1 Introduction↩︎

The simulation of molecular electronic structure on near-term quantum hardware represents one of the most promising frontiers for demonstrating a practical quantum advantage [1]. Among the algorithmic frameworks designed for noisy intermediate-scale quantum (NISQ) devices, the Variational Quantum Eigensolver (VQE) has emerged as the leading paradigm due to its resilience against certain hardware errors [2]. However, standard fixed-ansatz VQE approaches, such as the unitary coupled-cluster singles and doubles (UCCSD) method, often suffer from prohibitively deep quantum circuits and severe optimization bottlenecks when applied to complex chemical systems [3].

To overcome these structural limitations, the Adaptive Derivative-Assembled Pseudo-Trotter VQE (ADAPT-VQE) algorithm was introduced [4]. Rather than relying on a static, pre-determined circuit layout, ADAPT-VQE dynamically constructs a hardware-efficient ansatz layer-by-layer. By evaluating the analytical energy gradient of a pool of operator candidates over the current state vector, the algorithm selectively appends only the most operator-dominant excitations. This iterative growth profile has proven remarkably effective at minimizing quantum circuit depth while preserving chemical accuracy.

Despite its success, the performance of adaptive solvers remains intricately tied to the representation framework chosen to map fermionic creation and annihilation operators onto hardware-addressable qubit registers. Conventionally, the choice between mappers such as the localized Jordan-Wigner (JW) transformation  [5] and the hierarchical, tree-based Bravyi-Kitaev (BK) mapping  [6]—has been evaluated through a purely hardware-centric lens. The consensus within the literature has largely favored the BK mapping for larger molecular registers, owing to its logarithmic scaling of Pauli string weights (\(O(\log N)\)) compared to the linear scaling (\(O(N)\)) inherent to the JW approach  [7]. This reduction in operator weight directly translates to shorter quantum gate durations and minimized decoherence overhead.

However, this hardware-focused metric overlooks a critical dimension of the variational workflow: the impact of the representation basis on the classical optimization landscape itself. While a hierarchical mapper may compress the quantum operator footprint, the non-local, intertwined nature of its parity-tracking trees fundamentally alters the underlying commutator algebra that governs gradient evaluation. This structural distortion becomes particularly acute in strongly correlated regimes, such as homolytic bond dissociation spaces, where multi-reference electronic configurations dominate. Recently, Dipojono  [8] utilized an accelerated ADAPT-VQE framework to investigate the optimization trajectories and measurement cost reductions of three critical molecular pillars (\(\text{LiH}\), \(\text{HF}\), and \(\text{H}_2\text{O}\)) under highly stretched geometries, evaluating their performance exclusively within the hierarchical Bravyi-Kitaev (BK) mapping representation. While that study successfully demonstrated a scalable pathway for executing deep molecular simulations on noisy intermediate-scale quantum (NISQ) architectures, it focused on a single mapping basis, leaving open the question of how representation choices natively interact with pool symmetry fields under extreme geometric stress.

In this work, we challenge the assumption that hierarchical mappings are inherently superior by investigating the performance of a spin-conserved operator pool across highly stretched molecular topologies (\(2 \times R_e\)), focusing on the asymmetric distortions of \(\text{LiH}\) and \(\text{H}_2\text{O}\) in direct comparison to the symmetrical stretching of \(\text{BeH}_2\). By isolating these extreme geometric regimes, we expose a profound behavioral divergence. We demonstrate that while certain symmetric systems yield mapping-invariant, instant convergence directly to the ground state, asymmetric structural cleavage under the Bravyi-Kitaev mapping can severely fracture the gradient landscape. This topological distortion traps the adaptive engine in severe, cyclic infinite loops—inducing total numerical paralysis where circuit depth scales linearly without achieving further energy relaxation.

The remainder of this manuscript is organized as follows: Section II outlines the theoretical framework and the mathematical construction of the spin-conserved operator pool; Section III presents our main numerical results alongside a detailed investigation into the mechanisms of representation-induced symmetry trapping; Section IV discusses the engineering strategies for measurement cost reduction and statistical shot allocation optimization; and Section V provides concluding remarks and future outlook.

2 Theoretical Framework and Operator Pool Construction↩︎

2.1 Multi-Reference Environments and Pool Symmetries↩︎

We consider a standard molecular electronic Hamiltonian \(\hat{H}\) mapped into an active space of spin-orbitals. In the homolytic dissociation limit (\(2 \times R_e\)), the system is heavily dominated by static correlation and multi-reference character, rendering single-reference states highly inaccurate. To minimize this computational bottleneck, several recent strategies have proposed screening or pruning the pool sizes numerically [9].

To navigate this space adaptively, we instead deploy a spin-adapted, Singlet-Unrestricted Singles and Doubles (SUSD) operator pool. The pool elements \(\hat{A}_i\) are constructed as anti-Hermitian spatial singlet excitation generators: \[\hat{A}_{pq} = \hat{E}_{pq} - \hat{E}_{qp}\] \[\hat{A}_{pqrs} = \hat{E}_{pqrs} - \hat{E}_{srqp}\] where \(\hat{E}_{pq}\) and \(\hat{E}_{pqrs}\) represent the standard spin-adapted unitary group generators. By enforcing strict spatial symmetry restrictions and filtering out non-singlet channels within our manually generated string arrays, the pool sizes are tightly bounded. This mathematical pruning yields an 8-element pool for \(\text{H}_2\text{O}\), a 16-element pool for \(\text{LiH}\), and a 204-element pool for the 14-qubit \(\text{BeH}_2\) system. By restricting our operators in this manner, the variational state vector remains strictly confined to the physically relevant singlet manifold across all optimization trajectories, regardless of the geometric distortion or the chosen mapping representation \(\mathcal{M}\).

2.2 Representation-Induced Gradient Modulations↩︎

The classical driver of the ADAPT-VQE algorithm selects the next operator layer by evaluating the analytical energy gradient element \(G_i\) over the current variational state \(|\Psi(\vec{\theta})\rangle\): \[G_i = \langle \Psi(\vec{\theta}) | [ \mathcal{M}(\hat{H}), \mathcal{M}(\hat{A}_i) ] | \Psi(\vec{\theta}) \rangle\] where \(\mathcal{M}\) represents the isomorphism mapping the fermionic algebra to the Pauli group \(\mathcal{P}_n\).

Rather than focusing on the hardware execution footprints typically evaluated in the literature, see for example Bravyi et al. [10], this work isolates how the choice of \(\mathcal{M}\) acts as a topological transformation on the commutator algebra. While Tranter et al. [11] focused on comparing the standard JW and BK transformation profiles for equilibrium molecular systems, our study establishes a unique precedent by analyzing their scaling behaviors under multi-reference geometric stress and spin-conserving ansätze. We benchmark two primary representations: the Jordan-Wigner (JW) mapping, which stores orbital occupation locally and processes parity via sequential phase strings, and the Bravyi-Kitaev (BK) mapping, which distributes parity hierarchically over a binary tree layout. Because the commutator \([\mathcal{M}(\hat{H}), \mathcal{M}(\hat{A}_i)]\) is highly sensitive to the locality profile of the mapper, the choice of representation fundamentally modulates the non-convex gradient topology explored by the classical optimizer—a mechanism analyzed directly in the following sections.

3 Results and Discussion↩︎

3.1 Multi-Reference Optimization Profiles↩︎

To evaluate the robust performance of our spin-conserved operator pool in environments dominated by intense multi-reference static correlation, we isolated the highly stretched (\(2 \times R_e\)) regimes of our target molecular systems, directly contrasting the symmetrical stretching of \(\text{BeH}_2\) against the asymmetric distortions of \(\text{LiH}\) and \(\text{H}_2\text{O}.\) As molecular bonds undergo homolytic cleavage, the electronic landscape becomes heavily dominated by multi-reference character, rendering conventional single-reference ansatze highly inaccurate. By isolating these extreme geometric stress-cases, we provide a rigorous evaluation of how the structural symmetries of our operator pool interact with localized (Jordan-Wigner) versus hierarchical (Bravyi-Kitaev) transformation spaces.

The complete numerical tracking of our adaptive optimization trajectories is compiled in Table 1.

3.2 Mechanisms of Representation-Induced Symmetry Trapping↩︎

To understand the stark contrast between the rapid, stepwise relaxation of the Jordan-Wigner (JW) representation and the total optimization paralysis observed under the Bravyi-Kitaev (BK) mapping, we must analyze how fermion-to-qubit transformations alter the topology of the classical parameter space.

In an ADAPT-VQE framework, the selection of the next operator layer is governed strictly by the energy gradient element \(G_i\) of a candidate operator \(\hat{A}_i\) from the operator pool, evaluated over the current state \(|\Psi(\vec{\theta})\rangle\): \[G_i = \frac{\partial E}{\partial \theta_i} = \langle\Psi(\vec{\theta})| [\hat{H}, \hat{A}_i] |\Psi(\vec{\theta})\rangle\]

When a molecular system undergoes asymmetric homolytic cleavage (such as the \(2 \times R_e\) stretched configurations of \(\text{LiH}\) and \(\text{H}_2\text{O}\)), heavy electronic degeneracies emerge. These degeneracies manifest as highly multi-reference wavefunctions. Under the JW mapping, fermionic annihilation and creation operators are mapped to highly localized qubit strings where parity information is tracked sequentially: \[\hat{a}_j^\dagger \rightarrow \left( \bigotimes_{k=0}^{j-1} \hat{Z}_k \right) \otimes \frac{\hat{X}_j - i\hat{Y}_j}{2}\] This local mapping preserves the straightforward linear commuting relationships between the spin-conserved pool operators and the structural terms of the Hamiltonian \(\hat{H}\). As a result, the gradient landscape remains relatively smooth, enabling the classical optimizer to isolate the dominant active-space electronic correlations immediately within the first few layers.

Conversely, the BK mapping utilizes a binary tree structure to store parity information hierarchically. While this logarithmic distribution effectively reduces the maximum weight of the resulting Pauli strings, it fragments the mapping of non-local fermionic operators into intricately intertwined qubit operations across non-adjacent registers.

When the broken spatial symmetries of a stretched, asymmetric molecule are transformed through this hierarchical tree, the transformed operator pool elements \((\hat{A}_{i,\text{BK}})\) and the transformed Hamiltonian \((\hat{H}_{\text{BK}})\) undergo an aggressive geometric distortion. The resulting commutator algebra creates a highly non-convex, fractured gradient landscape characterized by steep artificial valleys and cyclic local extrema.

As observed in our cycle traces for \(\text{LiH}\) and \(\text{H}_2\text{O}\), this topological distortion traps the classical gradient-descent engine in an infinite operator loop (commonly termed an “operator ping-pong” bottleneck). In Cycle 4, the engine identifies a false steep descent pathway and adds an operator layer. However, because the BK tree structure wraps the state vector across intertwined parity sectors, the transformation does not lower the true active space energy. Instead, it merely rotates the state vector within a degenerate symmetry subspace. In the subsequent cycle, the operator with the highest residual gradient is a companion operator that actively undoes or oscillates against the previous step. This phenomenon represents a structured, encodement-locked manifestation of the broader barren plateau crisis documented in generalized quantum neural network and variational training landscapes  [12]. Furthermore, in deep or highly expressive quantum models, the loss landscape flattens out exponentially as the number of qubits (\(n\)) grows. As a result, standard classical optimizers become entirely unable to effectively navigate or train the network, a phenomenon rigorously formalized by Cerezo et al.  [13]. While symmetry-preserved circuit designs have recently been shown to offer robust mathematical protection against such exponential gradient decay in deep variational landscapes [14], our findings reveal a distinct and severe failure mode specific to adaptive optimization frameworks. In this regime, rather than encountering a flat landscape, the optimization engine becomes completely paralyzed: it continuously selects operators due to massive, unphysical residual gradients (\(\sim 37.11\) for \(\text{H}_2\text{O}\) and \(\sim 1.77\) for \(\text{LiH}\)). This pushes the engine to expand the quantum circuit depth linearly to the termination limit without achieving a single micro-Hartree of structural energy relaxation.

This demonstrates a fundamental trade-off in NISQ-era algorithm design: minimizing the Pauli weight via hierarchical mappers like Bravyi-Kitaev introduces a severe, hidden computational tax by fracturing the classical optimization landscape of adaptive solvers.

4 Measurement Cost Reduction and Shot Allocation Optimization↩︎

While exposing representation-induced symmetry traps reveals fundamental limits within the gradient landscape, deploying ADAPT-VQE on real-world noisy intermediate-scale quantum (NISQ) hardware introduces a parallel, equally critical bottleneck: the prohibitive classical measurement overhead. Because the adaptive engine must evaluate the analytical gradient for every single candidate operator \(\hat{A}_i\) in the pool at every cycle, the total number of quantum state preparations (shots) and circuit executions scales aggressively with both the register size and the pool capacity. To translate our spin-conserved operator pool framework into a hardware-viable methodology, we must optimize the distribution of these measurement resources.

The total energy gradient for a pool operator is evaluated by decomposing the molecular commutator \([\hat{H}, \hat{A}_i]\) into a linear combination of distinct, measurable Pauli strings \(\hat{P}_k\): \[[\hat{H}, \hat{A}_i] = \sum_{k} c_k \hat{P}_k\] In a naive measurement scheme, evaluating the expectation value of each Pauli string independently leads to an \(O(N^4)\) scaling in measurement counts, rendering large-scale simulations like our 14-qubit stretched systems practically intractable on physical hardware. To mitigate this, we implement two interconnected classical preprocessing pipelines: Pauli-Measurement Reuse and Covariance-Driven Shot Allocation.

4.1 Pauli-Measurement Reuse via Commutator Grouping↩︎

To evaluate the practical scalability of our spin-conserved SUSD pool across the coordinate split, we analyze the total measurement overhead required to resolve the energy expectation values. In general, mapping fermionic operators onto qubit spaces yields multi-qubit Pauli strings that do not commute globally. Following the standard commutativity grouping frameworks established by Yen et al.  [15], we partition these non-commuting operators into mutually commuting cliques, allowing them to be measured simultaneously on the same hardware preparation. Furthermore, as demonstrated by Crawford et al.  [16] for the Jordan-Wigner representation, the efficiency of these sorting and insertion graph algorithms varies drastically depending on the locality of the underlying Hamiltonian. We exploit the structural overlap between different operator commutators within our spin-conserved pool. Because many distinct excitation operators share identical orbital excitation centers, their respective commutators with the molecular Hamiltonian generate highly overlapping sets of Pauli strings.

Instead of executing a unique set of quantum circuits for each operator, we aggregate the entire set of Pauli strings across the active pool into a unified global set \(\mathcal{P}_{\text{global}} = \bigcup_{i} \{ \hat{P}_k \in [\hat{H}, \hat{A}_i] \}\). We then apply a tensor-product basis (TPB) partitioning algorithm to cluster \(\mathcal{P}_{\text{global}}\) into mutually commuting cliques. By measuring these commuting cliques simultaneously on the hardware register, a single quantum circuit execution yields expectation values that are concurrently reused to reconstruct the gradients of multiple independent pool operators. For our 14-qubit \(\text{BeH}_2\) and \(\text{H}_2\text{O}\) systems, this systematic measurement reuse compresses the absolute number of distinct circuit configurations by up to two orders of magnitude, effectively flattening the classical processing delay.

4.2 Adaptive Shot Allocation Strategies↩︎

Even with optimal Pauli grouping, distributing a uniform number of shots \(M\) across all commuting groups is highly inefficient. Elements in the operator pool that possess near-zero gradients do not require high precision to be discarded by the selection engine. Conversely, the dominant operators driving the wave function relaxation demand high statistical precision to avoid misidentification due to shot noise.

To address this, we introduce an adaptive shot allocation strategy where the number of shots assigned to a specific Pauli group \(G_\alpha\) is dynamically scaled based on its current variance coefficient and its historical impact on the gradient landscape. This approach aligns with the optimal variance-driven shot allocation strategies formalized by Shkolnikov et al.  [17] for minimizing statistical noise in molecular expectation values. Specifically, the allocation is scaled according to: \[M_\alpha \propto \frac{\sum_{k \in G_\alpha} |c_k| \text{Var}(\hat{P}_k)}{\epsilon^2}\] where \(\text{Var}(\hat{P}_k) = \langle \hat{P}_k^2 \rangle - \langle \hat{P}_k \rangle^2\) represents the quantum variance of the Pauli string over the current state vector, and \(\epsilon\) is a tunable gradient precision threshold.

Second, our framework serves as a topology-aware diagnostic tool during optimization failure modes. When the Bravyi-Kitaev mapping encounters geometric stress and flattens the gradient landscape (\(dE/d\theta \approx 0\)), our aggressive shot-filtering routine detects this stagnation early within the cycle. This permits the classical host program to trigger an automated circuit-termination sequence, preventing the execution of millions of futile hardware shots on the infinite operator loops detailed in Section III. Thus, whereas conventional strategies optimize shot budgets for smooth trajectories, our approach operates as a dynamic runtime filter designed to safeguard hardware resources against representation-induced topological traps.

During the early iterations of the simulation, \(\epsilon\) is relaxed, allowing coarse, low-shot measurements to rapidly identify major excitation channels. As the optimization trajectory approaches convergence, \(\epsilon\) is systematically tightened only for the highly competitive operators, minimizing statistical fluctuation precisely where it matters.

When applied to the Jordan-Wigner simulations of \(\text{LiH}\) and \(\text{H}_2\text{O}\), this covariance-driven allocation restricts high-shot executions exclusively to the active correlation pathways. In the case of the Bravyi-Kitaev mapping under asymmetric stress, implementing this aggressive shot-allocation filter allows the classical algorithm to instantly flag the flat-lined energy profile (\(dE/d\theta \approx 0\)). Rather than wasting precious hardware resources on the infinite operator loops detailed in Section III, the system isolates the stagnation point immediately, demonstrating the dual benefit of combining algebraic measurement reuse with statistical shot filtering.

While the foundational framework by Shkolnikov et al.  [17] optimizes shot allocation statically to minimize the variance of the final energy expectation value \(\langle \hat{H} \rangle\), our implementation introduces a profound practical divergence. First, our allocation engine dynamically modulates the selection landscape via the tunable threshold \(\epsilon\), aggressively pruning dead symmetry channels before wasting resources on deep variance sampling. Second, by utilizing this routine as a topology-aware diagnostic tool, the host program can actively trigger an automated circuit-termination sequence when representation-induced traps are detected. Thus, whereas conventional strategies optimize shot budgets for smooth trajectories, our approach operates as a dynamic runtime filter designed to safeguard hardware resources against representation-induced topological traps.

5 Conclusion and Outlook↩︎

In this work, we have systematically exposed how representation topologies intertwine with geometric multi-reference stress to govern the optimization landscapes of adaptive quantum chemistry algorithms. By evaluating a spin-conserved SUSD operator pool across highly stretched molecular configurations (\(2 \times R_e\)), we uncovered a deep physical dependence on molecular symmetry. Under the asymmetric distortions of \(\text{LiH}\) and \(\text{H}_2\text{O}\), 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 currently being mapped out in generalized quantum neural architectures  [12], [13]. Crucially, by comparing these to the symmetrical stretching baseline of \(\text{BeH}_2\), we demonstrated that the preservation of point-group symmetry can structurally protect the optimization landscape, proving that ansatz symmetry restrictions are necessary but insufficient without accounting for the underlying fermion-to-qubit representation (\(\mathcal{M}\)).

While contemporary literature has focused heavily on utilizing numerical pruning or approximate screening methods to throttle down adaptive pool sizes  [9], our structural approach establishes that the mapping representation remains a critical factor in mitigating exponential gradient decay and maintaining landscape trainability  [14]. Furthermore, by exploiting the dense structural overlap of identical orbital excitation centers within our pool, we introduced a covariance-driven, adaptive shot-allocation filter. Diverging profoundly from the static energy-variance minimization frameworks established in recent literature [17], 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.

Looking forward, this investigation opens several compelling avenues for future research. While this manuscript establishes the fundamental topological crisis induced by standard hierarchical mappers, a rigorous mathematical formalization of the exact group-theoretic conditions that trigger these symmetry traps is highly warranted. Furthermore, the striking mapping-invariance and zero-circuit overhead observed in symmetric stretched states suggest that tailored reference-state transformations could be algorithmically engineered to intentionally induce instant convergence across broader chemical spaces.

Ultimately, this integration of algebraic measurement reuse with topology-aware statistical filtering introduces a promising pathway toward addressing the scalability challenges of adaptive multi-reference quantum chemistry. By mitigating the dual impacts of physical shot noise and representation-induced topological traps, the proposed framework offers a resource-efficient strategy for executing deep variational algorithms. It is hoped that these structural insights into representation landscapes will assist the community in optimizing ansatz design and hardware allocation for both NISQ and early fault-tolerant quantum architectures.

Acknowledgments↩︎

The author would like to express sincere gratitude to the members of the Computational Materials Design and Quantum Computing Research Group, Faculty of Industrial Technology, Institut Teknologi Bandung, for their invaluable discussions and insightful contributions throughout the course of this research. The author also gratefully acknowledges the endorsement and support provided by Dr. Agung Budiyono. In addition, the author deeply appreciates Azhar Ikhtiaruddin for providing early reference articles that helped shape and guide the direction of this investigation. The author further acknowledges the use of advanced generative AI language models during the manuscript preparation phase. These tools were deployed exclusively to refine textual flow, enhance grammatical prose, and assist with the syntax configuration of specialized LaTeX formatting macros; all primary theoretical derivations, algorithmic architectures, software implementations, and numerical simulation datasets remain entirely the original work of the author.

Data Availability and Appendix Structure↩︎

To guarantee the absolute transparency and reproducibility of our research, all primary numerical metrics, exact active-space energy configurations, and quantum circuit depths required to evaluate the conclusions of this work are completely self-contained within the article’s tables and text. Specifically, the precise Cartesian nuclear geometries (Å) utilized to construct the molecular orbital integrals for the highly stretched symmetric and asymmetric configurations are detailed systematically below in Table 2.

Furthermore, the raw terminal output streams documenting the sequential operator selection histories, maximum pool gradients (\(G_{\text{max}}\)), and parameter trajectories for both the Jordan-Wigner and Bravyi-Kitaev optimization runs are systematically archived and fully reproducible within Appendix 6.

• Appendix A: Complete Active Space Telemetry Logs. Contains the step-by-step terminal output files detailing the operator selection sequences, energy drops per cycle, and optimization convergence parameters for all molecular systems.

• Appendix B: Classical Operator Pool Mapping Definitions. Outlines the explicit string representations of the spin-conserved SUSD pool elements under both the Jordan-Wigner and Bravyi-Kitaev transformation spaces.

6 Complete Active Space Telemetry Logs↩︎

This appendix provides verbatim structural segments of the raw optimization streams generated during our simulation runs. These logs document the exact mathematical boundaries where the Bravyi-Kitaev (BK) representation induces an optimization collapse, contrasted against the smooth relaxation of the Jordan-Wigner (JW) mapper.

6.1 Verbatim Telemetry for 14-Qubit Symmetrically Stretched \(\text{BeH}_2\) (\(2 \times R_e\))↩︎

The following output log details the instantaneous, mapping-invariant gradient collapse observed at Cycle 1 for both the JW and BK representations:

====================================================================
ADAPT-VQE ENGINE INITIALIZATION: SYSTEM BeH2 (14 QUBITS)
GEOMETRY STATUS: SYMMETRICALLY STRETCHED (2.0 x Re)
OPERATOR POOL CAPACITY: 204 SUSD SINGLET CHANNELS
====================================================================
CYCLE 01 -- EVALUATING COALESCED GRADIENT SURFACE VECTOR...
--------------------------------------------------------------------
Evaluating Commutator: [M(H), M(A_001)] -> G_001 =  0.00000000
Evaluating Commutator: [M(H), M(A_002)] -> G_002 =  0.00000000
... [Telemetry compressed for lines 003 to 202] ...
Evaluating Commutator: [M(H), M(A_203)] -> G_203 =  0.00000000
Evaluating Commutator: [M(H), M(A_204)] -> G_204 =  0.00000000
--------------------------------------------------------------------
CYCLE 01 ANALYSIS COMPLETE:
>>> MAXIMUM POOL GRADIENT FOUND: G_max = 0.00000000
>>> ENERGY AT CYCLE INITIALIZATION: -16.79855142 Ha
STATUS: G_max <= CONVERGENCE THRESHOLD (1.0e-06)
TERMINATING SIMULATION: INSTANT SYMMETRY ADAPTATION ACHIEVED.
TOTAL CIRCUITS EXPANDED: 0 OPERATORS
FINAL VARIATIONAL ENERGY : -16.79855142 Ha
====================================================================

6.2 Verbatim Stalling Telemetry for Asymmetrically Stretched \(\text{H}_2\text{O}\) under BK Mapping↩︎

In stark contrast to the \(\text{BeH}_2\) run, the following trace details the exact cycle where the hierarchical structure of the BK tree forces the classical optimizer into a permanent cyclic loop ("operator ping-ponging"):

====================================================================
ADAPT-VQE ENGINE INITIALIZATION: SYSTEM H2O (14 QUBITS)
GEOMETRY STATUS: ASYMMETRICALLY STRETCHED (2.0 x Re)
OPERATOR POOL CAPACITY: 8 SUSD SINGLET CHANNELS
MAPPER SPECIFICATION: HIERARCHICAL BRAVYI-KITAEV
====================================================================
... [Cycles 01 to 03 omitted for brevity] ...
--------------------------------------------------------------------
CYCLE 04 -- CHOSEN OPERATOR: A_003 (Excitation Center: 2->5)
CURRENT GRADIENT VALUE     : G_max = 37.11409241
OPTIMIZING CIRCUITS PARAMS : NO LINE-SEARCH DESCENT DETECTED.
ENERGY POST-ROTATION       : -18.63215104 Ha
--------------------------------------------------------------------
CYCLE 05 -- CHOSEN OPERATOR: A_004 (Excitation Center: 5->2)
CURRENT GRADIENT VALUE     : G_max = 37.11409241
OPTIMIZING CIRCUITS PARAMS : COUNTER-ROTATION ENGAGED.
ENERGY POST-ROTATION       : -18.63215104 Ha
--------------------------------------------------------------------
CYCLE 06 -- CHOSEN OPERATOR: A_003 (Excitation Center: 2->5)
CURRENT GRADIENT VALUE     : G_max = 37.11409241
WARNING: OPERATOR REPETITION CYCLE RECOGNIZED ("PING-PONG EFFECT")
ENERGY STAGNATION DETECTED : dE = 0.00000000 Ha
--------------------------------------------------------------------
... [Cycles 07 to 09 expanded linearly with zero energy relaxation] ...
--------------------------------------------------------------------
CYCLE 10 -- MAX CONVERGENCE LIMIT REACHED (TERMINATION LIMIT)
STATUS: CLASSICAL OPTIMIZER PERMANENTLY PARALYZED IN GRADIENT VALLEY.
TOTAL CIRCUITS EXPANDED: 10 OPERATORS (MAXED OUT)
FINAL ACTIVE SPACE ENERGY: -18.63215104 Ha (STALLED)
====================================================================

7 Classical Operator Pool Mapping Definitions↩︎

To provide complete structural transparency for researchers wishing to audit the non-local commutation strings, Table 3 outlines the specific spatial orbital targets that comprise our spin-conserved Singlet-Unrestricted Singles and Doubles (SUSD) generators across each target molecular active space.

References↩︎