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:2409.16194 · New Journal of Physics

Preparing ground and excited states using adiabatic CoVaR

Wooseop Hwang, Bálint Koczor

Published Feb 1, 2025• 1 citation
arXivHTMLPDF

Abstract

Abstract CoVariance Root finding with classical shadows (CoVaR) was recently introduced as a new paradigm for training variational quantum circuits. Common approaches, such as variants of the variational quantum eigensolver, aim to optimise a non-linear classical cost function and thus suffer from, e.g. poor local minima, high shot requirements and barren plateaus. In contrast, CoVaR fully exploits powerful classical shadows and finds joint roots of a very large number of covariances using only a logarithmic number of shots and linearly scaling classical computing resources. As a result, CoVaR has been demonstrated to be particularly robust against local traps, however, its main limitation has been that it requires a sufficiently good initial state. We address this limitation by introducing an adiabatic morphing of the target Hamiltonian and demonstrate in a broad range of application examples that CoVaR can successfully prepare eigenstates of the target Hamiltonian when no initial warm start is known. CoVaR succeeds even when Hamiltonian energy gaps are very small—this is in stark contrast to adiabatic evolution and phase estimation algorithms where circuit depths scale inversely with the Hamiltonian energy gaps. On the other hand, when the energy gaps are relatively small then adiabatic CoVaR may converge to higher excited states as opposed to a targeted specific low-lying state. Nevertheless, we exploit this feature of adiabatic CoVaR and demonstrate that it can be used to map out the low lying spectrum of a Hamiltonian which can be useful in practical applications, such as estimating thermal properties or in high-energy physics.

DOI
10.1088/1367-2630/adb173

Related papers

Training Variational Quantum Circuits with CoVaR: Covariance Root Finding with Classical Shadows

Gregory Boyd, Bálint Koczor

Exploiting near-term quantum computers and achieving practical value is a considerable and exciting challenge.Most prominent candidates as variational algorithms typically aim to find the ground state of a Hamiltonian by minimizing a single classical (energy) surface which is sampled from by a quantum computer.Here, we introduce a method we call CoVaR, an alternative means to exploit the power of variational circuits: We find eigenstates by finding joint roots of a polynomially growing number of properties of the quantum state as covariance functions between the Hamiltonian and an operator pool of our choice.The most remarkable feature of our CoVaR approach is that it allows us to fully exploit the extremely powerful classical shadow techniques; i.e., we simultaneously estimate a very large number > 10 4 -10 7 of covariances.We randomly select covariances and estimate analytical derivatives at each iteration applying a stochastic Levenberg-Marquardt step via a large but tractable linear system of equations that we solve with a classical computer.We prove that the cost in quantum resources per iteration is comparable to a standard gradient estimation; however, we observe in numerical simulations a very significant improvement by many orders of magnitude in convergence speed.CoVaR is directly analogous to stochastic gradient-based optimizations of paramount importance to classical machine learning while we also offload significant but tractable work onto the classical processor.As we demonstrate numerically, the approach shares features with phase-estimation protocols that prepare eigenstates with a dominant initial fidelity contribution.

arXiv:2204.08494•Physical Review X•Nov 28, 2022•25 citations

Warm Starts, Cold States: Exploiting Adiabaticity for Variational Ground-States

Ricard Puig, Berta Casas, Alba Cervera-Lierta +2 more

Reliable preparation of many-body ground states is an essential task in quantum computing, with applications spanning areas from chemistry and materials modeling to quantum optimization and benchmarking. A variety of approaches have been proposed to tackle this problem, including variational methods. However, variational training often struggle to navigate complex energy landscapes, frequently encountering suboptimal local minima or suffering from barren plateaus. In this work, we introduce an iterative strategy for ground-state preparation based on a stepwise (discretized) Hamiltonian deformation. By complementing the Variational Quantum Eigensolver (VQE) with adiabatic principles, we demonstrate that solving a sequence of intermediate problems facilitates tracking the ground-state manifold toward the target system, even as we scale the system size. We provide a rigorous theoretical foundation for this approach, proving a lower bound on the loss variance that suggests trainability throughout the deformation, provided the system remains away from gap closings. Numerical simulations, including the effects of shot noise, confirm that this path-dependent tracking consistently converges to the target ground state.

quant-pharXiv:2602.06137•Feb 9, 2026•0 citations

Adiabatic-Inspired Hybrid Quantum-Classical Methods for Molecular Ground State Preparation

Sean Thrasher, Ioannis Kolotouros, Julien Michel +1 more

Quantum computing promises to efficiently and accurately solve many important problems in quantum chemistry which elude classical solvers, such as the electronic structure problem of highly correlated materials. Two leading methods in solving the ground state problem are the Variational Quantum Eigensolver (VQE) and Adiabatic Quantum Computing (AQC) algorithms. VQE often struggles with convergence due to the energy landscape being highly non-convex and the existence of barren plateaux, and implementing AQC is beyond the capabilities of current quantum devices as it requires deep circuits. Adiabatically-inspired algorithms aim to fill this gap. In this paper, we first present a unifying framework for these algorithms and then benchmark the following methods: the Adiabatically Assisted VQE (AAVQE) (Garcia-Saez and Latorre (2018)), the Variational Adiabatic Quantum Computing (VAQC) (Harwood et al (2022)), and the Adiabatic Quantum Computing with Parametrized Quantum Circuits (AQC-PQC) (Kolotouros et al (2025)) algorithms. Second, we introduce a novel hybrid approach termed G-AQC-PQC, which generalizes the AQC-PQC method, and combines adiabatic-inspired initialization with the low-memory BFGS optimizer, reducing the quantum computational cost of the method. Third, we compare the accuracy of the methods for chemistry applications using the beryllium hydride molecule (BeH$_2$). We compare the approaches across a number of different choices (ansätze types, depth, discretization steps, initial Hamiltonian, adiabatic schedules and method used). Our results show that the G-AQC-PQC outperforms conventional VQE. We further discuss limitations such as the zero-gradient problem and identify regimes where adiabatically-inspired methods offer a tangible advantage for near-term quantum chemistry applications.

quant-pharXiv:2512.14449•Dec 17, 2025•0 citations

Universal Parent Hamiltonians for Adiabatic Warm Starts

Feng Qian, Peter J. Love

Computing the ground state properties of quantum systems is an important potential application of quantum computing. The success probability of quantum phase estimation approaches to ground state problems is proportional to the overlap of the input state with the ground state. Local ansatz state approaches suffer from the orthogonality catastrophe, whereas adiabatic state preparation (ASP) can prepare good approximations with a cost growing with the inverse square of the minimum spectral gap along the adiabatic path. When the initial and target Hamiltonians lie in quantum phases separated by a first order phase transition, the minimum spectral gap along the adiabatic path becomes exponentially small as a function of system size. We pursue a solution to this problem based on choosing an initial Hamiltonian for adiabatic state preparation whose ground state lies in the same quantum phase as the target ground state. We develop a protocol for universal adiabatic warm starts with universal parent Hamiltonians (UPHAWS) that can initialize ASP in any state whose preparation circuit is known. We use the Feynman--Kitaev clock Hamiltonian as a universal parent Hamiltonian, for preparation circuits with and without mid circuit measurement. We benchmark the framework on a $\mathbb{Z}_2$-symmetric matrix product state (MPS) family interpolating to a target GHZ Hamiltonian, and on the linear $H_6$ chain under symmetric bond stretching. For the $H_6$ system a bond-dimension-$4$ matrix product state warm-start increases the minimum gap on the adiabatic path by a factor of two relative to the Hartree-Fock initialization. To perform these classical benchmark simulations we develop a momentum-space truncation of the adiabatic Hamiltonian that may be of independent interest.

quant-pharXiv:2607.16857•Jul 21, 2026•0 citations

Algorithmic Shadow Spectroscopy

Hans Hon Sang Chan, Richard Meister, Matthew L. Goh +1 more

We present shadow spectroscopy as a simulator-agnostic quantum algorithm for estimating energy gaps using very few circuit repetitions (shots) and no extra resources (ancilla qubits) beyond performing time evolution and measurements. The approach builds on the fundamental feature that every observable property of a quantum system must evolve according to the same harmonic components: we can reveal them by postprocessing classical shadows of time-evolved quantum states to extract a large number of time-periodic signals N o ∝ 10 8 , whose frequencies correspond to Hamiltonian energy differences with precision limited as ϵ ∝ 1 / T for simulation time T . We provide strong analytical guarantees that (a) quantum resources scale as O ( log  N o ) , while the classical computational complexity is linear O ( N o ) , (b) the signal-to-noise ratio increases with the number of processed signals as ∝ N o , and (c) spectral peak positions are immune to reasonable levels of noise. We demonstrate our approach on model spin systems and the excited-state conical intersection of molecular CH 2 and verify that our method is indeed intuitively easy to use in practice, robust against gate noise, amiable to a new type of algorithmic-error mitigation technique, and uses relatively few shots given a reasonable initial state is supplied—we demonstrate that even 10 shots per time step can be sufficient. Finally, we measured a high-quality, experimental shadow spectrum of a spin chain on readily available IBM quantum computers, achieving the same precision as in noise-free simulations without using any advanced error mitigation, and verified scalability in tensor-network simulations of up to 100-qubit systems.

arXiv:2212.11036•PRX Quantum•Mar 17, 2025•19 citations

Trainable, self-verifying variational quantum eigensolver using adiabatic warm starts

Bojan Žunkovič, Marco Ballarin, Lewis Wright +1 more

We study an adiabatic variant of the variational quantum eigensolver (VQE) in which VQE is performed iteratively for a sequence of Hamiltonians along an adiabatic path. We derive the conditions under which gradient-based optimization successfully prepares the adiabatic ground states. These conditions define a trainable tracking region. Additionally, when combined with a gap lower bound, energy-standard-deviation measurements certify eigenstate accuracy at run-time and verify convergence to the global optimum.

quant-pharXiv:2602.17612•Physical Review Research•Feb 20, 2026•0 citations
Abstract CoVariance Root finding with classical shadows (CoVaR) was recently introduced as a new paradigm for training variational quantum circuits. Common approaches, such as variants of the variational quantum eigensolver, aim to optimise a non-linear classical cost function and thus suffer from, e.g. poor local minima, high shot requirements and barren plateaus. In contrast, CoVaR fully exploits powerful classical shadows and finds joint roots of a very large number of covariances using only a logarithmic number of shots and linearly scaling classical computing resources. As a result, CoVaR has been demonstrated to be particularly robust against local traps, however, its main limitation has been that it requires a sufficiently good initial state. We address this limitation by introducing an adiabatic morphing of the target Hamiltonian and demonstrate in a broad range of application examples that CoVaR can successfully prepare eigenstates of the target Hamiltonian when no initial warm start is known. CoVaR succeeds even when Hamiltonian energy gaps are very small—this is in stark contrast to adiabatic evolution and phase estimation algorithms where circuit depths scale inversely with the Hamiltonian energy gaps. On the other hand, when the energy gaps are relatively small then adiabatic CoVaR may converge to higher excited states as opposed to a targeted specific low-lying state. Nevertheless, we exploit this feature of adiabatic CoVaR and demonstrate that it can be used to map out the low lying spectrum of a Hamiltonian which can be useful in practical applications, such as estimating thermal properties or in high-energy physics.
Exploiting near-term quantum computers and achieving practical value is a considerable and exciting challenge.Most prominent candidates as variational algorithms typically aim to find the ground state of a Hamiltonian by minimizing a single classical (energy) surface which is sampled from by a quantum computer.Here, we introduce a method we call CoVaR, an alternative means to exploit the power of variational circuits: We find eigenstates by finding joint roots of a polynomially growing number of properties of the quantum state as covariance functions between the Hamiltonian and an operator pool of our choice.The most remarkable feature of our CoVaR approach is that it allows us to fully exploit the extremely powerful classical shadow techniques; i.e., we simultaneously estimate a very large number > 10 4 -10 7 of covariances.We randomly select covariances and estimate analytical derivatives at each iteration applying a stochastic Levenberg-Marquardt step via a large but tractable linear system of equations that we solve with a classical computer.We prove that the cost in quantum resources per iteration is comparable to a standard gradient estimation; however, we observe in numerical simulations a very significant improvement by many orders of magnitude in convergence speed.CoVaR is directly analogous to stochastic gradient-based optimizations of paramount importance to classical machine learning while we also offload significant but tractable work onto the classical processor.As we demonstrate numerically, the approach shares features with phase-estimation protocols that prepare eigenstates with a dominant initial fidelity contribution.
Reliable preparation of many-body ground states is an essential task in quantum computing, with applications spanning areas from chemistry and materials modeling to quantum optimization and benchmarking. A variety of approaches have been proposed to tackle this problem, including variational methods. However, variational training often struggle to navigate complex energy landscapes, frequently encountering suboptimal local minima or suffering from barren plateaus. In this work, we introduce an iterative strategy for ground-state preparation based on a stepwise (discretized) Hamiltonian deformation. By complementing the Variational Quantum Eigensolver (VQE) with adiabatic principles, we demonstrate that solving a sequence of intermediate problems facilitates tracking the ground-state manifold toward the target system, even as we scale the system size. We provide a rigorous theoretical foundation for this approach, proving a lower bound on the loss variance that suggests trainability throughout the deformation, provided the system remains away from gap closings. Numerical simulations, including the effects of shot noise, confirm that this path-dependent tracking consistently converges to the target ground state.
Quantum computing promises to efficiently and accurately solve many important problems in quantum chemistry which elude classical solvers, such as the electronic structure problem of highly correlated materials. Two leading methods in solving the ground state problem are the Variational Quantum Eigensolver (VQE) and Adiabatic Quantum Computing (AQC) algorithms. VQE often struggles with convergence due to the energy landscape being highly non-convex and the existence of barren plateaux, and implementing AQC is beyond the capabilities of current quantum devices as it requires deep circuits. Adiabatically-inspired algorithms aim to fill this gap. In this paper, we first present a unifying framework for these algorithms and then benchmark the following methods: the Adiabatically Assisted VQE (AAVQE) (Garcia-Saez and Latorre (2018)), the Variational Adiabatic Quantum Computing (VAQC) (Harwood et al (2022)), and the Adiabatic Quantum Computing with Parametrized Quantum Circuits (AQC-PQC) (Kolotouros et al (2025)) algorithms. Second, we introduce a novel hybrid approach termed G-AQC-PQC, which generalizes the AQC-PQC method, and combines adiabatic-inspired initialization with the low-memory BFGS optimizer, reducing the quantum computational cost of the method. Third, we compare the accuracy of the methods for chemistry applications using the beryllium hydride molecule (BeH2_22​). We compare the approaches across a number of different choices (ansätze types, depth, discretization steps, initial Hamiltonian, adiabatic schedules and method used). Our results show that the G-AQC-PQC outperforms conventional VQE. We further discuss limitations such as the zero-gradient problem and identify regimes where adiabatically-inspired methods offer a tangible advantage for near-term quantum chemistry applications.
Computing the ground state properties of quantum systems is an important potential application of quantum computing. The success probability of quantum phase estimation approaches to ground state problems is proportional to the overlap of the input state with the ground state. Local ansatz state approaches suffer from the orthogonality catastrophe, whereas adiabatic state preparation (ASP) can prepare good approximations with a cost growing with the inverse square of the minimum spectral gap along the adiabatic path. When the initial and target Hamiltonians lie in quantum phases separated by a first order phase transition, the minimum spectral gap along the adiabatic path becomes exponentially small as a function of system size. We pursue a solution to this problem based on choosing an initial Hamiltonian for adiabatic state preparation whose ground state lies in the same quantum phase as the target ground state. We develop a protocol for universal adiabatic warm starts with universal parent Hamiltonians (UPHAWS) that can initialize ASP in any state whose preparation circuit is known. We use the Feynman--Kitaev clock Hamiltonian as a universal parent Hamiltonian, for preparation circuits with and without mid circuit measurement. We benchmark the framework on a Z2\mathbb{Z}_2Z2​-symmetric matrix product state (MPS) family interpolating to a target GHZ Hamiltonian, and on the linear H6H_6H6​ chain under symmetric bond stretching. For the H6H_6H6​ system a bond-dimension-444 matrix product state warm-start increases the minimum gap on the adiabatic path by a factor of two relative to the Hartree-Fock initialization. To perform these classical benchmark simulations we develop a momentum-space truncation of the adiabatic Hamiltonian that may be of independent interest.
We present shadow spectroscopy as a simulator-agnostic quantum algorithm for estimating energy gaps using very few circuit repetitions (shots) and no extra resources (ancilla qubits) beyond performing time evolution and measurements. The approach builds on the fundamental feature that every observable property of a quantum system must evolve according to the same harmonic components: we can reveal them by postprocessing classical shadows of time-evolved quantum states to extract a large number of time-periodic signals N o ∝ 10 8 , whose frequencies correspond to Hamiltonian energy differences with precision limited as ϵ ∝ 1 / T for simulation time T . We provide strong analytical guarantees that (a) quantum resources scale as O ( log  N o ) , while the classical computational complexity is linear O ( N o ) , (b) the signal-to-noise ratio increases with the number of processed signals as ∝ N o , and (c) spectral peak positions are immune to reasonable levels of noise. We demonstrate our approach on model spin systems and the excited-state conical intersection of molecular CH 2 and verify that our method is indeed intuitively easy to use in practice, robust against gate noise, amiable to a new type of algorithmic-error mitigation technique, and uses relatively few shots given a reasonable initial state is supplied—we demonstrate that even 10 shots per time step can be sufficient. Finally, we measured a high-quality, experimental shadow spectrum of a spin chain on readily available IBM quantum computers, achieving the same precision as in noise-free simulations without using any advanced error mitigation, and verified scalability in tensor-network simulations of up to 100-qubit systems.
We study an adiabatic variant of the variational quantum eigensolver (VQE) in which VQE is performed iteratively for a sequence of Hamiltonians along an adiabatic path. We derive the conditions under which gradient-based optimization successfully prepares the adiabatic ground states. These conditions define a trainable tracking region. Additionally, when combined with a gap lower bound, energy-standard-deviation measurements certify eigenstate accuracy at run-time and verify convergence to the global optimum.