VQE-PDFT: Hybrid Quantum-Classical MC-PDFT

• VQE-PDFT is a hybrid quantum-classical method that replaces the costly CASSCF step in MC-PDFT with a VQE-based CASCI approach to capture static correlation.
• It uses quantum-prepared multiconfigurational active-space wavefunctions and their reduced density matrices, which are post-processed by a tPBE on-top density functional to recover dynamic correlation.
• Benchmark studies on charge-transfer dimers and QM/MM models show that VQE-PDFT achieves near-classical MC-PDFT accuracy while significantly reducing quantum resource demands.

Searching arXiv for the primary VQE-PDFT paper and closely related MC-PDFT/VQE context papers. First, I’ll look up the exact paper by arXiv id, then I’ll search for closely related work on MC-PDFT and VQE to support contextual statements. VQE-PDFT is a hybrid quantum-classical realization of multiconfiguration pair-density functional theory in which the usual classical multiconfigurational solver is replaced by a variational quantum eigensolver used in a CASCI-like active-space calculation. Its defining division of labor is specific: the quantum computer prepares a multiconfigurational active-space wavefunction and provides its reduced density matrices, while the classical MC-PDFT layer evaluates the final energy through an on-top density functional. In this formulation, static correlation is carried by the multiconfigurational reference and dynamic correlation is recovered in the PDFT layer, with the stated goal of avoiding both the full classical cost of CASSCF and the full quantum cost of deep multireference circuits. The framework was benchmarked on the CT7/04 charge-transfer dataset, extended to a QM/MM treatment of electron transfer in European robin cryptochrome ErCRY4, and demonstrated on superconducting quantum hardware (Chen et al., 10 Aug 2025).

1. Conceptual definition and scope

VQE-PDFT was introduced to address the standard partition of electronic correlation in strongly correlated systems. Static correlation is treated through a multiconfigurational active-space wavefunction, whereas dynamic correlation is often added afterward by post-MCSCF methods such as CASPT2. MC-PDFT already provides a lower-scaling alternative to such post-SCF treatments by combining a multiconfigurational reference with an on-top density functional, but standard MC-PDFT still depends on a classical CASSCF reference, and that CASSCF step scales exponentially with active-space size. VQE-PDFT alters this architecture by replacing the classical active-space solver with VQE-based CASCI while retaining the usual MC-PDFT energy expression (Chen et al., 10 Aug 2025).

The method is therefore neither a generic “VQE plus DFT” construction nor a quantum implementation of full CASSCF. The authors state that VQE-PDFT “replaces the computationally expensive CASSCF optimization in MC-PDFT with VQE-based CASCI calculations,” thereby “reducing computational costs by limiting quantum calculations mainly to the CASCI level while preserving MC-PDFT’s accurate treatment of static and dynamic correlation.” A common misconception is that the quantum computer supplies the final molecular energy directly. In this framework it does not: it supplies an active-space state and its reduced density matrices, and those quantities are classically postprocessed into the MC-PDFT energy.

A second conceptual boundary concerns orbital optimization. VQE-PDFT in this work does not replace the entire CASSCF procedure including orbital optimization. It replaces the expensive CI-like active-space wavefunction optimization step with a quantum variational solver for fixed orbitals. This fixed-orbital character is central to its relationship to CASCI and remains one of the method’s stated limitations.

2. Formal structure and workflow

The workflow begins with an active-space partitioning of the molecular problem, as in CASCI and MC-PDFT. Within that active space, the electronic Hamiltonian is encoded on qubits after fermion-to-qubit mapping; the implementation explicitly mentions parity transformation. A parameterized circuit prepares the active-space state

$|\psi(\theta)\rangle = U(\theta)|0\rangle .$

The VQE stage then minimizes the active-space expectation value in a CASCI framework,

$$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$

with fixed orbitals rather than orbital optimization on the quantum computer. In the conventional second-quantized form implied by the manuscript’s CASCI language,

$$\hat H_{\mathrm{act}} = \sum_{pq} h_{pq}\, a_p^\dagger a_q + \frac12 \sum_{pqrs} g_{pqrs}\, a_p^\dagger a_r^\dagger a_s a_q .$$

After VQE convergence, the optimized parameters are frozen and reused for reduced-density-matrix measurements. The one-body reduced density matrix is given explicitly as

$$\gamma_{pq} = \langle \psi(\theta)| a_p^\dagger a_q |\psi(\theta)\rangle ,$$

and the same measurement procedure is used for the two-body reduced density matrix, written in standard notation as

$$\Gamma_{pqrs} = \langle \psi| a_p^\dagger a_r^\dagger a_s a_q |\psi\rangle$$

up to ordering convention. These RDMs constitute the interface between the quantum solver and the classical PDFT layer (Chen et al., 10 Aug 2025).

From the measured RDMs, the classical stage forms the total density and on-top pair density,

$$\rho(\mathbf r)=\sum_{pq}\gamma_{pq}\,\phi_p(\mathbf r)\phi_q(\mathbf r),$$

$$\Pi(\mathbf r)=\sum_{pqrs}\Gamma_{pqrs}\,\phi_p(\mathbf r)\phi_q(\mathbf r)\phi_r(\mathbf r)\phi_s(\mathbf r),$$

modulo the precise spin-summed convention used in the implementation. The 1-RDM provides the ingredients for $T$, $V_{ne}$, $\rho$, and $$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$0; the 2-RDM provides $$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$1, which enters the on-top functional. The total energy is then evaluated through the standard MC-PDFT expression

$$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$2

The named on-top functional used throughout the work is the translated PBE functional, tPBE. The manuscript does not print the translated or fully translated functional equations, so the technically precise statement is that VQE-PDFT uses the standard MC-PDFT energy form with a tPBE on-top functional and a VQE-generated multiconfigurational reference.

3. Position relative to MC-PDFT, CASCI, and plain VQE

Relative to standard MC-PDFT, VQE-PDFT changes only the source of the multiconfigurational reference. Standard MC-PDFT typically employs a classical multiconfigurational reference, often from CASSCF, and then evaluates the on-top functional from that reference’s density and on-top pair density. VQE-PDFT retains the same final PDFT functional form but replaces the classical reference solver with a VQE-based CASCI state (Chen et al., 10 Aug 2025).

Relative to CASCI, the methodological analogy is direct. Both approaches use fixed orbitals. The difference is that classical exact diagonalization or CI within the active space is replaced by VQE for wavefunction optimization and RDM extraction. This suggests that VQE-PDFT should be understood as a quantum-assisted active-space solver embedded inside the established MC-PDFT formalism, not as a reformulation of the PDFT layer itself.

Relative to plain VQE, the distinction is the recovery of dynamic correlation. Bare active-space VQE provides only the variational energy of the chosen active-space ansatz and can miss dynamic correlation severely. The CT7/04 benchmark reported physically unreasonable negative dissociation energies for plain VQE in $$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$3 and $$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$4. In VQE-PDFT, those deficiencies are repaired by the subsequent MC-PDFT correction.

Relative to standalone UCCSD- or ROUCCSD-based VQE strategies, the framework permits a more modest active-space quantum state while delegating dynamic-correlation recovery to PDFT. Relative to DFT-only descriptions of the biological application, the method explicitly targets multiconfigurational effects in radical and open-shell states. These distinctions are central to the paper’s interpretation of reduced quantum-resource demand.

4. Benchmark validation on charge-transfer dimers

For the CT7/04 dissociation-energy benchmark, the implementation used a UCCSD ansatz in both plain VQE and VQE-PDFT calculations with the jul-cc-pVTZ basis. The supplementary active spaces include $$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$5 for $$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$6, $$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$7, $$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$8, $$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$9, $$\hat H_{\mathrm{act}} = \sum_{pq} h_{pq}\, a_p^\dagger a_q + \frac12 \sum_{pqrs} g_{pqrs}\, a_p^\dagger a_r^\dagger a_s a_q .$$0, and $$\hat H_{\mathrm{act}} = \sum_{pq} h_{pq}\, a_p^\dagger a_q + \frac12 \sum_{pqrs} g_{pqrs}\, a_p^\dagger a_r^\dagger a_s a_q .$$1, with monomer spaces such as $$\hat H_{\mathrm{act}} = \sum_{pq} h_{pq}\, a_p^\dagger a_q + \frac12 \sum_{pqrs} g_{pqrs}\, a_p^\dagger a_r^\dagger a_s a_q .$$2 and $$\hat H_{\mathrm{act}} = \sum_{pq} h_{pq}\, a_p^\dagger a_q + \frac12 \sum_{pqrs} g_{pqrs}\, a_p^\dagger a_r^\dagger a_s a_q .$$3, while $$\hat H_{\mathrm{act}} = \sum_{pq} h_{pq}\, a_p^\dagger a_q + \frac12 \sum_{pqrs} g_{pqrs}\, a_p^\dagger a_r^\dagger a_s a_q .$$4 uses $$\hat H_{\mathrm{act}} = \sum_{pq} h_{pq}\, a_p^\dagger a_q + \frac12 \sum_{pqrs} g_{pqrs}\, a_p^\dagger a_r^\dagger a_s a_q .$$5. The benchmark was designed to test the central formal claim that a VQE-generated active-space reference can be fed into MC-PDFT and recover essentially the same quality as conventional MC-PDFT (Chen et al., 10 Aug 2025).

The reported dissociation energies, in kcal/mol, were: $$\hat H_{\mathrm{act}} = \sum_{pq} h_{pq}\, a_p^\dagger a_q + \frac12 \sum_{pqrs} g_{pqrs}\, a_p^\dagger a_r^\dagger a_s a_q .$$6, VQE $$\hat H_{\mathrm{act}} = \sum_{pq} h_{pq}\, a_p^\dagger a_q + \frac12 \sum_{pqrs} g_{pqrs}\, a_p^\dagger a_r^\dagger a_s a_q .$$7, VQE-PDFT $$\hat H_{\mathrm{act}} = \sum_{pq} h_{pq}\, a_p^\dagger a_q + \frac12 \sum_{pqrs} g_{pqrs}\, a_p^\dagger a_r^\dagger a_s a_q .$$8, MC-PDFT $$\hat H_{\mathrm{act}} = \sum_{pq} h_{pq}\, a_p^\dagger a_q + \frac12 \sum_{pqrs} g_{pqrs}\, a_p^\dagger a_r^\dagger a_s a_q .$$9, reference $$\gamma_{pq} = \langle \psi(\theta)| a_p^\dagger a_q |\psi(\theta)\rangle ,$$0; $$\gamma_{pq} = \langle \psi(\theta)| a_p^\dagger a_q |\psi(\theta)\rangle ,$$1, VQE $$\gamma_{pq} = \langle \psi(\theta)| a_p^\dagger a_q |\psi(\theta)\rangle ,$$2, VQE-PDFT $$\gamma_{pq} = \langle \psi(\theta)| a_p^\dagger a_q |\psi(\theta)\rangle ,$$3, MC-PDFT $$\gamma_{pq} = \langle \psi(\theta)| a_p^\dagger a_q |\psi(\theta)\rangle ,$$4, reference $$\gamma_{pq} = \langle \psi(\theta)| a_p^\dagger a_q |\psi(\theta)\rangle ,$$5; $$\gamma_{pq} = \langle \psi(\theta)| a_p^\dagger a_q |\psi(\theta)\rangle ,$$6, VQE $$\gamma_{pq} = \langle \psi(\theta)| a_p^\dagger a_q |\psi(\theta)\rangle ,$$7, VQE-PDFT $$\gamma_{pq} = \langle \psi(\theta)| a_p^\dagger a_q |\psi(\theta)\rangle ,$$8, MC-PDFT $$\gamma_{pq} = \langle \psi(\theta)| a_p^\dagger a_q |\psi(\theta)\rangle ,$$9, reference $$\Gamma_{pqrs} = \langle \psi| a_p^\dagger a_r^\dagger a_s a_q |\psi\rangle$$0; $$\Gamma_{pqrs} = \langle \psi| a_p^\dagger a_r^\dagger a_s a_q |\psi\rangle$$1, VQE $$\Gamma_{pqrs} = \langle \psi| a_p^\dagger a_r^\dagger a_s a_q |\psi\rangle$$2, VQE-PDFT $$\Gamma_{pqrs} = \langle \psi| a_p^\dagger a_r^\dagger a_s a_q |\psi\rangle$$3, MC-PDFT $$\Gamma_{pqrs} = \langle \psi| a_p^\dagger a_r^\dagger a_s a_q |\psi\rangle$$4, reference $$\Gamma_{pqrs} = \langle \psi| a_p^\dagger a_r^\dagger a_s a_q |\psi\rangle$$5; $$\Gamma_{pqrs} = \langle \psi| a_p^\dagger a_r^\dagger a_s a_q |\psi\rangle$$6, VQE $$\Gamma_{pqrs} = \langle \psi| a_p^\dagger a_r^\dagger a_s a_q |\psi\rangle$$7, VQE-PDFT $$\Gamma_{pqrs} = \langle \psi| a_p^\dagger a_r^\dagger a_s a_q |\psi\rangle$$8, MC-PDFT $$\Gamma_{pqrs} = \langle \psi| a_p^\dagger a_r^\dagger a_s a_q |\psi\rangle$$9, reference $$\rho(\mathbf r)=\sum_{pq}\gamma_{pq}\,\phi_p(\mathbf r)\phi_q(\mathbf r),$$0; $$\rho(\mathbf r)=\sum_{pq}\gamma_{pq}\,\phi_p(\mathbf r)\phi_q(\mathbf r),$$1, VQE $$\rho(\mathbf r)=\sum_{pq}\gamma_{pq}\,\phi_p(\mathbf r)\phi_q(\mathbf r),$$2, VQE-PDFT $$\rho(\mathbf r)=\sum_{pq}\gamma_{pq}\,\phi_p(\mathbf r)\phi_q(\mathbf r),$$3, MC-PDFT $$\rho(\mathbf r)=\sum_{pq}\gamma_{pq}\,\phi_p(\mathbf r)\phi_q(\mathbf r),$$4, reference $$\rho(\mathbf r)=\sum_{pq}\gamma_{pq}\,\phi_p(\mathbf r)\phi_q(\mathbf r),$$5; and $$\rho(\mathbf r)=\sum_{pq}\gamma_{pq}\,\phi_p(\mathbf r)\phi_q(\mathbf r),$$6, VQE $$\rho(\mathbf r)=\sum_{pq}\gamma_{pq}\,\phi_p(\mathbf r)\phi_q(\mathbf r),$$7, VQE-PDFT $$\rho(\mathbf r)=\sum_{pq}\gamma_{pq}\,\phi_p(\mathbf r)\phi_q(\mathbf r),$$8, MC-PDFT $$\rho(\mathbf r)=\sum_{pq}\gamma_{pq}\,\phi_p(\mathbf r)\phi_q(\mathbf r),$$9, reference $$\Pi(\mathbf r)=\sum_{pqrs}\Gamma_{pqrs}\,\phi_p(\mathbf r)\phi_q(\mathbf r)\phi_r(\mathbf r)\phi_s(\mathbf r),$$0.

The mean unsigned error was reported as $$\Pi(\mathbf r)=\sum_{pqrs}\Gamma_{pqrs}\,\phi_p(\mathbf r)\phi_q(\mathbf r)\phi_r(\mathbf r)\phi_s(\mathbf r),$$1 kcal/mol for plain VQE, $$\Pi(\mathbf r)=\sum_{pqrs}\Gamma_{pqrs}\,\phi_p(\mathbf r)\phi_q(\mathbf r)\phi_r(\mathbf r)\phi_s(\mathbf r),$$2 kcal/mol for VQE-PDFT, and $$\Pi(\mathbf r)=\sum_{pqrs}\Gamma_{pqrs}\,\phi_p(\mathbf r)\phi_q(\mathbf r)\phi_r(\mathbf r)\phi_s(\mathbf r),$$3 kcal/mol for classical MC-PDFT. The paper explicitly highlights the negative VQE dissociation energies for $$\Pi(\mathbf r)=\sum_{pqrs}\Gamma_{pqrs}\,\phi_p(\mathbf r)\phi_q(\mathbf r)\phi_r(\mathbf r)\phi_s(\mathbf r),$$4 and $$\Pi(\mathbf r)=\sum_{pqrs}\Gamma_{pqrs}\,\phi_p(\mathbf r)\phi_q(\mathbf r)\phi_r(\mathbf r)\phi_s(\mathbf r),$$5 as physically unreasonable. The benchmark therefore supports the intended role separation: VQE generates the multiconfigurational reference, while the PDFT correction supplies enough remaining correlation to reproduce conventional MC-PDFT accuracy to within the reported numerical differences.

5. QM/MM realization for electron transfer in ErCRY4

The biological extension embeds VQE-PDFT in a QM/MM framework. The QM region contains two adjacent tryptophan residues, while the remaining protein and solvent are treated at the MM level. The interface uses electrostatic embedding and link atoms. The software stack is specified: local modifications of PySCF and TenCirChem implement VQE-PDFT; ASH is used for QM/MM link-atom and electrostatic embedding; and TensorCircuit is used for measurement grouping, communication with the quantum device, and readout error mitigation (Chen et al., 10 Aug 2025).

The target process is electron transfer between adjacent tryptophans TrpB (W372) and TrpC (W318) in European robin cryptochrome ErCRY4. The initial state is cationic TrpB / neutral TrpC, and the final state is neutral TrpB / cationic TrpC. Because the cationic tryptophan states are open-shell, a standard paired-electron UCCSD ansatz is inadequate. The study therefore compared a restricted open-shell UCCSD reference ansatz, ROUCCSD, with shallow hardware-efficient ansätze denoted CHEA for closed-shell states and OHEA for open-shell states.

The active spaces for these tryptophan calculations are $$\Pi(\mathbf r)=\sum_{pqrs}\Gamma_{pqrs}\,\phi_p(\mathbf r)\phi_q(\mathbf r)\phi_r(\mathbf r)\phi_s(\mathbf r),$$6 for closed-shell and $$\Pi(\mathbf r)=\sum_{pqrs}\Gamma_{pqrs}\,\phi_p(\mathbf r)\phi_q(\mathbf r)\phi_r(\mathbf r)\phi_s(\mathbf r),$$7 for open-shell states in the 6-31G basis. Under parity mapping these spaces would require 6 qubits, but $$\Pi(\mathbf r)=\sum_{pqrs}\Gamma_{pqrs}\,\phi_p(\mathbf r)\phi_q(\mathbf r)\phi_r(\mathbf r)\phi_s(\mathbf r),$$8- and $$\Pi(\mathbf r)=\sum_{pqrs}\Gamma_{pqrs}\,\phi_p(\mathbf r)\phi_q(\mathbf r)\phi_r(\mathbf r)\phi_s(\mathbf r),$$9-electron number conservation reduces the requirement to 4 qubits because qubits 3 and 6, corresponding to total $T$0-electron and total-electron parity, remain fixed. The empirical circuit design is based on the observation that only 9 computational basis states contribute significantly in both shell types. For the open-shell $T$1 case with 2 $T$2 and 1 $T$3 electrons, the first two qubits must avoid $T$4 and qubits 3–4 must avoid $T$5, restricting the accessible space to $T$6 states. The circuits use $T$7 and controlled-$T$8 gates, preserve particle number for each spin, and have depths 4 for CHEA and 6 for OHEA, compared with depth 45 for ROUCCSD.

These shallow ansätze were validated against active-space CASCI energies for eight single-point calculations corresponding to the four-point electron-transfer protocol across open-shell and closed-shell states. The reported comparison shows that ROUCCSD matches FCI within at least $T$9 Hartree, whereas HEA differs from FCI by $V_{ne}$0 to $V_{ne}$1 Hartree. The examples given are state 1, with $V_{ne}$2, $V_{ne}$3, $V_{ne}$4, so $V_{ne}$5 Hartree, and state 6, with a 0.0036-Hartree deviation. In the manuscript’s interpretation, this establishes that the shallow ansatz is accurate enough for the intended QM/MM electron-transfer application.

The rate model is based on Marcus theory. The manuscript prints a typographically malformed rate expression; the physically corresponding standard nonadiabatic form is described as

$V_{ne}$6

Within the four-point scheme,

$V_{ne}$7

$V_{ne}$8

The electronic coupling is computed separately, not by VQE-PDFT, using Boys-localized orbitals and the direct-coupling scheme

$V_{ne}$9

where $\rho$0 is the Hamiltonian coupling element, $\rho$1 is the overlap integral, and $\rho$2 are localized orbital energies.

Twenty conformations were sampled from molecular dynamics of ErCRY4. Using tPBE and 6-31G for the QM calculations, the averaged Marcus parameters from VQE-PDFT with the empirical HEA are

$\rho$3

while ROUCCSD-based calculations give

$\rho$4

The average coupling from the direct-coupling calculation is

$\rho$5

The manuscript states that $\rho$6 agrees well with the 5 meV value reported experimentally by Timmer et al. The final transfer rates are

$\rho$7

for HEA and

$\rho$8

for ROUCCSD, both close to the experimental ultrafast-transient-absorption estimate

$\rho$9

The paper further notes that these values are roughly an order of magnitude slower than earlier DFT predictions, $$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$00, and explicitly presents this as an improvement over those DFT results.

6. Quantum-hardware realization, error behavior, and limitations

The hardware demonstration uses a customized 13-qubit superconducting device with single-qubit gate fidelity 99.93%, two-qubit CZ fidelity 99.13%, coherence times $$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$01, $$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$02, and readout fidelities $$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$03, $$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$04. Only 4 qubits, indexed $$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$05, are used. A critical procedural distinction is that VQE optimization itself is not run on hardware. To avoid noise accumulation during parameter updates, optimization is carried out on a noiseless classical simulator, after which the optimized parameters are transferred to hardware for measurement of the 1-RDM and 2-RDM matrix elements (Chen et al., 10 Aug 2025).

The hardware RDM protocol has six stated steps: construction of Pauli operators by parity transformation; grouping of compatible measurement operators; compilation of HEA circuits and grouped measurements to the hardware topology; execution of each circuit with 2048 shots; readout error mitigation via a correlated Markovian noise model using calibration circuits with 8192 shots per single-point energy; and assembly of the full 1-RDM and 2-RDM from the measured matrix elements. Measurement grouping is presented as a practical resource-saving method, with the explicit example that $$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$06 and $$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$07 can be measured together through $$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$08.

For a randomly selected conformation, frame 2071, the hardware-derived Marcus parameters are

$$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$09

leading to

$$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$10

These values are compared with noiseless HEA and ROUCCSD references for the same conformation,

$$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$11

$$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$12

The hardware result thus remains in the same range as the classical or noiseless references for that conformation despite hardware noise.

The error analysis attributes this behavior to partial cancellation in difference-based observables. Relative to ROUCCSD, the eight hardware single-point energies exhibit a systematic positive shift rather than random scatter, differing by roughly 61–86 mHartree, with an average positive deviation of $$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$13 mHartree according to the figure discussion. The manuscript argues that this points to correlated measurement or RDM errors rather than purely random fluctuations. Because both $$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$14 and $$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$15 are built from energy differences,

$$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$16

systematic errors may partially cancel. The reported consequence is that, although absolute single-point energies carry large errors, $$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$17 differs from the ROUCCSD reference by only 2.0 mHartree and $$E_{\mathrm{VQE}}(\theta)=\langle \psi(\theta)|\hat H_{\mathrm{act}}|\psi(\theta)\rangle ,$$18 by 0.9 mHartree.

The paper also states several limitations. Active-space size remains restricted by current hardware. Optimization and measurement costs are still substantial, and more efficient optimization and measurement-grouping methods are identified as necessary if substantially more protein conformations are to be sampled on hardware. The biological model uses a separate-ring approximation for indole rings on the two tryptophans, justified by weak coupling but still approximate. The empirical HEA circuits are system specific and were tailored from dominant-state analysis of tryptophan active spaces, so transfer to other systems would require redesign. By construction, the method is CASCI-like and uses fixed orbitals rather than orbital-optimized CASSCF. Functional dependence is not explored beyond tPBE. For the four-point Marcus treatment, the paper additionally notes the instantaneous-response assumption and the neglect of possible non-Markovian environmental effects.

The stated outlook is correspondingly pragmatic: improved optimization and measurement-grouping strategies, larger active spaces as hardware improves, constrained-population VQE schemes using penalty functions or Lagrange multipliers informed by Mulliken analysis to treat multiple simultaneous transfer pathways, and movement beyond the separate-ring approximation toward more complete multi-fragment treatments. This suggests a framework intended to inherit the formal structure of MC-PDFT while relocating only the active-space multiconfigurational solver to the quantum device, so that future hardware improvements can be incorporated without conceptual redesign.