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:2406.18474

Near-Term Quantum Spin Simulation of the Spin-$\frac{1}{2}$ Square $J_{1}-J_{2}$ Heisenberg Model

Dylan Sheils, Trevor David Rhone

Published Jun 26, 2024• 1 citation
arXivHTMLPDF

Abstract

Simulating complex spin systems, known for high frustration and entanglement, presents significant challenges due to their intricate energy landscapes. This study focuses on the $J_{1}-J_{2}$ Heisenberg model, renowned for its rich phase behavior on the square lattice, to investigate strongly correlated spin systems. We conducted the first experimental quantum computing study of this model using the 127-qubit IBM Rensselear Eagle processor and the Variational Quantum Eigensolver (VQE) algorithm. By employing classical warm-starting ($+40\%$ ground state energy approximation) and a newly developed ansatz ($+9.31\%$ improvement compared to prior best), we improved ground state approximation accuracy on the 16-site variant, achieving usable results with approximately $10^{3}$ iterations, significantly fewer than the $10^{4}-10^{5}$ steps proposed by previous theoretical studies. We utilized existing error mitigation strategies and introduced a novel Classically-Reinforced VQE error mitigation scheme, achieving $93\%$ ground state accuracy, compared to $83.7\%$ with the Quantum Moments algorithm and $60\%$ with standard error mitigation. These strategies reduced the average error of observable prediction from $\approx 20\%$ to $5\%$, enhancing phase prediction from qualitative to quantitative alignment. Additionally, we explored an experimental implementation of the Quantum Lanczos (QLanczos) algorithm using Variational-Fast Forwarding (VFF) on a 4-qubit site, achieving $\approx 97\%$ ground state approximation. Theoretical simulations indicated that Krylov-based methods outperform VQE, with the Lanczos basis converging faster than the real-time basis. Our study demonstrates that near-term quantum devices can predict phase-relevant observables for the $J_1-J_2$ Heisenberg model, transitioning focus from theoretical to experimental, and suggesting general improvements to VQE-based methods.

DOI
10.48550/arxiv.2406.18474

Related papers

Ground-state estimation of the Heisenberg model on frustrated lattices with Sample-based Krylov Quantum Diagonalization

Calvin Brooks, Henry Zou, Trevor David Rhone

Quantum spin simulations of frustrated lattices remain challenging for both classical and quantum algorithms, particularly in parameter regimes relevant to quantum spin liquid (QSL) phases. In this work, we apply Sample-based Krylov Quantum Diagonalization (SKQD) to estimate the ground state of the antiferromagnetic XXZ Heisenberg model on the $J_1$--$J_2$ square lattice, the Kagome lattice, and a 1D chain, studying system sizes from 12 to 72 spins. In our application of SKQD, we identify a ZZ deformation of $Δ=2$ as a sufficiently sparse Hamiltonian and introduce two modifications to the SKQD framework tailored to spin models: a canonical bitstring compression scheme that preserves the effectiveness of configuration recovery under spin-flip degeneracy, and the use of multiple Krylov subspaces to improve ground state coverage without any increase in quantum resources. For the 1D chain and Kagome lattice, SKQD achieves sub-percent ground-state energy errors at system sizes up to 24 spins, including a relative error of $0.002\%$ on the 12-site Kagome lattice, surpassing the best prior VQE result of $0.01\%$ on the same system while requiring no variational optimization. SKQD further extends to system sizes well beyond the reach of prior quantum algorithm studies, reaching 72 spins across all three geometries. Beyond 24 spins, accuracy degrades to relative errors of $19\%$--$36\%$ at 72 sites, but the gradual scaling of error with system size suggests these limits are set by available shot budgets and circuit depth rather than fundamental algorithmic constraints. Although classical tensor network methods remain state-of-the-art for these models, this work establishes a new benchmark for quantum simulation of the frustrated Heisenberg model and demonstrates SKQD as a scalable, hardware-compatible approach for studying strongly correlated spin systems.

quant-pharXiv:2605.29521•May 29, 2026•0 citations

Variational quantum eigensolver ansatz for the J1−J2 -model

Verena Feulner, Michael J. Hartmann

The ground state properties of the two-dimensional ${J}_{1}\ensuremath{-}{J}_{2}$-model are very challenging to analyze via classical numerical methods due to the high level of frustration. This makes the model a promising candidate for which quantum computers could be helpful to possibly explore regimes that classical computers cannot reach. The ${J}_{1}\ensuremath{-}{J}_{2}$-model is a quantum spin model composed of Heisenberg interactions along the rectangular lattice edges and along diagonal edges between next-nearest-neighbor spins. We propose an ansatz for the variational quantum eigensolver to approximate the ground state of an antiferromagnetic ${J}_{1}\ensuremath{-}{J}_{2}$ Hamiltonian for different lattice sizes and different ratios of ${J}_{1}$ and ${J}_{2}$. Moreover, we demonstrate that this ansatz can work without the need for gates along the diagonal next-nearest-neighbor interactions. This simplification is of great importance for solid-state-based hardware with qubits on a rectangular grid, where it eliminates the need for swap gates. In addition, we provide an extrapolation for the number of gates and parameters needed for larger lattice sizes, showing that these are expected to grow linearly in the qubit number up to lattice sizes which eventually can no longer be treated with classical computers.

arXiv:2205.11198•Physical review. B./Physical review. B•Oct 21, 2022•10 citations

U(1) -symmetric infinite projected entangled-pair states study of the spin-1/2 square J1−J2 Heisenberg model

Reza Haghshenas, D. N. Sheng

We develop an improved variant of $U(1)$-symmetric infinite projected entangled-pair states (iPEPS) ansatz to investigate the ground-state phase diagram of the spin-$1/2$ square ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the $U(1)$-symmetric iPEPS ansatz (for a fixed bond dimension $D)$ set a better upper bound, improving previous tensor-network-based results. By studying the finite-$D$ scaling of the magnetically order parameter, we find a N\'eel phase for ${J}_{2}/{J}_{1}<0.53$. For $0.53<{J}_{2}/{J}_{1}<0.61$, a nonmagnetic columnar valence bond solid (VBS) state is established as observed by the pattern of local bond energy. The divergent behavior of correlation length $\ensuremath{\xi}\ensuremath{\sim}{D}^{1.2}$ and vanishing order parameters are consistent with a deconfined N\'eel-to-VBS transition at ${J}_{2}^{{c}_{1}}/{J}_{1}=0.530(5)$, where estimated critical anomalous exponents are ${\ensuremath{\eta}}_{s}\ensuremath{\sim}0.6$ and ${\ensuremath{\eta}}_{d}\ensuremath{\sim}1.9$ for spin and dimer correlations, respectively. We show that the associated VBS order parameter monotonically increases with ${J}_{2}/{J}_{1}$ and finally a first-order quantum phase transition takes place at ${J}_{2}^{{c}_{2}}/{J}_{1}=0.610(2)$ to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.

arXiv:1711.07584•Physical review. B./Physical review. B•May 8, 2018•87 citations

The $U(1)$-symmetric iPEPS study of the spin-$1/2$ square $J_{1}$-$J_{2}$ Heisenberg model

Reza Haghshenas, D. N. Sheng

We develop an improved variant of $U(1)$-symmetric infinite projected entangled-pair states (iPEPS) ansatz to investigate the ground state phase diagram of the spin-$1/2$ square $J_{1}$-$J_{2}$ Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the $U(1)$-symmetric iPEPS ansatz (for a fixed bond dimension $D$) set a better upper bound, improving previous tensor-network-based results. By studying N\'{e}el order parameter and correlation function in the large-$D$ limit, we find that a deconfined quantum-critical point occurs at $J^{c_1}_{2}=0.530(5)$, where the correlation length of the system diverges as $\xi \sim D^{1.2}$. The estimated critical anomalous exponents $\eta_{s}=0.65(5)$ and $\eta_{d}=1.9(1)$ for spin and dimer correlations respectively show strong deviation from those of the well-known $\mathcal{J}$-$\mathcal{Q}$ model, suggesting possibly a different universality class. The pattern of local bond energy for the intermediate phase is compatible with the columnar valence bond solid (VBS) state, with a quite competitive variational ground-state energy in comparison with plaquette VBS one predicted by the density matrix renormalization group. We show that the associated VBS order parameter first monotonically increases with $J_2$ and finally a first-order quantum phase transition takes place at $J^{c_2}_{2}=0.610(3)$ to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.

•Nov 21, 2017•0 citations

Variational Quantum Simulations of a Two-Dimensional Frustrated Transverse-Field Ising Model on a Trapped-Ion Quantum Computer

Ammar A. Kirmani, Elijah Pelofske, Andreas Bärtschi +2 more

Quantum computers are an ideal platform to study the ground state properties of strongly correlated systems due to the limitation of classical computing techniques particularly for systems exhibiting quantum phase transitions. While the error rates of Noisy Intermediate-Scale Quantum (NISQ) computers are still high, simulating strongly correlated systems on such devices and extracting information of possible phases may be within reach. The frustrated transverse-field Ising model (TFIM) is such a system with multiple ordered magnetic phases. In this study, we simulate a two-dimensional frustrated TFIM with next-nearest-neighbor spin-exchange interactions at zero temperature. The competition between the nearest-neighbor ferromagnetic and next-nearest-neighbor antiferromagnetic coupling gives rise to frustration in the system. Moreover, the presence of quantum fluctuations makes the ground-state phase profile even richer. We use the Variational Quantum Eigensolver (VQE) to compute the phases on a square lattice with periodic boundary conditions for a system of 16 sites (qubits). The trained VQE circuits are compared to exact diagonalization, allowing us to extract error measures of VQE. We focus on the ground-state phase transitions of this model, where VQE succeeds in finding the dominant magnetic phases. The optimized VQE circuits are then executed on the Quantinuum H1-1 trapped-ion quantum computer without using any error mitigation techniques. Our experiments show near perfect recovery of the magnetic phases of the frustrated model through ground-state energy, the energy derivative, and the spin correlation functions. Thus, we show that the trapped-ion quantum processor is able to achieve reliable simulations of a strongly correlated system within the limitations of the VQE approach.

arXiv:2505.22932•May 28, 2025•0 citations

Basis Adaptive Algorithm for Quantum Many-Body Systems on Quantum Computers

Anutosh Biswas, Sayan Ghosh, Ritajit Majumdar +3 more

We introduce a Basis Adaptive (BA) algorithm for hybrid quantum-classical simulation of correlated quantum many-body systems. Starting from a small set of physically motivated bitstrings, the algorithm iteratively applies a single-step first-order Trotterized circuit on a quantum processor, filters the sampled configurations by enforcing $U(1)$ spin conservation and lattice reflection symmetry, and classically diagonalizes the Hamiltonian in the resulting reduced Hilbert space. This design avoids the variational optimization overhead of VQE, the deep coherent circuits required by QPE, and the symmetry-violating subspaces that arise in SKQD. The ground-state energy error is bounded analytically by $\sqrt{8}\,\|H\|\left(1-\sqrt{α_{D_T}}\right)^{1/2}$, where $α_{D_T}$ is the probability weight captured by the $D_T$ sampled basis states. This bound connects algorithm performance directly to ground-state sparsity and explains the observed accuracy hierarchy across different phases. Benchmarked on the spin-$1/2$ Heisenberg XXZ chain (up to $N=62$ qubits on the IBM Heron processor), the algorithm achieves a $3.5\%$ energy error in the gapped Neel phase ($Δ=2.0$) and below $0.5\%$ at the ferromagnetic boundary ($Δ=-1.0$). The accuracy degrades to $28.7\%$ in the strongly quasi-long-range-ordered regime ($Δ=0.5$). Spin-spin correlation functions are reproduced across all regimes, confirming that symmetry-filtered real-time sampling provides a practical and noise-resilient pathway to ground-state properties on near-term quantum hardware.

cond-mat.str-elCross-listedarXiv:2512.12753•Dec 16, 2025•0 citations
Near-Term Quantum Spin Simulation of the Spin-12\frac{1}{2}21​ Square J1−J2J_{1}-J_{2}J1​−J2​ Heisenberg Model
Simulating complex spin systems, known for high frustration and entanglement, presents significant challenges due to their intricate energy landscapes. This study focuses on the J1−J2J_{1}-J_{2}J1​−J2​ Heisenberg model, renowned for its rich phase behavior on the square lattice, to investigate strongly correlated spin systems. We conducted the first experimental quantum computing study of this model using the 127-qubit IBM Rensselear Eagle processor and the Variational Quantum Eigensolver (VQE) algorithm. By employing classical warm-starting (+40%+40\%+40% ground state energy approximation) and a newly developed ansatz (+9.31%+9.31\%+9.31% improvement compared to prior best), we improved ground state approximation accuracy on the 16-site variant, achieving usable results with approximately 10310^{3}103 iterations, significantly fewer than the 104−10510^{4}-10^{5}104−105 steps proposed by previous theoretical studies. We utilized existing error mitigation strategies and introduced a novel Classically-Reinforced VQE error mitigation scheme, achieving 93%93\%93% ground state accuracy, compared to 83.7%83.7\%83.7% with the Quantum Moments algorithm and 60%60\%60% with standard error mitigation. These strategies reduced the average error of observable prediction from ≈20%\approx 20\%≈20% to 5%5\%5%, enhancing phase prediction from qualitative to quantitative alignment. Additionally, we explored an experimental implementation of the Quantum Lanczos (QLanczos) algorithm using Variational-Fast Forwarding (VFF) on a 4-qubit site, achieving ≈97%\approx 97\%≈97% ground state approximation. Theoretical simulations indicated that Krylov-based methods outperform VQE, with the Lanczos basis converging faster than the real-time basis. Our study demonstrates that near-term quantum devices can predict phase-relevant observables for the J1−J2J_1-J_2J1​−J2​ Heisenberg model, transitioning focus from theoretical to experimental, and suggesting general improvements to VQE-based methods.
Quantum spin simulations of frustrated lattices remain challenging for both classical and quantum algorithms, particularly in parameter regimes relevant to quantum spin liquid (QSL) phases. In this work, we apply Sample-based Krylov Quantum Diagonalization (SKQD) to estimate the ground state of the antiferromagnetic XXZ Heisenberg model on the J1J_1J1​--J2J_2J2​ square lattice, the Kagome lattice, and a 1D chain, studying system sizes from 12 to 72 spins. In our application of SKQD, we identify a ZZ deformation of Δ=2Δ=2Δ=2 as a sufficiently sparse Hamiltonian and introduce two modifications to the SKQD framework tailored to spin models: a canonical bitstring compression scheme that preserves the effectiveness of configuration recovery under spin-flip degeneracy, and the use of multiple Krylov subspaces to improve ground state coverage without any increase in quantum resources. For the 1D chain and Kagome lattice, SKQD achieves sub-percent ground-state energy errors at system sizes up to 24 spins, including a relative error of 0.002%0.002\%0.002% on the 12-site Kagome lattice, surpassing the best prior VQE result of 0.01%0.01\%0.01% on the same system while requiring no variational optimization. SKQD further extends to system sizes well beyond the reach of prior quantum algorithm studies, reaching 72 spins across all three geometries. Beyond 24 spins, accuracy degrades to relative errors of 19%19\%19%--36%36\%36% at 72 sites, but the gradual scaling of error with system size suggests these limits are set by available shot budgets and circuit depth rather than fundamental algorithmic constraints. Although classical tensor network methods remain state-of-the-art for these models, this work establishes a new benchmark for quantum simulation of the frustrated Heisenberg model and demonstrates SKQD as a scalable, hardware-compatible approach for studying strongly correlated spin systems.
The ground state properties of the two-dimensional ${J}_{1}\ensuremath{-}{J}_{2}$-model are very challenging to analyze via classical numerical methods due to the high level of frustration. This makes the model a promising candidate for which quantum computers could be helpful to possibly explore regimes that classical computers cannot reach. The ${J}_{1}\ensuremath{-}{J}_{2}$-model is a quantum spin model composed of Heisenberg interactions along the rectangular lattice edges and along diagonal edges between next-nearest-neighbor spins. We propose an ansatz for the variational quantum eigensolver to approximate the ground state of an antiferromagnetic ${J}_{1}\ensuremath{-}{J}_{2}$ Hamiltonian for different lattice sizes and different ratios of J1{J}_{1}J1​ and J2{J}_{2}J2​. Moreover, we demonstrate that this ansatz can work without the need for gates along the diagonal next-nearest-neighbor interactions. This simplification is of great importance for solid-state-based hardware with qubits on a rectangular grid, where it eliminates the need for swap gates. In addition, we provide an extrapolation for the number of gates and parameters needed for larger lattice sizes, showing that these are expected to grow linearly in the qubit number up to lattice sizes which eventually can no longer be treated with classical computers.
We develop an improved variant of U(1)U(1)U(1)-symmetric infinite projected entangled-pair states (iPEPS) ansatz to investigate the ground-state phase diagram of the spin-1/21/21/2 square ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the U(1)U(1)U(1)-symmetric iPEPS ansatz (for a fixed bond dimension D)D)D) set a better upper bound, improving previous tensor-network-based results. By studying the finite-DDD scaling of the magnetically order parameter, we find a N\'eel phase for J2/J1<0.53{J}_{2}/{J}_{1}<0.53J2​/J1​<0.53. For 0.53<J2/J1<0.610.53<{J}_{2}/{J}_{1}<0.610.53<J2​/J1​<0.61, a nonmagnetic columnar valence bond solid (VBS) state is established as observed by the pattern of local bond energy. The divergent behavior of correlation length $\ensuremath{\xi}\ensuremath{\sim}{D}^{1.2}$ and vanishing order parameters are consistent with a deconfined N\'eel-to-VBS transition at J2c1/J1=0.530(5){J}_{2}^{{c}_{1}}/{J}_{1}=0.530(5)J2c1​​/J1​=0.530(5), where estimated critical anomalous exponents are ${\ensuremath{\eta}}_{s}\ensuremath{\sim}0.6$ and ${\ensuremath{\eta}}_{d}\ensuremath{\sim}1.9$ for spin and dimer correlations, respectively. We show that the associated VBS order parameter monotonically increases with J2/J1{J}_{2}/{J}_{1}J2​/J1​ and finally a first-order quantum phase transition takes place at J2c2/J1=0.610(2){J}_{2}^{{c}_{2}}/{J}_{1}=0.610(2)J2c2​​/J1​=0.610(2) to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.
The U(1)U(1)U(1)-symmetric iPEPS study of the spin-1/21/21/2 square J1J_{1}J1​-J2J_{2}J2​ Heisenberg model
We develop an improved variant of U(1)U(1)U(1)-symmetric infinite projected entangled-pair states (iPEPS) ansatz to investigate the ground state phase diagram of the spin-1/21/21/2 square J1J_{1}J1​-J2J_{2}J2​ Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the U(1)U(1)U(1)-symmetric iPEPS ansatz (for a fixed bond dimension DDD) set a better upper bound, improving previous tensor-network-based results. By studying N\'{e}el order parameter and correlation function in the large-DDD limit, we find that a deconfined quantum-critical point occurs at J2c1=0.530(5)J^{c_1}_{2}=0.530(5)J2c1​​=0.530(5), where the correlation length of the system diverges as ξ∼D1.2\xi \sim D^{1.2}ξ∼D1.2. The estimated critical anomalous exponents ηs=0.65(5)\eta_{s}=0.65(5)ηs​=0.65(5) and ηd=1.9(1)\eta_{d}=1.9(1)ηd​=1.9(1) for spin and dimer correlations respectively show strong deviation from those of the well-known J\mathcal{J}J-Q\mathcal{Q}Q model, suggesting possibly a different universality class. The pattern of local bond energy for the intermediate phase is compatible with the columnar valence bond solid (VBS) state, with a quite competitive variational ground-state energy in comparison with plaquette VBS one predicted by the density matrix renormalization group. We show that the associated VBS order parameter first monotonically increases with J2J_2J2​ and finally a first-order quantum phase transition takes place at J2c2=0.610(3)J^{c_2}_{2}=0.610(3)J2c2​​=0.610(3) to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.
Quantum computers are an ideal platform to study the ground state properties of strongly correlated systems due to the limitation of classical computing techniques particularly for systems exhibiting quantum phase transitions. While the error rates of Noisy Intermediate-Scale Quantum (NISQ) computers are still high, simulating strongly correlated systems on such devices and extracting information of possible phases may be within reach. The frustrated transverse-field Ising model (TFIM) is such a system with multiple ordered magnetic phases. In this study, we simulate a two-dimensional frustrated TFIM with next-nearest-neighbor spin-exchange interactions at zero temperature. The competition between the nearest-neighbor ferromagnetic and next-nearest-neighbor antiferromagnetic coupling gives rise to frustration in the system. Moreover, the presence of quantum fluctuations makes the ground-state phase profile even richer. We use the Variational Quantum Eigensolver (VQE) to compute the phases on a square lattice with periodic boundary conditions for a system of 16 sites (qubits). The trained VQE circuits are compared to exact diagonalization, allowing us to extract error measures of VQE. We focus on the ground-state phase transitions of this model, where VQE succeeds in finding the dominant magnetic phases. The optimized VQE circuits are then executed on the Quantinuum H1-1 trapped-ion quantum computer without using any error mitigation techniques. Our experiments show near perfect recovery of the magnetic phases of the frustrated model through ground-state energy, the energy derivative, and the spin correlation functions. Thus, we show that the trapped-ion quantum processor is able to achieve reliable simulations of a strongly correlated system within the limitations of the VQE approach.
We introduce a Basis Adaptive (BA) algorithm for hybrid quantum-classical simulation of correlated quantum many-body systems. Starting from a small set of physically motivated bitstrings, the algorithm iteratively applies a single-step first-order Trotterized circuit on a quantum processor, filters the sampled configurations by enforcing U(1)U(1)U(1) spin conservation and lattice reflection symmetry, and classically diagonalizes the Hamiltonian in the resulting reduced Hilbert space. This design avoids the variational optimization overhead of VQE, the deep coherent circuits required by QPE, and the symmetry-violating subspaces that arise in SKQD. The ground-state energy error is bounded analytically by 8 ∥H∥(1−αDT)1/2\sqrt{8}\,\|H\|\left(1-\sqrt{α_{D_T}}\right)^{1/2}8​∥H∥(1−αDT​​​)1/2, where αDTα_{D_T}αDT​​ is the probability weight captured by the DTD_TDT​ sampled basis states. This bound connects algorithm performance directly to ground-state sparsity and explains the observed accuracy hierarchy across different phases. Benchmarked on the spin-1/21/21/2 Heisenberg XXZ chain (up to N=62N=62N=62 qubits on the IBM Heron processor), the algorithm achieves a 3.5%3.5\%3.5% energy error in the gapped Neel phase (Δ=2.0Δ=2.0Δ=2.0) and below 0.5%0.5\%0.5% at the ferromagnetic boundary (Δ=−1.0Δ=-1.0Δ=−1.0). The accuracy degrades to 28.7%28.7\%28.7% in the strongly quasi-long-range-ordered regime (Δ=0.5Δ=0.5Δ=0.5). Spin-spin correlation functions are reproduced across all regimes, confirming that symmetry-filtered real-time sampling provides a practical and noise-resilient pathway to ground-state properties on near-term quantum hardware.