All linear scaling methods of quantum chemistry are based on a common principle: the problem of the quantum chemical representation of a large molecule is approached by algorithms setting direct or indirect distance limits on the interactions considered between molecular parts, that allows proportionality of the computational work with molecular size. This principle applies if some fragmentation scheme is considered, or if some thresholds are applied to molecular integrals, directly or indirectly dependent on distance. Ultimately, the success of linear scaling methods depends on how well the local representations and the approximations used for the interactions between them represent the actual, complete molecule. The relations between the parts and the whole of molecules are unavoidable constraints on all linear scaling methods. In this study one special aspect of these constraints is discussed in terms of the local site‐dominance of various electron density contributions, and the local—global aspects of macromolecules are discussed within the context of linear scaling methods.
All linear scaling methods of quantum chemistry are based on a common principle: the problem of the quantum chemical representation of a large molecule is approached by algorithms setting direct or indirect distance limits on the interactions considered between molecular parts, that allows proportionality of the computational work with molecular size. This principle applies if some fragmentation scheme is considered, or if some thresholds are applied to molecular integrals, directly or indirectly dependent on distance. Ultimately, the success of linear scaling methods depends on how well the local representations and the approximations used for the interactions between them represent the actual, complete molecule. The relations between the parts and the whole of molecules are unavoidable constraints on all linear scaling methods. In this study one special aspect of these constraints is discussed in terms of the local site‐dominance of various electron density contributions, and the local—global aspects of macromolecules are discussed within the context of linear scaling methods.
The field of quantum chemistry is concerned with the modelling and simulation of the behaviour of molecular systems on the basis of the fundamental equations of quantum mechanics. Since these equations exhibit an extreme case of the curse of dimensionality (the Schrödinger equation for N electrons being a partial differential equation on \mathbb R^{3N} ), the quantum-chemical simulation of even moderate-size molecules already requires highly sophisticated model-reduction, approximation, and simulation techniques. The workshop brought together selected quantum chemists and physicists, and the growing community of mathematicians working in the area, to report and discuss recent advances on topics such as coupled-cluster theory, direct approximation schemes in full configuration-interaction (FCI) theory, interacting Green’s functions, foundations and computational aspects of density-functional theory (DFT), low-rank tensor methods, quantum chemistry in the presence of a strong magnetic field, and multiscale coupling of quantum simulations.
We first give a brief, incomplete overview over historic milestones leading to the emergence of quantum chemistry, starting from John Dalton's earliest attempts to describe the atom in the early 19th century. After the formulation of the Schrödinger equation in 1926 and the first successful description of covalent bonding using the new theory, it became soon clear that the main challenge ahead was to find efficient approximations to the Schrödinger equation, as was famously stated by Paul A. M. Dirac in 1929. Since then, many quantum-chemical approximations have been introduced, with a key problem being the exponential increase of the computational cost with the system size when approaching the exact solution of the Schrödinger equation. In the second part, we will hence focus on selected techniques to overcome the scaling problem. Finally, we close with some insights into the new and challenging field of reaction network exploration, offering a glimpse into potential future directions of quantum chemistry.
Hybrid quantum-classical algorithms have begun to leverage quantum devices to efficiently represent many-electron wavefunctions, enabling early demonstrations of molecular simulations on real hardware. A key prerequisite for scalable quantum chemistry, however, is size consistency: the energy of non-interacting subsystems must scale linearly with system size. While many algorithms are theoretically size-consistent, noise on quantum devices may couple nominally independent subsystems and degrade this fundamental property. Here, we systematically evaluate size consistency on quantum hardware by simulating systems composed of increasing numbers of non-interacting H2 molecules using optimally shallow unitary circuits. We find that molecular energies remain size-consistent within chemical accuracy for an estimated 118 and 71 H2 subsystems for one- and two-qubit unitary designs, respectively, demonstrating that current quantum devices preserve size consistency over chemically relevant system sizes and supporting the feasibility of scalable, noise-resilient simulation of strongly correlated molecules and materials.
Significant progress in the development of efficient and fast algorithms for quantum chemical calculations has been made in the past two decades. The main focus has al-ways been the desire to be able to treat ever larger molecules or molecular assemblies— especially linear and sub-linear scaling techniques are devoted to the accomplishment of this goal. However, as many chemical reactions are rather local, they usually involve only a limited number of atoms so that models of about two hundred (or even less) atoms embedded in a suitable environment are sufficient to study their mechanisms. Thus, the system size does not need to be enlarged, but remains constant for reactions of this type that can be described by less than two hundred atoms. The question then arises how fast one can obtain the quantum chemical results. This question is not directly answered by linear-scaling techniques. In fact, ideas such as haptic quantum chemistry or interactive quantum chemistry require an immediate provision of quan-tum chemical information which demands the calculation of data in “real time”. In this perspective, we aim at a definition of real-time quantum chemistry, explore its realm and eventually discuss applications in the field of haptic quantum chemistry. For the latter we elaborate whether a direct approach is possible by virtue of real-time quantum chemistry. Submitted for publication as a perspective article in Int. J. Quantum Chem.
In ab-initio quantum chemistry, solving the electronic structure problem of a molecular system of interest to obtain an approximation of the ground and, in some cases, the excited state energies is a core and central task in various quantum chemistry processes. These include, but certainly not limited to, the calculation of the activation energy–that is the minimal energy required to launch a given chemical reaction–, the dipole and multipole moments which describe the polarity of the molecule, performing the optimization of the geometry of the molecule and searching for transition states. Furthermore, many quantum chemistry applications are predicated, not merely on the approximation of the ground state energy, but also on the corresponding first order derivatives. The body of literature of classical algorithms approximating the ground state energy and its first order derivative is extensively rich with a plethora of methods such as self-consistent mean-field methods, coupled-cluster methods, density matrix renormalization group and many more, each of which has intrinsic set of properties and characteristics that makes it suitable for specific tasks and cases. However, classical methods face a practically insurmountable challenge: the efficient computation of high accuracy approximation of ground state energies of large system sizes. This is fundamentally because the electronic structure problem provably belongs to the complexity class of QMA-hard problems. Which also means that it is out of reach for quantum algorithms as well in general. However, with certain assumptions that are expected to hold molecules of interest–that so far proved to be classically challenging because the required accuracy set by the standard of chemical accuracy threshold (generally held in quantum chemistry)– quantum algorithms can efficiently treat such cases. While originally theorized by Richard Feynmann as a potentially efficient way of simulating some classically intractable many-body systems, quantum computing has since proven to have a stronger and richer potential paving the way for the inception of new algorithms such as Shor’s algorithm for integer factorization, HHL for solving linear system of equation and Grover’s algorithm for constrained binary search, just to name a few. These algorithms provably exhibit an improvement rate that goes respectively exponentially and quadratically in the time complexity. Quantum chemistry is considered a prominent candidate to benefit from quantum computing given the variety of quantum algorithms put forth to solve the electronic structure problem of particular systems of interest. These quantum algorithm proposals can be broadly categorized in three main categories that span the spectrum: from noisy intermediate scale quantum (NISQ) computers to early and full scale fault tolerant quantum ((E)FTQC) computers. Firstly, the variational quantum eigensolver (VQE) family of quantum algorithms which variationally search for the ground state energy through the optimization of the energy of a parametrized state prepared in a quantum computer using an ansatz. Secondly, the quantum Krylov subspace diagonalization (QKSD) methods, which generate a subspace of the original Hilbert space and efficiently produce the lowest energy in said subspace as its output. Lastly, we have the quantum phase estimation (QPE) algorithms, which provably produce the ground state energy in Heisenberg scaling presenting hence a provable efficient solution. In this thesis, we firstly set to work on the problem of efficient Hamiltonian linear combination of unitaries (LCU) representation. This is mainly motivated by the wide-range impact an efficient compression of the Hamiltonian can potentially have on several quantum algorithms such as VQE, QPE...etc. We designed a method that efficiently compress the Hamiltonian representation in a way that is beneficially impactful across the set of the aforementioned quantum algorithms. Namely, we motivate and present in Sections 1.3 and 1.4 respectively, the regularized compressed double-factorization (RC-DF) and show how it reduces the energy measurement complexity both in terms of number of distinct observables and number of samples. Moreover, it also reduces the total depth of the quantum phase estimation circuit thanks to its low normalization factor. Secondly, given the importance of the energy derivatives, in Sections 2.1 and 2.2, we introduce and present an efficient quantum algorithm for the calculation of the analytic gradient of VQE energies of double-factorized Hamiltonians. Furthermore, in Section 2.1, we expand on the utility of RC-DF by presenting a quantum algorithm that determines the analytic gradient of energies measured with respect to RC-DF Hamiltonains. Third, in Section 3.1, we summarize the theoretical results of purification error-mitigation techniques, namely echo verification and virtual distillation. In Section 3.2, we report on the results of the experimental bench mark, of said techniques, performed on on Google’s Sycamore device to validate the theoretical scaling of error suppression and extrapolate the resources needed for classically intractable system sizes. Fourth, in Section 4 we establish the challenges of computing energy gradients in a quantum Krylov frame work. To overcome the problem, we provide, in Section 4.3, an efficient quantum algorithm for calculating energy derivative obtained through QKSD procedures. Our approach shows how the QKSD ground state can be prepared using quantum signal processing (QSP) and reduces the number of observables to O(1) per reduced density matrix (RDM). Lastly, in light of the importance of quantum RAMs due to their utility in a variety of quantum algorithms that require efficient initial state preparation, as explained in Section 5.5, we address the problematic exponential depth of the bucket-brigade QRAM (when decomposed). We present in Section 5.6 a very efficient Clifford+T decomposition of the bucket-brigade QRAM, which preserves the linear scaling of the depth of the circuit without introducing any additional ancilla.