Distributed photonic variational quantum eigensolver with parameterized weak measurements

npj | quantum information Article Published in partnership with The University of New South Wales https://doi.org/10.1038/s41534-025-01163-1 Distributed photonic variational quantum eigensolver with parameterized weak measurements Check for updates 1 1,2 1,2 1,2 1,3 1,4 1234567890():,; 1234567890():,; Donghwa Lee , Bohdan Bilash , Jaehak Lee , Hyang-Tag Lim , Yosep Kim , Seung-Woo Lee Yong-Su Kim1,2 & We demonstrate a two-qubit variational quantum eigensolver (VQE) implementation using two spatially separated single-photon processors connected via a 3 km optical fiber network. Our approach leverages local operations on pre-shared entanglement to evaluate two-qubit Hamiltonians. By incorporating parameterized weak measurement operations within the local operations framework, we enable access to the complete Hilbert space across distributed quantum processors – a capability typically requiring complex non-local operations. Our experimental results show accurate ground state energy estimation for Hamiltonians including H-He+ cation and the Schwinger model, validating both the necessity of weak measurements and high-quality entanglement in distributed quantum computing. This work establishes a promising direction for resource-efficient, scalable quantum network architectures that maintain full computational capabilities through local operations and controlled entanglement manipulation. Implementing large-scale quantum systems presents a major challenge, requiring scalable architectures that ensure high-fidelity state preparation and operations1,2. One promising approach is distributed quantum computing (DQC), where small, well-controlled qubit modules are interconnected via pre-shared entangled states, known as network qubits3–6. Within each module, circuit qubits execute local quantum operations, while network qubits mediate inter-module interactions, enabling cooperative large-scale quantum computation. To fully exploit the computational space of distributed circuit qubits, pre-shared entanglements across network qubits is essential for facilitating inter-module connectivity control. In traditional DQC approaches, non-local operations based on quantum gate teleportation allow the implementation of a universal gate set across the network7–10. However, these methods typically require complicated protocols involving multiple gates, additional ancilla qubits, and classical communication, which can introduce errors and reduce fidelity as the system scales. This raises the question: can we simplify the distributed quantum computing paradigm while maintaining access to the full computational Hilbert space? Weak measurement offers a promising alternative for controlling entanglement in distributed quantum systems11–13. Unlike projective measurements that collapse quantum states entirely, weak measurements induce partial collapses, allowing for the tuning of entanglement between distant qubits. By controlling the measurement strength, one can access quantum states that would otherwise require non-local operations. In this paper, we demonstrate the feasibility of distributed quantum computing by employing parameterized weak measurement. Using a basic configuration in which two modules each host a network qubit, we show that the non-local operations traditionally required for establishing arbitrary entangled states can be effectively replaced by local operations combined with weak measurement. Specifically, we prove that this approach enables the preparation of any arbitrary two-qubit state, thus accessing the full Hilbert space necessary for quantum algorithms. We integrate this technique into a variational quantum eigensolver (VQE) framework14–16, which relies on parameterized quantum state preparation and measurement, validating that high-fidelity quantum computations remain achievable in distributed architectures. Our experimental implementation using photonic qubits demonstrates that weak measurement can serve as a practical tool for entanglement manipulation in networked quantum processors, potentially simplifying the requirements for scaling up quantum computing resources. Results Variational quantum eigensolver The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm designed to find the ground state energy of a quantum system 1 Center for Quantum Technology, Korea Institute of Science and Technology (KIST), Seoul, Republic of Korea. 2Division of Quantum Information, KIST School, Korea University of Science and Technology, Seoul, Republic of Korea. 3Department of Physics, Korea University, Seoul, Republic of Korea. 4 Department of e-mail: Physics, Pohang University of Technology (POSTECH)P, Pohang, Republic of Korea. npj Quantum Information | (2026)12:20 1 Article https://doi.org/10.1038/s41534-025-01163-1 the expressibility of the ansatz, which determines the algorithm’s ability to represent the true ground state within the parameterized family of states. While VQE has been primarily implemented on single quantum processors with various physical platforms17–24, the growing interest in quantum networks opens new possibilities for distributed quantum computing25,26. In a networked environment, quantum processors at different locations can collaborate to solve computational problems that might be intractable for individual nodes. This distributed approach to VQE requires careful consideration of how entanglement and quantum operations are managed across the network. Distributed quantum processor units (QPUs) N – Bell states, |Ψ⟩ Quantum channels Alice Bob ( ⃗) (⃗ ) (⃗ ) {P } { ⃗ } +1 {P } Classical channels VQE on quantum network Figure 1 presents the schematic diagram of VQE between  distant parties. Let us begin by a central party distributes a Bell state ∣Ψþ ¼ p1ffiffi2 ð∣01i þ ∣10iÞ to two distant parties, Alice and Bob, via quantum channels. Without loss of generality, we can present the shared state between Alice and Bob in the form of { ⃗ , ⃗ } +1    1 ∣Ψi ¼ ðI A  C B Þ∣Ψþ ¼ pffiffiffi ð∣0i  ∣ψ b þ ∣1i  ∣ψ ? b Þ; 2 CPU ∑ = Linear calculation, ,{ ⃗, ⃗} ) → { ⃗ , ⃗} Classical Optimizer, O( +1 Fig. 1 | Schematic diagram of distributed VQE. Two spatially separated QPUs ! share N pairs of Bell states, with Alice performing local unitary operations U A ð θ A Þ ! and Bob implementing both local unitaries U B ð θ B Þ and weak measurements ! WMð p Þ. Measurement outcomes are sent to classical processing units (CPU) which computes the expectation value of the Hamiltonian and updates the parameters for subsequent iterations. described by a Hamiltonian H14–16. VQE leverages the variational principle! of Equantum mechanics, which states that for any trial wavefunction ∣Ψð θ Þ , the expectation value of the Hamiltonian provides an upper bound to the ground state energy: D ! ! !E Eð θ Þ ¼ Ψð θ Þ∣H∣Ψð θ Þ ≥ Eg ; ð1Þ ! where Eg is the true ground state energy, and θ represents a set of variational parameters that character (...truncated)