Evaluating the trainability of adaptive quantum chemistry algorithms under multi-reference static correlation requires understanding how representation topologies intertwine with molecular geometry. We systematically expose a deep physical dependence on point-group symmetry by evaluating a spin-conserved SUSD operator pool across highly stretched configurations (2 x Re) of asymmetric LiH, symmetric BeH2, and asymmetric H2O. Under asymmetric distortions, the non-local mapping constraints of the Bravyi-Kitaev transformation create an optimization trapping effect--an encodement-locked manifestation of the broader barren plateau crisis. Crucially, by comparing these to the symmetrical stretching baseline of BeH2, we demonstrate that the preservation of point-group symmetry structurally protects the optimization landscape, proving that ansatz symmetry restrictions are necessary but insufficient without accounting for the underlying fermion-to-qubit representation. While current methods rely on numerical pruning to throttle pool sizes, our structural approach establishes that the mapping representation remains a critical factor in maintaining landscape trainability. Furthermore, exploiting structural overlap within our pool, we introduce a covariance-driven, adaptive shot-allocation filter. Diverging from static energy-variance minimization frameworks, our allocation engine operates as a dynamic runtime diagnostic tool. By continuously monitoring the gradient precision threshold epsilon, it aggressively prunes dead symmetry channels and triggers an automated circuit-termination sequence upon detecting representation-induced flat-lined states (dE/dtheta approx 0). This integration of algebraic measurement reuse with topology-aware statistical filtering provides a promising, resource-efficient strategy for executing deep variational algorithms on early fault-tolerant architectures.
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11 pages include Data Availability and Appendix Structure
Evaluating the trainability of adaptive quantum chemistry algorithms under multi-reference static correlation requires understanding how representation topologies intertwine with molecular geometry. We systematically expose a deep physical dependence on point-group symmetry by evaluating a spin-conserved SUSD operator pool across highly stretched configurations (2 x Re) of asymmetric LiH, symmetric BeH2, and asymmetric H2O. Under asymmetric distortions, the non-local mapping constraints of the Bravyi-Kitaev transformation create an optimization trapping effect--an encodement-locked manifestation of the broader barren plateau crisis. Crucially, by comparing these to the symmetrical stretching baseline of BeH2, we demonstrate that the preservation of point-group symmetry structurally protects the optimization landscape, proving that ansatz symmetry restrictions are necessary but insufficient without accounting for the underlying fermion-to-qubit representation. While current methods rely on numerical pruning to throttle pool sizes, our structural approach establishes that the mapping representation remains a critical factor in maintaining landscape trainability. Furthermore, exploiting structural overlap within our pool, we introduce a covariance-driven, adaptive shot-allocation filter. Diverging from static energy-variance minimization frameworks, our allocation engine operates as a dynamic runtime diagnostic tool. By continuously monitoring the gradient precision threshold epsilon, it aggressively prunes dead symmetry channels and triggers an automated circuit-termination sequence upon detecting representation-induced flat-lined states (dE/dtheta approx 0). This integration of algebraic measurement reuse with topology-aware statistical filtering provides a promising, resource-efficient strategy for executing deep variational algorithms on early fault-tolerant architectures.
The Variational Quantum Eigensolver (VQE) is a leading algorithm for noisy intermediate-scale quantum (NISQ) devices, but its adaptive variants (e.g., ADAPT-VQE) suffer from severe classical simulation bottlenecks during the ansatz growth phase. Representing and exponentiating pool operators for multi-qubit systems constructs massive sparse matrices that quickly scale to millions of elements, choking classical memory bandwidth and CPU/GPU cycle capacity. In this work, we present a resource-efficient software-layer framework that completely bypasses dense matrix exponentiation by evaluating state updates through a deterministic fifth-order (O(5)) Taylor series expansion. This approach reframes the costly unitary evolution into a chained sequence of five lightweight, successive sparse matrix-vector multiplications scaling strictly as O(Nz), where Nz is the number of non-zero elements. We validate our framework using equilibrium BeH2 (14 qubits), equilibrium H2O (12 qubits), and strongly correlated asymmetrically stretched H2O molecular profiles under both Jordan-Wigner (JW) and Bravyi-Kitaev (BK) transformations. The simulation results demonstrate that the O(5) truncation maintains exceptional numerical fidelity, retaining a state fidelity > 0.999999 and matching absolute ground state energies down to subchemical accuracy, while effortlessly navigating matrix spaces exceeding 268 million structural elements. This framework provides a scalable, high-performance pathway for executing deep variational simulations on hardware platforms with constrained computational budgets.
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.
Fixed-ansatz Variational Quantum Eigensolvers (VQE), such as the Unitary Coupled Cluster with Singles and Doubles (UCCSD) framework, frequently suffer from severe initialization paralyzation and zero-gradient traps when evaluated using the non-local Bravyi-Kitaev (BK) fermion-to-qubit mapping. In this work, we systematically demonstrate how the Adaptive Derivative-Assembled Pseudo-Trotter (ADAPT-VQE) framework shatters these structural limitations across three distinct electronic and geometric molecular pillars: Lithium Hydride (LiH), Hydrogen Fluoride (HF), and Water (H2O), under heavily stretched or asymmetric multi-reference configurations. While conventional UCCSD-VQE flatlines completely at a zero energy shift (0.000000~Ha) due to global phase cancellations inherent to the BK tree structures, our dynamic ADAPT-VQE loop successfully isolates the dominant symmetry-breaking operators using analytical commutator gradients. To bypass the severe O(N3) computational bottlenecks of dense matrix exponentiation and Singular Value Decomposition on larger registers, we implement a highly optimized, vector-based Taylor series expansion state-evolution engine. Our numerical results show that the accelerated ADAPT-VQE framework achieves instant, exact Full Configuration Interaction (FCI) convergence within the very first macro-cycle across all three molecular systems, maintaining absolute numerical stability up to a 12-qubit register space. This study establishes a robust, hardware-efficient path for simulating strongly correlated, highly polarized triatomic chemical environments on near-term local architectures.
Accurate determination of ground-state energies for molecules remains a challenge in quantum chemistry and a cornerstone for progress in fields such as drug discovery and materials design. The Variational Quantum Eigensolver (VQE) represents a leading hybrid quantum-classical paradigm for addressing this challenge; however, its widespread realization is limited by noise and the restricted scalability of current quantum hardware. Achieving efficient simulations on Noisy Intermediate-Scale Quantum (NISQ) devices and forthcoming Fault-Tolerant Application-Scalable Quantum (FASQ) systems demands a detailed understanding of how computational resources scale with molecular complexity and fermion-to-qubit encodings. In this study, resource requirements for VQE implementations employing the Unitary Coupled Cluster Singles and Doubles (UCCSD) ansatz are systematically analyzed. The molecular Hamiltonian is formulated in second quantization and mapped to qubit operators through the Jordan-Wigner (JW), Bravyi-Kitaev (BK), and Parity (Pa) transformations. Hamiltonian reduction strategies, including Z2 tapering and frozen-core approximations, are examined to assess their effect on quantum resource scaling. The analysis reveals that appropriate transformations, when combined with symmetry-based reductions, can substantially reduce qubit counts by up to ≈50% and quantum gate counts by up to ≈27.5× and Hamiltonian Pauli string counts by up to ≈2.75×, relative to the corresponding unreduced Hamiltonian representations within the same active-space configuration for the representative set of molecular systems under study. These findings provide practical circuit-level insights for executing chemically relevant simulations on NISQ hardware, while establishing physical-resource baselines that may inform future logical-level analyses targeting FASQ systems.
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.
The Variational Quantum Eigensolver (VQE) and its adaptive variants, such as ADAPT-VQE, are central to the study of strongly correlated quantum systems. However, the classical simulation of the ansatz growth process remains constrained by the exponential scaling of operator space and the associated computational cost of unitary evolution. We introduce the Greedy Operator Commutativity Partitioning (GOCP) framework, an analytical methodology designed to optimize both operator selection and state evolution. By reformulating complex unitary rotations as a chained sequence of fifth-order O(5) Taylor series expansions, GOCP bypasses the need for explicit matrix exponentiation, reducing the computational task to a sequence of sparse matrix-vector operations. We evaluate the performance of this framework across diverse molecular systems, including BeH2 and strongly correlated H2O geometries, utilizing both Jordan-Wigner and Bravyi-Kitaev mappings. Our results demonstrate that the GOCP framework maintains exceptional numerical fidelity-exceeding 1 - 10^-6 in state fidelity-while achieving sub-chemical accuracy in ground-state energy calculations. By enabling the simulation of operator manifolds exceeding 2.68 x 10^8 elements with high efficiency, this approach provides a scalable and rigorous pathway for exploring deep variational circuits in complex quantum many-body systems.