A balanced description of both static and dynamic correlations in electronic systems with nearly degenerate low-lying states presents a challenge for multiconfigurational methods on classical computers.We present here a quantum algorithm utilizing the action of correlating cluster operators to provide highquality wave-function ansätze employing a nonorthogonal multireference basis that captures a significant portion of the exact wave function in a highly compact manner and that allows computation of the resulting energies and wave functions at polynomial cost with a quantum computer.This enables a significant improvement over the corresponding classical nonorthogonal solver, which incurs an exponential cost when evaluating off-diagonal matrix elements between the ansatz states and is therefore intractable.We implement the nonorthogonal quantum eigensolver (NOQE) here with an efficient ansatz parametrization inspired by classical quantum chemistry methods that succeed in capturing significant amounts of electronic correlation accurately.Crucially, we avoid the need to perform any optimization of the ansatz on the quantum device.By taking advantage of such classical approaches, NOQE provides a flexible, compact, and rigorous description of both static and dynamic electronic correlation, making it an attractive method for the calculation of electronic states of a wide range of molecular systems.
A balanced description of both static and dynamic correlations in electronic systems with nearly degenerate low-lying states presents a challenge for multiconfigurational methods on classical computers.We present here a quantum algorithm utilizing the action of correlating cluster operators to provide highquality wave-function ansätze employing a nonorthogonal multireference basis that captures a significant portion of the exact wave function in a highly compact manner and that allows computation of the resulting energies and wave functions at polynomial cost with a quantum computer.This enables a significant improvement over the corresponding classical nonorthogonal solver, which incurs an exponential cost when evaluating off-diagonal matrix elements between the ansatz states and is therefore intractable.We implement the nonorthogonal quantum eigensolver (NOQE) here with an efficient ansatz parametrization inspired by classical quantum chemistry methods that succeed in capturing significant amounts of electronic correlation accurately.Crucially, we avoid the need to perform any optimization of the ansatz on the quantum device.By taking advantage of such classical approaches, NOQE provides a flexible, compact, and rigorous description of both static and dynamic electronic correlation, making it an attractive method for the calculation of electronic states of a wide range of molecular systems.
We present a shadow-tomography-enhanced nonorthogonal quantum eigensolver (NOQE) for more efficient and accurate electronic structure calculations on near-term quantum devices. By integrating shadow tomography into the NOQE, the measurement cost scales linearly rather than quadratically with the number of reference states, while also reducing the required qubits and circuit depth by half. This approach enables extraction of all matrix elements via randomized measurements and classical postprocessing. We analyze its sample complexity and show that, for small systems, it remains constant in the high-precision regime, while for larger systems, it scales linearly with the system size. We further apply shadow-based error mitigation to suppress noise-induced bias without increasing quantum resources. Demonstrations on the hydrogen molecule in the strongly correlated regime achieve chemical accuracy under realistic noise, showing that our method is both resource-efficient and noise-resilient for practical quantum chemistry simulations in the near term.
Quantum subspace diagonalization methods are promising algorithms for quantum chemistry on near-term quantum computers. These methods can estimate low-lying energies of molecular systems using shallow quantum circuits, at the cost of many circuit repetitions to estimate the projected matrix elements. Errors in these matrix elements can be converted into much larger eigenvalue errors by an ill-conditioned overlap matrix. We study this bottleneck for the nonorthogonal quantum eigensolver (NOQE), which constructs a compact multireference subspace from dressed unrestricted Hartree-Fock states. We prove a finite-shot perturbation bound showing that, after overlap thresholding, the eigenvalue sensitivity is controlled by the condition number of the retained overlap matrix rather than by a worst-case dimension factor. With a scalable thresholding scheme, the upper bound on the per-matrix-element shot count required to reach a target accuracy scales as O(M), improving on the previously known O(M3) bound, where M is the number of reference states. Numerical experiments on hydrogen chains and rings suggest that, in practice, the measurement cost of structured NOQE instances can grow even more slowly than this linear bound.
Computer simulation is central to understanding the behaviour of matter. Yet, more than a century after the development of quantum theory, first-principles simulation of chemistry remains a daunting challenge, owing to the exponential growth in memory required to represent interacting quantum particles such as electrons in molecules. Quantum computing offers, in principle, a fundamentally more efficient paradigm for simulating quantum systems, with quantum chemistry often cited as a flagship application. However, many quantum algorithms face intrinsic limitations, and any genuine quantum advantage must surpass the strong baseline set by decades of progress in classical algorithms and hardware. While classical methods often suffice for simulating weakly correlated systems, systems with many strongly correlated electrons remain a challenge for both classical and quantum approaches. In this thesis, we develop quantum algorithms for solving the time-independent Schrödinger equation, with a focus on molecular systems with strongly correlated electrons. We introduce a framework for identifying a compact set of states that parametrise the many-body wavefunction by capturing the correlation due to spin coupling between localised electrons. This representation requires very few variational parameters but incurs a very high memory cost on classical computers. To overcome this, we design quantum circuits that efficiently prepare such states by exploiting their symmetry, encoding the dominant entanglement structure of electronic wavefunctions into the initial qubit states. Through theoretical analysis and classical simulations of quantum computers, we show that this substantially reduces the quantum resources required to accurately compute ground- and excitedstate properties. Finally, we develop an approach for approximating the optimal ansatz within the variational quantum eigensolver, an algorithm designed for near-term quantum devices. Our method constructs wavefunctions from products of symmetrypreserving unitary operators and couples discrete and continuous optimisation to yield highly compact representations of strongly correlated electrons using shallow quantum circuits. Together, these results establish complementary directions for embedding problemspecific structure into quantum simulation algorithms, advancing the practical simulation of strongly correlated electrons.
The Non-Orthogonal Quantum Eigensolver (NOQE) provides an accurate framework for electronic-structure calculations, but the estimation of its Hamiltonian and overlap matrix elements relies on sampling and requires O(1/ε2) circuit repetitions to achieve additive precision ε. Here, we reformulate this matrix-element estimation step as a collection of amplitude-estimation tasks and integrate iterative quantum amplitude estimation into the NOQE workflow. The resulting protocol achieves near-Heisenberg query complexity O(1/ε) for these estimation tasks, by replacing incoherent statistical averaging with coherent amplitude amplification. We present explicit circuit constructions and the corresponding implementation procedure. Numerical simulations for the electronic states of the hydrogen molecule show that the proposed method reaches chemical accuracy with substantially fewer total queries than the original sampling-based protocol. Overall, this work provides a measurement-efficient route to high-precision energy estimation and illustrates how sampling-limited quantum algorithms can be systematically reformulated to leverage quantum coherence and achieve lower measurement costs.
We propose a modification of the Variational Quantum Eigensolver algorithm for electronic structure optimization using quantum computers, named nonunitary Variational Quantum Eigensolver (nu-VQE), in which a nonunitary operator is combined with the original system Hamiltonian leading to a new variational problem with a simplified wave function ansatz. In the present work, as nonunitary operator, we use the Jastrow factor, inspired from classical Quantum Monte Carlo techniques for simulation of strongly correlated electrons. The method is applied to prototypical molecular Hamiltonians for which we obtain accurate ground-state energies with shallower circuits, at the cost of an increased number of measurements. Finally, we also show that this method achieves an important error mitigation effect that drastically improves the quality of the results for VQE optimizations on today's noisy quantum computers. The absolute error in the calculated energy within our scheme is 1 order of magnitude smaller than the corresponding result using traditional VQE methods, with the same circuit depth.
Accurately describing strong electron correlation in complex systems remains a prominent challenge in computational chemistry as near-term quantum algorithms treating total correlation often require prohibitively deep circuits. Here we present a hybrid strategy combining the Variational Quantum Eigensolver with Multiconfiguration Pair-Density Functional Theory to efficiently decouple correlation effects. This approach confines static correlation to a compact multireference quantum state while recovering dynamic correlation through a classical on-top density functional using reduced-density information. By enabling self-consistent orbital optimization, the method significantly reduces quantum resource overheads without sacrificing physical rigor. We demonstrate chemical accuracy on standard benchmarks by reproducing C2 equilibrium bond lengths and benzene excitation energies with mean absolute errors of 0.006 Å and 0.048 eV respectively. Most notably, for the strongly correlated Cr2 dimer requiring a large complete active space (48e, 42o), the framework yields a bound potential-energy curve and recovers qualitative dissociation behavior despite realistic hardware noise. These results establish that separating correlation types provides a practical route to reliable predictions on near-term quantum hardware.