Photonic variational quantum eigensolver for NISQ-compatible quantum technology

TL;DR

The paper argues that VQE is a practical, NISQ-friendly approach to quantum-enabled problems that require shallow circuits. It presents a comprehensive theory of VQE—covering Hamiltonian mapping, ansatz design, qubitization, measurement strategies, and error mitigation—alongside a broad survey of photonic implementations. Experimental demonstrations span quantum chemistry (H2, HeH+, LiH), many-body physics (Schwinger model), and even integer factorization, employing both multi-qubit and high-dimensional qudit encodings. Key advances include Pauli measurement grouping, Bell-measurement schemes, quantum natural gradient optimizers, and zero-noise extrapolation, underscoring photonics as a versatile platform for scalable, noise-resilient VQE on near-term devices.

Abstract

Quantum computers have the potential to deliver speed-ups for solving certain important problems that are intractable for classical counterparts, making them a promising avenue for advancing modern computation. However, many quantum algorithms require deep quantum circuits, which are challenging to implement on current noisy devices. To address this limitation, variational quantum algorithms (VQAs) have been actively developed, enabling practical quantum computing in the noisy intermediate-scale quantum (NISQ) era. Among them, the variational quantum eigensolver (VQE) stands out as a leading approach for solving problems in quantum chemistry, many-body physics, and even integer factorization. The VQE algorithm can be implemented on various quantum hardware platforms, including photonic systems, quantum dots, trapped ions, neutral atoms, and superconducting circuits. In particular, photonic platforms offer several advantages: they operate at room temperature, exhibit low decoherence, and support multiple degrees of freedom, making them suitable for scalable, high-dimensional quantum computation. Here we present methodologies for realizing VQE on photonic systems, highlighting their potential for practical quantum computing. We first provide a theoretical overview of the VQE framework, focusing on the procedure for variationally estimating ground state energies. We then explore how photonic systems can implement these processes, showing that a wide variety of problems can be addressed using either multiple qubit states or a single qudit state.

Figures (16)

• Figure 1: Overview of variational quantum eigensolver (VQE). First, in the encoding process, an original Hamiltonian for representing a molecular system, which is composed of position and momentum of a nuclei and electrons, is rewritten by Hartree-Fock configuration in terms of occupied or vacant orbitals. Then, this configuration is transformed to qubitized form, in which both annihilation and creation operators are represented in terms of $\hat{X}-i\hat{Y}$ and $\hat{X}+i\hat{Y}$ using Pauli operators $\hat{X}$ and $\hat{Y}$, respectively. The transformed structure is subsequently encoded into the quantum processing unit (QPU) parameterized by $\boldsymbol{\theta}_{n-1}$, with $n$ denoting the iteration number. After measuring the ansatz state $|\psi(\boldsymbol{\theta}_{n-1})\rangle=\hat{U}(\boldsymbol{\theta}_{n-1})|\psi\rangle$, the measurement outcome is used to update the parameter in the classical processing unit (CPU). Here, $\boldsymbol{\theta}_{n-1}$ changes to $\boldsymbol{\theta}_n$ such that the expectation value of the energy is minimized, and then $\boldsymbol{\theta}_n$ is updated to the QPU.
• Figure 2: Simulated results of quantum chemistry problems. Red points are energy values estimated by the VQE, and black curved lines show the theoretical values of ground energies. (a) $\mathrm{H}_2$ model. (b) ${\rm HeH^+}$ model. (c) LiH model.
• Figure 3: Description of classical optimizers. (a) Gradient-descent method, in which parameters are updated according to the iteration rule $\boldsymbol{\theta}_n=\boldsymbol{\theta}_{n-1}-\eta\vec{\nabla}E(\boldsymbol{\theta}_{n-1})$. (b) Nelder-Mead method, in which a simplex $A_{n-1}B_{n-1}C_{n-1}D_{n-1}$ iteratively contracts or expands to $A_{n-1}B_{n-1}C_{n-1}D_{n}$ according to predefined iteration rules. (c) Particle swarm optimization, in which parameters are updated simultaneously, and the ground state energy is typically estimated at the location where the largest number of parameters converge. (d) Trust region method that includes COBYLA method, in which parameters iteratively move within each trusted area.
• Figure 4: Concept of the zero-noise extrapolation method (ZNE), as illustrated in Ref. borzenkova2024. The blue curve describes the expectation value $E(\varepsilon)$ as a function of the noise strength $\varepsilon$. The orange line indicates a linear extrapolation between two measurements taken at different noise levels, $\varepsilon_1$ and $\varepsilon_2$, to estimate the zero-noise value. The inset shows the corresponding measurement probability distributions. Here, the blue, orange, and cyan curves correspond to the distributions under noise, in the ideal (noise-free) case, and the ZNE estimator $E_{\rm{est}}$, respectively.
• Figure 5: (a) Experimental scheme for implementing the VQE to estimate the ground state energy of $\mathrm{HeH^+}$ using two path qubits on a photonic integrated circuit peruzzo2014skryabin2023. In the QPU, initial two-qubit states $|00\rangle$, generated via spontaneous parametric down-conversion (SPDC), are modulated by phase shifters controlled by a CPU. A beam-splitter network is used for entangling these qubits. Finally, the two-qubit state entangled by the beam-splitter network is measured by photodetectors that extract the which-way information, thereby estimating expectation value $\langle\psi|\hat{A}\otimes\hat{B}|\psi\rangle$ with $\hat{A},\hat{B}\in\{\hat{X},\hat{Y},\hat{Z}\}$. (b) Quantum circuit implemented by the photonic integrated circuit in (a). The blue boxes represent single-qubit gates controlled by phases $\varphi_j$.