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arXiv:2306.03227

How to really measure operator gradients in ADAPT-VQE

Panagiotis G. Anastasiou, Nicholas J. Mayhall, Edwin Barnes, Sophia E. Economou

Published Jun 5, 2023• 7 citations
arXivHTMLPDF

Abstract

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 $\mathcal{O}(N^8)$ 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 $\mathcal{O}(N^3)$-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 $\mathcal{O}(N^4)$, 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.

DOI
10.48550/arxiv.2306.03227

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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.
The adaptive derivative-assembled problem-tailored variational quantum eigensolver (ADAPT-VQE) provides a promising approach for simulating highly correlated quantum systems on quantum devices, as it strikes a balance between hardware efficiency, trainability, and accuracy. Although ADAPT-VQE avoids many of the shortcomings of other VQEs, it is sometimes hindered by a phenomenon known as gradient troughs. This refers to a non-monotonic convergence of the gradients, which may become very small even though the minimum energy has not been reached. This results in difficulties finding the right operators to add to the ansatz, due to the limited number of shots and statistical uncertainties, leading to stagnation in the circuit structure optimization. In this paper, we propose ways to detect and mitigate this phenomenon. Leveraging the non-commutative algebra of the ansatz, we develop heuristics for determining where to insert new operators into the circuit. We find that gradient troughs are more likely to arise when the same locations are used repeatedly for new operator insertions. Our novel protocols, which add new operators in different ansatz positions, allow us to escape gradient troughs and thereby lower the measurement cost of the algorithm. This approach achieves an effective balance between cost and efficiency, leading to faster convergence without compromising the low circuit depth and gate count of ADAPT-VQE.
Abstract The adaptive variational quantum eigensolver (ADAPT-VQE) is a promising approach for quantum simulation in the noisy intermediate-scale quantum era, offering advantages over traditional VQE methods by reducing circuit depth and mitigating challenges in classical optimisation. Although ADAPT-VQE can require fewer total measurements than fixed ansätze for matched precision, the combined cost of repeated energy and gradient evaluations still leads to a substantial measurement overhead in practice, motivating strategies to reduce it further. In this work, we study two complementary strategies to reduce this cost using only standard projective Pauli measurements. First, we apply variance-based shot allocation (variance-minimised shot assignment and variance-preserved shot reduction (VPSR)) to both Hamiltonian energy and operator-gradient measurements, extending techniques previously studied only for energy estimation in standard VQE. VPSR achieves shot reductions of up to 43% for H 2 and 51% for LiH relative to uniform allocation while maintaining chemical accuracy. Second, we introduce a Pauli-measurement reuse protocol that stores outcomes from VQE optimisation and reuses them for gradient evaluation in the next ADAPT iteration. Across six molecular benchmarks (4–14 qubits), measurement reuse yields only marginal savings confined to the smallest systems and is effectively negligible for larger sizes, pivot-based grouping alone provides the dominant reduction and should be considered the practical baseline. These results characterise the interplay between shot allocation, measurement grouping, and cross-iteration reuse, offering practical guidance on when each strategy is most beneficial.
While fermion-to-qubit isomorphisms are mathematically invariant under exact unitary statevector transformations, their physical implementation on noisy intermediate-scale quantum (NISQ) devices introduces severe, practical measurement bottlenecks. In this work, we systematically investigate the measurement efficiency of adaptive derivative-assembled pseudo-trotter variational quantum eigensolver (ADAPT-VQE) algorithms across three critical molecular benchmarks (LiH, BeH2, and H2O) at stretched bond lengths (2xRe) covering distinct point-group symmetries. First, we establish exact representation invariance across all target multi-reference geometries, demonstrating that operator pool gradients and statevector electronic energies remain strictly isomorphic under exact mathematical execution (E(LiH) = -7.691469 Ha, E(BeH2) = -15.110110 Ha, and E(H2O) = -74.566874 Ha). Second, we quantify the hardware measurement overhead by partitioning operator pool observables into qubit-wise commuting (QWC) tensor-product basis (TPB) cliques. We reveal that while both Jordan-Wigner (JW) and Bravyi-Kitaev (BK) encodings yield identical total Pauli term counts across the singlet excitation manifold (128 terms for H2O and BeH2), the hierarchical binary-tree layout of the BK mapping dramatically compresses the required measurement circuits---achieving up to a 45.31% reduction in total hardware execution steps compared to 12.50% under JW. Our findings demonstrate that tree-structured mappers serve as implicit, highly efficient measurement-packing engines for adaptive quantum simulations without altering the underlying variational trajectory.
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.