Quantum processing units boost entanglement at the level of hardware and enable physical simulations of highly correlated electron states in molecules and intermolecular chemical bonds. The variational quantum eigensolver provides a hardware-efficient toolbox for ground state simulation; however, with limitations in precision. Even in the absence of noise, the algorithm may result into a biased energy estimation, particularly with some shallower ansatz types. Noise additionally degrades entanglement and hinders the ground state energy estimation (especially if the noise is not fully characterized). Here we develop a method to exploit the quantum-classical interface provided by informationally complete measurements to use classical software on top of the hardware entanglement booster for ansatz- and noise-related error reduction. We use the tensor network representation of a quantum channel that drives the noisy state toward the ground one. The tensor network is a completely positive map by construction, but we elaborate on making the trace preservation condition local so as to activate the sweeping variational optimization. This method brings into reach energies below the noiseless ansatz by creating additional correlations among the qubits and denoising them. Analyzing the example of the stretched water molecule with a tangible entanglement, we argue that a hybrid strategy of using the quantum hardware together with the classical software outperforms a purely classical strategy provided the classical parts have the same bond dimension. The proposed optimization algorithm extends the variety of noise mitigation methods and facilitates the more accurate study of the energy landscape for deformed molecules. The algorithm can be applied as the final postprocessing step in the quantum hardware simulation of protein-ligand complexes in the context of drug design.
Quantum processing units boost entanglement at the level of hardware and enable physical simulations of highly correlated electron states in molecules and intermolecular chemical bonds. The variational quantum eigensolver provides a hardware-efficient toolbox for ground state simulation; however, with limitations in precision. Even in the absence of noise, the algorithm may result into a biased energy estimation, particularly with some shallower ansatz types. Noise additionally degrades entanglement and hinders the ground state energy estimation (especially if the noise is not fully characterized). Here we develop a method to exploit the quantum-classical interface provided by informationally complete measurements to use classical software on top of the hardware entanglement booster for ansatz- and noise-related error reduction. We use the tensor network representation of a quantum channel that drives the noisy state toward the ground one. The tensor network is a completely positive map by construction, but we elaborate on making the trace preservation condition local so as to activate the sweeping variational optimization. This method brings into reach energies below the noiseless ansatz by creating additional correlations among the qubits and denoising them. Analyzing the example of the stretched water molecule with a tangible entanglement, we argue that a hybrid strategy of using the quantum hardware together with the classical software outperforms a purely classical strategy provided the classical parts have the same bond dimension. The proposed optimization algorithm extends the variety of noise mitigation methods and facilitates the more accurate study of the energy landscape for deformed molecules. The algorithm can be applied as the final postprocessing step in the quantum hardware simulation of protein-ligand complexes in the context of drug design.
As a hybrid quantum-classical algorithm, the variational quantum eigensolver is widely applied in quantum chemistry simulations, especially in computing the electronic structure of complex molecular systems. However, on existing noisy intermediate-scale quantum devices, some factors such as quantum decoherence, measurement errors, and gate operation imprecisions are unavoidable. To overcome these challenges, this study proposes an efficient noise-mitigating variational quantum eigensolver for accurate computation of molecular ground state energies in noisy environments. We design the quantum circuit with reference to the structure of matrix product states and utilize it to pre-train the circuit parameters, which ensures circuit stability and mitigates fluctuations caused by initialization. We also employ zero-noise extrapolation to mitigate quantum noise and combine it with neural networks to improve the accuracy of the noise-fitting function, which significantly eliminates noise interference. Furthermore, we implement an intelligent grouping strategy for measuring Hamiltonian Pauli strings, which not only reduces measurement errors but also improves sampling efficiency. We perform numerical simulations to solve the ground state energy of the H4molecule by using MindSpore Quantum framework, and the results demonstrate that our algorithm can constrain noise errors within the range of O(10−2)∼O(10−1), outperforming mainstream variational quantum eigensolvers. This work provides a new strategy for high-precision quantum chemistry calculations on near-term noisy quantum hardware.
The variational quantum eigensolver (VQE) is a promising algorithm for demonstrating quantum advantage in the noisy intermediate-scale quantum (NISQ) era. However, optimizing VQE from random initial starting parameters is challenging due to a variety of issues including barren plateaus, optimization in the presence of noise, and slow convergence. While simulating quantum circuits classically is generically difficult, classical computing methods have been developed extensively, and powerful tools now exist to approximately simulate quantum circuits. This opens up various strategies that limit the amount of optimization that needs to be performed on quantum hardware. Here we present and benchmark an approach where we find good starting parameters for parameterized quantum circuits by classically simulating VQE by approximating the parameterized quantum circuit (PQC) as a matrix product state (MPS) with a limited bond dimension. Calling this approach the variational tensor network eigensolver (VTNE), we apply it to the 1D and 2D Fermi-Hubbard model with system sizes that use up to 32 qubits. We find that in 1D, VTNE can find parameters for PQC whose energy error is within 0.5% relative to the ground state. In 2D, the parameters that VTNE finds have significantly lower energy than their starting configurations, and we show that starting VQE from these parameters requires non-trivially fewer operations to come down to a given energy. The higher the bond dimension we use in VTNE, the less work needs to be done in VQE. By generating classically optimized parameters as the initialization for the quantum circuit one can alleviate many of the challenges that plague VQE on quantum computers.
Quantum hardware noise remains a major obstacle to accurate molecular simulations using the variational quantum eigensolver (VQE). We introduce a novel error-mitigation strategy based on the extended random phase approximation (ERPA), which reconstructs the two-particle reduced density matrix (2-RDM) from approximate transition density matrices, effectively filtering errors in quantum measurements. Unlike quantum subspace expansion, the approach requires only the one- and two-particle RDMs. Since ERPA preserves the input 1-RDM, we further propose a hybrid quantum-classical scheme employing an inexpensive density matrix renormalization group (DMRG) calculation to provide a noise-free approximation to this quantity. Benchmarks on water double-bond stretching and cyclobutadiene automerization, performed on IBM quantum hardware, demonstrate reductions in energy errors by orders of magnitude. Combined with multireference adiabatic connection (AC0) corrections, the approach additionally recovers dynamical correlation beyond the active space, yielding accurate molecular potential energy curves and reaction barriers. Our results establish ERPA as a promising, measurement-efficient route toward reliable quantum-classical electronic structure calculations.
Abstract Electronic ground states are of central importance in chemical simulations, but have remained beyond the reach of efficient classical algorithms except in cases of weak electron correlation or one-dimensional spatial geometry. We introduce a hybrid quantum-classical eigenvalue solver that constructs a wavefunction ansatz from a linear combination of matrix product states in rotated orbital bases, enabling the characterization of strongly correlated ground states with arbitrary spatial geometry. The energy is converged via a gradient-free generalized sweep algorithm based on quantum subspace diagonalization, with a potentially exponential speedup in the off-diagonal matrix element contractions upon translation into compact quantum circuits of linear depth in the number of qubits. Chemical accuracy is attained in numerical experiments for both a stretched water molecule and an octahedral arrangement of hydrogen atoms, achieving substantially better correlation energies compared to a unitary coupled-cluster benchmark, with orders of magnitude reductions in quantum resource estimates and a surprisingly high tolerance to shot noise. This proof-of-concept study suggests a promising new avenue for scaling up simulations of strongly correlated chemical systems on near-term quantum hardware.
molecule, emphasizing practical implementation and performance on current quantum hardware. Our research presents a comparative study of HEA and UCCSD ansätze on noiseless and noisy simulations and implements VQE on recent IBM quantum computer noise models and a real quantum computer, IBM Fez, providing a fully functional code employing Qiskit 1.2. Our experiments confirm UCCSD's reliability in ideal conditions, while the HEA demonstrates greater robustness to hardware noise, achieving chemical accuracy on state-vector simulation (SVS). The results reveal that achieving ground-state energy within chemical accuracy is feasible without error mitigation during VQE convergence. We demonstrate that current quantum devices effectively optimize circuit parameters despite misestimating simulated energies. The SVS-evaluated energies provide a more accurate representation of the solution quality compared to QPU-estimated energy values, indicating that VQE converges to the correct ground state despite quantum noise. Our study also applies noise mitigation as a postprocessing technique, using zero-noise extrapolation (ZNE) on a real quantum computer. The detailed methodologies presented in this study, including Hamiltonian construction and Fermionic-to-qubit transformations, facilitate flexible adaptation of the VQE approach for various algorithm variants and across different levels of algorithmic implementation.
We propose a hybrid quantum-classical algorithm for approximating the ground state of two-dimensional quantum systems using an isometric tensor network ansatz, which maps naturally to quantum circuits. Inspired by the density matrix renormalization group, we optimize tensors sequentially by diagonalizing a series of effective Hamiltonians. These are constructed using a tomography-inspired method on a qubit subset whose size depends only on the bond dimension. Our approach leverages quantum computers to enable accurate solutions without relying on approximate contractions, circumventing the exponential complexity faced by classical techniques. We demonstrate our method on the two-dimensional (2D) transverse-field Ising model, achieving ground-state optimization on up to 25 qubits with modest quantum overhead -- significantly less than standard solutions based on variational quantum eigensolvers. Overall, our results offer a path towards scalable variational quantum algorithms in both noisy and fault-tolerant regimes.