Exact simulations of quantum circuits (QCs) are currently limited to ∼50 qubits because the memory and computational cost required to store the QC wave function scale exponentially with qubit number. Therefore, developing efficient schemes for approximate QC simulations is a current research focus. Here, we show simulations of QCs with a method inspired by density functional theory (DFT), a widely used approach for studying many-electron systems. Our calculations can predict marginal single-qubit probabilities (SQPs) with over 90% accuracy in several classes of QCs with universal gate sets, using memory and computational resources linear in qubit number despite the formal exponential cost of the SQPs. This is achieved by developing a mean-field description of QCs and formulating optimal single- and two-qubit gate functionals─analogues of exchange-correlation functionals in DFT─to evolve the SQPs without computing the QC wave function. Current limitations and future extensions of this formalism are discussed.
Exact simulations of quantum circuits (QCs) are currently limited to ∼50 qubits because the memory and computational cost required to store the QC wave function scale exponentially with qubit number. Therefore, developing efficient schemes for approximate QC simulations is a current research focus. Here, we show simulations of QCs with a method inspired by density functional theory (DFT), a widely used approach for studying many-electron systems. Our calculations can predict marginal single-qubit probabilities (SQPs) with over 90% accuracy in several classes of QCs with universal gate sets, using memory and computational resources linear in qubit number despite the formal exponential cost of the SQPs. This is achieved by developing a mean-field description of QCs and formulating optimal single- and two-qubit gate functionals─analogues of exchange-correlation functionals in DFT─to evolve the SQPs without computing the QC wave function. Current limitations and future extensions of this formalism are discussed.
Classical simulations of quantum circuits are essential for verifying and benchmarking quantum algorithms, particularly for large circuits, where computational demands increase exponentially with the number of qubits. Among available methods, the classical simulation of quantum circuits inspired by density functional theory -- the so-called QC-DFT method, shows promise for large circuit simulations as it approximates the quantum circuits using single-qubit reduced density matrices to model multi-qubit systems. However, the QC-DFT method performs very poorly when dealing with multi-qubit gates. In this work, we introduce a novel CNOT "functional" that leverages neural networks to generate unitary transformations, effectively mitigating the simulation errors observed in the original QC-DFT method. For random circuit simulations, our modified QC-DFT enables efficient computation of single-qubit marginal measurement probabilities, or single-qubit probability (SQPs), and achieves lower SQP errors and higher fidelities than the original QC-DFT method. Despite some limitations in capturing full entanglement and joint probability distributions, we find potential applications of SQPs in simulating Shor's and Grover's algorithms for specific solution classes. These findings advance the capabilities of classical simulations for some quantum problems and provide insights into managing entanglement and gate errors in practical quantum computing.
Accurate descriptions of the electronic properties of molecules, solids and other materials are crucial in the development of many modern technologies, and are therefore the subject of much scientific research. In tandem with experimental research, computational simulations play a huge role. Density functional theory (DFT) is the most popular method for performing ground-state electronic structure calculations due to its accuracy in many situations along with its computational efficiency. Its time-dependent extension (TDDFT) allows the modelling of the dynamics of systems of interacting electrons. (TD)DFT is an exact theory in principle, but requires approximations in practice, specifically to the many-body effects of electron exchange and correlation (xc), which are described by some unknown universal functional. Despite much success, it often proves inadequate in many scenarios of increasing interest. Therefore, we aim to contribute to the development of improved approximate density functionals. \n \nIn this thesis, we take the approach of studying small model systems for which we can compute the exact functionals. These systems are designed to exhibit the key features present in realistic quantum systems, such as atoms and molecules, which the exact universal (TD)DFT functional is capable of describing, but are often ill-described by the commonly used approximations. By analysing the nature of these exact functionals, we gain valuable insight into their fundamental properties, together with identifying where the current approximations are lacking, thereby allowing us to suggest how these approximations may be improved.
Quantum computers (QC) could harbor the potential to significantly advance materials simulations, particularly at the atomistic scale involving strongly correlated fermionic systems, where an accurate description of quantum many-body effects scales unfavorably with size. While a full-scale treatment of condensed matter systems with currently available noisy quantum computers remains elusive, quantum embedding schemes like dynamical mean-field theory (DMFT) allow the mapping of an effective, reduced subspace Hamiltonian to available devices to improve the accuracy of ab initio calculations such as density functional theory (DFT). Here, we report on the development of a hybrid quantum-classical DFT + DMFT simulation framework which relies on a quantum impurity solver based on the Lehmann representation of the impurity Green’s function. Hardware experiments with up to 14 qubits on the IBM Quantum system are conducted, using advanced error mitigation methods and a novel calibration scheme for an improved zero-noise extrapolation to effectively reduce adverse effects from inherent noise on current quantum devices. We showcase the utility of our quantum DFT + DMFT workflow by assessing the correlation effects on the electronic structure of a real material, Ca 2 CuO 2 Cl 2 , which is mapped to an effective single-band Hubbard Hamiltonian and the subsequently derived Anderson impurity model solved with up to 6 bath sites on available quantum hardware. Further, we carefully benchmark our quantum results with respect to exact reference solutions and experimental spectroscopy measurements. While challenges remain to scale our approach to larger, multi-orbital and multi-site systems with more bath sites, the present work marks an important milestone towards achieving utility-scale quantum computation in materials simulation.
While quantum computers have shown significant promise for electronic structure calculations, their potential to accelerate density functional theory (DFT) calculations remains unclear. In this work, we present a qubit-efficient encoding scheme for wavefunctions in Kohn--Sham (KS) DFT, together with a quantum algorithm that computes all occupied orbitals simultaneously. We further show that our algorithm is particularly well suited to the Harris functional, enabling the total energy to be evaluated with a potential exponential speedup over classical approaches by entirely avoiding the costly readout of the electronic density. In addition, we propose a second method for achieving self-consistent DFT calculations using multiple copies of the wavefunction, which likewise circumvents density readout. The applicability of our algorithms is demonstrated through several numerical examples, and their efficiency is compared with that of existing approaches.
An extension of the Variational Quantum Eigensolver (VQE) method is presented where a quantum computer generates an accurate exchange-correlation potential for a Density Functional Theory (DFT) simulation on classical hardware. The method enables efficient simulations of quantum systems by interweaving calculations on classical and quantum resources. DFT is implemented on classical hardware, which enables the efficient representation of and operation on quantum systems while being formally exact. The portion of the simulation operating on quantum hardware produces an accurate exchange-correlation potential but only requires relatively short depth quantum circuits.
Density-functional theory (DFT) has revolutionized computer simulations in chemistry and material science. A faithful implementation of the theory requires self-consistent calculations. However, this effort involves repeatedly diagonalizing the Hamiltonian, for which a classical algorithm typically requires a computational complexity that scales cubically with respect to the number of electrons. This limits DFT's applicability to large-scale problems with complex chemical environments and microstructures. This article presents a quantum algorithm that has a linear scaling with respect to the number of atoms, which is much smaller than the number of electrons. Our algorithm leverages the quantum singular value transformation (QSVT) to generate a quantum circuit to encode the density-matrix, and an estimation method for computing the output electron density. In addition, we present a randomized block coordinate fixed-point method to accelerate the self-consistent field calculations by reducing the number of components of the electron density that needs to be estimated. The proposed framework is accompanied by a rigorous error analysis that quantifies the function approximation error, the statistical fluctuation, and the iteration complexity. In particular, the analysis of our self-consistent iterations takes into account the measurement noise from the quantum circuit. These advancements offer a promising avenue for tackling large-scale DFT problems, enabling simulations of complex systems that were previously computationally infeasible.