, calculated at a high-level of theory. Using the data set, we compared noiseless VQE simulations to conventionally performed density functional and wavefunction theory-based methods to understand the quality of results. We also investigated the effectiveness of a quantum state tomography-based error mitigation technique in applications of VQE under noise (simulated and real). Our findings reveal that the use of quantum error mitigation is crucial in the NISQ era and advantageous to yield almost noiseless quality results.
, calculated at a high-level of theory. Using the data set, we compared noiseless VQE simulations to conventionally performed density functional and wavefunction theory-based methods to understand the quality of results. We also investigated the effectiveness of a quantum state tomography-based error mitigation technique in applications of VQE under noise (simulated and real). Our findings reveal that the use of quantum error mitigation is crucial in the NISQ era and advantageous to yield almost noiseless quality results.
Minimizing the effect of noise is essential for quantum computers. The conventional method to protect qubits against noise is through quantum error correction. However, for current quantum hardware in the so-called noisy intermediate-scale quantum (NISQ) era, noise presents in these systems and is too high for error correction to be beneficial. Quantum error mitigation is a set of alternative methods for minimizing errors, including error extrapolation, probabilistic error cancellation, measurement error mitigation, subspace expansion, symmetry verification, virtual distillation, etc. The requirement for these methods is usually less demanding than error correction. Quantum error mitigation is a promising way of reducing errors on NISQ quantum computers. This paper gives a comprehensive introduction to quantum error mitigation. The state-of-art error mitigation methods are covered and formulated in a general form, which provides a basis for comparing, combining and optimizing different methods in future work.
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.
Quantum computing offers a potential for algorithmic speedups for applications, such as large-scale simulations in chemistry and physics. However, these speedups must yield results that are sufficiently accurate to predict realistic outcomes of experiments precisely. Delivering on the promise of high accuracy and precision requires methods to evaluate the computational accuracy of the quantum computing devices. We develop a framework to estimate the computational accuracy of near-term noisy, intermediate scale quantum (NISQ) computing devices using a quantum chemistry application. Application benchmarks that run on NISQ devices require techniques for mitigating errors to improve accuracy and precision. We use device agnostic error-mitigation schemes, quantum error detection and readout error detection, with post-selection to mitigate the dominant sources of noise. We evaluate the framework by simulating the ground state of molecular hydrogen with the variational quantum eigensolver (VQE) algorithm, estimating the energy and calculating the precision of the estimate using numerical simulations with realistic noise models. We first quantify the improvement in the logical error rate and state fidelity of the VQE application when encoded with the [[4,2,2]] quantum error detection code. When additionally encoded with readout error detection, we show that compared to the unencoded simulation, the encoded simulation yields a more accurate estimate by more than 1 mHa (0.027 eV) with comparable precision and higher state fidelity. Additionally, unlike the best estimate from the unencoded simulations, the results from the encoded simulation fall within the chemical accuracy threshold of 1.6 mHa of the exact energy. The estimated accuracy and precision indicate that current quantum computers can achieve error rates that yield useful outcomes for chemical applications.
ABSTRACT The Variational Quantum Eigensolver (VQE) is a widely studied hybrid classical‐quantum algorithm for approximating ground‐state energies in molecular and materials systems. This study benchmarks the performance of the VQE for calculating ground‐state energies of small aluminum clusters (, , and ) within a quantum‐density functional theory (DFT) embedding framework, systematically varying key parameters: (I) classical optimizers, (II) circuit types, (III) number of repetitions, (IV) simulator types, (V) basis sets, and (VI) noise models. All calculations were performed using quantum simulators to evaluate VQE performance under both idealized and noise‐augmented conditions. Our findings demonstrate that certain optimizers converge efficiently, while circuit choice and basis set selection have a marked impact on energy estimates, with higher‐level basis sets closely matching classical computation data from Numerical Python Solver (NumPy) and Computational Chemistry Comparison and Benchmark DataBase (CCCBDB). To approximate realistic conditions, we employed IBM noise models to simulate the effects of hardware noise. The results showed close agreement with CCCBDB benchmarks, with percent errors consistently below 0.2%. The results demonstrate that VQE can approximate energy estimates under simulated conditions for small aluminum clusters and highlight the importance of optimizing quantum‐DFT parameters to balance computational cost and precision. This work contributes to ongoing efforts to benchmark VQE in practical settings and lays the groundwork for future benchmarking tools for quantum chemistry and materials applications.
Quantum computers, currently in the noisy intermediate-scale quantum (NISQ) era, have started to provide scientists with a novel tool to explore quantum physics and chemistry. While several electronic systems have been extensively studied, Frenkel excitons, as prototypical optical excitations, remain among the less-explored applications. Here, we first use variational quantum deflation to calculate the eigenstates of the Frenkel Hamiltonian and evaluate the observables based on the oscillator strength for each eigenstate. Furthermore, using NISQ quantum computers requires performing error mitigation techniques alongside simulations. To deal with noisy qubits, we developed a deep-learning-based framework combined with a post-selection technique to learn the noise pattern and mitigate the error. Our mitigation methods work well and outperform the conventional post-selection and remain valid on real hardware.
Reducing the impact of errors and decoherence in near-term quantum computers, such as noisy intermediate-scale quantum (NISQ) devices, is critical for their practical implementation. These factors significantly limit the applicability of quantum algorithms, necessitating a comprehensive understanding of their physical origins to establish effective error mitigation strategies. In this study, we present a non-Markovian model of quantum state evolution and a quantum error mitigation cost function tailored for NISQ devices interacting with an environment represented by a set of simple harmonic oscillators as a noise source. Employing the projection operator formalism and both advanced and retarded propagators in time, we derive the reduced-density operator for the output quantum states in a time-convolutionless form by solving the quantum Liouville equation. We examine the output quantum state fluctuations for both identity and controlled-NOT (CNOT) gate operations in two-qubit operations using a range of input states. Subsequently, these results are compared with experimental data from ion-trap and superconducting quantum computing systems to estimate the crucial parameters of the cost functions for quantum error mitigation. Our findings reveal that the cost function for quantum error mitigation increases as the coupling strength between the quantum system and its environment intensifies. This study underscores the significance of non-Markovian models in understanding quantum state evolution and highlights the practical implications of the quantum error mitigation cost function when assessing experimental results from NISQ devices.