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The simulation of molecular electronic structure is an important application of quantum devices. Recently, it has been shown that quantum devices can be effectively combined with classical supercomputing centers in the context of the sample-based quantum diagonalization (SQD) algorithm. This allowed the largest electronic structure quantum simulation to date (77 qubits) and opened near-term devices to practical use cases in chemistry toward the hundred-qubit mark. However, the description of many important physical and chemical properties of those systems, such as photo-absorption/-emission, requires a treatment that goes beyond the ground state alone. In this work, we extend the SQD algorithm to determine low-lying molecular excited states. The extended-SQD method improves over the original SQD method in accuracy, at the cost of an additional computational step. It also improves over quantum subspace expansion based on single and double electronic excitations, a widespread approach to excited states on pre-fault-tolerant quantum devices, in both accuracy and efficiency. We employ the extended SQD method to compute the first singlet (S$_1$) and triplet (T$_1$) excited states of the nitrogen molecule with a correlation-consistent basis set, and the ground- and excited-state properties of the [2Fe-2S] cluster., Comment: 15 pages, 7 figures, comments are welcome!

We propose a variational scheme to represent composite quantum systems using multiple parameterized functions of varying accuracies on both classical and quantum hardware. The approach follows the variational principle over the entire system, and is naturally suited for scenarios where an accurate description is only needed in a smaller subspace. We show how to include quantum devices as high-accuracy solvers on these correlated degrees of freedom, while handling the remaining contributions with a classical device. We demonstrate the effectiveness of the protocol on spin chains and small molecules and provide insights into its accuracy and computational cost., Comment: 15 pages, 7 figures. The framework has been extended to include embeddings of classical variational methods

We propose an adaptive quantum algorithm to prepare accurate variational time evolved wave functions. The method is based on the projected Variational Quantum Dynamics (pVQD) algorithm, that performs a global optimization with linear scaling in the number of variational parameters. Instead of fixing a variational ansatz at the beginning of the simulation, the circuit is grown systematically during the time evolution. Moreover, the adaptive step does not require auxiliary qubits and the gate search can be performed in parallel on different quantum devices. We apply the new algorithm, named Adaptive pVQD, to the simulation of driven spin models and fermionic systems, where it shows an advantage when compared to both Trotterized circuits and non-adaptive variational methods. Finally, we use the shallower circuits prepared using the Adaptive pVQD algorithm to obtain more accurate measurements of physical properties of quantum systems on hardware., Comment: 11 pages, 9 figures

We propose an extension of the Variational Quantum Eigensolver (VQE) that leads to more accurate energy estimations and can be used to study excited states. The method is based on the introduction of a sequence of increasing penalties in the cost function. This approach does not require circuit modifications and thus can be applied with no additional depth cost. Through numerical simulations, we show that we are able to produce variational states with desired physical properties, such as total spin and charge. We assess its performance both on classical simulators and on currently available quantum devices, calculating the potential energy curves of small molecular systems in different physical configurations. Finally, we compare our method to the original VQE and to another extension, obtaining a better agreement with exact simulations for both energy and targeted physical quantities., Comment: 11 pages, 9 figures, published in IOP Quantum Science and Technology

We propose a variational quantum algorithm to study the real time dynamics of quantum systems as a ground-state problem. The method is based on the original proposal of Feynman and Kitaev to encode time into a register of auxiliary qubits. We prepare the Feynman-Kitaev Hamiltonian acting on the composed system as a qubit operator and find an approximate ground state using the Variational Quantum Eigensolver. We apply the algorithm to the study of the dynamics of a transverse field Ising chain with an increasing number of spins and time steps, proving a favorable scaling in terms of the number of two qubit gates. Through numerical experiments, we investigate its robustness against noise, showing that the method can be use to evaluate dynamical properties of quantum systems and detect the presence of dynamical quantum phase transitions by measuring Loschmidt echoes., Comment: 11 pages, 8 figures. Published on Phys. Rev. Research

We introduce a novel hybrid algorithm to simulate the real-time evolution of quantum systems using parameterized quantum circuits. The method, named "projected - Variational Quantum Dynamics" (p-VQD) realizes an iterative, global projection of the exact time evolution onto the parameterized manifold. In the small time-step limit, this is equivalent to the McLachlan's variational principle. Our approach is efficient in the sense that it exhibits an optimal linear scaling with the total number of variational parameters. Furthermore, it is global in the sense that it uses the variational principle to optimize all parameters at once. The global nature of our approach then significantly extends the scope of existing efficient variational methods, that instead typically rely on the iterative optimization of a restricted subset of variational parameters. Through numerical experiments, we also show that our approach is particularly advantageous over existing global optimization algorithms based on the time-dependent variational principle that, due to a demanding quadratic scaling with parameter numbers, are unsuitable for large parameterized quantum circuits., Comment: 7+4 pages, 8 figures; Manuscript revised for publication. Method: added Section 2.2, Results: added Figure 6, Appendix: added Appendix E with Figure 8

Quantum simulations of molecular systems on quantum computers often employ minimal basis sets of Gaussian orbitals. In comparison with more realistic basis sets, quantum simulations employing minimal basis sets require fewer qubits and quantum gates, but yield results of lower accuracy. A natural strategy to achieve more accurate results is to increase the basis set size, which in turn requires increasing the number of qubits and quantum gates. Here we explore the use of intrinsic atomic orbitals (IAOs) in quantum simulations of molecules, to improve the accuracy of energies and properties at the same computational cost required by a minimal basis. We investigate ground-state energies and one- and two-body density operators in the framework of the variational quantum eigensolver, employing and comparing different Ans\"{a}tze. We also demonstrate the use of this approach in the calculation of ground- and excited-states energies of small molecules by a combination of quantum algorithms, using IBM Quantum computers., Comment: 14 pages, 14 figures. Published in Physical Review A