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arXiv:2404.15579 · Quantum Science and Technology

Photonic variational quantum eigensolver using entanglement measurements

Jinil Lee, Wooyeong Song, Donghwa Lee, Yosep Kim, Seung-Woo Lee, Hyang‐Tag Lim, Hojoong Jung, Sang-Wook Han, Yong‐Su Kim

Published Aug 9, 2024• 4 citations
arXivHTMLPDF

Abstract

Abstract Variational quantum eigensolver (VQE), which combines quantum systems with classical computational power, has been arisen as a promising candidate for near-term quantum computing applications. However, the experimental resources such as the number of measurements to implement VQE rapidly increases as the Hamiltonian problem size grows. Applying entanglement measurements to reduce the number of measurement setups has been proposed to address this issue, but, entanglement measurements themselves can introduce additional resource demands. Here, we apply entanglement measurements to the photonic VQE utilizing polarization and path degrees of freedom of a single-photon. In our photonic VQE, entanglement measurements can be deterministically implemented using linear optics, so it takes full advantage of introducing entanglement measurements without additional experimental demands. Moreover, we show that such a setup can mitigate errors in measurement apparatus for a certain Hamiltonian.

DOI
10.1088/2058-9565/ad6d87

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Abstract Variational quantum eigensolver (VQE), which combines quantum systems with classical computational power, has been arisen as a promising candidate for near-term quantum computing applications. However, the experimental resources such as the number of measurements to implement VQE rapidly increases as the Hamiltonian problem size grows. Applying entanglement measurements to reduce the number of measurement setups has been proposed to address this issue, but, entanglement measurements themselves can introduce additional resource demands. Here, we apply entanglement measurements to the photonic VQE utilizing polarization and path degrees of freedom of a single-photon. In our photonic VQE, entanglement measurements can be deterministically implemented using linear optics, so it takes full advantage of introducing entanglement measurements without additional experimental demands. Moreover, we show that such a setup can mitigate errors in measurement apparatus for a certain Hamiltonian.
Variational quantum algorithms, a representative class of modern quantum algorithms, provide practical uses of near-term quantum processors. The size of the problem that can be encoded and solved on a quantum processor is limited by the dimension of the Hilbert space associated with the processor. One common approach for increasing the system dimension is to utilize a larger number of quantum systems. Here, we adopt an alternative approach to utilize multiple degrees of freedom of individual quantum systems to experimentally resource-efficiently increase the Hilbert space. We report experimental implementation of the variational quantum eigensolver (VQE) using four-dimensional photonic quantum states of single photons. The four-dimensional quantum states are implemented by utilizing polarization and path degrees of freedom of a single photon. Our photonic VQE is equipped with a quantum error mitigation protocol that efficiently reduces the effects of Pauli noise in the quantum processing unit. We apply our photonic VQE to estimate the ground state energy of the H e − H + cation. Simulation and experimental results demonstrate that our experimental resource-efficient photonic VQE can accurately estimate the bond dissociation curve, even in the presence of large noise in the quantum processing unit. We also discuss further possible resource-efficient enhancement of the Hilbert space in photonic quantum processors. Our results propose that photonic systems utilizing multiple degrees of freedom can provide a resource-efficient avenue to implement practical near-term quantum processors.
Abstract 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.
Variational quantum eigensolver (VQE) is a promising algorithm suitable for near-term quantum computers. VQE aims to approximate solutions to exponentially-sized optimization problems by executing a polynomial number of quantum subproblems. However, the number of subproblems scales as N4for typical problems of interest-a daunting growth rate that poses a serious limitation for emerging applications such as quantum computational chemistry. We mitigate this issue by exploiting the simultaneous measurability of subproblems corresponding to commuting terms. Our technique transpiles VQE instances into a format optimized for simultaneous measurement, ultimately yielding 8-30x lower cost. Our work also encompasses a synthesis tool for compiling simultaneous measurement circuits with minimal overhead. We demonstrate experimental validation of our techniques by estimating the ground state energy of deuteron with a quantum computer. We also investigate the underlying statistics of simultaneous measurement and devise an adaptive strategy for mitigating harmful covariance terms.
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.
Current implementations of the Variational Quantum Eigensolver (VQE) technique for solving the electronic structure problem involve splitting the system qubit Hamiltonian into parts whose elements commute within their single qubit subspaces. The number of such parts rapidly grows with the size of the molecule, this increases the uncertainty in the measurement of the energy expectation value because elements from different parts need to be measured independently. To address this problem we introduce a more efficient partitioning of the qubit Hamiltonian using fewer parts that need to be measured separately. The new partitioning scheme is based on two ideas: 1) grouping terms into parts whose eigenstates have a single-qubit product structure, and 2) devising multi-qubit unitary transformations for the Hamiltonian or its parts to produce less entangled operators. The first condition allows the new parts to be measured in the number of involved qubit consequential one-particle measurements. Advantages of the new partitioning scheme resulting in several-fold reduction of separately measured terms are illustrated on the H2 and LiH problems.
Abstract We discuss the procedure for obtaining measurement-based implementations of quantum algorithms given by quantum circuit diagrams and how to reduce the required resources needed for a given measurement-based computation. This forms the foundation for quantum computing on photonic systems in the near term. To demonstrate that these ideas are well grounded we present three different problems which are solved by employing a measurement-based implementation of the variational quantum eigensolver algorithm (MBVQE). We show that by utilising native measurement-based gates rather than standard gates, such as the standard controlled not gate (CNOT), measurement-based quantum computations may be obtained that are both shallow and have simple connectivity while simultaneously exhibiting a large expressibility. We conclude that MBVQE has promising prospects for resource states that are not far from what is already available today.