Circuit depth and measurement count are the main limitations for chemical applications of the leading variational quantum algorithms on quantum computers. We show that both resource requirements can be reduced by augmenting adaptive variational quantum algorithms with post-optimization corrective techniques. We focus on two such techniques---Hamiltonian-moment Krylov subspace diagonalization and the identity-augmented quantum self-consistent equation of motion (q-sc-EOM)---applied to optimized ADAPT-VQE states without further ansatz growth or nonlinear optimization. Across molecular systems studied with up to 20 qubits, first-order Krylov correction reduces the optimized iterations (and the corresponding number of adaptive layers) needed to reach the $1.6~\mathrm{m}E_h$ threshold by up to approximately sixfold, with a general trend toward larger reductions at higher determinant Rényi entropy. We also find measurement-cost reductions of up to approximately fortyfold using a suitable resource-cost model. Identity-augmented q-sc-EOM improves molecular potential-energy curves obtained from truncated ADAPT-VQE ansätze with limited expressivity and recovers accuracy from states produced by sampled operator ranking without additional gradient-based operator selection, providing an advantage when small pool gradients make selection measurement intensive. Together, these post-optimization corrections offer routes to improve energy accuracy and reduce adaptive circuit growth in quantum algorithms as quantum hardware advances.
Circuit depth and measurement count are the main limitations for chemical applications of the leading variational quantum algorithms on quantum computers. We show that both resource requirements can be reduced by augmenting adaptive variational quantum algorithms with post-optimization corrective techniques. We focus on two such techniques---Hamiltonian-moment Krylov subspace diagonalization and the identity-augmented quantum self-consistent equation of motion (q-sc-EOM)---applied to optimized ADAPT-VQE states without further ansatz growth or nonlinear optimization. Across molecular systems studied with up to 20 qubits, first-order Krylov correction reduces the optimized iterations (and the corresponding number of adaptive layers) needed to reach the 1.6mEh threshold by up to approximately sixfold, with a general trend toward larger reductions at higher determinant Rényi entropy. We also find measurement-cost reductions of up to approximately fortyfold using a suitable resource-cost model. Identity-augmented q-sc-EOM improves molecular potential-energy curves obtained from truncated ADAPT-VQE ansätze with limited expressivity and recovers accuracy from states produced by sampled operator ranking without additional gradient-based operator selection, providing an advantage when small pool gradients make selection measurement intensive. Together, these post-optimization corrections offer routes to improve energy accuracy and reduce adaptive circuit growth in quantum algorithms as quantum hardware advances.
Recently, adaptive variational quantum algorithms, e.g., Adaptive Derivative-Assembled Pseudo-Trotter-Variational Quantum Eigensolver (ADAPT-VQE) and Iterative Qubit-Excitation Based-Variational Quantum Eigensolver (IQEB-VQE), have been proposed to optimize the circuit depth, while a huge number of additional measurements make these algorithms highly inefficient. In this work, we reformulate the ADAPT-VQE with reduced density matrices (RDMs) to avoid additional measurement overhead. With Valdemoro's reconstruction of the three-electron RDM, we present a revised ADAPT-VQE algorithm, termed ADAPT-V, without any additional measurements but at the cost of increasing variational parameters compared to the ADAPT-VQE. Furthermore, we present an ADAPT-Vx algorithm by prescreening the anti-Hermitian operator pool with this RDM-based scheme. ADAPT-Vx requires almost the same variational parameters as ADAPT-VQE but a significantly reduced number of gradient evaluations. Numerical benchmark calculations for small molecules demonstrate that ADAPT-V and ADAPT-Vx provide an accurate description of the ground- and excited-state potential energy curves. In addition, to minimize the quantum resource demand, we generalize this RDM-based scheme to circuit-efficient IQEB-VQE algorithm and achieve significant measurement reduction.
Adaptive ansatz construction has emerged as a powerful technique for reducing circuit depth and improving optimization efficiency in variational quantum eigensolvers. However, existing adaptive methods, including ADAPT-VQE, rely solely on first-order gradients and therefore ignore the underlying geometry of the quantum state space, limiting both convergence behavior and operator-selection efficiency. We introduce Geo-ADAPT-VQE, a geometry-aware adaptive VQE algorithm that selects operators from a pool using the natural gradient rule. The geometric operator-selection rule enables the ansatz to grow along directions aligned with the underlying quantum-state geometry, thereby improving convergence and reducing the algorithm's susceptibility to shallow local minima and saddle-point regions. We further provide an asymptotic convergence result. We present numerical simulations involving five molecules, which demonstrate that Geo-ADAPT-VQE achieves faster and more stable convergence compared to existing methods, while producing significantly shorter ansatz. In particular, Geo-ADAPT achieves up to 100-fold reduction in energy error compared to existing methods.
Adaptive derivative-assembled problem-tailored ansatz variational quantum eigensolver (ADAPT-VQE) stands out as a robust algorithm for constructing compact ansätze for molecular simulation. It enables to significantly reduce the circuit depth with respect to other methods, such as unitary coupled cluster singles and doubles, while achieving higher accuracy and not suffering from so-called barren plateaus that hinder the variational optimization of many hardware-efficient ansätze. In its standard implementation, however, it introduces a considerable measurement overhead in the form of gradient evaluations through estimations of many commutator operators. In this work, we mitigate this measurement overhead by exploiting a recently introduced method for energy evaluation relying on adaptive informationally complete generalized measurements (AIMs). Besides offering an efficient way to measure the energy itself, informationally complete (IC) measurement data can be reused to estimate all the commutators of the operators in the operator pool of ADAPT-VQE, using only classically efficient postprocessing. The framework is generic for any IC positive operator-valued measures (POVMs) implementation, and in this work, we demonstrate it’s effectiveness with dilation POVMs. We present the AIM-ADAPT-VQE scheme in detail, and investigate its performance with H 4 , N 2 , and 1,3,5,7-octatetraene Hamiltonians and several operator pools. Our numerical simulations indicate that the measurement data obtained to evaluate the energy can be reused to implement ADAPT-VQE with no additional measurement overhead for the systems considered here. In addition, we show that, if the energy is measured within chemical precision, the count in the resulting circuits is close to the ideal one. With scarce measurement data, AIM-ADAPT-VQE still converges to the ground state with high probability, albeit with an increased circuit depth in some cases.
Predicting many molecular properties requires accurate ground-state energies of molecular Hamiltonians. Quantum-selected configuration interaction (QSCI) obtains such energies by measuring an input state to select important electronic configurations and classically diagonalizing the Hamiltonian in the subspace they span. ADAPT-QSCI, an adaptive extension of QSCI, improves the input state through iterative circuit growth. For single-reference molecules, the sampling distribution is dominated by the Hartree-Fock (HF) configuration, so most measurements reproduce a configuration already known. We reduce this redundancy by placing the HF configuration in the diagonalization subspace classically, before any measurement. This invalidates the operator-selection criterion of ADAPT-QSCI, and we derive a consistent replacement with energy and gradient in closed form, adding no query to the quantum device. At the per-iteration shot budgets previously used for ADAPT-QSCI, the proposed scheme reaches chemical accuracy with approximately 63% and 43% of the CNOT gates for H4 and H6, and 44% and 40% of the total shots. The advantage depends on the budget, however: in the infinite-shot limit the H4 CNOT count increases to 125% of that of ADAPT-QSCI. These results identify the per-iteration shot budget as a key design variable governing resource savings in QSCI-based algorithms.
Problems in quantum chemical simulations, especially achieving accurate excited-state potential energy surfaces, are among the primary applications to achieve quantum utility. On near-term quantum hardware, variants of the variational quantum eigensolver (VQE) algorithms are the primary choice for chemistry simulation. In this study, a combination of leading ground and excited state quantum algorithms for general excited states, namely, ADAPT-VQE/LUCJ and q-sc-EOM, are utilized to calculate accurate excited state potential energy surfaces in challenging bond-breaking scenarios and compared with the classical scalable EOM-CCSD method. This work investigates avenues toward quantum utility in excited-state quantum chemistry using the q-sc-EOM approach. We assess its accuracy while mitigating major scaling bottlenecks through the Davidson algorithm and basis rotation grouping, reducing the measurement scaling from O(N12) to O(N5), and implementing the method on quantum hardware with various error mitigation strategies to reduce gate and measurement errors in excited states. The hardware implementation of the q-sc-EOM algorithm, augmented by mitigation of M3 readout error and symmetry projection, produces reasonably accurate excited-state energies with gate noise identified as the predominant source of error. This paves the way for accurate and scalable, generally applicable quantum excited-state methods with potential for quantum utility while identifying critical problems that require advancements.
Abstract Adaptive variational quantum algorithms arguably offer the best prospects for quantum advantage in the Noisy Intermediate-Scale Quantum era. Since the inception of the first such algorithm, the Adaptive Derivative-Assembled Problem-Tailored Variational Quantum Eigensolver (ADAPT-VQE), many improvements have appeared in the literature. We combine the key improvements along with a novel operator pool—which we term Coupled Exchange Operator (CEO) pool—to assess the cost of running state-of-the-art ADAPT-VQE on hardware in terms of measurement counts and circuit depth. We show a dramatic reduction of these quantum computational resources compared to the early versions of the algorithm: CNOT count, CNOT depth and measurement costs are reduced by up to 88%, 96% and 99.6%, respectively, for molecules represented by 12 to 14 qubits (LiH, H 6 and BeH 2 ). We also find that our state-of-the-art CEO-ADAPT-VQE outperforms the Unitary Coupled Cluster Singles and Doubles ansatz, the most widely used static VQE ansatz, in all relevant metrics, and offers a five order of magnitude decrease in measurement costs as compared to other static ansätze with competitive CNOT counts.