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arXiv:2610.03989

Certification failure in variational-principle observables

Ganga Singh Manchanda

quant-phSubmitted Oct 2, 2026• Announced Oct 6, 2026• 0 citations
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Abstract

Algorithms which prepare states via the variational principle assume that closeness in energy implies closeness in state; we formalise how this assumption can fail by showing that a ground-state energy estimate converged to within $ε$ of the true value is compatible with observable expectation values in error by up to $\sqrt{2εχ_O}$ at leading order, where $χ_O$ is the static susceptibility of the observable. As a result, the observable error can be orders of magnitude larger than the energy error, a mechanism we call non-linear error amplification, and because a variational algorithm cannot guarantee the correctness of these properties from energy alone, energy does not necessarily certify an observable. In light of our findings, we examine three algorithms: the variational quantum eigensolver on the transverse-field Ising chain, the density-matrix renormalisation group on a pair of weakly coupled hydrogen chains, and Krylov quantum diagonalisation on lithium fluoride. In each case we demonstrate that a well-converged energy can leave observables uncertified.

Comments
10 pages, 4 figures

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Algorithms which prepare states via the variational principle assume that closeness in energy implies closeness in state; we formalise how this assumption can fail by showing that a ground-state energy estimate converged to within εεε of the true value is compatible with observable expectation values in error by up to 2εχO\sqrt{2εχ_O}2εχO​​ at leading order, where χOχ_OχO​ is the static susceptibility of the observable. As a result, the observable error can be orders of magnitude larger than the energy error, a mechanism we call non-linear error amplification, and because a variational algorithm cannot guarantee the correctness of these properties from energy alone, energy does not necessarily certify an observable. In light of our findings, we examine three algorithms: the variational quantum eigensolver on the transverse-field Ising chain, the density-matrix renormalisation group on a pair of weakly coupled hydrogen chains, and Krylov quantum diagonalisation on lithium fluoride. In each case we demonstrate that a well-converged energy can leave observables uncertified.
We propose a variational quantum algorithm for estimating microcanonical expectation values in models obeying the eigenstate thermalization hypothesis. Using a relaxed criterion for convergence of the variational optimization loop, the algorithm generates weakly entangled superpositions of eigenstates at a given target energy density. An ensemble of these variational states is then used to estimate microcanonical averages of local operators, with an error whose dominant contribution decreases initially as a power law in the size of the ensemble and is ultimately limited by a small bias. We apply the algorithm to the one-dimensional mixed-field Ising model, where it converges for ansatz circuits of depth roughly linear in system size. The most accurate thermal estimates are produced for intermediate energy densities. In our error analysis, we find connections with recent works investigating the underpinnings of the eigenstate thermalization hypothesis. In particular, the failure of energy-basis matrix elements of local operators to behave as independent random variables is a potential source of error that the algorithm can overcome by averaging over an ensemble of variational states.
Abstract Quantum computers have the potential to help solve a range of physics and chemistry problems, but noise in quantum hardware currently limits our ability to obtain accurate results from the execution of quantum-simulation algorithms. Various methods have been proposed to mitigate the impact of noise on variational algorithms, including several that model the noise as damping expectation values of observables. In this work, we benchmark various methods, including a new method proposed here. We compare their performance in estimating the ground-state energies of several instances of the 1D mixed-field Ising model using the variational-quantum-eigensolver algorithm with up to 20 qubits on two of IBM’s quantum computers. We find that several error-mitigation techniques allow us to recover energies to within 10% of the true values for circuits containing up to about 25 ansatz layers, where each layer consists of CNOT gates between all neighboring qubits and Y -rotations on all qubits.
Quantum annealing and the variational quantum eigensolver are two promising quantum algorithms to find the ground state of complicated Hamiltonians on near-term quantum devices. However, it is necessary to limit the evolution time or the circuit depth as much as possible since otherwise decoherence will degrade the computation. Even when this is done, there always exists a non-negligible estimation error in the ground state energy. Here we propose a scalable extrapolation approach to mitigate this error. With an appropriate regression, we can significantly improve the estimation accuracy for quantum annealing and variational quantum eigensolver for fixed quantum resources. The inference is achieved by extrapolating the annealing time to infinity or extrapolating the variance to zero. The only additional overhead is an increase in the number of measurements by a constant factor. We verified the validity of our method with the transverse-field Ising model. The method is robust to noise, and the techniques are applicable to other physics problems. Analytic derivations for the quadratic convergence feature of the residual energy in quantum annealing and the linear convergence feature of energy variance are given.
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Abstract The variational quantum eigensolver (VQE) is a hybrid quantum–classical algorithm for quantum simulation that can be run on near-term quantum hardware. A challenge in VQE—as well as any other heuristic algorithm for finding ground states of Hamiltonians—is to know how close the algorithm’s output solution is to the true ground state, when we do not know what the true ground state or ground-state energy is. This is especially important in iterative algorithms such as VQE, where we need to decide when to stop iterating. Stopping when the current solution energy no longer changes rapidly from one iteration to the next is a common choice but in many situations can lead to stopping too early and outputting an incorrect result. Recent developments in Hamiltonian reconstruction—the inference of a Hamiltonian given an eigenstate—give a tool that can be used to assess the quality of a variational solution to a Hamiltonian-eigensolving problem: a metric quantifying the distance between the problem Hamiltonian and the reconstructed Hamiltonian can give an indication of whether the variational solution is an eigenstate, which can (as a special case) indicate if the variational solution is not the ground state. Crucially, computing the Hamiltonian-reconstruction (HR) distance does not rely on knowing the true ground state or ground-state energy. We propose and study using the HR distance as a metric for assessing the success of VQE eigensolving. In numerical simulations and in demonstrations on a cloud-based trapped-ion quantum computer, we show that for examples of both one-dimensional transverse-field-Ising (11 qubits) and two-dimensional J 1 – J 2 transverse-field-Ising (6 qubits) spin problems, the HR distance gives a helpful indication of whether VQE has yet found the ground state or not. Our demonstrations included cases where the energy plateaus as a function of the VQE iteration, which could have resulted in erroneous early stopping of the VQE algorithm, but where the HR distance correctly suggests to continue iterating. We find that HR distance has a useful correlation with the fidelity (i.e. state overlap) between the VQE solution and the true ground state, and with the energy difference between VQE solution’s energy and the true ground-state energy. Our work suggests that HR distance may be a useful tool for assessing success in VQE, including on noisy quantum processors in practice.
Solutions to many-body problem instances often involve an intractable number of degrees of freedom and admit no known approximations in general form. In practice, representing quantum-mechanical states of a given Hamiltonian using available numerical methods, in particular those based on variational Monte Carlo simulations, become exponentially more challenging with increasing system size. Recently quantum algorithms implemented as variational models have been proposed to accelerate such simulations. The variational ansatz states are characterized by a polynomial number of parameters devised in a way to minimize the expectation value of a given Hamiltonian, which is emulated by local measurements. In this study, we develop a means to certify the termination of variational algorithms. We demonstrate our approach by applying it to three models: the transverse field Ising model, the model of one-dimensional spinless fermions with competing interactions, and the Schwinger model of quantum electrodynamics. By means of comparison, we observe that our approach shows better performance near critical points in these models. We hence take a further step to improve the applicability and to certify the results of variational quantum simulators.