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.
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.
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.
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.
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.
Adaptive Derivative-Assembled Problem-Tailored variational quantum eigensolvers (ADAPT-VQE) represent one of the most promising approaches for quantum chemistry on near-term quantum devices. However, their optimization is slow and may stall due to vanishing parameters and redundant operators in the ansatz. In this work, we propose a simple strategy of operator elimination that removes non-contributing operators from the pool once they are detected, enabling the optimization to continue progressing toward convergence. We examine two variants, with and without pool restoration after elimination, and find that the former converges more smoothly and faster than the latter and the standard ADAPT-VQE. To capture dynamical correlations between the active space and its environment, we combine ADAPT-VQE with our recently developed downfolding approach, the one-body downfolding framework (OBDF). In OBDF, the bare molecular Hamiltonian in the active space is replaced by a correlated effective Hamiltonian that incorporates dynamical correlation effects outside the active space. We benchmark our implementation on a linear \ce{H_6} chain, an \ce{H_6} lattice, an \ce{H_6} ring, and the \ce{N_2} molecule using the OpenFermion simulator. Our results show that operator elimination significantly reduces circuit depth and iteration count, and that OBDF-ADAPT-VQE yields energies closer to the full configuration interaction (FCI) reference than the standard approach within the same active space.
Adaptive variational quantum simulation algorithms use information from a quantum computer to dynamically create optimal trial wave functions for a given problem Hamiltonian. A key ingredient in these algorithms is a predefined operator pool from which trial wave functions are constructed. Finding suitable pools is critical for the efficiency of the algorithm as the problem size increases. Here, we present a technique called operator pool tiling that facilitates the construction of problem-tailored pools for arbitrarily large problem instances. By first performing an Adaptive Derivative-Assembled Problem-Tailored Ansatz Variational Quantum Eigensolver (ADAPT-VQE) calculation on a smaller instance of the problem using a large, but computationally inefficient, operator pool, we extract the most relevant operators and use them to design more efficient pools for larger instances. We demonstrate the method here on strongly correlated quantum spin models in one and two dimensions, finding that ADAPT automatically finds a highly effective ansatz for these systems. Given that many problems, such as those arising in condensed matter physics, have a naturally repeating lattice structure, we expect the pool tiling method to be a widely applicable technique apt for such systems. Published by the American Physical Society 2024
ADAPT-VQE is a leading variational quantum algorithm as it circumvents the choice-of-ansatz conundrum by iteratively growing compact and arbitrarily accurate problem-tailored ansätze. However, for molecular Hamiltonians and hardware-efficient operator pools, the gradient-measurement step of the algorithm requires the estimation of O(N8) observables, which may represent a bottleneck for relevant system sizes on real devices. We present an efficient strategy for measuring the gradients of the three best-performing hardware-efficient operator pools by partitioning the required observables into O(N3)-sized sets of commuting Paulis that can be simultaneously measured with an at most N−3 CNOT overhead. We argue that our approach is robust to shot-noise effects and show that measuring the pool gradients is, in fact, not O(N4), but only about 4N times as expensive as measuring the energy once. Our proposed measurement strategy significantly ameliorates the measurement overhead of ADAPT-VQE and brings us one step closer to practical implementations on real devices.