Róbert Izsák

We have developed a low-cost method using the extended version of the second-order algebraic diagrammatic construction method for core-ionization potential calculations (IP-ADC(2)-x). The efficiency of the method is enhanced by the incorporation of frozen natural orbitals (FNOs) and natural auxiliary functions (NAFs) in combination with the core–valence separation (CVS) and density fitting (DF) approximations. This approach balances computational efficiency and precision remarkably well. Employing natural orbitals significantly reduces the size of the virtual space while maintaining a systematically controlled accuracy, delivering a substantial speed-up compared to the standard method. Furthermore, the error due to the truncation of the virtual orbital space can be reduced by adding a simple correction to the FNO-IP-ADC(2)-x results. The overall accuracy of the method is governed by setting up FNO and NAF thresholds. The use of the spin-free exact two-component Hamiltonian (SFX2C1e) improves the agreement with experimental values. We have also investigated the use of projector-based embedding theory to calculate core-ionization potentials for large molecules. Using the embedded FNO-IP-ADC(2)-x method, we have successfully computed the core-ionization energies of N-doped fullerene (azafullerene) and chlorophyll-a molecules, with approximately 2600 and 4000 basis functions, respectively, with an impressive speed-up.

Quantum simulations of strongly interacting fermionic systems, such as those described by the Hubbard model, are promising candidates for useful early fault-tolerant quantum computing applications. This paper presents tile Trotterization, a generalization of plaquette Trotterization, which uses a set of tiles to construct Trotter decompositions of arbitrary lattice Hubbard models. The tile Trotterization scheme also enables the simulation of more complex models, including the extended Hubbard model. We improve previous Hubbard-model commutator bounds, further provide tight commutator bounds for periodic extended Hubbard models, and demonstrate the use of tensor network methods for this task. We consider applications of tile Trotterization to simulate hexagonal lattice Hubbard models and compare the resource requirements of tile Trotterization for performing quantum phase estimation to a qubitization-based approach, demonstrating that tile Trotterization scales more efficiently with system size. These advancements significantly broaden the potential applications of early fault-tolerant quantum computers to models of practical interest in materials research and organic chemistry.

Quantum simulation of materials is a promising application area of quantum computers. To practically realize this promise, we must reduce quantum resources while maintaining accuracy. In electronic structure calculations on classical computers, resource reduction has been achieved by using the projector augmented-wave method (PAW) and plane wave basis sets. However, the PAW method generalized for many-body states introduces nonorthogonality effects which impede its direct application to quantum computing. In this work, we develop a unitary variant of the PAW (UPAW) that preserves the orthogonality constraints. We provide a linear-combination-of-unitaries decomposition of the UPAW Hamiltonian to enable ground state estimation using qubitized quantum phase estimation. Additionally, we further improve algorithmic efficiency by extending classical down-sampling techniques into the quantum setting. We then estimate quantum resources for crystalline solids to estimate the energy within chemical accuracy with respect to the full basis set limit, and also consider a supercell approach which is more suitable for calculations of defect states. We provide the quantum resources for energy estimation of a nitrogen-vacancy defect center in diamond which is a challenging system for classical algorithms and a quintessential problem in the studies of quantum point defects.

In this educational paper, we will discuss calculations on the hydrogen molecule on both classical and quantum computers. In the former case, we will discuss the calculation of molecular integrals that can then be used to calculate potential energy curves at the Hartree-Fock level and to correct them by obtaining the exact results for all states in the minimal basis. Some aspects of spin-symmetry will also be discussed. In the case of quantum computing, we will start out from the second-quantized Hamiltonian and qubit mappings. We then provide circuits for quantum phase estimation using two different algorithms: Trotterization and qubitization. Finally, the significance of quantum error correction will be briefly discussed.

Quantum phase estimation (QPE) is a key quantum algorithm, which has been widely studied as a method to perform chemistry and solid-state calculations on future fault-tolerant quantum computers. Recently, several authors have proposed statistical alternatives to QPE that have benefits on early fault-tolerant devices, including shorter circuits and better suitability for error-mitigation techniques. However, experimental investigations of the algorithm on real quantum processors are lacking. Here, we implement statistical phase estimation on Rigetti’s superconducting processors. Specifically, we use a modification of the Lin and Tong [PRX Quantum 3, 010318 (2022)] algorithm with the improved Fourier approximation of Wan et al. [Phys. Rev. Lett. 129, 030503 (2022)] and apply a variational-compilation technique to reduce the circuit depth. We then incorporate error-mitigation strategies including zero-noise extrapolation and readout-error mitigation with bit-flip averaging. We propose a new method to estimate energies from the statistical phase estimation data, which is found to improve the accuracy in the final energy estimates by 1–2 orders of magnitude with respect to prior theoretical bounds, reducing the cost of performing accurate phase-estimation calculations. We apply these methods to chemistry problems for active spaces up to four electrons in four orbitals, including the application of a quantum embedding method, and use them to correctly estimate energies within chemical precision. Our work demonstrates that statistical phase estimation has a natural resilience to noise, particularly after mitigating coherent errors, and can achieve far higher accuracy than suggested by previous analysis, demonstrating its potential as a valuable quantum algorithm for early fault-tolerant devices.

In this perspective, the various measures of electron correlation used in wavefunction theory, density functional theory and quantum information theory are briefly reviewed. We then focus on a more traditional metric based on dominant weights in the full configuration solution and discuss its behaviour with respect to the choice of the $N$-electron and the one-electron basis. The impact of symmetry is discussed and we emphasize that the distinction between determinants, configuration state functions and configurations as reference functions is useful because the latter incorporate spin-coupling into the reference and should thus reduce the complexity of the wavefunction expansion. The corresponding notions of single determinant, single spin-coupling and single configuration wavefunctions are discussed and the effect of orbital rotations on the multireference character is reviewed by analysing a simple model system. In molecular systems, the extent of correlation effects should be limited by finite system size and in most cases the appropriate choices of one-electron and $N$-electron bases should be able to incorporate these into a low-complexity reference function, often a single configurational one.

Computational chemistry is an essential tool in the pharmaceutical industry. Quantum computing is a fast evolving technology that promises to completely shift the computational capabilities in many areas of chemical research by bringing into reach currently impossible calculations. This perspective illustrates the near-future applicability of quantum computation of molecules to pharmaceutical problems. We briefly summarize and compare the scaling properties of state-of-the-art quantum algorithms and provide novel estimates of the quantum computational cost of simulating progressively larger embedding regions of a pharmaceutically relevant covalent protein-drug complex involving the drug Ibrutinib. Carrying out these calculations requires an error-corrected quantum architecture that we describe. Our estimates showcase that recent developments on quantum phase estimation algorithms have dramatically reduced the quantum resources needed to run fully quantum calculations in active spaces of around 50 orbitals and electrons, from estimated over 1000 years using the Trotterization approach to just a few days with sparse qubitization, painting a picture of fast and exciting progress in this nascent field.

A second quantised formulation for the evaluation of spin-dependent properties in the UHF basis will be introduced. The configuration interaction singles ansatz will be used as an illustration of the practical usage of this formulation and the results will be compared to spin-restricted cases of the same ansatz. On a more tentative basis, we will also discuss the notion of quasi-spin-adaptation which ensures that the ansatz becomes spin-adapted as the unrestricted orbitals approach the restricted ones. GRAPHICAL ABSTRACT

Quantum computers are special purpose machines that are expected to be particularly useful in simulating strongly correlated chemical systems. The quantum computer excels at treating a moderate number of orbitals within an active space in a fully quantum mechanical manner. We present a quantum phase estimation calculation on F₂ in a (2,2) active space on Rigetti's Aspen‐11 QPU. While this is a promising start, it also underlines the need for carefully selecting the orbital spaces treated by the quantum computer. In this work, a scheme for selecting such an active space automatically is described and simulated results obtained using both the quantum phase estimation (QPE) and variational quantum eigensolver (VQE) algorithms are presented and combined with a subtractive method to enable accurate description of the environment. The active occupied space is selected from orbitals localized on the chemically relevant fragment of the molecule, while the corresponding virtual space is chosen based on the magnitude of interactions with the occupied space calculated from perturbation theory. This protocol is then applied to two chemical systems of pharmaceutical relevance: the enzyme [Fe] hydrogenase and the photosenzitizer temoporfin. While the sizes of the active spaces currently amenable to a quantum computational treatment are not enough to demonstrate quantum advantage, the procedure outlined here is applicable to any active space size, including those that are outside the reach of classical computation.