A Quantum Strategy for the Simulation of Large Proteins: From Fragmentation in Small Proteins to Scalability in Complex Systems

Abstract

We present a scalable and resource-aware framework for the quantum simulation of large proteins, grounded in systematic molecular fragmentation, analytical Toffoli gate modeling, and empirical validation. The ground-state energy of a target biomolecule is reconstructed from capped amino acid fragments, with fixed corrections to account for artificial boundaries. Analytical cost estimates—derived from reduced Hamiltonians—are benchmarked against empirical Toffoli counts using PennyLane’s resource estimation module. Our model maintains predictive accuracy across biologically relevant systems of up to 1852 electrons, capturing consistent patterns across diverse fragments. This framework enables early-stage feasibility assessments for achieving quantum advantage in biochemical simulation pipelines.

1. Introduction

The quantum simulation of biomolecular systems remains a major computational challenge due to the exponential scaling associated with solving the electronic Schrödinger equation. Proteins, in particular, present a formidable case, as they involve a large number of electrons and complex many-body interactions [1,2]. Fragmentation techniques, which decompose macromolecules into smaller subsystems such as amino acids or peptides, have emerged as a viable approach to reducing computational overhead while preserving physical accuracy [3,4,5,6].

Quantum computing offers theoretical advantages in simulating electronic structure problems, where classical methods rapidly become intractable with system size [7,8]. By encoding quantum states directly in qubit-based representations, quantum computers avoid the combinatorial explosion characteristic of classical full-configuration interaction methods. Despite the availability of approximations such as DFT [9] and CAS-CI [10], these approaches face well-known limitations in accuracy or scalability, particularly for systems with strong electronic correlation [11,12].

Exact quantum chemical simulations remain out of reach for full protein systems, even with quantum hardware. As such, methodologically principled reductions in problem size are essential. Our objective is to evaluate a fragmentation-based framework for scalable quantum simulation of proteins, balancing error control with computational feasibility. The methodology builds on our prior work [3], which introduced a reassembly scheme for independently simulated fragments with chemical corrections. That work demonstrated high accuracy on small peptides, achieving relative errors of approximately $0.005\%$ for amino acid-level fragmentation and $0.27\%$ for finer subdivisions.

In this study, we extend the methodology to larger and biologically relevant peptides, including Glucagon, Oxytocin, Vasopressin, and Angiotensin II, which vary in electron counts and structural complexity. Among them, Glucagon stands out as a critical benchmark due to its physiological role and its size, comprising 29 amino acids and over 1800 electrons. Its simulation requires addressing more than 10⁴⁸ coefficients, posing a stringent test for the scalability of our approach.

By systematically evaluating prediction errors and quantum resource estimates across these molecules, we aim to assess the practical applicability of fragmentation-based strategies for quantum chemistry. This work contributes to the broader effort of making accurate electronic structure simulations tractable for systems of biological and chemical relevance. Beyond molecular simulation, the methodology has potential implications in areas such as drug discovery and materials design, where electronic structure accuracy is essential, and classical methods face intrinsic limitations.

The structure of the paper is as follows: Section 2 reviews related work in quantum simulation and fragmentation-based methods. Section 3 introduces our methodology, including regression models, Toffoli gate estimation, and the proposed multi-level fragmentation and reassembly strategy. Section 4 presents experimental validation on a range of peptides and proteins, from small dipeptides to large systems such as Glucagon, along with detailed error and scalability analyses. Section 5 discusses limitations, compares with alternative techniques, and outlines practical implications and future research. Section 6 summarizes our main findings and perspectives. Additional technical derivations and regression diagnostics are provided in the appendices.

2. Related Works

Quantum simulations of biomolecules [5,6,13] have progressed along two converging lines: (i) classical fragment-based electronic-structure methods that exploit locality to curb cost, and (ii) quantum algorithms that aggressively compress qubit and gate resources [14,15]. Our work positions itself at this intersection, extending classical fragmentation ideas into the quantum era and knitting together the most effective resource-optimization tools reported to date.

Fragmentation approaches such as the Fragment Molecular Orbital (FMO) method [16], Our N-Layered Integrated Molecular Orbital and Molecular Mechanics (ONIOM) [17], and adaptive Quantum Mechanics/Molecular Mechanics (QM/MM) [18] mitigate the exponential scaling bottleneck by decomposing proteins into chemically intuitive fragments. However, the high-level electronic treatment of each block is still limited to density-functional or semi-empirical accuracy.

$$E_{\mathrm{protein}}=\sum _{i=1}^n{E}_{f_i}\pm \sum _{j=1}^k\Delta E_{\mathrm{coupling},j},$$

(1)

$$\Delta E_{\mathrm{coupling},j}=\left\{\begin{array}{c}{E}_{a{m}_j},\mathrm{capping}/\mathrm{missing}\mathrm{groups},\hfill \\ {\displaystyle \sum _{n=2}^N{E}_{n\text{-body}},\text{many-body}\mathrm{terms}.}\hfill \end{array}\right.$$

(2)

In Equation (1), $E_{\mathrm{protein}}$ denotes the total ground-state energy of the reassembled molecule, computed from a set of n fragments. Each $E_{f_i}$ is the energy of fragment i, typically obtained from an independent quantum simulation. The term $\Delta E_{\mathrm{coupling},j}$ represents an additional correction associated with inter-fragment effects or artificial modifications introduced via fragmentation.

Equation (2) defines two types of corrections that $\Delta E_{\mathrm{coupling},j}$ may include the following:

• $E_{a{m}_j}$: the energy of a group assembled or removed during fragmentation (e.g., a capping hydrogen atom to preserve chemical valency);
• ${\sum }_{n=2}^N{E}_{n\text{-body}}$: a higher-order many-body interaction correction involving n fragments simultaneously, as in the fragment molecular orbital (FMO) method [16].

The index k denotes the total number of such coupling corrections considered. The ± sign in Equation (1) reflects the fact that these contributions can either increase or decrease the total energy, depending on whether the correction represents an additive or subtractive effect (e.g., insertion/removal of atoms or stabilizing/destabilizing couplings). By introducing the general symbol $\Delta E_{\mathrm{coupling},j}$, we unify the treatment of both structural and energetic corrections into a single formalism. This allows Equations (1) and (2) to remain valid across a wide range of fragmentation strategies, including those that incorporate post-reassembly many-body expansions (MBE).

Bowling et al. [19] exemplify this approach by combining single-residue fragmentation, minimal hydrogen capping, and screened two-body terms within a convergent many-body expansion (MBE) scheme:

$$E=\sum _I{E}_I+\sum _{I<J}\Delta E_{IJ}+\sum _{I<J<K}\Delta E_{IJK}+\cdots$$

(3)

Each correction term accounts for interactions omitted at lower orders of the expansion. In particular, truncating the series at the n-body level defines the degree of approximation. For instance, the three-body correction is given by the following:

$$\Delta E_{IJK}=E_{IJK}-(E_{IJ}+E_{IK}+E_{JK})+(E_I+E_J+E_K)$$

(4)

Although many-body expansions offer chemically accurate results, their combinatorial scaling limits practical applicability. To address this, our method introduces resource-aware fragmentation, statistical estimation, and circuit-level compression to ensure scalability. Prior works, such as MFCC-MBE(2) [20], have improved classical accuracy by incorporating fragment and cap interactions. Extending these ideas, our framework replaces classical subroutines with quantum solvers while preserving compatibility with post-fragmentation corrections, thus enabling seamless integration into hybrid quantum–classical approaches.

Three breakthroughs underpin our scalable workflow:

• Local qubit tapering. Extending the symmetry-based tapering of Bravyi et al. [21], we identify ${\mathbb{Z}}_2$ symmetries within each fragment, removing ∼4–6 logical qubits on average.
• SelectSwap oracle synthesis. The SelectSwap network of Zhu et al. [22] prepares fragment phase oracles at a cost of $\mathcal{O}\left(\sqrt{2^{n_f}log(1/\epsilon )}\right)$ T gates, where $n_f=\lceil {log}_2{N}_{\mathrm{coeff}}\rceil$ is the number of logical qubits required to represent the $N_{\mathrm{coeff}}$ diagonal coefficients of the fragment.
• Optimal state preparation [23]. Diagonal-unitary synthesis plus exact amplitude amplification [24,25] reduces the non-Clifford depth by 20–50% in published benchmarks (22% for QAOA, 50% for random diagonals) [24,25].

Together, these optimizations shrink the space–time volume of a 400-orbital active site by nearly two orders of magnitude versus the double-factorized algorithm of von Burg et al. [26].

Most prior quantum–chemistry demonstrations target small peptides ($<200{\mathrm{e}}^{-}$) or model chromophores. To probe genuine scalability, we select four bio-relevant hormones spanning two decades in electron count and 30 orders in the Hamiltonian-coefficient space:

• Glucagon (29aa, $1852{\mathrm{e}}^{-}$) — $4.33\times {10}^{48}$ coefficients; 2679 logical qubits after tapering.
• Oxytocin (9aa, $536{\mathrm{e}}^{-}$) — $8.85\times {10}^{17}$ coefficients; 778 qubits.
• Vasopressin (9aa, $1134{\mathrm{e}}^{-}$) — $7.81\times {10}^{31}$ coefficients; 1641 qubits.
• Angiotensin II (8aa, $558{\mathrm{e}}^{-}$) — $2.88\times {10}^{18}$ coefficients; 809 qubits.

These systems fill the gap between toy peptides and full enzymes—precisely the scale at which the existing methods begin to break down and where our integrated pipeline is explicitly designed to operate.

Tensor-network simulation methods (e.g., DMRG [27], MPS [28], and TTN [29]) are highly efficient for systems with low or structured entanglement. However, when applied to quantum circuits that generate strong global entanglement—such as GHZ preparation, quantum Fourier transform, or Hamiltonian evolution—the bond dimension grows exponentially, making contraction intractable [29,30]. This limits their applicability to deep circuits or realistic biomolecules.

Most current approaches to quantum simulation remain limited to small peptides with fewer than 150 electrons, where accurate reassembly has been demonstrated with sub-1% errors. However, no previous method integrates fragmentation-aware oracle synthesis and tapering into a scalable pipeline for biomolecules with 500–2000 electrons. Our work addresses this gap by combining these techniques into a unified, resource-efficient framework capable of operating at the hormone scale. While challenges remain—such as chemically informed fragmentation, correlation beyond MP2 [31], and cross-fragment error mitigation—recent advances in entanglement-guided heuristics [32] suggest promising directions to extend the approach.

3. Methodology

Our methodology addresses the challenges of simulating large protein systems on quantum computers by combining fragmentation strategies with advanced quantum algorithms. Below, we outline the key components of our approach, from data modeling and resource estimation to fragmentation and reassembly techniques.

3.1. Fragmentation and Recombination Strategy

Our methodology builds upon the fragmentation protocol introduced in the QMProt framework [33], extending it to enable scalable quantum simulations of large biomolecules. The core idea is to approximate the ground-state energy of a full protein or peptide by summing the energies of chemically meaningful fragments, primarily based on individual amino acid residues. Each fragment is saturated with hydrogen atoms or small chemical groups to preserve valency and molecular stability.

In our framework, the reassembled ground–state energy $E_m$ is given by

$$E_m=\sum _{i=1}^n{E}_{f_i}\pm \sum _{j=1}^k{E}_{a{m}_j},$$

(5)

where the following applies:

• n denotes the total number of fragments generated;
• $E_{f_i}$ is the ground-state energy (GSE) of fragment i;
• k is the number of small molecules added or removed;
• $E_{a{m}_j}$ is the GSE of each such molecule;
• $E_m$ is the final GSE of the reassembled molecule.

Here, the index i runs from 1 to n, enumerating each fragment, and j runs from 1 to k, enumerating each added or removed molecule.

To correct the artificial effects introduced at the cutting points, we apply fixed energy corrections based on the capping strategy:

• If a methyl group (CH₃) is added to complete valency, we add the energy of a methane molecule as a corrective term.
• If a water molecule is implicitly removed during fragmentation (e.g., in peptide bonds), its reference energy is subtracted.
• If no group is added or removed, no correction is applied.

These correction values are precomputed and consistently reused. This avoids the need for iterative many-body corrections or self-consistent inter-fragment coupling, drastically reducing computational complexity compared to traditional fragmentation schemes.

We apply this fragmentation strategy at two levels:

• Amino acid level: Each amino acid is split into radical and backbone groups. We benchmark the ground-state energy (GSE) and resource estimates (e.g., qubits, Toffoli gates) of the full amino acid against the sum of its fragments, quantifying the reduction factor.
• Peptide/protein level: For representative peptides such as Oxytocin, Angiotensin II, and Glucagon, we compute total energy as the sum of fragment energies plus corrective terms, as expressed in Equation (1). We compare this estimate to the full molecular simulation to evaluate accuracy and resource savings.

This approach maintains high computational efficiency while ensuring that GSE estimates remain within acceptable error bounds. The residual recombination error grows with the number of fragments and is influenced by the precision of the energy evaluation for each fragment. In our current implementation, calculations are performed at the Hartree–Fock (HF) level, which introduces some limitations for large systems. However, the modular nature of the method makes it compatible with more accurate solvers (e.g., DFT, MP2, or quantum algorithms), which are expected to further reduce the error as these techniques mature. Thus, our strategy provides a robust foundation for efficient and scalable quantum simulations of biomolecular systems.

3.2. Modeling Based on Experimental Data

To support our modeling and prediction efforts, we use a dataset—QMProt [33]. This dataset encompasses 45 carefully selected organic molecules, with a particular focus on the 20 canonical amino acids essential to human biology. Each molecule is decomposed into chemically meaningful subunits, including amino and carboxyl termini, central $\alpha$-carbon atoms, and characteristic side chains.

The molecules in QMProt are composed primarily of non-hydrogen atoms such as carbon, nitrogen, oxygen, and sulfur, and they contain up to 15 heavy atoms. For each molecular entry, the dataset provides the following:

• The total number of electrons and molecular orbitals.
• The corresponding number of logical qubits required for simulation.
• The full Hamiltonian encoded as a set of quantum coefficients.
• Ground-state energy estimates derived from quantum mechanical methods
• Additional physicochemical attributes relevant for simulation benchmarking.

This dataset bridges quantum chemical characterization with quantum resource modeling, offering a representative basis for scaling predictions to larger biomolecules. Using this foundation, we develop regression models to forecast quantum resource needs—including qubits and gate counts—based on fundamental descriptors such as the electron number, enabling extrapolation to peptides and protein fragments well beyond the initial dataset.

3.2.1. Linear Model for Qubits

We assume a linear relationship between the number of qubits and the number of electrons:

$$n_{\mathrm{qubits}}=\alpha +\beta ·n_{\mathrm{electrons}}+\epsilon ,$$

(6)

where $\alpha$ is the intercept, $\beta$ is the slope, and $\epsilon$ is the residual error. The model is fitted using ordinary least squares (OLS), minimizing the squared residuals:

$$\underset{\alpha ,\beta }{min}\sum _{i=1}^n{\left(n_{\mathrm{qubits}}^{\left(i\right)}-(\alpha +\beta n_{\mathrm{electrons}}^{\left(i\right)})\right)}^2.$$

(7)

The analytical solution involves solving the normal equations:

$$\left[\begin{array}{cc}n& \sum n_{\mathrm{electrons}}\\ \sum n_{\mathrm{electrons}}& \sum n_{\mathrm{electrons}}^2\end{array}\right]\left[\begin{array}{c}\alpha \\ \beta \end{array}\right]=\left[\begin{array}{c}\sum n_{\mathrm{qubits}}\\ \sum n_{\mathrm{electrons}}{n}_{\mathrm{qubits}}\end{array}\right].$$

(8)

3.2.2. Log-Linear Robust Model for Qubits

While the linear model offers a simple baseline, an empirical evaluation revealed that the relationship between electron count and qubit requirements is more accurately captured using a log-transformed regression. Accordingly, we consider the following:

$$log\left(n_{\mathrm{qubits}}\right)=\alpha +\beta ·n_{\mathrm{electrons}}+\epsilon ,$$

(9)

where the parameters $\alpha$ and $\beta$ are estimated using robust methods, such as the Huber Regressor [34] and the Theil–Sen Estimator [35], to mitigate the influence of outliers.

To account for structural diversity, we partition the dataset into three molecular complexity segments: Small (electron count ≤ 150), medium (151–500), and large (>500), enabling localized regressions adapted to each scale. Predictions are returned to the original scale via the following:

$${\widehat{n}}_{\mathrm{qubits}}=exp(\alpha +\beta ·n_{\mathrm{electrons}}).$$

(10)

3.2.3. Exponential Model for Hamiltonian Coefficients

We hypothesize an exponential growth in the number of coefficients concerning the electron count:

$$n_{\mathrm{coef}}=a·exp(b·n_{\mathrm{electrons}}),$$

(11)

with a and b as model parameters. Taking the natural logarithm yields a linearized version:

$$log\left(n_{\mathrm{coef}}\right)=log\left(a\right)+b·n_{\mathrm{electrons}}+\epsilon ,$$

(12)

which is fitted using OLS. The predicted values in the original scale are obtained via exponentiation:

$${\widehat{n}}_{\mathrm{coef}}=exp(log\left(a\right)+b·n_{\mathrm{electrons}}).$$

(13)

3.2.4. Confidence Intervals

For the linear model, the 95% confidence interval for a prediction $\widehat{y}$ is as follows:

$$\widehat{y}\pm t_{n-2,0.975}·s_{\mathrm{fit}}\left(x\right),$$

(14)

where x is the number of electrons for which the confidence interval is being computed, and $\overline{x}$ denotes the sample mean of the observed electron counts.

$$s_{\mathrm{fit}}\left(x\right)=\widehat{\sigma }·\sqrt{{\displaystyle \frac{1}{n}}+{\displaystyle \frac{{(x-\overline{x})}^2}{\sum {(x_i-\overline{x})}^2}}}.$$

(15)

$\widehat{\sigma }$ denotes the estimated standard deviation of the residuals, calculated as $\widehat{\sigma }=\sqrt{{\displaystyle \frac{1}{n-2}}{\sum }_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}$, assuming the model is fitted using ordinary least squares, and where $t_{n-2,0.975}$ denotes the critical value from the Student’s t-distribution with $n-2$ degrees of freedom, corresponding to a two-tailed confidence level of 95%.

  For the log-linear robust model, the 95% confidence interval is computed using the delta method. Let $\widehat{y}=\alpha +\beta x$ denote the log-qubit prediction. The Jacobian vector with respect to the parameters is as follows:

$$J\left(x\right)=\left[\begin{array}{cc}{\displaystyle \frac{\partial \widehat{y}}{\partial \alpha }}& {\displaystyle \frac{\partial \widehat{y}}{\partial \beta }}\end{array}\right]=[1,x],$$

(16)

and the variance of the prediction is as follows:

$$\mathrm{Var}\left(\widehat{y}\right)=J\left(x\right)·\Sigma ·J{\left(x\right)}^{\top },$$

(17)

where $\Sigma$ is the estimated covariance matrix of the fitted parameters. The resulting interval is given by the following:

$$\widehat{y}\pm t_{n-2,0.975}·\sqrt{\mathrm{Var}\left(\widehat{y}\right)},$$

(18)

and transformed back to the original scale as follows:

$${\widehat{n}}_{\mathrm{qubits}}\in \left[exp(\widehat{y}-\Delta ),exp(\widehat{y}+\Delta )\right],$$

(19)

where $\Delta =t_{n-2,0.975}·\sqrt{\mathrm{Var}\left(\widehat{y}\right)}$. A complete derivation of these expressions and additional justifications for the segment-wise log-qubit modeling can be found in Appendix D.

3.2.5. Error Metrics

We evaluate model performance using standard regression error metrics commonly employed in predictive modeling. The root mean squared error (RMSE) and the mean absolute error (MAE) are used to quantify the average magnitude of residuals, with RMSE giving greater weight to larger errors and MAE providing a more robust central tendency measure less affected by outliers [36].

To contextualize the errors concerning the scale of the observed values, we also report the mean relative error (MRE), expressed as a percentage. Additionally, we compute the relative standard deviation (${\sigma }_{\mathrm{rel}}$), which captures the dispersion of predicted values normalized by their mean. These metrics provide a consistent basis for comparing model outputs across different experimental conditions.

3.3. Estimation of Toffoli Gate Count

To estimate the quantum cost, we employ a function inspired by the SelectSwap algorithm [15]. This algorithm achieves a quadratic improvement in gate counts by reducing the number of Toffoli gates to a scale of $f·\sqrt{M}$, where M is the number of coefficients, and f is a multiplicative factor. Without SelectSwap, the cost grows proportionally to M and requires $log\left(M\right)$ ancillary qubits.

The Toffoli gate count is estimated using the following function:

$$T_{\mathrm{toffoli}}=C_1\sqrt{2^n·{log}_2\left({\displaystyle \frac{n}{ϵ}}\right)}+C_2{log}_2\left({\displaystyle \frac{n}{ϵ}}\right),$$

(20)

where

$$n=\lceil {log}_2\left(n_{\mathrm{coef}}\right)\rceil ,C_1,\mathrm{and}{C}_2\in \mathbb{R}.$$

We define the target precision $ϵ$ as a function of fragment size, setting $ϵ={\displaystyle \frac{n}{2^b}}$ for a fragment comprising n logical qubits. Here, b denotes the resolution parameter of the quantum hardware, capturing the logarithmic relationship between numerical accuracy and the size of the Hamiltonian coefficient vector. In our experiments, we fix $b=20$, corresponding to a 20-bit precision level that reflects realistic thresholds for fault-tolerant quantum architectures. This formulation is consistent with practical accuracy requirements and aligns with resource scaling trends reported in prior studies [22].

Equation (20) expresses the asymptotic scaling of the SelectSwap algorithm and serves as a practical estimator for Toffoli gate counts in fault-tolerant quantum simulations. For all experiments, we assume constant factors $C_1=C_2=3$ unless otherwise stated. Additional implementation details and derivations are provided in Appendix B.

3.4. Experimental Validation

To validate our analytical Toffoli-gate estimates, we employ PennyLane’s resource estimation module, available through the pennylane.labs.resource_estimation interface (Commit 5fc5d02308d25831594ecb41e4acc6f74ce2da30 on GitHub; not yet part of the stable release or fully documented at the time of our study). Using this framework, we instantiate both QROM-based state preparation circuits [37] and qubitization routines [38] for each fragment and benchmark peptide, adapting the templates to match the structure of our reduced Hamiltonians.

For each system, we extract empirical Toffoli counts and compare them against the theoretical predictions obtained from Equations (6)–(20). This experimental procedure serves to verify the internal consistency of our cost model and ensure compatibility with circuit-level resource estimators in current quantum toolchains.

4. Results

We begin by evaluating the accuracy of our fragmentation strategy via regression analysis. Figure 1 shows an exponential fit of the total number of Hamiltonian coefficients versus the electron count, including a 95% confidence interval. In addition, Figure 2 provides a complementary linear interpolation for the smallest molecules in our dataset.

Next, we quantify recombination and estimation errors.

Table 1. Summary of electrons, orbitals, theoretical energy (GT), calculated energy (Em), and relative error (%RE) for different peptides.

Peptides	Electrons	Orbitals	GT	Em	%RE
Gly-Gly	70	53	$-4.83\times {10}^2$	$-4.83\times {10}^2$	$4.00\times {10}^{-3}$
Gly-Ala	78	60	$-5.22\times {10}^2$	$-5.22\times {10}^2$	$3.16\times {10}^{-3}$
Glu-Gly	108	82	$-7.45\times {10}^2$	$-7.45\times {10}^2$	$2.52\times {10}^{-3}$
Ser-Cys	110	81	$-1.03\times {10}^3$	$-1.03\times {10}^3$	$1.58\times {10}^{-3}$
Carnosine (Ala-His)	120	94	$-7.81\times {10}^2$	$-7.81\times {10}^2$	$1.94\times {10}^{-3}$
Gly-Ser	86	65	$-5.96\times {10}^2$	$-5.96\times {10}^2$	$2.62\times {10}^{-3}$
Pro-Gly	92	72	$-5.98\times {10}^2$	$-5.98\times {10}^2$	$3.33\times {10}^{-3}$
Cystine (Cys-Cys)	126	90	$-1.42\times {10}^3$	$-1.42\times {10}^3$	$5.40\times {10}^{-4}$
Leu-Thr	126	100	$-7.89\times {10}^2$	$-7.89\times {10}^2$	$1.82\times {10}^{-3}$
Gly-Val-Ala	132	104	$-8.42\times {10}^2$	$-8.42\times {10}^2$	$3.59\times {10}^{-3}$
Thr-Lys	134	106	$-8.43\times {10}^2$	$-8.43\times {10}^2$	$1.88\times {10}^{-3}$
Val-Ala-Ser	148	116	$-9.54\times {10}^2$	$-9.54\times {10}^2$	$3.09\times {10}^{-3}$
Phe-Ile	150	122	$-9.03\times {10}^2$	$-9.03\times {10}^2$	$1.60\times {10}^{-3}$
Ser-Gly-Glu	154	117	$-1.06\times {10}^3$	$-1.06\times {10}^3$	$3.40\times {10}^{-3}$
Aspartame (Asp-Phe)	156	123	$-1.01\times {10}^3$	$-1.01\times {10}^3$	$5.00\times {10}^{-2}$
Tyr-Asp	156	121	$-1.05\times {10}^3$	$-1.05\times {10}^3$	$1.47\times {10}^{-3}$
Glutathione (Cys-Glu-Gly)	162	121	$-1.38\times {10}^3$	$-1.38\times {10}^3$	$2.34\times {10}^{-3}$
Arg-Met	164	127	$-1.31\times {10}^3$	$-1.31\times {10}^3$	$1.02\times {10}^{-3}$
Val-Asp-Ser	170	131	$-1.14\times {10}^3$	$-1.14\times {10}^3$	$2.71\times {10}^{-3}$
Gly-His-Lys	182	144	$-1.16\times {10}^3$	$-1.16\times {10}^3$	$2.94\times {10}^{-3}$
Trp-His	180	144	$-1.14\times {10}^3$	$-1.14\times {10}^3$	$1.84\times {10}^{-3}$
Tyr-Arg	180	143	$-1.14\times {10}^3$	$-1.14\times {10}^3$	$1.43\times {10}^{-3}$
His-Arg-Val	220	175	$-1.38\times {10}^3$	$-1.38\times {10}^3$	$2.01\times {10}^{-3}$
Tuftsin (Thr-Lys-Pro-Arg)	270	215	$-1.64\times {10}^3$	$-1.68\times {10}^3$	$2.84$
Methionine-enkephalin (Tyr-Gly-Gly-Phe-Met)	304	239	$-2.16\times {10}^3$	$-2.21\times {10}^3$	$2.06$
Leucine-enkephalin (Tyr-Gly-Gly-Phe-Leu)	296	237	$-1.81\times {10}^3$	$-1.85\times {10}^3$	$2.59$
Oxytocin (Cys-Tyr-Ile-Gln-Asn-Cys-Pro-Leu–Gly)	536	419	$-3.84\times {10}^3$	$-3.91\times {10}^3$	$1.85$
Opiorphin (Gln-Arg-Phe-Ser-Arg)	558	446	$-2.29\times {10}^3$	$-2.35\times {10}^3$	$2.53$
Bradykinin (Arg-Pro-Pro-Gly-Phe-Ser-Pro-Phe-Arg)	566	453	$-3.43\times {10}^3$	$-3.53\times {10}^3$	$2.92$
Neurotensin (Glu-Leu-Tyr-Glu-Asn-Lys-Pro-Arg-Arg-Pro-Tyr-Ile-Leu)	896	716	$-5.43\times {10}^3$	$-5.27\times {10}^3$	$2.83$
Gastrin-14 (Trp-Leu-Glu-Glu-Glu-Glu-Glu-Ala-Tyr-Gly-Trp-Met-Asp-Phe)	970	763	$-6.39\times {10}^3$	$-6.25\times {10}^3$	$2.31$
Angiotensin IV (Val-Tyr-Ile-His-Pro-Phe)	414	334	$-2.47\times {10}^3$	$-2.55\times {10}^3$	$2.98$
Angiotensin II (Asp-Arg-Val-Tyr-Ile-His-Pro-Phe)	558	446	$-3.40\times {10}^3$	$-3.49\times {10}^3$	$2.83$
Angiotensin I (Asp-Arg-Val-Tyr-Ile-His-Pro-Phe-His-Leu)	692	554	$-4.19\times {10}^3$	$-4.32\times {10}^3$	$2.97$
Glucagon (His-Ser-Gln-Gly-Thr-Phe-Thr-Ser-Asp-Tyr-Ser-Lys-Tyr-
Leu-Asp-Ser-Arg-Arg-Ala-Gln-Asp-Phe-Val-Gln-Trp-Leu-Met-Asn-Thr)	1852	1459	$-1.18\times {10}^4$	$-1.22\times {10}^4$	$2.80$

reports fragment recombination errors across all peptides, confirming minimal error accumulation for small fragments and identifying increased errors for larger systems.

We then examine resource-scaling trends. In Figure 3, we plot both Toffoli gate counts and qubit requirements against molecular size, comparing the original full-Hamiltonian approach to our base-structure method.

Table 2. Resource comparison for peptide simulations: theoretical Toffoli gate estimates using the full Hamiltonian ($C_1=C_2=3$) versus the proposed fragmentation–reassembly (base-structure) approach. For each molecule and version (Ver.: Orig. = Original; Prop. = Proposed), the table lists the number of Hamiltonian coefficients (# Coeffs.), the estimated Toffoli gate count, the Toffoli reduction factor (Red. (Toffoli)), the electron count (when available), and the coefficient reduction factor (Red. (Coeff.)).

Molecules	Ver.	# Coeffs.	Toffoli	Red. (Toffoli)	Electrons	Red. (Coeff.)
Alanine	Orig.	$2.73\times {10}^6$	$2.02\times {10}^4$	-	$4.80\times {10}^1$	-
R_ala + Base Structures	Prop.	$5.79\times {10}^4$	$2.59\times {10}^3$	$7.80$	–	$4.72\times {10}^1$
Histidine	Orig.	$2.38\times {10}^7$	$5.67\times {10}^4$	-	$8.20\times {10}^1$	-
R_his + Base Structures	Prop.	$2.03\times {10}^6$	$1.43\times {10}^4$	$3.96$	–	$1.17\times {10}^1$
Leucine	Orig.	$1.62\times {10}^7$	$4.02\times {10}^4$	-	$7.20\times {10}^1$	-
R_leu + Base Structures	Prop.	$5.76\times {10}^5$	$1.02\times {10}^4$	$3.96$	–	$2.81\times {10}^1$
Isoleucine	Orig.	$1.64\times {10}^7$	$4.02\times {10}^4$	-	$7.20\times {10}^1$	-
R_ile + Base Structures	Prop.	$5.76\times {10}^5$	$1.02\times {10}^4$	$3.96$	–	$2.84\times {10}^1$
Lysine	Orig.	$2.39\times {10}^7$	$5.67\times {10}^4$	-	$8.00\times {10}^1$	-
R_lys + Base Structures	Prop.	$2.25\times {10}^6$	$2.02\times {10}^4$	$2.82$	–	$1.06\times {10}^1$
Methionine	Orig.	$1.78\times {10}^7$	$5.67\times {10}^4$	-	$8.00\times {10}^1$	-
R_met + Base Structures	Prop.	$5.63\times {10}^5$	$1.02\times {10}^4$	$5.64$	–	$3.16\times {10}^1$
Phenylalanine	Orig.	$3.61\times {10}^7$	$8.01\times {10}^4$	-	$8.80\times {10}^1$	-
R_phe + Base Structures	Prop.	$3.78\times {10}^6$	$2.02\times {10}^4$	$3.99$	–	$9.56$
Threonine	Orig.	$8.36\times {10}^6$	$2.58\times {10}^4$	-	$6.40\times {10}^1$	-
R_thr + Base Structures	Prop.	$1.06\times {10}^5$	$3.90\times {10}^3$	$7.91$	–	$7.92\times {10}^1$
Tryptophan	Orig.	$9.24\times {10}^7$	$6.11\times {10}^4$	-	$1.08\times {10}^2$	-
R_trp + Base Structures	Prop.	$1.49\times {10}^7$	$9.54\times {10}^3$	$2.83$	–	$6.19$
Valine	Orig.	$9.82\times {10}^6$	$2.58\times {10}^4$	-	$6.40\times {10}^1$	-
R_val + Base Structures	Prop.	$3.98\times {10}^5$	$3.89\times {10}^3$	$5.63$	–	$2.47\times {10}^1$
Arginine	Orig.	$4.36\times {10}^7$	$8.01\times {10}^4$	-	$9.40\times {10}^1$	-
R_arg + Base Structures	Prop.	$5.47\times {10}^6$	$3.13\times {10}^4$	$2.56$	–	$7.97$
Cysteine	Orig.	$6.19\times {10}^6$	$2.38\times {10}^4$	-	$6.60\times {10}^1$	-
R_cys + Base Structures	Prop.	$1.56\times {10}^5$	$4.82\times {10}^3$	$5.62$	–	$3.97\times {10}^1$
Glutamine	Orig.	$1.83\times {10}^7$	$5.67\times {10}^4$	-	$7.80\times {10}^1$	-
R_gln + Base Structures	Prop.	$8.73\times {10}^5$	$1.43\times {10}^4$	$5.64$	–	$2.09\times {10}^1$
Asparagine	Orig.	$1.13\times {10}^7$	$3.79\times {10}^4$	-	$7.00\times {10}^1$	-
R_asn + Base Structures	Prop.	$3.45\times {10}^5$	$6.97\times {10}^3$	$5.63$	–	$3.28\times {10}^1$
Tyrosine	Orig.	$4.85\times {10}^7$	$8.00\times {10}^4$	-	$9.60\times {10}^1$	-
R_tyr + Base Structures	Prop.	$4.32\times {10}^6$	$3.17\times {10}^4$	$2.52$	–	$1.12\times {10}^1$
Serine	Orig.	$4.53\times {10}^6$	$2.00\times {10}^4$	-	$5.60\times {10}^1$	-
R_ser + Base Structures	Prop.	$9.70\times {10}^4$	$4.82\times {10}^3$	$7.91$	–	$4.67\times {10}^1$
Glycine	Orig.	$1.16\times {10}^6$	$1.43\times {10}^4$	-	$4.00\times {10}^1$	-
R_gly + Base Structures	Prop.	$5.60\times {10}^4$	$3.38\times {10}^3$	$5.58$	–	$2.08\times {10}^1$
Aspartic_Acid	Orig.	$1.05\times {10}^7$	$3.79\times {10}^4$	-	$7.00\times {10}^1$	-
R_asp + Base Structures	Prop.	$4.31\times {10}^5$	$6.97\times {10}^3$	$5.63$	–	$2.45\times {10}^1$
Glutamic_Acid	Orig.	$1.72\times {10}^7$	$5.67\times {10}^4$	-	$7.80\times {10}^1$	-
R_glu + Base Structures	Prop.	$1.22\times {10}^6$	$1.43\times {10}^4$	$3.99$	–	$1.41\times {10}^1$
Proline	Orig.	$8.37\times {10}^6$	$2.58\times {10}^4$	-	$6.20\times {10}^1$	-
R_pro + Base Structures	Prop.	$1.29\times {10}^5$	$4.82\times {10}^3$	$7.91$	–	$6.48\times {10}^1$
Glucagon	Orig.	$4.33\times {10}^{48}$	$3.24\times {10}^{25}$	-	$1.85\times {10}^3$	-
Amino acids - Glucagon	Prop.	$5.02\times {10}^8$	$3.11\times {10}^5$	$1.04\times {10}^{20}$	–	$8.63\times {10}^{39}$
Oxytocin	Orig.	$8.85\times {10}^{17}$	$1.44\times {10}^{10}$	-	$5.36\times {10}^2$	-
Amino acids - Oxytocin	Prop.	$1.31\times {10}^8$	$1.55\times {10}^5$	$9.26\times {10}^4$	–	$6.76\times {10}^9$
Vasopressin	Orig.	$7.81\times {10}^{31}$	$1.21\times {10}^{17}$	-	$1.13\times {10}^3$	-
Amino acids - Vasopressin	Prop.	$1.76\times {10}^8$	$2.20\times {10}^5$	$5.50\times {10}^{11}$	–	$4.44\times {10}^{23}$
Angiotensin II	Orig.	$2.88\times {10}^{18}$	$2.88\times {10}^{10}$	-	$5.58\times {10}^2$	-
Amino acids - Angiotensin II	Prop.	$1.93\times {10}^8$	$2.20\times {10}^5$	$1.31\times {10}^5$	–	$1.49\times {10}^{10}$
Kyotorphin	Orig.	$4.41\times {10}^9$	$1.24\times {10}^6$	-	$1.80\times {10}^2$	-
Amino acids - Kyotorphin	Prop.	$8.84\times {10}^7$	$1.55\times {10}^5$	$8.00$	–	$5.00\times {10}^1$
Methionine-enkephalin	Orig.	$3.44\times {10}^{12}$	$2.81\times {10}^7$	-	$3.04\times {10}^2$	-
Amino acids - Methionine-enkephalin	Prop.	$1.03\times {10}^8$	$1.55\times {10}^5$	$1.81\times {10}^2$	–	$3.34\times {10}^4$
Leucine-enkephalin	Orig.	$2.24\times {10}^{12}$	$2.81\times {10}^7$	-	$2.96\times {10}^2$	-
Amino acids - Leucine-enkephalin	Prop.	$1.01\times {10}^8$	$1.55\times {10}^5$	$1.81\times {10}^2$	–	$2.21\times {10}^4$
Tuftsin	Orig.	$5.54\times {10}^{11}$	$1.41\times {10}^7$	-	$2.70\times {10}^2$	-
Amino acids - Tuftsin	Prop.	$8.22\times {10}^7$	$1.55\times {10}^5$	$9.05\times {10}^1$	–	$6.74\times {10}^3$
Opiorphin	Orig.	$1.19\times {10}^{14}$	$1.59\times {10}^8$	-	$3.70\times {10}^2$	-
Amino acids - Opiorphin	Prop.	$1.42\times {10}^8$	$2.20\times {10}^5$	$7.24\times {10}^2$	–	$8.37\times {10}^5$
Angiotensin IV	Orig.	$1.26\times {10}^{15}$	$6.37\times {10}^8$	-	$4.14\times {10}^2$	-
Amino acids - Angiotensin IV	Prop.	$1.41\times {10}^8$	$2.20\times {10}^5$	$2.90\times {10}^3$	–	$8.95\times {10}^6$
Neurotensin	Orig.	$2.20\times {10}^{26}$	$2.36\times {10}^{14}$	-	$8.96\times {10}^2$	-
Amino acids - Neurotensin	Prop.	$3.12\times {10}^8$	$3.11\times {10}^5$	$7.59\times {10}^8$	–	$7.05\times {10}^{17}$
Bradykinin	Orig.	$4.43\times {10}^{18}$	$2.88\times {10}^{10}$	-	$5.66\times {10}^2$	-
Amino acids - Bradykinin	Prop.	$1.86\times {10}^8$	$2.20\times {10}^5$	$1.31\times {10}^5$	–	$2.38\times {10}^{10}$
Angiotensin I	Orig.	$3.84\times {10}^{21}$	$9.22\times {10}^{11}$	-	$6.92\times {10}^2$	-
Amino acids - Angiotensin I	Prop.	$2.33\times {10}^8$	$2.20\times {10}^5$	$4.19\times {10}^6$	–	$1.64\times {10}^{13}$
Gastrin-14	Orig.	$7.03\times {10}^{23}$	$1.02\times {10}^{13}$	-	$7.89\times {10}^2$	-
Amino acids - Gastrin-14	Prop.	$2.17\times {10}^8$	$2.20\times {10}^5$	$6.37\times {10}^7$	–	$3.23\times {10}^{15}$
GLU_CYS_GLY	Orig.	$5.47\times {10}^9$	$8.95\times {10}^5$	-	$1.62\times {10}^2$	-
Amino acids - GLU_CYS_GLY	Prop.	$2.46\times {10}^7$	$5.67\times {10}^4$	$1.58\times {10}^1$	–	$2.23\times {10}^2$
ALA_HIS	Orig.	$3.01\times {10}^8$	$2.25\times {10}^5$	-	$1.20\times {10}^2$	-
Amino acids - ALA_HIS	Prop.	$2.66\times {10}^7$	$5.67\times {10}^4$	$4.00$	–	$1.13\times {10}^1$
PRO_GLY_PRO	Orig.	$1.87\times {10}^9$	$4.49\times {10}^5$	-	$1.82\times {10}^2$	-
Amino acids - PRO_GLY_PRO	Prop.	$1.79\times {10}^7$	$4.49\times {10}^4$	$7.99$	–	$1.04\times {10}^2$
GLY_HIS_LYS	Orig.	$1.44\times {10}^{10}$	$1.26\times {10}^6$	-	$1.44\times {10}^2$	-
Amino acids - GLY_HIS_LYS	Prop.	$4.89\times {10}^7$	$8.01\times {10}^4$	$1.58\times {10}^1$	–	$2.94\times {10}^2$

summarizes these benchmarks— numbers of the coefficients, Toffoli estimates, reduction factors, and electron counts—for every fragment.

To validate regression robustness,

Table 3. Comparison of regression models (Huber and Theil–Sen) for predicting log-transformed qubit counts across molecular complexity segments. Metrics include $R^2$, the cross-validated $R^2$ (when applicable), the MAE, the RMSE, the standard deviation, and the coefficient of variation. Segment-specific modeling reveals performance variation by molecular size. Cross-validation scores are omitted (–) where data scarcity limits statistical significance.

Segment	Model	R2 Total	R2 CV	MAE (log)	RMSE (log)	Std Dev (log)	CV (%)
Small (≤150)	Huber $ϵ=1.1$	0.922	0.807	0.057	0.081	0.288	6.059
Small (≤150)	Huber $ϵ=1.35$	0.925	0.824	0.057	0.079	0.288	6.059
Small (≤150)	Huber $ϵ=1.75$	0.929	0.828	0.056	0.076	0.288	6.059
Small (≤150)	Huber $ϵ=2.0$	0.931	0.831	0.057	0.076	0.288	6.059
Small (≤150)	Theil–Sen	0.927	0.806	0.056	0.078	0.288	6.059
Medium (151–500)	Huber $ϵ=1.1$	0.978	–	0.039	0.051	0.347	5.741
Medium (151–500)	Huber $ϵ=1.35$	0.983	–	0.042	0.045	0.347	5.741
Medium (151–500)	Huber $ϵ=1.75$	0.983	–	0.042	0.046	0.347	5.741
Medium (151–500)	Huber $ϵ=2.0$	0.983	–	0.042	0.046	0.347	5.741
Medium (151–500)	Theil–Sen	0.977	–	0.041	0.052	0.347	5.741
Large (>500)	Huber $ϵ=1.1$	0.946	–	0.078	0.093	0.398	5.591
Large (>500)	Huber $ϵ=1.35$	0.948	–	0.078	0.091	0.398	5.591
Large (>500)	Huber $ϵ=1.75$	0.956	–	0.075	0.083	0.398	5.591
Large (>500)	Huber $ϵ=2.0$	0.956	–	0.075	0.083	0.398	5.591
Large (>500)	Theil–Sen	0.912	–	0.071	0.118	0.398	5.591

presents performance metrics for robust regression models (Huber with varying $ϵ$ and Theil–Sen) in predicting log-transformed qubit counts across small, medium, and large molecular complexity segments.

Finally, to ground our theoretical Toffoli formula (Equation (20)) in real circuit data, we measured gate counts using PennyLane’s resource_estimation module [39].

Table 4. Comparison of analytical Toffoli estimates [Equation (20), $C_1=C_2=3$] with empirical counts (Toffoli *) from PennyLane’s resource estimation module [39], applied to each fragment’s qubitization circuit. A global OLS fit gives $\langle C_1\rangle =2.64$, $\langle C_2\rangle =3.17$ ($|\Delta C_1|=0.36$, $|\Delta C_2|=0.17$), with a mean relative error of 11.9%. See Appendix C for $C_{1,i}$, $C_{2,i}$ derivations at different $\epsilon$. Ver.: Orig. = Original; Prop. = Proposed; # Coefs. = number of Hamiltonian coefficients.

Molecules	Ver.	# Coeffs	Toffoli	Toffoli *	${\mathit{C}}_1$	${\mathit{C}}_2$	$\Delta {\mathit{C}}_1$ (%)	$\Delta {\mathit{C}}_2$ (%)	$\Delta \mathit{T}$ (%)
Alanine	Orig.	$2.73\times {10}^6$	$2.02\times {10}^4$	$1.77\times {10}^4$	$2.62$	$3.27$	$3.78\times {10}^{-1}$	$2.68\times {10}^{-1}$	$1.26\times {10}^1$
R_ala + Base Structures	Prop.	$5.79\times {10}^4$	$2.59\times {10}^3$	$2.37\times {10}^3$	$2.74$	$3.05$	$2.56\times {10}^{-1}$	$4.66\times {10}^{-2}$	$8.41$
Histidine	Orig.	$2.38\times {10}^7$	$5.67\times {10}^4$	$5.10\times {10}^4$	$2.70$	$3.14$	$3.02\times {10}^{-1}$	$1.37\times {10}^{-1}$	$1.01\times {10}^1$
R_his + Base Structures	Prop.	$2.03\times {10}^6$	$1.43\times {10}^4$	$1.27\times {10}^4$	$2.67$	$3.06$	$3.31\times {10}^{-1}$	$5.99\times {10}^{-2}$	$1.10\times {10}^1$
Leucine	Orig.	$1.62\times {10}^7$	$4.02\times {10}^4$	$3.45\times {10}^4$	$2.58$	$3.30$	$4.24\times {10}^{-1}$	$2.97\times {10}^{-1}$	$1.41\times {10}^1$
R_leu + Base Structures	Prop.	$5.76\times {10}^5$	$1.02\times {10}^4$	$8.73\times {10}^3$	$2.58$	$3.22$	$4.24\times {10}^{-1}$	$2.21\times {10}^{-1}$	$1.41\times {10}^1$
Isoleucine	Orig.	$1.64\times {10}^7$	$4.02\times {10}^4$	$3.42\times {10}^4$	$2.56$	$3.13$	$4.45\times {10}^{-1}$	$1.28\times {10}^{-1}$	$1.48\times {10}^1$
R_ile + Base Structures	Prop.	$5.76\times {10}^5$	$1.02\times {10}^4$	$9.24\times {10}^3$	$2.73$	$3.04$	$2.74\times {10}^{-1}$	$4.33\times {10}^{-2}$	$9.10$
Lysine	Orig.	$2.39\times {10}^7$	$5.67\times {10}^4$	$5.05\times {10}^4$	$2.67$	$3.12$	$3.30\times {10}^{-1}$	$1.22\times {10}^{-1}$	$1.10\times {10}^1$
R_lys + Base Structures	Prop.	$2.25\times {10}^6$	$2.02\times {10}^4$	$1.81\times {10}^4$	$2.69$	$3.13$	$3.07\times {10}^{-1}$	$1.26\times {10}^{-1}$	$1.02\times {10}^1$
Methionine	Orig.	$1.78\times {10}^7$	$5.67\times {10}^4$	$4.82\times {10}^4$	$2.55$	$3.26$	$4.53\times {10}^{-1}$	$2.55\times {10}^{-1}$	$1.51\times {10}^1$
R_met + Base Structures	Prop.	$5.63\times {10}^5$	$1.02\times {10}^4$	$9.31\times {10}^3$	$2.75$	$3.23$	$2.52\times {10}^{-1}$	$2.26\times {10}^{-1}$	$8.35$
Mhenylalanine	Orig.	$3.61\times {10}^7$	$8.01\times {10}^4$	$7.26\times {10}^4$	$2.72$	$3.31$	$2.81\times {10}^{-1}$	$3.05\times {10}^{-1}$	$9.36$
R_phe + Base Structures	Prop.	$3.78\times {10}^6$	$2.02\times {10}^4$	$1.74\times {10}^4$	$2.59$	$3.17$	$4.12\times {10}^{-1}$	$1.73\times {10}^{-1}$	$1.37\times {10}^1$
Threonine	Orig.	$8.36\times {10}^6$	$2.85\times {10}^4$	$2.45\times {10}^4$	$2.58$	$3.06$	$4.19\times {10}^{-1}$	$6.11\times {10}^{-2}$	$1.39\times {10}^1$
R_thr + Base Structures	Prop.	$1.06\times {10}^5$	$3.64\times {10}^3$	$3.14\times {10}^3$	$2.58$	$3.25$	$4.18\times {10}^{-1}$	$2.50\times {10}^{-1}$	$1.37\times {10}^1$
Tryptophan	Orig.	$9.24\times {10}^7$	$1.13\times {10}^5$	$9.82\times {10}^4$	$2.61$	$3.26$	$3.93\times {10}^{-1}$	$2.65\times {10}^{-1}$	$1.31\times {10}^1$
R_trp + Base Structures	Prop.	$1.49\times {10}^7$	$4.02\times {10}^4$	$3.56\times {10}^4$	$2.65$	$3.20$	$3.46\times {10}^{-1}$	$2.02\times {10}^{-1}$	$1.15\times {10}^1$
Valine	Orig.	$9.82\times {10}^6$	$4.02\times {10}^4$	$3.53\times {10}^4$	$2.63$	$3.27$	$3.66\times {10}^{-1}$	$2.68\times {10}^{-1}$	$1.22\times {10}^1$
R_val + Base Structures	Prop.	$3.98\times {10}^5$	$7.21\times {10}^3$	$6.27\times {10}^3$	$2.60$	$3.18$	$3.95\times {10}^{-1}$	$1.80\times {10}^{-1}$	$1.31\times {10}^1$
Arginine	Orig.	$4.36\times {10}^7$	$8.01\times {10}^4$	$7.13\times {10}^4$	$2.67$	$3.19$	$3.28\times {10}^{-1}$	$1.89\times {10}^{-1}$	$1.23\times {10}^1$
R_arg + Base Structures	Prop.	$5.47\times {10}^6$	$2.85\times {10}^4$	$2.44\times {10}^4$	$2.57$	$3.16$	$4.28\times {10}^{-1}$	$1.59\times {10}^{-1}$	$1.42\times {10}^1$
Cysteine	Orig.	$6.19\times {10}^6$	$2.85\times {10}^4$	$2.47\times {10}^4$	$2.60$	$3.03$	$3.95\times {10}^{-1}$	$3.12\times {10}^{-2}$	$1.32\times {10}^1$
R_cys + Base Structures	Prop.	$1.56\times {10}^5$	$5.12\times {10}^3$	$4.48\times {10}^3$	$2.62$	$3.06$	$3.80\times {10}^{-1}$	$5.74\times {10}^{-2}$	$1.26\times {10}^1$
Glutamine	Orig.	$1.83\times {10}^7$	$5.67\times {10}^4$	$4.99\times {10}^4$	$2.64$	$3.03$	$3.61\times {10}^{-1}$	$3.31\times {10}^{-2}$	$1.20\times {10}^1$
R_gln + Base Structures	Prop.	$8.73\times {10}^5$	$1.02\times {10}^4$	$9.18\times {10}^3$	$2.71$	$3.23$	$2.91\times {10}^{-1}$	$2.25\times {10}^{-1}$	$9.65$
Asparagine	Orig.	$1.13\times {10}^7$	$4.02\times {10}^4$	$3.46\times {10}^4$	$2.59$	$3.12$	$4.15\times {10}^{-1}$	$1.23\times {10}^{-1}$	$1.38\times {10}^1$
R_asn + Base Structures	Prop.	$3.45\times {10}^5$	$7.21\times {10}^3$	$6.38\times {10}^3$	$2.65$	$3.18$	$3.48\times {10}^{-1}$	$1.85\times {10}^{-1}$	$1.15\times {10}^1$
Tyrosine	Orig.	$4.85\times {10}^7$	$8.01\times {10}^4$	$7.12\times {10}^4$	$2.67$	$3.31$	$3.32\times {10}^{-1}$	$3.12\times {10}^{-1}$	$1.25\times {10}^1$
R_tyr + Base Structures	Prop.	$4.32\times {10}^6$	$2.85\times {10}^4$	$2.42\times {10}^4$	$2.55$	$3.10$	$4.47\times {10}^{-1}$	$1.02\times {10}^{-1}$	$1.49\times {10}^1$
Serine	Orig.	$4.53\times {10}^6$	$2.85\times {10}^4$	$2.54\times {10}^4$	$2.67$	$3.15$	$3.29\times {10}^{-1}$	$1.54\times {10}^{-1}$	$1.09\times {10}^1$
R_ser + Base Structures	Prop.	$9.70\times {10}^4$	$3.64\times {10}^3$	$3.14\times {10}^3$	$2.58$	$3.26$	$4.21\times {10}^{-1}$	$2.63\times {10}^{-1}$	$1.38\times {10}^1$
Glycine	Orig.	$1.16\times {10}^6$	$1.43\times {10}^4$	$1.22\times {10}^4$	$2.56$	$3.10$	$4.43\times {10}^{-1}$	$9.58\times {10}^{-2}$	$1.47\times {10}^1$
R_gly + Base Structures	Prop.	$5.60\times {10}^4$	$2.59\times {10}^3$	$2.37\times {10}^3$	$2.74$	$3.05$	$2.57\times {10}^{-1}$	$4.76\times {10}^{-2}$	$8.42$
Aspartic_acid	Orig.	$1.05\times {10}^7$	$4.02\times {10}^4$	$3.68\times {10}^4$	$2.75$	$3.12$	$2.53\times {10}^{-1}$	$1.15\times {10}^{-1}$	$8.42$
R_asp + Base Structures	Prop.	$4.31\times {10}^5$	$7.21\times {10}^3$	$6.53\times {10}^3$	$2.71$	$3.07$	$2.86\times {10}^{-1}$	$7.44\times {10}^{-2}$	$9.48$
Glutamic_acid	Orig.	$1.72\times {10}^7$	$5.67\times {10}^4$	$4.93\times {10}^4$	$2.61$	$3.32$	$3.93\times {10}^{-1}$	$3.19\times {10}^{-1}$	$1.31\times {10}^1$
R_glu + Base Structures	Prop.	$1.22\times {10}^6$	$1.43\times {10}^4$	$1.22\times {10}^4$	$2.56$	$3.28$	$4.36\times {10}^{-1}$	$2.80\times {10}^{-1}$	$1.45\times {10}^1$
Proline	Orig.	$8.37\times {10}^6$	$2.85\times {10}^4$	$2.55\times {10}^4$	$2.69$	$3.22$	$3.12\times {10}^{-1}$	$2.24\times {10}^{-1}$	$1.04\times {10}^1$
R_pro + Base Structures	Prop.	$1.29\times {10}^5$	$3.64\times {10}^3$	$3.21\times {10}^3$	$2.64$	$3.30$	$3.64\times {10}^{-1}$	$3.00\times {10}^{-1}$	$1.19\times {10}^1$
Glucagon	Orig.	$4.33\times {10}^{48}$	$2.15\times {10}^{25}$	$1.84\times {10}^{25}$	$2.57$	$3.28$	$4.31\times {10}^{-1}$	$2.79\times {10}^{-1}$	$1.44\times {10}^1$
Amino acids – Glucagon	Prop.	$5.02\times {10}^8$	$2.25\times {10}^5$	$1.99\times {10}^5$	$2.65$	$3.08$	$3.52\times {10}^{-1}$	$8.24\times {10}^{-2}$	$1.17\times {10}^1$
Oxytocin	Orig.	$8.85\times {10}^{17}$	$1.01\times {10}^{10}$	$8.56\times {10}^9$	$2.55$	$3.31$	$4.50\times {10}^{-1}$	$3.07\times {10}^{-1}$	$1.50\times {10}^1$
Amino acids – Oxytocin	Prop.	$1.31\times {10}^8$	$1.13\times {10}^5$	$1.03\times {10}^5$	$2.74$	$3.19$	$2.65\times {10}^{-1}$	$1.95\times {10}^{-1}$	$8.82$
Vasopressin	Orig.	$7.81\times {10}^{31}$	$8.20\times {10}^{16}$	$7.10\times {10}^{16}$	$2.60$	$3.28$	$4.02\times {10}^{-1}$	$2.80\times {10}^{-1}$	$1.34\times {10}^1$
Amino acids – Vasopressin	Prop.	$1.76\times {10}^8$	$1.60\times {10}^5$	$1.43\times {10}^5$	$2.68$	$3.31$	$3.17\times {10}^{-1}$	$3.08\times {10}^{-1}$	$1.06\times {10}^1$
Angiotensin II	Orig.	$2.88\times {10}^{18}$	$2.01\times {10}^{10}$	$1.75\times {10}^{10}$	$2.61$	$3.12$	$3.91\times {10}^{-1}$	$1.24\times {10}^{-1}$	$1.30\times {10}^1$
Amino acids – Angiotensin II	Prop.	$1.93\times {10}^8$	$1.60\times {10}^5$	$1.41\times {10}^5$	$2.65$	$3.06$	$3.47\times {10}^{-1}$	$5.81\times {10}^{-2}$	$1.16\times {10}^1$
Kyotorphin	Orig.	$4.41\times {10}^9$	$8.95\times {10}^5$	$7.93\times {10}^5$	$2.66$	$3.10$	$3.41\times {10}^{-1}$	$9.56\times {10}^{-2}$	$1.14\times {10}^1$
Amino acids – Kyotorphin	Prop.	$8.84\times {10}^7$	$1.13\times {10}^5$	$9.73\times {10}^4$	$2.58$	$3.16$	$4.18\times {10}^{-1}$	$1.59\times {10}^{-1}$	$1.39\times {10}^1$
Metionina encefalina	Orig.	$3.44\times {10}^{12}$	$2.00\times {10}^7$	$1.83\times {10}^7$	$2.75$	$3.28$	$2.52\times {10}^{-1}$	$2.83\times {10}^{-1}$	$8.40$
Amino acids – Metionina encefalina	Prop.	$1.03\times {10}^8$	$1.13\times {10}^5$	$1.02\times {10}^5$	$2.71$	$3.30$	$2.93\times {10}^{-1}$	$2.97\times {10}^{-1}$	$9.77$
Leucina encefalina	Orig.	$2.24\times {10}^{12}$	$2.00\times {10}^7$	$1.83\times {10}^7$	$2.74$	$3.03$	$2.58\times {10}^{-1}$	$2.53\times {10}^{-2}$	$8.62$
Amino acids – Leucina encefalina	Prop.	$1.01\times {10}^8$	$1.13\times {10}^5$	$1.03\times {10}^5$	$2.73$	$3.19$	$2.68\times {10}^{-1}$	$1.85\times {10}^{-1}$	$8.93$
Tuftsin	Orig.	$5.54\times {10}^{11}$	$1.00\times {10}^7$	$8.93\times {10}^6$	$2.67$	$3.16$	$3.31\times {10}^{-1}$	$1.56\times {10}^{-1}$	$1.10\times {10}^1$
Amino acids – Tuftsin	Prop.	$8.22\times {10}^7$	$1.13\times {10}^5$	$1.03\times {10}^5$	$2.74$	$3.09$	$2.62\times {10}^{-1}$	$9.37\times {10}^{-2}$	$8.74$
Opiorfina	Orig.	$1.19\times {10}^{14}$	$1.13\times {10}^8$	$9.62\times {10}^7$	$2.56$	$3.06$	$4.38\times {10}^{-1}$	$6.12\times {10}^{-2}$	$1.46\times {10}^1$
Amino acids – Opiorfina	Prop.	$1.42\times {10}^8$	$1.60\times {10}^5$	$1.37\times {10}^5$	$2.58$	$3.13$	$4.16\times {10}^{-1}$	$1.30\times {10}^{-1}$	$1.38\times {10}^1$
Angiotensina IV	Orig.	$1.26\times {10}^{15}$	$4.49\times {10}^8$	$3.82\times {10}^8$	$2.55$	$3.32$	$4.47\times {10}^{-1}$	$3.23\times {10}^{-1}$	$1.49\times {10}^1$
Amino acids – Angiotensina IV	Prop.	$1.41\times {10}^8$	$1.60\times {10}^5$	$1.39\times {10}^5$	$2.61$	$3.13$	$3.88\times {10}^{-1}$	$1.26\times {10}^{-1}$	$1.29\times {10}^1$
Neurotensina	Orig.	$2.20\times {10}^{26}$	$1.62\times {10}^{14}$	$1.41\times {10}^{14}$	$2.63$	$3.19$	$3.75\times {10}^{-1}$	$1.88\times {10}^{-1}$	$1.25\times {10}^1$
Amino acids – Neurotensina	Prop.	$3.12\times {10}^8$	$2.25\times {10}^5$	$1.95\times {10}^5$	$2.60$	$3.25$	$4.00\times {10}^{-1}$	$2.47\times {10}^{-1}$	$1.33\times {10}^1$
Bradicinina	Orig.	$4.43\times {10}^{18}$	$2.01\times {10}^{10}$	$1.82\times {10}^{10}$	$2.72$	$3.14$	$2.82\times {10}^{-1}$	$1.39\times {10}^{-1}$	$9.40$
Amino acids – Bradicinina	Prop.	$1.86\times {10}^8$	$1.60\times {10}^5$	$1.39\times {10}^5$	$2.62$	$3.33$	$3.82\times {10}^{-1}$	$3.32\times {10}^{-1}$	$1.27\times {10}^1$
Angiotensina I	Orig.	$3.84\times {10}^{21}$	$6.38\times {10}^{11}$	$5.54\times {10}^{11}$	$2.60$	$3.33$	$3.98\times {10}^{-1}$	$3.29\times {10}^{-1}$	$1.33\times {10}^1$
Amino acids – Angiotensina I	Prop.	$2.33\times {10}^8$	$1.60\times {10}^5$	$1.41\times {10}^5$	$2.66$	$3.10$	$3.42\times {10}^{-1}$	$1.03\times {10}^{-1}$	$1.14\times {10}^1$
Gastrin-14	Orig.	$7.03\times {10}^{23}$	$1.02\times {10}^{13}$	$8.71\times {10}^{12}$	$2.57$	$3.18$	$4.27\times {10}^{-1}$	$1.81\times {10}^{-1}$	$1.42\times {10}^1$
Amino acids – Gastrin-14	Prop.	$2.17\times {10}^8$	$1.60\times {10}^5$	$1.44\times {10}^5$	$2.71$	$3.12$	$2.87\times {10}^{-1}$	$1.19\times {10}^{-1}$	$9.58$
GLU_CYS_GLY	Orig.	$5.47\times {10}^9$	$8.95\times {10}^5$	$7.64\times {10}^5$	$2.56$	$3.11$	$4.41\times {10}^{-1}$	$1.14\times {10}^{-1}$	$1.47\times {10}^1$
Amino acids – GLU_CYS_GLY	Prop.	$2.46\times {10}^7$	$5.67\times {10}^4$	$5.20\times {10}^4$	$2.75$	$3.03$	$2.48\times {10}^{-1}$	$3.48\times {10}^{-2}$	$8.28$
ALA_HIS	Orig.	$3.01\times {10}^8$	$2.25\times {10}^5$	$2.03\times {10}^5$	$2.71$	$3.22$	$2.94\times {10}^{-1}$	$2.17\times {10}^{-1}$	$9.79$
Amino acids – ALA_HIS	Prop.	$2.66\times {10}^7$	$5.67\times {10}^4$	$4.89\times {10}^4$	$2.58$	$3.18$	$4.15\times {10}^{-1}$	$1.83\times {10}^{-1}$	$1.38\times {10}^1$
PRO_GLY_PRO	Orig.	$1.87\times {10}^9$	$4.49\times {10}^5$	$3.81\times {10}^5$	$2.54$	$3.04$	$4.56\times {10}^{-1}$	$3.95\times {10}^{-2}$	$1.52\times {10}^1$
Amino acids – PRO_GLY_PRO	Prop.	$1.79\times {10}^7$	$5.67\times {10}^4$	$5.13\times {10}^4$	$2.72$	$3.11$	$2.85\times {10}^{-1}$	$1.12\times {10}^{-1}$	$9.48$
GLY_HIS_LYS	Orig.	$1.44\times {10}^{10}$	$1.26\times {10}^6$	$1.13\times {10}^6$	$2.69$	$3.31$	$3.08\times {10}^{-1}$	$3.12\times {10}^{-1}$	$1.03\times {10}^1$
Amino acids – GLY_HIS_LYS	Prop.	$4.89\times {10}^7$	$8.01\times {10}^4$	$7.20\times {10}^4$	$2.70$	$3.10$	$3.03\times {10}^{-1}$	$9.93\times {10}^{-2}$	$1.01\times {10}^1$

reports these empirical gate counts alongside the derived coefficients $C_1$ and $C_2$ and their relative errors.

Using our regression models—linear for qubits and exponential for coefficients—we predicted the resources required to simulate a variety of proteins and peptides, including both well-known and novel examples. Our models provide the following key insights:

• Number of coefficients ($n_{\mathrm{coef}}$): The number of Hamiltonian coefficients exhibits exponential growth with the number of electrons, as captured by the exponential model in Equation (11).
• Number of Qubits ($n_{\mathrm{qubits}}$): the number of qubits grows moderately linearly with the number of electrons, as described by the linear model in Equation (6).
• Number of Toffoli Gates: While fragmentation occasionally introduces overhead for small amino acids—due to duplicated setup costs and additional reassembly steps—it proves advantageous at the peptide scale, where monolithic encodings become intractable. This trade-off is acceptable, given the preservation of accuracy and the exponential savings in larger systems. However, for small systems, fragmentation maintains extremely low errors, supporting the method’s accuracy and feasibility. This suggests that, while fragmentation introduces a slight overhead in gate count for small systems, it remains a viable strategy for reducing resource requirements in larger systems.

These predictions were validated on a diverse set of intermediate systems and then extrapolated to full hormone peptides—such as Glucagon—demonstrating that the model retains predictive power across four orders of magnitude in electron count and 30 orders of Hamiltonian complexity.

To further analyze the scalability of our approach, we generated log-log plots that relate the following quantities:

• $n_{\mathrm{coef}}$ versus $n_{\mathrm{electrons}}$,
• Toffoli gates versus $n_{\mathrm{electrons}}$,
• Reduction factor versus $n_{\mathrm{electrons}}$,
• Total qubits versus $n_{\mathrm{electrons}}$.

These plots reveal that, although the quantum cost increases with system size, the application of fragmentation and the SelectSwap algorithm significantly mitigates this increase. Specifically, the reduction factor achieved through fragmentation keeps the gate count and qubit requirements within acceptable ranges, even for large systems. This demonstrates the effectiveness of our approach in managing the exponential scaling of quantum resources. Further details of these affirmations can be corroborated in Table 2, where the reduction in terms of coefficients and Toffoli gates of our approach is presented, both in the case of protein and amino acid fragmentation levels. These results confirm that, while quantum resource scaling remains exponential, targeted fragmentation combined with modern quantum oracles offers a tractable and chemically accurate pathway to simulate biologically meaningful molecules on future quantum hardware.

To bring the strategy of our previous work to the next level  [3], we compute the accuracy of our fragmentation strategy by comparing reference energies ($E_{\mathrm{GT}}$) with calculated energies ($E_m$) for a set of larger peptides. Table 1 summarizes the number of electrons, orbitals, theoretical energy, computed energy, and relative error ($\%RE$), starting with some small peptides included in our previous work and continuing with much larger ones to compare the change in accuracy as the system scales up. Key observations include the following:

• Small peptides: Relative errors of 0.0005–0.0065% in dipeptides (e.g., Gly-Gly, Pro-Gly, Gly-Ala). This confirms the high accuracy of our fragmentation strategy for small systems.
• Intermediate peptides: Some (e.g., Aspartame and Phe-Ile) exhibit slightly higher errors (up to  0.065%), confirming again the accuracy of the strategy.
• Large systems: In molecules with hundreds of electrons (e.g., Angiotensin II and IV, Oxytocin, Glucagon), the relative error increases (between 2 and 3%), highlighting the need for further optimization strategies, even though the errors remain within acceptable limits for practical applications.

Our combined methodology—statistical modeling, hierarchical fragmentation, and quantum resource optimization—enables the accurate simulation of complex biomolecules at an unprecedented scale. These results lay the groundwork for near-term beyond applications in quantum biochemistry and offer a scalable framework compatible with fault-tolerant quantum computing architectures.

Experimental Validation

Table 4 presents a detailed comparison between theoretical Toffoli gate counts—computed via Equation (20) with fixed parameters $C_1=C_2=3$—and empirical counts (Toffoli*) obtained using the pennylane.labs.resource_estimation [40] module for each fragment’s qubitization circuit. Additionally, we perform an ordinary least squares (OLS) fit across all samples to extract empirical scaling coefficients $\langle C_1\rangle$ and $\langle C_2\rangle$, along with their deviations. Per-fragment parameter variations under different precision settings $({\epsilon }_1,{\epsilon }_2)$ are detailed in Appendix C.

5. Discussion

The results presented demonstrate that our fragmentation–reassembly protocol effectively decomposes large protein Hamiltonians into tractable subunits while preserving overall accuracy. By applying Equation (20) with fixed parameters ($C_1=C_2=3$) and validating against PennyLane’s resource_estimation [39] (Table 4), we achieve sub-3% relative energy errors alongside reductions of up to 20 orders of magnitude in Toffoli gate counts (Table 2). This dual attainment of precision and scalability is essential for hybrid quantum–classical workflows on near-term devices.

A key strength of our methodology lies in its modular error propagation: The overall simulation error results from the additive contributions of each independently simulated fragment. Consequently, any improvement in fragment accuracy, whether through higher-order quantum algorithms (e.g., quantum phase estimation [41] or improved state preparation circuits [23], directly results in a lower global error. As logical qubit fidelity advances, we therefore anticipate convergence toward chemical precision without altering the core fragmentation scheme.

Our resource-prediction framework is underpinned by robust regression, with Huber estimators ($ϵ\simeq 2.0$) delivering superior outlier resistance and fit quality across small, medium, and large fragment regimes (Table 3). Compared to Theil–Sen, Huber regression maintains higher $R^2$ values and lower MAE in log space, justifying its selection as the primary model for qubit- and gate-count estimation. The trends illustrated in Figure 3 further corroborate the consistency of these predictions across molecular sizes.

Empirical calibration refines our theoretical scaling: fitting Equation (20) to measured data yields global corrections $\langle C_1\rangle =2.64$, $\langle C_2\rangle =3.17$, reducing the mean Toffoli-count error to 11.9%. These adjustments preserve the validity of the analytical model while aligning it with practical hardware characteristics, thereby bridging theory and experiment with minimal overhead.

When benchmarked against classical fragmentation approaches (e.g. ONIOM, many-body expansions), our pipeline offers comparable or improved accuracy for small peptides and more favorable scaling for larger biomolecules. Unlike global symmetry tapering [21], which is limited due to the scarcity of protein-wide symmetries, our fragment-wise active-space identification applies universally, enhancing adaptability across diverse protein architectures.

Nonetheless, several limitations remain. Current noise and decoherence on real quantum hardware constrain immediate fault-tolerant implementation. Fragments containing atypical bonds or motifs may introduce systematic deviations; targeted corrections—guided by the QMProt dataset [33]—and refined state-preparation protocols are required to address these edge cases. Moreover, the integration of advanced subroutines will improve fragment precision at the expense of greater circuit complexity, necessitating a careful trade-off between accuracy and resource availability.

In summary, our fragmentation framework provides a scientifically rigorous, modular, and scalable blueprint for quantum simulation of biomolecules. By combining analytical modeling, empirical validation, and error modularity, we propose a scientifically grounded and practically oriented pathway toward achieving quantum advantage in real-world biochemical simulations.

6. Conclusions and Perspectives

We have presented a scalable, resource-efficient protocol for the quantum simulation of large protein systems, based on molecular fragmentation, regression-driven resource estimation, and circuit optimization via the SelectSwap algorithm.

Starting from small peptides and extending to Glucagon, our method maintains relative energy errors below 3% while reducing Toffoli gate counts by up to 20 orders of magnitude. Regression models accurately predict resource requirements and support pre-optimization of quantum workloads for both current and near-term devices.

Future work includes expanding the molecular dataset to improve model robustness, integrating advanced subroutines (e.g. QPE and optimized state preparation) to approach chemical accuracy, and benchmarking on actual hardware to evaluate noise resilience. Comparative analysis with techniques such as Hamiltonian tapering and active-space reduction will further refine the reassembly pipeline, and the modular nature of our framework ensures that hardware improvements map directly to simulation accuracy without altering the core methodology.

Our head-to-head comparison with PennyLane’s resource_estimation module [39] confirms that, once validated, it can serve as a reliable standalone estimator of circuit cost. Simulating each fragment’s qubitization circuit under the chosen precision settings can replace the explicit extraction of empirical coefficients $C_1$ and $C_2$, simplifying future resource estimates without sacrificing rigor.

Moreover, implementing quantum phase estimation at the fragment level is expected to reduce individual fragment errors to near–chemical precision. Because the total energy error accumulates fragment-wise, improving each fragment’s precision directly lowers the global error. This property is well suited to near-term quantum processors with limited numbers of high-fidelity logical qubits.

Overall, the proposed strategy offers a practical and verifiable route to simulate increasingly large biomolecules, aligning with the current trajectory of quantum hardware development and providing a clear framework for ongoing methodological enhancement.