Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2603.02305

TIMES-ADAPT: A Quantum algorithm for real-time evolution in low-energy subspaces using fixed-depth circuits

Bharath Sambasivam, Kyle Sherbert, Karunya Shirali, Nicholas J. Mayhall, Edwin Barnes, Sophia E. Economou

quant-phSubmitted Mar 2, 2026• Announced Mar 4, 2026• 0 citations
arXivHTMLPDF

Abstract

We propose a new variational quantum algorithm, which we refer to as TIMES-ADAPT, that prepares time-evolved states in a low-energy or symmetric subspace of a time-independent Hamiltonian on a quantum computer. Using a specially trained unitary that diagonalizes the Hamiltonian in a subspace, we construct fixed-depth circuits for real-time evolution in the subspace, where time only enters as a circuit parameter. We present two versions of the algorithm depending on whether the initial state is specified in the energy eigenbasis or computational basis. We consider two important applications of our methods: wave packet evolution and energy transport in spin systems. We benchmark our algorithms using variants of the Heisenberg XXZ model.

Comments
6 pages of main text, 5 pages of appendices

Related papers

Variational Quantum Circuits to Prepare Low Energy Symmetry States

Raja Selvarajan, Manas Sajjan, Sabre Kais

We explore how to build quantum circuits that compute the lowest energy state corresponding to a given Hamiltonian within a symmetry subspace by explicitly encoding it into the circuit. We create an explicit unitary and a variationally trained unitary that maps any vector output by ansatz A(α→) from a defined subspace to a vector in the symmetry space. The parameters are trained varitionally to minimize the energy, thus keeping the output within the labelled symmetry value. The method was tested for a spin XXZ Hamiltonian using rotation and reflection symmetry and H2 Hamiltonian within Sz=0 subspace using S2 symmetry. We have found the variationally trained unitary gives good results with very low depth circuits and can thus be used to prepare symmetry states within near term quantum computers.

arXiv:2112.12857•Symmetry•Feb 24, 2022•11 citations

Low-Depth Hamiltonian Simulation by an Adaptive Product Formula

Zijian Zhang, Jinzhao Sun, Xiao Yuan +1 more

Various Hamiltonian simulation algorithms have been proposed to efficiently study the dynamics of quantum systems on a quantum computer. The existing algorithms generally approximate the time evolution operators, which may need a deep quantum circuit that is beyond the capability of near-term noisy quantum devices. Here, focusing on the time evolution of a fixed input quantum state, we propose an adaptive approach to construct a low-depth time evolution circuit. By introducing a measurable quantifier that characterizes the simulation error, we use an adaptive strategy to learn the shallow quantum circuit that minimizes that error. We numerically test the adaptive method with electronic Hamiltonians of the H_{2}O and H_{4} molecules, and the transverse field Ising model with random coefficients. Compared to the first-order Suzuki-Trotter product formula, our method can significantly reduce the circuit depth (specifically the number of two-qubit gates) by around two orders while maintaining the simulation accuracy. We show applications of the method in simulating many-body dynamics and solving energy spectra with the quantum Krylov algorithm. Our work sheds light on practical Hamiltonian simulation with noisy-intermediate-scale-quantum devices.

arXiv:2011.05283•Physical Review Letters•Jan 25, 2023•26 citations

Adaptive projected variational quantum dynamics

David Linteau, Stefano Barison, Netanel H. Lindner +1 more

We propose an adaptive quantum algorithm to prepare accurate variational time evolved wave functions. The method is based on the projected variational quantum dynamics (pVQD) algorithm, that performs a global optimization with linear scaling in the number of variational parameters. Instead of fixing a variational ansatz at the beginning of the simulation, the circuit is grown systematically during the time evolution. Moreover, the adaptive step does not require auxiliary qubits and the gate search can be performed in parallel on different quantum devices. We apply the algorithm, named adaptive pVQD, to the simulation of driven spin models and fermionic systems, where it shows an advantage when compared to both Trotterized circuits and nonadaptive variational methods. Finally, we use the shallower circuits prepared using the adaptive pVQD algorithm to obtain more accurate measurements of physical properties of quantum systems on hardware. Published by the American Physical Society 2024

arXiv:2307.03229•Physical Review Research•May 6, 2024•11 citations

Adaptive Variational Quantum Computing Approaches for Green’s Functions and Nonlinear Susceptibilities

Martin Mootz, Thomas Iadecola, Yongxin Yao

We present and benchmark quantum computing approaches for calculating real-time single-particle Green's functions and nonlinear susceptibilities of Hamiltonian systems. The approaches leverage adaptive variational quantum algorithms for state preparation and propagation. Using automatically generated compact circuits, the dynamical evolution is performed over sufficiently long times to achieve adequate frequency resolution of the response functions. We showcase accurate Green's function calculations using a statevector simulator on classical hardware for Fermi-Hubbard chains of 4 and 6 sites, with maximal ansatz circuit depths of 65 and 424 layers, respectively, and for the molecule LiH with a maximal ansatz circuit depth of 81 layers. Additionally, we consider an antiferromagnetic quantum spin-1 model that incorporates the Dzyaloshinskii-Moriya interaction to illustrate calculations of the third-order nonlinear susceptibilities, which can be measured in two-dimensional coherent spectroscopy experiments. These results demonstrate that real-time approaches using adaptive parametrized circuits to evaluate linear and nonlinear response functions can be feasible with near-term quantum processors.

arXiv:2407.01313•Journal of Chemical Theory and Computation•Sep 27, 2024•5 citations

Subspace Variational Quantum Simulation: Fidelity Lower Bounds as Measures of Training Success

Seung Park, Dongkeun Lee, Jeongho Bang +2 more

We propose an iterative variational quantum algorithm to simulate the time evolution of arbitrary initial states within a given subspace. The algorithm compresses the Trotter circuit into a shorter-depth parameterized circuit, which is optimized simultaneously over multiple initial states in a single training process using fidelity-based cost functions. After the whole training procedure, we provide an efficiently computable lower bound on the fidelities for arbitrary states within the subspace, which guarantees the performance of the algorithm in the worst-case training scenario. We also show our cost function exhibits a barren-plateau-free region near the initial parameters at each iteration in the training landscape. The experimental demonstration of the algorithm is presented through the simulation of a 2-qubit Ising model on an IBMQ processor. As a demonstration for a larger system, a simulation of a 10-qubit Ising model is also provided.

arXiv:2509.06360•Sep 8, 2025•0 citations

Hamiltonian simulation for low-energy states with optimal time dependence

Alexander Zlokapa, Rolando D. Somma

We consider the task of simulating time evolution under a Hamiltonian H within its low-energy subspace. Assuming access to a block-encoding of H ′ := ( H − E ) / λ , for some λ > 0 and E ∈ R , the goal is to implement an ϵ -approximation to the evolution operator e − i t H when the initial state is confined to the subspace corresponding to eigenvalues [ − 1 , − 1 + Δ / λ ] of H ′ , for Δ ≤ λ . We present a quantum algorithm that requires O ( t λ Γ + λ / Γ log ⁡ ( 1 / ϵ ) ) queries to the block-encoding for any choice of Γ such that Δ ≤ Γ ≤ λ . When the parameters satisfy log ⁡ ( 1 / ϵ ) = o ( t λ ) and Δ / λ = o ( 1 ) , this result improves over generic methods with query complexity Ω ( t λ ) . Our quantum algorithm leverages spectral gap amplification and the quantum singular value transform.For a given H , the block-encoding of its H ′ must be prepared efficiently to achieve an asymptotic speedup in simulating the low-energy subspace; we refer to these Hamiltonians as g a p − a m p l i f i a b l e . We show necessary and sufficient conditions for gap amplifiability in terms of an operationally useful decomposition of H

arXiv:2404.03644•Quantum•Aug 27, 2024•12 citations
We propose a new variational quantum algorithm, which we refer to as TIMES-ADAPT, that prepares time-evolved states in a low-energy or symmetric subspace of a time-independent Hamiltonian on a quantum computer. Using a specially trained unitary that diagonalizes the Hamiltonian in a subspace, we construct fixed-depth circuits for real-time evolution in the subspace, where time only enters as a circuit parameter. We present two versions of the algorithm depending on whether the initial state is specified in the energy eigenbasis or computational basis. We consider two important applications of our methods: wave packet evolution and energy transport in spin systems. We benchmark our algorithms using variants of the Heisenberg XXZ model.
We explore how to build quantum circuits that compute the lowest energy state corresponding to a given Hamiltonian within a symmetry subspace by explicitly encoding it into the circuit. We create an explicit unitary and a variationally trained unitary that maps any vector output by ansatz A(α→) from a defined subspace to a vector in the symmetry space. The parameters are trained varitionally to minimize the energy, thus keeping the output within the labelled symmetry value. The method was tested for a spin XXZ Hamiltonian using rotation and reflection symmetry and H2 Hamiltonian within Sz=0 subspace using S2 symmetry. We have found the variationally trained unitary gives good results with very low depth circuits and can thus be used to prepare symmetry states within near term quantum computers.
Various Hamiltonian simulation algorithms have been proposed to efficiently study the dynamics of quantum systems on a quantum computer. The existing algorithms generally approximate the time evolution operators, which may need a deep quantum circuit that is beyond the capability of near-term noisy quantum devices. Here, focusing on the time evolution of a fixed input quantum state, we propose an adaptive approach to construct a low-depth time evolution circuit. By introducing a measurable quantifier that characterizes the simulation error, we use an adaptive strategy to learn the shallow quantum circuit that minimizes that error. We numerically test the adaptive method with electronic Hamiltonians of the H_{2}O and H_{4} molecules, and the transverse field Ising model with random coefficients. Compared to the first-order Suzuki-Trotter product formula, our method can significantly reduce the circuit depth (specifically the number of two-qubit gates) by around two orders while maintaining the simulation accuracy. We show applications of the method in simulating many-body dynamics and solving energy spectra with the quantum Krylov algorithm. Our work sheds light on practical Hamiltonian simulation with noisy-intermediate-scale-quantum devices.
We propose an adaptive quantum algorithm to prepare accurate variational time evolved wave functions. The method is based on the projected variational quantum dynamics (pVQD) algorithm, that performs a global optimization with linear scaling in the number of variational parameters. Instead of fixing a variational ansatz at the beginning of the simulation, the circuit is grown systematically during the time evolution. Moreover, the adaptive step does not require auxiliary qubits and the gate search can be performed in parallel on different quantum devices. We apply the algorithm, named adaptive pVQD, to the simulation of driven spin models and fermionic systems, where it shows an advantage when compared to both Trotterized circuits and nonadaptive variational methods. Finally, we use the shallower circuits prepared using the adaptive pVQD algorithm to obtain more accurate measurements of physical properties of quantum systems on hardware. Published by the American Physical Society 2024
We present and benchmark quantum computing approaches for calculating real-time single-particle Green's functions and nonlinear susceptibilities of Hamiltonian systems. The approaches leverage adaptive variational quantum algorithms for state preparation and propagation. Using automatically generated compact circuits, the dynamical evolution is performed over sufficiently long times to achieve adequate frequency resolution of the response functions. We showcase accurate Green's function calculations using a statevector simulator on classical hardware for Fermi-Hubbard chains of 4 and 6 sites, with maximal ansatz circuit depths of 65 and 424 layers, respectively, and for the molecule LiH with a maximal ansatz circuit depth of 81 layers. Additionally, we consider an antiferromagnetic quantum spin-1 model that incorporates the Dzyaloshinskii-Moriya interaction to illustrate calculations of the third-order nonlinear susceptibilities, which can be measured in two-dimensional coherent spectroscopy experiments. These results demonstrate that real-time approaches using adaptive parametrized circuits to evaluate linear and nonlinear response functions can be feasible with near-term quantum processors.
We propose an iterative variational quantum algorithm to simulate the time evolution of arbitrary initial states within a given subspace. The algorithm compresses the Trotter circuit into a shorter-depth parameterized circuit, which is optimized simultaneously over multiple initial states in a single training process using fidelity-based cost functions. After the whole training procedure, we provide an efficiently computable lower bound on the fidelities for arbitrary states within the subspace, which guarantees the performance of the algorithm in the worst-case training scenario. We also show our cost function exhibits a barren-plateau-free region near the initial parameters at each iteration in the training landscape. The experimental demonstration of the algorithm is presented through the simulation of a 2-qubit Ising model on an IBMQ processor. As a demonstration for a larger system, a simulation of a 10-qubit Ising model is also provided.
We consider the task of simulating time evolution under a Hamiltonian H within its low-energy subspace. Assuming access to a block-encoding of H ′ := ( H − E ) / λ , for some λ > 0 and E ∈ R , the goal is to implement an ϵ -approximation to the evolution operator e − i t H when the initial state is confined to the subspace corresponding to eigenvalues [ − 1 , − 1 + Δ / λ ] of H ′ , for Δ ≤ λ . We present a quantum algorithm that requires O ( t λ Γ + λ / Γ log ⁡ ( 1 / ϵ ) ) queries to the block-encoding for any choice of Γ such that Δ ≤ Γ ≤ λ . When the parameters satisfy log ⁡ ( 1 / ϵ ) = o ( t λ ) and Δ / λ = o ( 1 ) , this result improves over generic methods with query complexity Ω ( t λ ) . Our quantum algorithm leverages spectral gap amplification and the quantum singular value transform.For a given H , the block-encoding of its H ′ must be prepared efficiently to achieve an asymptotic speedup in simulating the low-energy subspace; we refer to these Hamiltonians as g a p − a m p l i f i a b l e . We show necessary and sufficient conditions for gap amplifiability in terms of an operationally useful decomposition of H