Practical challenges in simulating quantum systems on classical computers have been widely recognized in the quantum physics and quantum chemistry communities over the past century. Although many approximation methods have been introduced, the complexity of quantum mechanics remains hard to appease. The advent of quantum computation brings new pathways to navigate this challenging and complex landscape. By manipulating quantum states of matter and taking advantage of their unique features such as superposition and entanglement, quantum computers promise to efficiently deliver accurate results for many important problems in quantum chemistry, such as the electronic structure of molecules. In the past two decades, significant advances have been made in developing algorithms and physical hardware for quantum computing, heralding a revolution in simulation of quantum systems. This Review provides an overview of the algorithms and results that are relevant for quantum chemistry. The intended audience is both quantum chemists who seek to learn more about quantum computing and quantum computing researchers who would like to explore applications in quantum chemistry.

In the last decade, the quantum chemical version of the density matrix renormalization group (DMRG) method has established itself as the method of choice for calculations of strongly correlated molecular systems. Despite its favourable scaling, it is in practice not suitable for computations of dynamic correlation. We present a novel method for accurate "post-DMRG" treatment of dynamic correlation based on the tailored coupled cluster (CC) theory in which the DMRG method is responsible for the proper description of non-dynamic correlation, whereas dynamic correlation is incorporated through the framework of the CC theory. We illustrate the potential of this method on prominent multireference systems, in particular N2, Cr2 molecules and also oxo-Mn(Salen) for which we have performed the first "post-DMRG" computations in order to shed light on the energy ordering of the lowest spin states.The coupled cluster (CC) approach, introduced to quantum chemistry (QC) by Čížek 1 , is one of the most accurate ab initio methods for the treatment of dynamic electron correlation. The advantages of this scheme include a compact description of the wave function, size-extensivity, invariance to orbital rotations together with a systematic hierarchy of approximations converging towards the full configuration interaction (FCI) limit 2 . Despite the great success of QC and in particular the CC methodology 3 in standard (single-reference) cases, the situation is dramatically different for strongly correlated (multireference) systems 4 , where the usual single-reference approaches become inaccurate or even completely break down. One category of methods designed for the treatment of such systems are multireference coupled cluster (MRCC) approaches, which generalize the CC exponential parameterization of the wave function 5-7 . Out of many formulations of MRCC theories, the class of methods relevant to this work are externally corrected CC, which extract information about the most important higher excitations or active space single and double excitations from an "external" calculation performed by a different method like complete active space self-consistent field (CASSCF) or multireference configuration interaction (MRCI) [8][9][10][11][12][13][14][15][16][17][18][19] . In this letter, we present a further development in this field concerning the tailored CC (TCC) method, where the information for external correction is obtained from a density matrix renormalization group (DMRG) calculation.DMRG is a very powerful approach suitable for treatment of strongly correlated systems originally developed in solid state physics [20][21][22] . The success of DMRG in this field motivated its application to QC problems [23][24][25][26][27][28][29][30][31] where it has proven the potential to outperform traditional QC methods for systems which require very large active spaces, like molecules containing several transition metal atoms 32,33 . Despite the favourable scaling of the DMRG method, it is computationally prohibitive to treat the dynamic cor...

Quantum simulations are bound to be one of the main applications of near-term quantum computers. Quantum chemistry and condensed matter physics are expected to benefit from these technological developments. Several quantum simulation methods are known to prepare a state on a quantum computer and measure the desired observables. The most resource economic procedure is the variational quantum eigensolver (VQE), which has traditionally employed unitary coupled cluster as the ansatz to approximate ground states of many-body fermionic Hamiltonians. A significant caveat of the method is that the initial state of the procedure is a single reference product state with no entanglement extracted from a classical Hartree-Fock calculation. In this work, we propose to improve the method by initializing the algorithm with a more general fermionic Gaussian state, an idea borrowed from the field of nuclear physics. We show how this Gaussian reference state can be prepared with a linear-depth circuit of quantum matchgates. By augmenting the set of available gates with nearest-neighbor phase coupling, we generate a low-depth circuit ansatz that can accurately prepare the ground state of correlated fermionic systems. This extends the range of applicability of the VQE to systems with strong pairing correlations such as superconductors, atomic nuclei, and topological materials.

Quantum computers are appealing for their ability to solve some tasks much faster than their classical counterparts. It was shown in [Aspuru-Guzik et al., Science 309, 1704 (2005)] that they, if available, would be able to perform the full configuration interaction (FCI) energy calculations with a polynomial scaling. This is in contrast to conventional computers where FCI scales exponentially. We have developed a code for simulation of quantum computers and implemented our version of the quantum FCI algorithm. We provide a detailed description of this algorithm and the results of the assessment of its performance on the four lowest lying electronic states of CH(2) molecule. This molecule was chosen as a benchmark, since its two lowest lying (1)A(1) states exhibit a multireference character at the equilibrium geometry. It has been shown that with a suitably chosen initial state of the quantum register, one is able to achieve the probability amplification regime of the iterative phase estimation algorithm even in this case.

The quantum mechanical description of the chemical bond is generally given in terms of delocalized bonding orbitals, or, alternatively, in terms of correlations of occupations of localised orbitals. However, in the latter case, multiorbital correlations were treated only in terms of two-orbital correlations, although the structure of multiorbital correlations is far richer; and, in the case of bonds established by more than two electrons, multiorbital correlations represent a more natural point of view. Here, for the first time, we introduce the true multiorbital correlation theory, consisting of a framework for handling the structure of multiorbital correlations, a toolbox of true multiorbital correlation measures, and the formulation of the multiorbital correlation clustering, together with an algorithm for obtaining that. These make it possible to characterise quantitatively, how well a bonding picture describes the chemical system. As proof of concept, we apply the theory for the investigation of the bond structures of several molecules. We show that the non-existence of well-defined multiorbital correlation clustering provides a reason for debated bonding picture.

Tensor network states and specifically matrix-product states have proven to be a powerful tool for simulating ground states of strongly correlated spin models. Recently, they have also been applied to interacting fermionic problems, specifically in the context of quantum chemistry. A new freedom arising in such non-local fermionic systems is the choice of orbitals, it being far from clear what choice of fermionic orbitals to make. In this work, we propose a way to overcome this challenge. We suggest a method intertwining the optimisation over matrix product states with suitable fermionic Gaussian mode transformations. The described algorithm generalises basis changes in the spirit of the Hartree-Fock method to matrix-product states, and provides a black box tool for basis optimisation in tensor network methods.Capturing strongly correlated quantum systems is one of the major challenges of modern theoretical and computational physics. Recent years have seen a surge of interest in the development of potent numerical methods based on tensor networks to approximate ground states of interacting lattice models [1][2][3][4][5][6][7], building upon the success of the density-matrix renormalisation group (DMRG) [1]. It has become clear that such ideas are also applicable to fermionic systems [8][9][10], and even to systems of quantum chemistry [11][12][13][14][15][16][17][18][19], lacking the locality present in lattice models in condensed-matter systems. Such tools allow in principle to approximate the full configuration interaction solution to good accuracy with reasonable effort, going in instances beyond conventional approaches to quantum chemistry, such as coupled cluster [20], configuration interaction or density-functional theory [21,22], as convincingly shown by first implementations of DMRG algorithms in quantum chemistry (QC-DMRG) [11][12][13][14][15].Yet, there is a new obstacle to be overcome: Tensor network methods have originally been tailored to capture local interactions, and consequently ground states exhibiting shortrange correlations and entanglement area laws [7]. Systems in quantum chemistry pose new challenges due to the inherent long-ranged interactions, which are present no matter in what basis the systems are expressed. New questions hence arise concerning the optimal topology and physical (orbital) basis used to construct the tensor network state [13][14][15][16][17][18][19][23][24][25].In this work, we propose a novel approach towards making use of tensor network methods in quantum chemistry, by suggesting an adaptive scheme of updating basis transformations "on the fly" in conjunction with tensor network updates. In this way, we bring together advantages of matrix product states -which can capture strongly correlated states, but are tailored to short-ranged correlations and low entanglement -and fermionic Gaussian mode transformations -for which entanglement is no obstacle, but non-Gaussian correlations are. We hence go significantly beyond previous approaches towards optimising fermionic base...

We have performed a full configuration interaction (FCI) quality benchmark calculation for the tetramethyleneethane molecule in the cc-pVTZ basis set employing a subset of complete active space second order perturbation theory, CASPT2(6,6), natural orbitals for the FCI quantum Monte Carlo calculation. The results are in an excellent agreement with the previous large scale diffusion Monte Carlo calculations by Pozun et al. and available experimental results. Our computations verified that there is a maximum on the potential energy surface (PES) of the ground singlet state (A) 45° torsional angle, and the corresponding vertical singlet-triplet energy gap is 0.01 eV. We have employed this benchmark for the assessment of the accuracy of Mukherjee's coupled clusters with up to triple excitations (MkCCSDT) and CCSD tailored by the density matrix renormalization group method (DMRG). Multireference MkCCSDT with CAS(2,2) model space, though giving good values for the singlet-triplet energy gap, is not able to properly describe the shape of the multireference singlet PES. Similarly, DMRG(24,25) is not able to correctly capture the shape of the singlet surface, due to the missing dynamic correlation. On the other hand, the DMRG-tailored CCSD method describes the shape of the ground singlet state with excellent accuracy but for the correct ordering requires computation of the zero-spin-projection component of the triplet state (B).

In this article, we investigate the numerical and theoretical aspects of the coupled-cluster method tailored by matrix-product states. We investigate formal properties of the used method, such as energy size consistency and the equivalence of linked and unlinked formulation. The existing mathematical analysis is here elaborated in a quantum chemical framework. In particular, we highlight the use of what we have defined as a complete active space-external space gap describing the basis splitting between the complete active space and the external part generalizing the concept of a HOMO−LUMO gap. Furthermore, the behavior of the energy error for an optimal basis splitting, i.e., an active space choice minimizing the density matrix renormalization group-tailored coupled-cluster singles doubles error, is discussed. We show numerical investigations on the robustness with respect to the bond dimensions of the single orbital entropy and the mutual information, which are quantities that are used to choose a complete active space. Moreover, the dependence of the groundstate energy error on the complete active space has been analyzed numerically in order to find an optimal split between the complete active space and external space by minimizing the density matrix renormalization group-tailored coupled-cluster error.