavelengths, which are then imaged to Spatial Light Modulators (SLMs). The photons are subsequently imaged to the interface of Single-Mode Fibers (SMF), thereafter coupled and detected by avalanche photodiodes (Det A and Det B) for coincidence counting. (b) Optimisation using the quantum computer. Quantum circuit ansatz prepared using the MATLAB Support Package for Quantum Computing. Circuits of variable depth were generated with either single-qubit $R_Y(\cdot)$ rotations or $R_Z(\cdot)R_Y(\cdot)R_Z(\cdot)$ blocks with adaptable rotation angles. The circuits were executed on IBM Quantum hardware via Qiskit Runtime primitives, using the Estimator primitive for energy minimization and the Sampler primitive for final state extraction.
• Figure 4: Implementation of the VQE-based state-reconstruction scheme. Energy expectation values evaluated at each iteration for antis-correlated (a) and correlated (b) OAM-entangled target states, shown for the different tested circuit architectures.
• Figure 5: Sampled bitstring statistics and reconstructed density matrices using VQE. Bar plots show sampled bitstring frequencies for each measurement input (with inset panels displaying the corresponding measurement configurations). (a)(i–ii) Bitstring distributions from simulated and experimental measurements for the antisymmetric state. (b)(i–ii) Bitstring distributions for the symmetric state, with the bitstring having the highest probability mapped onto the corresponding density matrix. In each panel, the reconstructed density matrix is obtained from the weighted sums of the reshaped bitstrings. The corresponding solutions obtained from the quadratic optimisation using the conventional MLE approach are shown for panel (ii),while those in panel (i) match the ideal case.