This study analyzes the expressibility characteristics and gradient distribution in the variational quantum eigensolver (VQE) algorithm using nine variations of hardware-efficient ansatz (HEA) in the hydrogen molecular system. The evaluation was conducted by measuring Kullback–Leibler divergence ( D KL ) value as an indicator of expressibility ansatz and the gradient variance as an indicator of the barren plateau phenomenon. The system was optimized using the L-BFGS algorithm, and each ansatz was tested to identify the extent to its parametric structure affected optimization performance and the potential for barren plateaus. The analysis results show differences among the nine ansatzes. Variations such as SR y, TR y, R y R x, and R y R z tend to produce high expressibility but are followed by a drastic decrease in gradient variance, thus showing symptoms of a barren plateau. Conversely, more complex ansatzes involving the mixing of rotation bases, such as HR y, R y, and XR y provide a more stable gradient distribution and more consistent optimization performance. These findings are consistent with the proposed linear mixing expressibility–trainability model, states that increased expressibility is not always directly proportional to the ease of the optimization process; homogenization of the state space actually reduces the directional information needed by the optimizer. These results provide an important basis for selecting more effective ansatzes in future implementations of variational quantum algorithms.
This study analyzes the expressibility characteristics and gradient distribution in the variational quantum eigensolver (VQE) algorithm using nine variations of hardware-efficient ansatz (HEA) in the hydrogen molecular system. The evaluation was conducted by measuring Kullback–Leibler divergence ( D KL ) value as an indicator of expressibility ansatz and the gradient variance as an indicator of the barren plateau phenomenon. The system was optimized using the L-BFGS algorithm, and each ansatz was tested to identify the extent to its parametric structure affected optimization performance and the potential for barren plateaus. The analysis results show differences among the nine ansatzes. Variations such as SR y, TR y, R y R x, and R y R z tend to produce high expressibility but are followed by a drastic decrease in gradient variance, thus showing symptoms of a barren plateau. Conversely, more complex ansatzes involving the mixing of rotation bases, such as HR y, R y, and XR y provide a more stable gradient distribution and more consistent optimization performance. These findings are consistent with the proposed linear mixing expressibility–trainability model, states that increased expressibility is not always directly proportional to the ease of the optimization process; homogenization of the state space actually reduces the directional information needed by the optimizer. These results provide an important basis for selecting more effective ansatzes in future implementations of variational quantum algorithms.
Variational Quantum Eigensolver (VQE) algorithms suffer from barren plateaus, where gradients vanish with system size and circuit depth. Although many mitigation strategies exist, their connection to convergence performance under different iteration budgets remains unclear. Moreover, a systematic analysis identifying which state-of-the-art mitigation techniques perform best under specific scenarios is also lacking. We benchmark four approaches, Local-Global, Adiabatic, State Efficient Ansatz (SEA), and Pretrained VQE, against standard VQE on molecular systems from 4 to 14 qubits, analyzing gradient variance up to 50 layers and convergence over 1000 iterations. Our results show that the impact of gradient preservation is iteration-dependent. In the 14-qubit BeH2 system, Pretrained VQE outperforms SEA at 100 iterations despite lower gradient variance, but SEA becomes 2.2x more accurate at 1000 iterations. For smaller systems, SEA achieves near-exact energies (H2: 10^-5 Ha, LiH: 2x10^-4 Ha) with fidelities 0.999, while standard methods plateau early. The results demonstrate that robust barren plateau mitigation depends on aligning the chosen strategy with both system size and available computational budget, rather than treating gradient variance as the sole predictor of performance.
Variational quantum algorithms (VQAs) rely on parameterized quantum circuits (PQCs), whose performance is governed by expressibility and trainability. Existing studies typically evaluate these properties at the logical circuit level, implicitly assuming that designed PQCs remain unchanged during hardware execution. In practice, however, hardware-aware transpilation modifies circuit structure through qubit mapping, routing, and basis decomposition, potentially altering PQC behavior. In this paper, we perform a systematic hardware-aware analysis of expressibility and trainability by comparing logical and transpiled PQCs across multiple ansatz families, qubit counts, and circuit depths. Expressibility is measured using fidelity-based KL divergence, while trainability is quantified through gradient variance. Our results show that transpilation acts as an implicit architectural perturbation, producing strongly ansatz-dependent effects. Expressibility deviations exceed upto 125% in some cases, while trainability variations reach up to 25%. Structured ansatzes are generally more robust, whereas highly entangled architectures are more sensitive to transpilation-induced transformations. We further show that transpilation can alter the commonly assumed expressibility-trainability trade-off, demonstrating that logical-level analyses may not reliably predict hardware-level behavior. These findings highlight the importance of hardware-aware evaluation for accurate characterization of VQAs.
Variational quantum algorithms are suitable for use on noisy quantum systems. One of the most important use-cases is the quantum simulation of materials, using the variational quantum eigensolver (VQE). To optimize VQE performance, a suitable parameterized quantum circuit (ansatz) must be selected. We investigate a class of ansatze that incorporates knowledge of the quantum hardware, namely the hardware efficient ansatze. The performance of hardware efficient ansatze is affected differently by noise, and our goal is to study the effect of noise on evaluating which ansatz gives more accurate results in practice. First, we study the effect of noise on the different hardware efficient ansatze by benchmarking and ranking the performance of each ansatz family (i) on a chemistry application using VQE and (ii) by the recently established metric of "expressibility". The results demonstrate the ranking of optimal circuits does not remain constant in the presence of noise. Second, we evaluate the suitability of the expressibility measure in this context by performing a correlation study between expressibility and the performance of the same circuits on a chemistry application using VQE. Our simulations reveal a weak correlation and therefore demonstrate that expressibility is not an adequate measure to quantify the effectiveness of parameterized quantum circuits for quantum chemistry. Third, we evaluate the effect of different quantum device noise models on the ordering of which ansatz family is best. Interestingly, we see that to decide which ansatz is optimal for use, one needs to consider the specific hardware used even within the same family of quantum hardware.
Ansatz selection is a key factor in the performance of variational quantum algorithms (VQAs). While much of the state-of-the-art still relies on heuristic choices, an inadequate circuit structure can compromise both the expressive power and the trainability of the resulting model. Recent results have also established theoretical connections between expressibility and the onset of barren plateaus, highlighting the need for systematic criteria for ansatz selection. In this work, the ansatz is treated as a design feature to be optimized rather than a fixed block, and a design space exploration (DSE) is performed over a diverse set of parametrized quantum circuits (PQCs). Three complementary metrics -- expressibility, trainability, and resource cost -- are evaluated and used to analyze the trade-offs that emerge across different PQCs. Beyond identifying Pareto-optimal candidates, this multi-objective perspective helps clarify the interplay between these metrics and contributes quantitative evidence toward understanding the expressibility--trainability tension in variational circuits.
Hardware-efficient ans¨atze (HEA) for the variational quantum eigensolver (VQE) suffer barren plateaus (BP), in which gradient variance decays exponentially for molecular Hamiltonians at chemically relevant scales. The present work tests whether wrapping the HEA into forward–inverse layer pairs (“fragmentation”) removes this observed exponential-decay signature over the tested depth range. Under the canonical McClean et al. random-uniform protocol, plain HEA and an architecture-only fragmented variant are compared on H2, LiH, BeH2, stretched N2 in a complete active space [CAS(10,6): 10 electrons in 6 orbitals], and a reduced-active-space [Fe2S2(SH)4]2− model. On the four molecules in the BP regime, plain HEA exhibits the canonical exponential decay with BP exponent α ∈ [0.32, 0.41], whereas the fragmented variant shows no clear exponential trend (|α| ≲ 0.05) and retains gradient variance roughly two to four orders of magnitude above plain HEA at the deepest tested depths. A separate identity-initialized fragmented variant shows a distinct negative-slope profile, reflecting variance growth from a near-identity starting point rather than the canonical BP protocol. Short-descent tests on LiH and N2 show that unconstrained HEA trajectories can leave the target particle-number sector; thus any below-Hartree–Fock (HF) claim requires explicit sector verification. On H2, where no BP is present, plain HEA reaches the full configuration interaction (FCI) energy while fragmented variants remain trapped near HF. These results indicate that forward–inverse fragmentation targets the BP regime specifically.
Recent advances in the variational quantum eigensolver (VQE) have strengthened its promise for molecular electronic-structure calculations on near-term noisy intermediate-scale quantum (NISQ) devices. However, barren plateaus (BPs) remain a major obstacle to scaling VQE, as vanishing gradients can severely obstruct optimization at large scale. While most BP studies focus on randomly initialized circuits and global gradient behavior, practical quantum chemistry VQE typically uses chemically motivated initializations and local optimization around HF- or MP2-derived states. Although chemically informed initializations are widely expected to improve VQE trainability, how they mitigate local barren plateaus remains unclear. We therefore study the occurrence of local barren plateaus in molecular UCCSD-VQE, considering both chemically motivated initialization points and iterates along the corresponding optimization trajectories. In the benchmarks considered here, chemically initialized starting points and variational trajectories show polynomial decay of local gradient variance with system size instead of the exponential BP-like decay observed for random controls, indicating that chemical initialization mitigates barren plateaus in the local optimization regions and highlights the trainability advantage of chemically motivated VQE strategies.