Quantum Ensemble Variational Optimization (QEVO)

• QEVO is a hybrid quantum-classical framework that utilizes parameterized quantum circuits to prepare superposed candidate solutions across complex, combinatorial spaces.
• It employs ensemble-based cost functions and iterative classical optimization to refine quantum sampling, ensuring efficient exploration in high-dimensional design spaces.
• The approach has significant applications in molecular inverse design and combinatorial optimization, enabling rapid identification of high-scoring drug-like molecules.

Quantum Ensemble Variational Optimization (QEVO) is a hybrid quantum-classical optimization framework in which the quantum computer prepares a parameterized superposition state representing an ensemble over a large, combinatorial solution space, while classical resources (optimizers, data postprocessing) adapt the quantum parameters to optimize ensemble-level properties. QEVO’s main advantage is efficiently sampling and refining high-dimensional design spaces by leveraging quantum superposition in conjunction with variational circuit architectures. Originally developed to address challenges in molecular inverse design on near-term quantum hardware, QEVO’s principled methodology and circuit efficiency have immediate relevance for applications in chemical discovery and other combinatorial optimization tasks (Calcagno et al., 21 Aug 2025).

1. Fundamental Principles of QEVO

QEVO formulates optimization problems by mapping complex, discrete candidate structures to orthonormal quantum basis states. For molecular design, structures are first encoded using, for example, the SELFIES symbolic representation, which is guaranteed to be bijective and robust. Each symbol (token) is converted to a binary string, and each molecule (a sequence of $k$ tokens) is mapped to a tensor product state:

$|m_i\rangle = \bigotimes_{j=1}^{k} |q_j\rangle$

where $|q_j\rangle$ encodes a single token. The orthonormal basis $\{|m_i\rangle\}$ spans the full chemical (or solution) space $\mathcal{M}$, allowing the quantum state to represent all possible candidates via superposition.

A parameterized quantum circuit (ansatz) prepares the ensemble state

$$|\psi(\vec{\theta})\rangle = U(\vec{\theta})|0\rangle$$

typically starting with a Hadamard-transformed, uniform superposition

$$H^{\otimes n}|0\rangle = \frac{1}{\sqrt{2^n}} \sum_{x} |x\rangle$$

and then refined through variational updates, where $n$ is the total number of encoding qubits.

The probability of sampling molecule $m_i$ is given by $|\omega_i(\vec{\theta})|^2$, where $\omega_i(\vec{\theta})$ is the amplitude for basis state $|m_i\rangle$ in $|\psi(\vec{\theta})\rangle$.

2. Ensemble-Based Cost Functions and Optimization Process

Distinct from traditional expectation-value VQAs, QEVO defines its objective function as an ensemble average of a target property over all sampled solutions. After sampling the circuit in the computational basis to obtain candidate molecules, each candidate is scored using a property function $p_i$ (e.g., a drug-likeness score, docking affinity, etc). The ensemble average is then

$$p_{\mathcal{M}}(\vec{\theta}) = \sum_{i} \omega_i(\vec{\theta})\,p_i$$

Optimization proceeds by minimizing a global loss function:

$$\mathcal{L}(\vec{\theta}) = |p_{\mathcal{M}}(\vec{\theta}) - p_0| + \epsilon(\vec{\theta})$$

where $p_0$ is the target property value and $\epsilon(\vec{\theta})$ a regularization for, e.g., ensemble purity or diversity.

A classical optimizer, e.g., gradient-free or finite-difference-based, adjusts $\vec{\theta}$ iteratively. After each update, the quantum ensemble is resampled and candidates with desired properties are preferentially amplified by tuning the amplitudes via ansatz parameter updates.

3. Circuit Architecture and Quantum Resource Efficiency

QEVO is designed for resource adaptivity and NISQ-compatibility. The standard implementation uses a RealAmplitudes (RA) ansatz with alternating layers of single-qubit $R_y(\theta)$ rotations and hardware-efficient entanglement, with circuit depth and number of parameters scaling linearly in $n$. The Bologna–Yale (BY) ansatz offers a qubit-frugal alternative, using iterative measurement/reset with as few as two qubits to encode high-dimensional states.

Typical circuit initialization (Uniform Superposition):

$|\psi_0\rangle = H^{\otimes n}|0\rangle$

Sampled amplitudes are manipulated by parameter updates, maintaining ensemble exploration with shallow circuits. For an $n$-qubit system, both depth and qubit count are chosen to fit hardware constraints; this enables practical use on modestly-sized quantum processors.

4. Applications: Inverse Design of Drug-like Molecules

A central application of QEVO is combinatorial molecular design. The approach encodes molecules as binary strings (tokens) in self-consistent representations (e.g., 40-character ASCII strings per molecule). QEVO’s ansatz samples this space, with each output string mapped back to a candidate molecule.

In benchmark studies, QEVO rapidly identifies high-scoring molecules (e.g., those with low plogP or maximizing binding affinity). An anticancer drug design use-case is highlighted, targeting Janus kinase 2 (JAK2) inhibition; by modifying the loss function to favor molecules structurally related to a reference (e.g., ruxolitinib), QEVO generates candidate ensembles with improved selectivity toward JAK2 over off-target kinases. After quantum sampling, classical postprocessing—such as molecular docking simulations—evaluates binding energetics, closing the quantum-classical workflow.

In both “unbiased” (uniform sampling) and “bias” (using an initialized distribution or rewarded property) modes, QEVO demonstrates the ability to efficiently navigate $>10^{40}$-dimensional chemical spaces, producing competitive candidate lists with very limited quantum resources.

5. Comparison to Related Variational Approaches

QEVO differs fundamentally from traditional VQA protocols such as VQE and QAOA. Standard VQA minimizes a quantum expectation value as a function of parameters tied to ground state or optimal solution wavefunction preparation, while QEVO explicitly leverages quantum sampling to optimize an ensemble property aggregated over superposed candidates. This distinction aligns QEVO with ensemble-averaged and distributional quantum optimization paradigms.

Unlike exhaustive screening, which is infeasible for high-dimensional design problems, QEVO leverages quantum parallelism: the measurement collapses the parameterized state to yield classical candidates, and iterative resampling drives the distribution toward increasingly favorable regions. In contrast to approaches that require explicit enumeration or brute-force search, QEVO’s quantum state access and shallow circuits enable practical advancement on current hardware.

6. Resource Requirements and Scalability

A key feature of QEVO is its scalability with respect to the chemical or combinatorial solution space. For a $k$-token molecule with $d$ discrete allowed tokens, the state space scales as $d^k$. Classical methods cannot systematically explore such a space past $k \sim 10$–$15$; in QEVO, this space is sampled in superposition on $n = k \log_2 d$ qubits, with circuit depth adapted to avoid NISQ noise bottlenecks. The RA and BY ansätze can be scaled down to fit available hardware, with the resource scaling summarized as

Encoding	Qubits	Circuit Depth
RA (standard)	$n$	$O(n)$
BY (holographic)	$O(1)$-few	$O(n)$

These shallow designs, together with measurement-based classical postselection, obviate the need for deep, error-prone circuits.

7. Outlook and Prospective Developments

QEVO’s ensemble-based approach and variational refinement provide a generic, device-agnostic toolkit for quantum-enhanced exploration of complex design spaces. The methodology is applicable to a wide range of problems, including drug discovery, material science, and combinatorial optimization where objective functions depend on properties aggregated over a solution distribution.

Potential extensions include:

• Generalizing the encoding/modulation scheme to more complex molecular descriptors or higher-order property objectives.
• Augmenting the variational ansatz to support alternative forms of quantum state connectivity, including error-mitigated or holographic circuits for deeply nested design spaces.
• Integrating more advanced loss functions encapsulating multiple, potentially conflicting objectives (multi-objective QEVO).
• Hybridizing QEVO with state-of-the-art classical surrogate modeling or molecular simulation to further reduce sampling complexity.

QEVO’s capacity to efficiently sample and optimize over large spaces on near-term hardware presents a practical advance toward quantum-enabled discovery in molecular and materials systems, offering a template for ensemble-based variational optimization in broader problem domains (Calcagno et al., 21 Aug 2025).