Surrogate Quantum Circuit (SQC)

• Surrogate Quantum Circuit (SQC) is a learned quantum or classical model that approximates complex quantum processes to reduce hardware evaluation costs.
• SQCs enable resource-efficient variational optimization, architecture search, and physics simulations by leveraging efficient surrogate models.
• They employ methodologies such as neural networks, MPS emulators, and graph-based regressors to achieve scalable, noise-resistant optimization in hybrid quantum-classical systems.

A Surrogate Quantum Circuit (SQC) is a learned, parameterized quantum circuit or classical neural model that stands in for a more expensive or analytically intractable quantum process, typically within variational optimization, architecture search, quantum machine learning, or physics simulation. The SQC framework either (a) learns a circuit-based approximation to a quantum process that is hard to implement exactly with quantum hardware (as in nonlinear, dissipative, or non-unitary dynamics), or (b) replaces expensive quantum evaluations during circuit design, optimization, or benchmarking with efficient, classically accessible surrogates. SQCs are a rapidly evolving toolkit instrumental in scalable quantum-classical hybrid algorithms, resource-efficient quantum experiments, and the practical application of quantum computing to scientific and engineering problems.

1. Foundational Principles and Definitions

The SQC paradigm arises from the need to: (i) sidestep the prohibitive computational or physical cost of direct quantum evaluation in variational algorithms or quantum simulation, and (ii) construct explicit, deployable models for dynamics or architectures that are otherwise implicit, non-unitary, or inaccessible on near-term quantum hardware.

In variational quantum circuit optimization, suppose one wishes to find parameters $\theta \in \mathbb{R}^N$ minimizing a quantum cost function

$$C(\theta) = \langle 0| U^\dagger(\theta) H U(\theta) |0 \rangle,$$

where $U(\theta)$ is a parameterized quantum circuit and $H$ a Hermitian operator. Computing gradients and Hessians directly on quantum hardware is subject to sampling noise and resource bottlenecks. The SQC approach leverages a classically tractable surrogate—such as a truncated state-vector simulator, a matrix product state (MPS), or an explicit regression model—to approximate the optimization landscape or map features to performance metrics, thus reducing required quantum resources and accelerating convergence (Gustafson et al., 2024, O'Leary et al., 8 Jan 2025).

In quantum algorithm or circuit architecture search, the SQC is often a classical regressor (e.g., a graph neural network or random forest) mapping circuit descriptors to performance metrics, trained on previously evaluated quantum circuits and applied to accelerate search or benchmarking (Martyniuk et al., 7 Jun 2025, Choudhary et al., 10 Dec 2025).

In physics simulation and quantum machine learning, circuit-based SQCs are explicitly trained to approximate non-unitary or nonlinear dynamics, such as the lattice Boltzmann BGK collision operator, or to construct explicit quantum circuits corresponding to quantum kernel models (Lăcătuş et al., 16 Jul 2025, Nakayama et al., 2024).

2. SQC Methodologies and Algorithmic Frameworks

Several distinct SQC methodologies have been developed, reflecting different domains of application:

• Surrogate-assisted Variational Quantum Circuit Optimization: Here, a classical surrogate (e.g., a sparse wavefunction simulator or an MPS emulator) is used to compute an approximate Hessian or cost landscape. Surrogate pre-optimization yields a good starting point and a surrogate Hessian, from which quasi-Newton updates or parallel line-search directions are derived. Noisy quantum hardware is then used only to perform sampled line searches in low-dimensional subspaces, with the conjugate search directions and windows informed by the surrogate (Gustafson et al., 2024).
• On-the-Fly Classical Surrogate Fitting: Online RBF models are iteratively fit to sparse hardware data (parameters and observed costs), with the surrogate minimum providing the acquisition function for subsequent quantum hardware queries. No pre-trained hyperparameters or GP uncertainty models are needed, enabling robust and resource-efficient optimization of parametrized quantum circuits up to 127 qubits (O'Leary et al., 8 Jan 2025).
• Surrogate-Guided Architecture Search: Classical regressors (graph neural networks, random forests) are trained to predict quantum circuit fidelity or test accuracy directly from features or circuit graphs. These surrogates enable rapid evaluation within automated quantum architecture search (QAS), such as SQuASH or GNN-guided Bayesian optimization, offering orders-of-magnitude speedups in candidate evaluation and prototype iteration (Martyniuk et al., 7 Jun 2025, Choudhary et al., 10 Dec 2025).
• Direct Quantum Surrogates for Physics/QML: In contexts requiring direct circuit-level surrogates, such as explicit quantum models for quantum kernel SVMs (Nakayama et al., 2024) or parameterized unitaries for non-unitary dynamical maps (Lăcătuş et al., 16 Jul 2025), the SQC is trained (classically or via hybrid quantum-classical optimization) to approximate a target operator, enforcing physical constraints such as conservation laws, symmetries, or kernel observables.

Examples of representative approaches across domains are outlined in the table below:

Domain	SQC Role	Reference
Variational optimization	Hessian estimation, line search	(Gustafson et al., 2024)
Circuit learning (QAOA/VQE)	RBF surrogate of hardware cost	(O'Leary et al., 8 Jan 2025)
Architecture search	GNN/RF surrogate for performance	(Martyniuk et al., 7 Jun 2025, Choudhary et al., 10 Dec 2025)
Quantum kernel models	Explicit circuit surrogate (EQS)	(Nakayama et al., 2024)
Quantum CFD (BGK collision)	Trained local unitary for collision	(Lăcătuş et al., 16 Jul 2025)
Hybrid neural operators	Quantum-circuit mixer in FNO	(Papierz et al., 6 Apr 2026)

3. Detailed Algorithmic Examples

Surrogate Optimization of Variational Quantum Circuits

A representative workflow is as follows (Gustafson et al., 2024):

1. Surrogate Pre-optimization: Classical optimizer finds initial $\theta^{(0)}$ on a noise-free state-vector surrogate.
2. Computing Hessian: Surrogate simulator is used to compute the full Hessian via central finite differences; diagonalization yields conjugate directions.
3. Line Search: For each significant positive-curvature direction, a window is established; parallelized line searches (each with $M$ sampled points) are run on hardware or high-level simulator.
4. Parameter Update: The parameter vector is updated along these directions using polynomial fits to the sampled data.
5. Convergence: Iterate until cost or parameter changes are below tolerance thresholds.

This approach reduces the number of noisy quantum function calls by a factor of 2–4× compared to standard optimizers and is compatible with parallel quantum resource allocation. The hardware implementation on 40 qubits of IBM hardware demonstrated robust convergence in three SQC macro-iterations under realistic noise (Gustafson et al., 2024).

Online RBF Surrogates for Quantum Circuit Learning

In this setting, surrogate $s(\theta)$ is an RBF interpolator trained on a growing set of $(\theta,f(\theta))$ pairs. At each iteration, the surrogate's minimizer becomes the next hardware query. No uncertainty modeling or pre-training is required, and the approach is highly shot-efficient. Benchmarks on the QAOA problem for Max-Cut and random Ising models up to 127 qubits show that this approach outperforms state-of-the-art GP-based optimizers and achieves improved parameter transferability (O'Leary et al., 8 Jan 2025).

GNN Surrogates for Circuit Architecture Search

Circuits are encoded as graphs with node and edge features, capturing gate types, timing, and qubit participation. A graph neural network surrogate is trained to predict performance metrics, including accuracy and fidelity. During Bayesian optimization, candidate circuits are ranked by expected improvement times a cost penalty, incorporating surrogate mean and uncertainty estimates (from MC dropout). Empirical results indicate faster convergence and significant wall-clock acceleration over classic MLP-based surrogates (Choudhary et al., 10 Dec 2025, Martyniuk et al., 7 Jun 2025).

Explicit Quantum Surrogate Circuits (EQS) for Quantum Kernels

This methodology turns an implicit quantum-kernel model into a single explicit n-qubit quantum circuit: the kernel observable is spectrally decomposed, the top eigenvectors are mapped to the computational basis by a gate-efficient circuit, and predictions are made by evaluating this surrogate circuit. The EQS construction largely mitigates barren plateaus due to deterministically structured initialization (Nakayama et al., 2024).

Surrogate Quantum Circuits for Non-Unitary Physics (QLBM BGK Example)

A learned 4-qubit unitary, structured as a stack of 15 parameterized blocks of single-qubit rotations and all-to-all CNOT entanglers, approximates the non-unitary BGK collision operator at a single lattice site. The SQC is trained to ensure mass and momentum conservation, D8 and scale equivariance, and data-driven accuracy. No ancillae or post-selection are needed; the circuit depth is independent of lattice size. On fluid benchmarks (Taylor–Green vortex, lid-driven cavity), the SQC accurately reproduces physical metrics within a few percent of the ideal result. The circuit compiles to 2,430 IBM Heron-native gates (Lăcătuş et al., 16 Jul 2025).

4. Surrogate Model Architectures and Encoding

SQC surrogates span both classical and quantum implementations:

• Classical surrogates: Graph neural networks for circuit topology, random forests on instruction sequences, RBF/GP models over parameter space; input featurizations include QASM3-to-graph conversion (for GNNs), flattened instruction lists, or convex combinations of architectures and performance metrics (Martyniuk et al., 7 Jun 2025, Choudhary et al., 10 Dec 2025, O'Leary et al., 8 Jan 2025).
• Quantum surrogates: Parameterized circuits with rotation and entangling layers, designed to match desired operator eigenbasis or enforce physical conservation laws, or as VQC mixers within neural operator architectures (Lăcătuş et al., 16 Jul 2025, Papierz et al., 6 Apr 2026, Nakayama et al., 2024).
• Loss and constraint enforcement: Training loss functions blend data-driven accuracy (mean-squared error or fidelity) with physical constraint penalties, e.g., mass/momentum conservation, equivariance under symmetry groups, or regularization for overfitting/barren plateaus.

5. Empirical Performance and Applications

SQC methods have demonstrated significant gains in:

• Resource efficiency: Function-call and wall-clock speedups of 2–25× in VQE and architecture search relative to direct quantum evaluation or classical benchmarks, confirmed across molecular, combinatorial, and QML tasks (Gustafson et al., 2024, Martyniuk et al., 7 Jun 2025, Choudhary et al., 10 Dec 2025).
• Scalability: Successful deployment up to 127-qubit QAOA problems with minimal hyperparameter tuning, and maintenance of solution quality under hardware noise profiles (O'Leary et al., 8 Jan 2025).
• Fidelity and physical accuracy: Explicit circuit surrogates for physics (BGK collision) achieve sub-percent errors on benchmark flows and enforce exact symmetry and conservation constraints, with circuit depth and resource footprint amenable to NISQ-era experimentation (Lăcătuş et al., 16 Jul 2025).
• Generalization and regularization: EQS approaches show near-identical generalization performance to full kernel-based SVM models while mitigitating barren plateau phenomena via deterministic circuit initialization (Nakayama et al., 2024).
• Hybrid quantum-classical modeling: VQC-mixer-based neural operators reduce parameter count by 15.6% and relative error by up to 26% compared to classical architectures, with an optimal trade-off between quantum and classical resources discovered via parameter sweeps (Papierz et al., 6 Apr 2026).

6. Limitations and Future Directions

Key challenges and open questions in SQC research include:

• Extension to generalized operators: Generalizing SQC design beyond D2Q9 (as in QLBM BGK) to higher-order or higher-dimensional physical models requires novel encoding strategies and deeper circuits, which may be challenging for current hardware (Lăcătuş et al., 16 Jul 2025).
• Integration with noise-aware training: Most current SQC pipelines train on noiseless surrogates or simulators; robust empirical validation on real quantum hardware, including noise-aware surrogate construction and error-mitigation integration, remains active research.
• Theoretical understanding of surrogate-induced error: While empirical results are strong, formal bounds on surrogate-induced bias and convergence rates, especially in iterative schemes, are not yet fully characterized.
• Resource-optimal hybrid architectures: In hybrid neural operators, ablations confirm that mode-sharing is the dominant principle for parameter efficiency; further exploration of quantum advantages (beyond compactness of VQC mixing) is necessary to justify quantum allocation versus classical controls (Papierz et al., 6 Apr 2026).
• Limits of transferability and generalization: The scalability and transfer of SQC-optimized parameters (for example, in QAOA) across problem instances and noise regimes are promising, but systematic studies—especially for "out-of-distribution" problems—are needed (O'Leary et al., 8 Jan 2025).
• End-to-end quantum implementations: While SQC surrogates have been validated in component steps (e.g., collision or spectral mixing), full end-to-end quantum pipelines incorporating both surrogate and non-surrogate steps remain an open target (Lăcătuş et al., 16 Jul 2025, Papierz et al., 6 Apr 2026).

7. Comparative Summary and Practical Impact

Surrogate Quantum Circuits—whether implemented as trained quantum unitaries or as classical regression/approximation models—enable practical, scalable, and resource-efficient quantum algorithm development, optimization, and benchmarking under current hardware constraints. The SQC framework brings together classical simulation, machine learning, and quantum hardware execution to realize (i) efficient variational optimization, (ii) scalable quantum circuit architecture search, (iii) explicit quantum surrogates for otherwise implicit models, and (iv) compact quantum-classical hybrid operators for scientific computing.

By strictly adhering to resource and hardware constraints, enforcing physical symmetries, and enabling rapid iteration, the SQC paradigm is central in advancing near-term quantum computing, with validated empirical performance across chemistry, combinatorial optimization, machine learning, and fluid dynamics on both simulated and actual quantum devices (Gustafson et al., 2024, O'Leary et al., 8 Jan 2025, Martyniuk et al., 7 Jun 2025, Nakayama et al., 2024, Choudhary et al., 10 Dec 2025, Lăcătuş et al., 16 Jul 2025, Papierz et al., 6 Apr 2026).