Abstract Variational Quantum Eigensolver (VQE) methods aim to maximize the resources of existing noisy devices. However, they encounter difficulties in simulating molecules of industrially-relevant sizes, such as constructing the efficient ansatz. Adaptive variational algorithms (ADAPT-VQE) can solve this problem but with a significant increase in the number of measurements. Here, we reduce the measurement overhead of ADAPT-VQE via adding operator batches to the ansatz while keeping it compact. We reformulate the previously proposed qubit pool completeness criteria for the tapered qubit space and propose an automated pool construction procedure. Our numerical results indicate that reducing the qubit pool size from polynomial to linear conversely increases the number of measurements. We simulate a set of molecules, participating in the carbon monoxide oxidation processes using the statevector simulator and compare the results with VQE-UCCSD and classical methods. Our results pave the way towards usage of variational approaches for solving practically relevant chemical problems.
Abstract Variational Quantum Eigensolver (VQE) methods aim to maximize the resources of existing noisy devices. However, they encounter difficulties in simulating molecules of industrially-relevant sizes, such as constructing the efficient ansatz. Adaptive variational algorithms (ADAPT-VQE) can solve this problem but with a significant increase in the number of measurements. Here, we reduce the measurement overhead of ADAPT-VQE via adding operator batches to the ansatz while keeping it compact. We reformulate the previously proposed qubit pool completeness criteria for the tapered qubit space and propose an automated pool construction procedure. Our numerical results indicate that reducing the qubit pool size from polynomial to linear conversely increases the number of measurements. We simulate a set of molecules, participating in the carbon monoxide oxidation processes using the statevector simulator and compare the results with VQE-UCCSD and classical methods. Our results pave the way towards usage of variational approaches for solving practically relevant chemical problems.
Quantum simulation of strongly correlated systems is potentially the most feasible useful application of near-term quantum computers. Minimizing quantum computational resources is crucial to achieving this goal. A promising class of algorithms for this purpose consists of variational quantum eigensolvers (VQEs). Among these, problem-tailored versions such as ADAPT-VQE that build variational ansätze step by step from a predefined operator pool perform particularly well in terms of circuit depths and variational parameter counts. However, this improved performance comes at the expense of an additional measurement overhead compared to standard VQEs. Here, we show that this overhead can be reduced to an amount that grows only linearly with the number n of qubits, instead of quartically as in the original ADAPT-VQE. We do this by proving that operator pools of size 2n−2 can represent any state in Hilbert space if chosen appropriately. We prove that this is the minimal size of such "complete" pools, discuss their algebraic properties, and present necessary and sufficient conditions for their completeness that allow us to find such pools efficiently. We further show that, if the simulated problem possesses symmetries, then complete pools can fail to yield convergent results, unless the pool is chosen to obey certain symmetry rules. We demonstrate the performance of such symmetry-adapted complete pools by using them in classical simulations of ADAPT-VQE for several strongly correlated molecules. Our findings are relevant for any VQE that uses an ansatz based on Pauli strings.
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.
We present a self-consistent field (SCF) approach within the adaptive derivative-assembled problem-tailored ansatz variational quantum eigensolver (ADAPT-VQE) framework for efficient quantum simulations of chemical systems on near-term quantum computers. To this end, our ADAPT-VQE-SCF approach combines the idea of generating an ansatz with a small number of parameters, resulting in shallow-depth quantum circuits with a direct minimization of an energy expression that is correct to second order with respect to changes in the molecular orbital basis. Our numerical analysis, including calculations for the transition-metal complex ferrocene [Fe (C 5 H 5 ) 2 ], indicates that convergence in the self-consistent orbital optimization loop can be reached without a considerable increase in the number of two-qubit gates in the quantum circuit by comparison to a VQE optimization in the initial molecular orbital basis. Moreover, the orbital optimization can be carried out simultaneously within each iteration of the ADAPT-VQE cycle. ADAPT-VQE-SCF thus allows us to implement a routine analogous to the complete active space SCF, a cornerstone of state-of-the-art computational chemistry, in a hardware-efficient manner on near-term quantum computers. Hence, ADAPT-VQE-SCF paves the way toward a paradigm shift for quantitative quantum-chemistry simulations on quantum computers by requiring fewer qubits and opening up for the use of large and flexible atomic orbital basis sets in contrast to earlier methods that are predominantly based on the idea of full active spaces with minimal basis sets.
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.
The variational quantum eigensolver (VQE) algorithm can simulate the chemical systems such as molecules in the noisy-intermediate-scale quantum devices and shows promising applications in quantum chemistry simulations. The accuracy and computational cost of the VQE simulations are determined by the underlying ansatz. Therefore, the most important issue is to generate a compact and accurate ansatz, which requires a shallower parametric quantum circuit and can achieve an acceptable accuracy. The newly developed adaptive algorithms (AAs) such as the adaptive derivative-assembled pseudo-Trotter VQE (ADAPT-VQE) can solve this issue via generating compact and accurate ansatzes. However, these AAs show very low computational efficiency because they require a large number of additional measurements. Here we propose an amplitude reordering (AR) strategy to accelerate the promising but expensive AAs by adding operators in a “batched” fashion in a way that their order is still quasi-optimal. We first introduce the AR method into ADAPT-VQE and build the AR-ADAPT-VQE algorithm. We then endow the energy-sorting VQE (ES-VQE) algorithm with the adaptive feature and introduce the AR into AES-VQE to form the AR-AES-VQE algorithm. To demonstrate the performance of these algorithms, we calculate the dissociation curves of three small molecules, LiH, linear BeH 2, and linear H 6, by using (AR-)ADAPT-VQE and (AR-)AES-VQE algorithms. It is found that all of the AR-equipped AAs (AR-AAs) can significantly reduce the number of iterations and subsequently accelerate the calculations with a speedup of up to more than ten times without the obvious loss of accuracy. The final ansatz generated by the AR-AAs not only avoids extra circuit depth but also maintains the computational accuracy; sometimes the AR-AAs even outperforms their original counterparts.
Designing compact ansätze in Variational Quantum Eigensolver (VQE) is crucial for solving energetic problems of practical molecules on near-term quantum devices. However, existing Adaptive Derivative-Assembled Pseudo-Trotter (ADAPT) ansätze face two challenges: improper operator selection and accumulation of degraded operators. In this paper, we propose the Hamiltonian-Aware (HA) ADAPT-VQE algorithm to address these issues. First, we establish a novel excitation operator selection criterion. It breaks the local constraint of existing criteria by incorporating Hamiltonian information, prioritizes physically meaningful excitation operators, and incurs no extra classical or quantum computational overhead. Furthermore, we develop a problem-adaptive method for discriminating and pruning redundant excitation operators stemming from improper selection and inevitable degradation. This method balances redundant operator pruning and convergence guarantee, and is applicable to ansätze with arbitrary scales. Systematic numerical experiments on typical strongly correlated molecular systems demonstrate that our HA-ADAPT-VQE avoids energy plateaus and outperforms baseline algorithms in terms of energy error, ansatz size, and measurement cost. This work offers an efficient, robust ansatz construction paradigm, facilitating the development and practical deployment of large-scale VQE in quantum chemistry.