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Cited by 224Quantum Computing Algorithms and ArchitectureQuantum many-body systemsAdvanced Chemical Physics Studies

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VQE method: a short survey and recent developments

Open access
Fedorov, Dmitry A., Peng, Bo, Govind, Niranjan, Alexeev, Yuri·2022·Materials Theory
DOI 10.1186/s41313-021-00032-6
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Abstract

The variational quantum eigensolver (VQE) is a method that uses a hybrid quantum-classical computational approach to find eigenvalues of a Hamiltonian. VQE has been proposed as an alternative to fully quantum algorithms such as quantum phase estimation (QPE) because fully quantum algorithms require quantum hardware that will not be accessible in the near future. VQE has been successfully applied to solve the electronic Schrödinger equation for a variety of small molecules. However, the scalability of this method is limited by two factors: the complexity of the quantum circuits and the complexity of the classical optimization problem. Both of these factors are affected by the choice of the variational ansatz used to represent the trial wave function. Hence, the construction of an efficient ansatz is an active area of research. Put another way, modern quantum computers are not capable of executing deep quantum circuits produced by using currently available ansatzes for problems that map onto more than several qubits. In this review, we present recent developments in the field of designing efficient ansatzes that fall into two categories—chemistry–inspired and hardware–efficient—that produce quantum circuits that are easier to run on modern hardware. We discuss the shortfalls of ansatzes originally formulated for VQE simulations, how they are addressed in more sophisticated methods, and the potential ways for further improvements.

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Fig. 1 Schematic of the variational quantum eigensolver (VQE) method that minimizes the energy of the Hamiltonian 〈 ψ( ⃗ θ) | ˆ Hel | ψ( ⃗ θ) 〉 by adjusting variational parameters θ . It uses classical computing resources denoted by green color and quantum computing resources denoted by blue. A simulation starts by constructing a fermionic Hamiltonian and finding mean-field solution | ψ HF 〉 . Next, the fermionic Hamiltonian is mapped into qubit Hamiltonian, represented as a sum of Pauli strings H = ∑ j α j ∏ i σ j i . Then the ansatz to represent the wave function is chosen and initialized with the initial set of parameters ⃗ θ 0. The trial state is prepared on a quantum computer as a quantum circuit consisting of parametrized gates. The rest of the procedure is performed repeatedly until the convergence criterion is met. At iteration k the energy of the Hamiltonian is computed by measuring every Hamiltonian term 〈 ψ( ⃗ θ k ) | Pj | ψ( ⃗ θ k ) 〉 on a quantum computer and adding them on a classical computer. The energy E ( ⃗ θ k ) is fed into the classical algorithm that updates parameters for the next step of optimization ⃗ θ k + 1 according to the chosen optimization algorithm
Fig. 1 Schematic of the variational quantum eigensolver (VQE) method that minimizes the energy of the Hamiltonian 〈 ψ( ⃗ θ) | ˆ Hel | ψ( ⃗ θ) 〉 by adjusting variational parameters θ . It uses classical computing resources denoted by green color and quantum computing resources denoted by blue. A simulation starts by constructing a fermionic Hamiltonian and finding mean-field solution | ψ HF 〉 . Next, the fermionic Hamiltonian is mapped into qubit Hamiltonian, represented as a sum of Pauli strings H = ∑ j α j ∏ i σ j i . Then the ansatz to represent the wave function is chosen and initialized with the initial set of parameters ⃗ θ 0. The trial state is prepared on a quantum computer as a quantum circuit consisting of parametrized gates. The rest of the procedure is performed repeatedly until the convergence criterion is met. At iteration k the energy of the Hamiltonian is computed by measuring every Hamiltonian term 〈 ψ( ⃗ θ k ) | Pj | ψ( ⃗ θ k ) 〉 on a quantum computer and adding them on a classical computer. The energy E ( ⃗ θ k ) is fed into the classical algorithm that updates parameters for the next step of optimization ⃗ θ k + 1 according to the chosen optimization algorithmCC-BY · From the PDF

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Cites

Research works printed but not linked: 12

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Metadata

Authors
Fedorov, D. A., Peng, B., Govind, N., & Alexeev, Y.
Affiliations
  • Oak Ridge Associated Universities (US)
  • Pacific Northwest National Laboratory (US)
  • Argonne National Laboratory (US)
Corresponding author
  • Fedorov, Dmitry A.
  • Peng, Bo
  • Govind, Niranjan
  • Alexeev, Yuri
ORCID
Venue
Materials Theory
Publisher
Springer Science and Business Media LLC
Published
2022-01-06
Volume
6
Issue
1
ISSN
2509-8012 (ISSN-L)
DOI
10.1186/s41313-021-00032-6
OpenAlex
W4206745400
Cited by
224
References
101
Citations by year
2026: 28 · 2025: 64 · 2024: 52 · 2023: 53 · 2022: 24
FWCI
15.87
Type
journal article
Language
en
Open access
Licence
Locations
  • Materials Theory (publishedVersion, cc-by)
  • arXiv (Cornell University) (submittedVersion)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
  • OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information) (submittedVersion)
  • arXiv (Cornell University)
Funders
Office of Science (DE-SC002024, DE-AC02-06CH11357)
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Advanced Chemical Physics Studies
Keywords
variational quantum eigensolver, variational ansatz, hybrid quantum-classical algorithms, hardware-efficient ansatz, electronic Schrödinger equation, molecular quantum chemistry, variational quantum algorithms, quantum chemistry on quantum computers

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