Truncated Variational Hamiltonian Ansatz

• The paper demonstrates that tVHA reduces quantum circuit depth and parameter count significantly, achieving 40–70% reduction in measurements and gate operations.
• tVHA is defined by a strategic truncation of Hamiltonian terms via coefficients and physics-based operator classification to balance resource savings with chemical accuracy.
• The framework integrates with VQE through staged optimizations and variational parameter transfer, enabling accurate simulations with fewer quantum resources.

The Truncated Variational Hamiltonian Ansatz (tVHA) is a variational quantum circuit framework designed for efficient quantum simulations of electronic structure and material systems on noisy intermediate-scale quantum (NISQ) devices, with significant reductions in parameter count and circuit depth relative to standard approaches. tVHA techniques leverage physically motivated truncation or partitioning of the quantum Hamiltonian to minimize quantum resources—measurement and gate complexity—while retaining or systematically improving accuracy via controlled restoration of neglected Hamiltonian terms. These methods enable practical quantum computations for both quantum chemistry and quantum materials modeling (Xu et al., 2 Feb 2024, Possel et al., 26 May 2025).

1. Theoretical Foundations and Hamiltonian Truncation

Truncated Variational Hamiltonian Ansatz approaches are fundamentally based on decomposing the quantum Hamiltonian—typically mapped to the qubit space for simulation—into dominant and subdominant term groups, and integrating variational strategies that only gradually restore the neglected interactions.

For molecular Hamiltonians,

$$H = \sum_{i=1}^K c_i\,P_i, \text{ with } P_i \in \{I, X, Y, Z\}^{\otimes N},\; c_i \in \mathbb{R},$$

one constructs a truncated Hamiltonian,

$H_{\mathrm{trunc}} = \sum_{i\in T} c_i\,P_i,$

by selecting a subset $T$ of “important” terms—either those above a coefficient threshold or categorized by physical significance. The remainder, $R = H - H_{\mathrm{trunc}}$, represents the neglected contributions initially omitted from the variational procedure (Xu et al., 2 Feb 2024).

In the circuit-ansatz context, tVHA leverages the adiabatic theorem with a Hamiltonian interpolation,

$$H(\tau) = (1 - \tau/T) H_0 + (\tau/T) H_f, \quad 0 \leq \tau \leq T,$$

leading, upon Trotter discretization and variational parameterization, to a state representation,

$$|\psi_{\mathrm{tVHA}}(\boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{\gamma})\rangle = \prod_{n=1}^N e^{-i\alpha_n \Delta\tau H_\alpha} e^{-i\beta_n \Delta\tau H_\beta} e^{-i\gamma_n \Delta\tau H_\gamma^{\text{cut}}} |{\psi}_0\rangle,$$

where $H_\alpha$ (one-body), $H_\beta$ (diagonal/Coulomb two-body), and a truncated $H_\gamma^{\text{cut}}$ (selected non-Coulomb two-body) blocks are exponentiated with independently tunable variational angles (Possel et al., 26 May 2025).

2. Operator Classification and Truncation Strategies

Two principal truncation strategies are employed:

• Coefficient Cutoff: Select Pauli terms $P_i$ with $|c_i| \geq c_\Lambda$ (Xu et al., 2 Feb 2024).
• Physics-Inspired Operator Classification: Decompose the second-quantized fermionic Hamiltonian into blocks:

$$H_{num},\; H_{cou},\; H_{exc},\; H_{nex},\; H_{dex},$$

ordered by physical relevance and norm, and build successively larger $H^{(m)}$ via nesting (Xu et al., 2 Feb 2024).

Within tVHA, the two-body part $H_\gamma$ is truncated via selection of the $M_\gamma(p)$ strongest integral terms $g_s$, so that the normalized cumulative weight $$p = (\sum_{s=1}^{M_\gamma(p)} |g_s|)/(\sum_{s=1}^{all} |g_s|)$$ controls the degree of truncation (Possel et al., 26 May 2025). By keeping the leading one-body and Coulomb blocks intact, and restoring non-Coulomb terms in order of strength, error and resource demands are strictly controlled.

3. Circuit Construction and Resource Scaling

The tVHA circuit construction is modular:

• $H_\alpha$: Realized as $m$ single-qubit $R_z$ rotations per Trotter step.
• $H_\beta$: Implemented via $O(m^2)$ two-qubit ZZ gates and local rotations.
• $H_\gamma^{\text{cut}}$: Each non-Coulomb term yields up to 8 Pauli strings (via Jordan–Wigner mapping), each requiring $\sim2(m-1)$ CNOTs and 2 rotations per four-body string.

Variational angles $\alpha_n$, $\beta_n$, $\gamma_n$ are attached to each block per Trotter step ($N$ typically $\mathcal{O}(1-10)$), leading to a total parameter count of $3N$, independent of molecular orbital count $m$ (Possel et al., 26 May 2025).

The circuit depth per Trotter step scales as $O(M_\gamma(p) m)$, with $M_\gamma(p)$ dependent on truncation threshold $p$. A moderate truncation ($p\approx 0.5$) already produces substantial resource reduction—hundreds to thousands of CNOTs for $m\lesssim 12$ (Possel et al., 26 May 2025).

By comparison, UCCSD requires $\sim \mathcal{O}(m^4)$ parameters and CNOT gates, and even hardware-efficient ansätze (HEA) require $2mL$ parameters for $L$ layers, typically with lower chemical accuracy (Possel et al., 26 May 2025).

4. Integration with VQE and Cost/Accuracy Trade-Offs

tVHA and related TVHA protocols are integrated within VQE by optimizing the parameterized state

$$|\psi(\boldsymbol{\theta})\rangle = U(\boldsymbol{\theta})|0\rangle^{\otimes N},$$

against staged Hamiltonians $H^{(m)}$ or the full $H$, as dictated by the truncation/interpolation schedule (Xu et al., 2 Feb 2024). At each stage,

$$E^{(m)}(\boldsymbol{\theta}) = \langle\psi(\boldsymbol{\theta})| H^{(m)} | \psi(\boldsymbol{\theta}) \rangle,$$

is minimized using a classical optimizer (e.g., SPSA, gradient-free SBPLX) (Xu et al., 2 Feb 2024, Possel et al., 26 May 2025).

Quantum resource cost is governed by the number of Pauli measurement groups and VQE iterations: $$M^{(m)}_{\mathrm{tot}} = N_{\mathrm{shots}} \times g_m \times I_m,$$ where $g_m$ is the number of commuting groups at stage $m$ (Xu et al., 2 Feb 2024).

Empirically, measurement and gate reductions of $40$–$70$\% are observed without loss in accuracy. For example, in LiH ($m=12$), $p=0.5,N=2$ tVHA achieves $\Delta E \approx 3$ mHa with $\sim 2,000$ CNOT gates and only $6$ parameters versus UCCSD's $4,900$ CNOTs and $200$ parameters (Possel et al., 26 May 2025).

Formal error bounds combine Trotter error ($\propto \|\left[H_j,H_k\right]\|\Delta\tau^2$) and truncation error ($\|\Delta H_\gamma\|$), yielding total bias that is a function of step size and omitted terms' norm (Possel et al., 26 May 2025).

5. Benchmark Results and Applications

Extensive numerical simulations confirm the practical efficacy of tVHA and TVHA approaches.

Table: Benchmark Results (select entries, all energies in Hartree, $\Delta E$ relative to FCI, CNOT counts approximate) (Possel et al., 26 May 2025, Xu et al., 2 Feb 2024)

System	Ansatz	CNOTs	Params	$\Delta E$
LiH, $m=12$	UCCSD	4900	200	2 mHa
	HEA (L=3)	1200	72	15 mHa
	tVHA ($p=0.5$)	2000	6	3 mHa
H$_2$, $m=4$	tVHA ($p=0.5$)	10	3	$<0.5$ mHa
H$_4$, $m=8$	tVHA ($p=0.5$)	500	3	8 mHa
CH$_2$ active	tVHA ($p=0.2$)	30	3	$<1.5$ mHa

Significant cost reductions are observed across both weakly and strongly correlated molecules, and comparable gains hold for active-space reductions as well as lattice/Hubbard-type materials models (Possel et al., 26 May 2025).

tVHA approaches retain chemical accuracy ($<$1–2 mHa) for practical truncation levels and Trotter steps, with circuit depth and parameter count well below those of traditional quantum chemistry ansätze.

6. Extensions, Generalizations, and Outlook

tVHA and TVHA protocols are extensible to a broad set of correlated electron models and materials Hamiltonians, notably:

• Active-Space Restrictions: tVHA applies unchanged to active-space Hamiltonians as in complete active space self-consistent field (CASSCF) methods (Possel et al., 26 May 2025).
• Materials and Lattice Models: For periodic or local-lattice contexts—e.g., transition metal oxides, extended Hubbard models, or spin Hamiltonians—truncation by strength or locality yields highly shallow circuits optimal for NISQ environments (Possel et al., 26 May 2025, Xu et al., 2 Feb 2024).

Hybrid strategies combining coefficient cutoffs with operator-classification, or introducing smooth interpolation ($\lambda$-schedule), further refine the trade-off between resources expended and accuracy regained (Xu et al., 2 Feb 2024).

Potential application domains include chemistry and drug-design pipelines where measurement cost is paramount, as well as solid-state quantum simulations where circuit depth and coherence times are limiting factors.

7. Discussion and Current Limitations

The principal advantage of the truncated variational Hamiltonian framework is the substantial reduction in quantum resource demands—both in measurement overhead and in quantum gate count—without significant loss in final energy accuracy for problems at the scale of 4–12 qubits. Introducing physical intuition into the term-selection process leads to rapid convergence even as neglected terms are reincorporated perturbatively in sequential VQE sub-runs (Xu et al., 2 Feb 2024).

A key trade-off is the replacement of a single, homogeneous VQE execution with multiple, staged, but shorter VQE optimizations; however, the transfer and fine-tuning of ansatz parameters across stages minimizes overhead. A plausible implication is that, as system size increases, the benefits (e.g., $\mathcal{O}(M^4)$ scaling in operator count for molecular electronic structure) become increasingly pronounced relative to brute-force full-Hamiltonian minimization (Xu et al., 2 Feb 2024, Possel et al., 26 May 2025).

tVHA and TVHA approaches are agnostic to ansatz circuit family and classical optimizer employed, making them readily compatible with emerging VQE workflow optimizations.

A possible limitation is that the degree of truncation and the choice of partitioning must preserve chemically and physically relevant correlations—over-aggressive truncation may compromise accuracy, especially for strongly correlated or multireference systems. Empirical studies suggest, however, that for chemistry-relevant targets, a truncation up to $p\approx0.5$ already recovers near-chemical accuracy with only a fraction of the original circuit cost (Possel et al., 26 May 2025).

---

For further technical details and implementation guidelines, see (Xu et al., 2 Feb 2024) and (Possel et al., 26 May 2025).