Towards reconstructing quantum structured light on a quantum computer

1. Problem Statement

Quantum State Tomography (QST) is essential for characterizing quantum systems, ensuring the reliability of quantum communication, computing, and imaging. However, reconstructing high-dimensional quantum states (such as structured light fields carrying Orbital Angular Momentum, or OAM) presents significant challenges:

• Scalability: The number of measurements and optimization parameters required for QST scales exponentially with the dimension of the Hilbert space ($O(d^n)$ for $n$ particles).
• Classical Bottlenecks: Traditional reconstruction methods (e.g., Maximum Likelihood Estimation, Least Squares) become computationally expensive for high-dimensional systems.
• Hardware Limitations: Existing quantum algorithms for solving linear systems (like HHL) require deep circuits and error correction, making them impractical for current Noisy Intermediate-Scale Quantum (NISQ) devices.

The authors aim to develop a Variational Quantum Algorithm (VQA) approach to reconstruct quantum states using experimental data as input, specifically targeting high-dimensional structured light on current noisy hardware.

2. Methodology

The authors propose a hybrid classical-quantum framework that reformulates the QST problem as a combinatorial optimization task solvable by a Variational Quantum Eigensolver (VQE).

A. Theoretical Framework: From QST to Ising Model

1. Least Squares Formulation: The reconstruction problem is initially defined as minimizing a quadratic cost function based on the difference between measured projection outcomes ($m$) and the theoretical predictions derived from a density matrix ($\rho$).
   $$f(\rho; m) = \|m - T\rho\|_2^2$$
   where $T$ is the measurement matrix.
2. Mapping to Ising Hamiltonian:
   • The density matrix elements are discretized and mapped onto binary spin variables ($s_j \in \{0, 1\}$).
   • An affine transformation maps these binary variables to the expectation values of Pauli-Z operators ($\langle Z_j \rangle$) on a quantum computer: $p_j \leftrightarrow \frac{1 - \langle Z_j \rangle}{2}$.
   • The quadratic cost function is algebraically transformed into an Ising Hamiltonian ($H$) composed of Pauli-Z strings:
     $$H = \sum_{j<k} J_{jk} Z_j Z_k + \sum_j h_j Z_j + \text{offset}$$
   • The coefficients $J_{jk}$ and $h_j$ are derived directly from the experimental measurement data and the measurement matrix $T$.

B. Variational Quantum Eigensolver (VQE) Implementation

• Ansatz: The quantum circuit uses parameterized single-qubit rotation gates without entangling gates (since the Ising Hamiltonian is diagonal in the computational basis). Three architectures were tested:
  1. Single-layer $R_y$ rotations.
  2. Depth-three $R_y$ rotations.
  3. Depth-three universal single-qubit rotations $\{R_z, R_y, R_z\}$.
• Workflow:
  1. Input: Experimental coincidence counts (classical data) are converted into the Ising Hamiltonian coefficients.
  2. Quantum Execution: The quantum processor prepares a state $|\psi(\theta)\rangle$ using the ansatz and measures the expectation value $\langle H \rangle$.
  3. Classical Optimization: A classical optimizer (Surrogate Optimizer in MATLAB) updates the parameters $\theta$ to minimize the energy (cost function).
  4. Reconstruction: Upon convergence, the optimized circuit is sampled to generate bitstrings. The bitstring with the highest probability (lowest energy) is reshaped to reconstruct the density matrix elements.

C. Experimental Setup

• Source: Entangled photon pairs were generated via Spontaneous Parametric Down-Conversion (SPDC) using a PPKTP crystal.
• States: Two types of OAM-entangled states were tested:
  • Anti-correlated: $|\ell\rangle_A |-\ell\rangle_B + |-\ell\rangle_A |\ell\rangle_B$
  • Correlated: $|\ell\rangle_A |\ell\rangle_B + |-\ell\rangle_A |-\ell\rangle_B$
• Measurement: Spatial Light Modulators (SLMs) and Single-Mode Fibers (SMF) were used to perform projective measurements in the OAM basis (Pauli X, Y, Z equivalents).
• Hardware: The algorithm was executed on IBM Quantum processors (ibmq_mumbai and ibmq_nazca) using the Qiskit Runtime primitives (Estimator and Sampler) with Level-1 error mitigation.

3. Key Contributions

1. Algebraic Mapping: The paper provides a rigorous algebraic derivation mapping the least-squares QST cost function directly to an Ising Hamiltonian, enabling the use of VQE for state reconstruction.
2. NISQ Compatibility: The method utilizes shallow circuits (single-qubit rotations) that are robust against noise, making it suitable for current hardware without requiring complex entangling gates or deep error correction.
3. Proof of Concept on Structured Light: It demonstrates the first successful reconstruction of OAM-entangled photon states using a hybrid quantum-classical approach on real quantum hardware.
4. Discretized Reconstruction: The approach effectively binarizes the density matrix, identifying the dominant logical components of the state. While it currently reconstructs real-valued components, the framework is generalizable to complex values.

4. Results

• Convergence: The VQE algorithm successfully minimized the energy function for both simulated and experimental data.
  • Simulated Data: Achieved fidelities ($F$) of 0.995 to 0.999 against the ideal target states.
  • Experimental Data: Achieved fidelities of 0.967 to 0.996 when compared to density matrices reconstructed via classical Maximum Likelihood Estimation (MLE).
• Architecture Performance:
  • The shallowest architecture (single $R_y$ layer) converged fastest for experimental data, suggesting that deeper circuits introduce more noise-induced instability on NISQ devices.
  • The $\{R_z, R_y, R_z\}$ architecture provided high-fidelity results in simulations but required more iterations to converge on noisy hardware.
• Reconstruction Quality: The reconstructed density matrices (visualized as 4x4 matrices for the two-qubit system) closely matched the theoretical and MLE-reconstructed states, correctly identifying the entangled nature of the photon pairs.

5. Significance and Future Outlook

• Complementary Approach: This work establishes VQAs as a viable complementary approach to classical QST, particularly for high-dimensional systems where classical optimization becomes a bottleneck.
• Scalability Potential: While the current demonstration is limited to a 2-qubit (4-dimensional) system and does not yet show a quantum advantage over classical methods for this specific size, the methodology provides a scalable platform. Future work can extend this to higher dimensions (e.g., qutrits or higher OAM modes) where the exponential scaling of classical methods becomes prohibitive.
• Noise Resilience: The success on noisy hardware demonstrates that variational approaches can extract useful quantum information even without full error correction.
• Limitations: The current implementation is restricted to real-valued density matrix elements. Future extensions would require decomposing real and imaginary parts separately or expanding the measurement set to handle complex encodings, which would increase the number of qubits required.

In conclusion, the paper successfully bridges the gap between quantum state tomography and variational quantum algorithms, offering a promising pathway for characterizing complex quantum states in the era of noisy quantum hardware.