Adaptive Circuit Knitting (ACK)

• Adaptive Circuit Knitting (ACK) is a quantum circuit partitioning technique that divides large circuits into hardware-realizable subcircuits using wire and gate cuts.
• ACK employs quasiprobability decomposition and adaptive, measurement-informed protocols to reduce sampling overhead and mitigate error propagation.
• ACK integrates hardware-aware strategies, entanglement resource engineering, and variational algorithm adaptations to enhance scalability and execution fidelity.

Adaptive Circuit Knitting (ACK) is a class of quantum circuit partitioning techniques designed to enable the scalable execution of large quantum algorithms by dividing them into smaller, hardware-realizable subcircuits and knitting the results together through classical communication, quasiprobability postprocessing, and entanglement-resource engineering. ACK generalizes fundamental circuit knitting strategies by introducing adaptive, measurement-informed protocols and error management mechanisms to optimize simulation overhead, resource utilization, and fidelity across distributed or resource-limited architectures.

1. Conceptual Foundations of Adaptive Circuit Knitting

ACK builds on the general notion of circuit knitting, which partitions a composite quantum circuit into smaller subcircuits such that each can be mapped onto available quantum hardware. The classical reconstruction phase synthesizes the outputs of each fragment into the final result, typically at the cost of increased sampling overhead related to the number and type of cuts introduced in the circuit topology (Brenner et al., 2023, Piveteau et al., 2022).

The partitioning modalities primarily fall into two categories:

• Wire cuts: Splitting the circuit along qubit trajectories, breaking quantum wires and enabling subcircuits that process partial quantum information.
• Gate cuts: Cutting through entangling gates, replacing nonlocal operations with simulated channels or resource states, often necessitating additional ancillas or error mitigation.

Quasiprobability decomposition (QPD) underpins ACK's theoretical foundation. Nonlocal unitaries or channels $\mathcal{U}$ are expanded as $$\mathcal{U} = \sum_k \alpha_k \mathcal{E}_k^1 \otimes \dots \otimes \mathcal{E}_k^N$$, where the cost is quantified by the overhead $\omega = (\sum_k |\alpha_k|)^2$. The sampling overhead, which grows exponentially with the number of cuts, fundamentally limits the scalability of circuit knitting (Brenner et al., 2023, Jing et al., 4 Apr 2024).

Adaptive protocols exploit measurement outcomes and classical communication to guide correction operations and post-select optimal reconstruction paths, allowing adaptive corrections to minimize overhead and mitigate errors arising from nonidealities (Brenner et al., 2023, Piveteau et al., 2022).

2. Entanglement Resource Engineering and Error Bounds

A central ACK insight is the explicit link between the sampling overhead required for reconstructing a nonlocal operation and the entanglement cost of the simulated channel (Jing et al., 4 Apr 2024). For a bipartite channel $\mathcal{N}$, the regularized sampling overhead $\gamma^{(\infty)}(\mathcal{N})$ is lower-bounded by the exponential of the exact entanglement cost $E_{C,0}(\mathcal{N})$: $$\gamma^{(\infty)}(\mathcal{N}) \geq 2^{E_{C,0}(\mathcal{N})}$$.

This bound, established via resource theory, implies that for highly entangling channels, the required number of Monte Carlo samples for virtual simulation via QPD becomes exponentially prohibitive. Benchmark metrics such as the $\kappa$-entanglement and max-Rains information serve as efficiently computable proxies for evaluating this overhead. These benchmarks directly connect ACK performance with quantum Shannon-theoretic constraints—the inherent entanglement in quantum information processing sets a hard limitation on the scalability of ACK (Jing et al., 4 Apr 2024).

3. Classical Communication and Overhead Reduction

Allowing classical communication between subcircuits can radically reduce the simulation overhead associated with cuts. In the absence of classical communication (LO operations only), the overhead for $n$ nonlocal CNOT gates is $O(9^n)$, but incorporating LOCC protocols enables bundled gate teleportation schemes that reduce this to $O(4^n)$ through submultiplicative bundling (Piveteau et al., 2022). Similar effects are shown for wire cuts, with $O(16^n)$ scaling reduced to $O(4^n)$ when adaptive classical communication is employed (Brenner et al., 2023).

The underlying mechanism is the joint simulation of multiple nonlocal resources, in which entangled states are shared across subcircuits and measurement-driven corrections are applied based on exchanged classical messages, optimizing the stitching process. While the per-gate overhead for Clifford gates may not decrease for a single gate, joint protocols yield improved asymptotic scaling.

4. Adaptive Methods in Variational Algorithms and Circuit Dynamics

ACK extends naturally to the simulation of variational quantum algorithms and quantum dynamics (Gentinetta et al., 2023, Wu et al., 5 Aug 2025). In these settings, the circuit's ansatz is partitioned such that entangling gates—especially those connecting weakly coupled blocks—are cut via QPD and classical postprocessing.

A constrained optimization framework enforces an upper bound $\tau$ on the multiplicative sampling overhead due to inter-block gate cuts, ensuring the viability of the simulation. In variational quantum dynamics, the projected variational quantum dynamics (PVQD) algorithm is employed to drive the evolution, and only weakly entangling inter-block terms are cut to keep the system tractable (Gentinetta et al., 2023). For variational quantum algorithms in the CKVQA framework, a quantum architecture search (QAS) jointly optimizes for expressibility and low sampling overhead, yielding superior trade-offs between efficiency and solution accuracy (Wu et al., 5 Aug 2025).

Subcircuit-level optimization strategies further accelerate training by updating parameters only within affected subcircuits, decoupling classical optimization from global circuit execution and reducing computational costs.

5. Hardware-Aware Partitioning and Scalability

An advanced dimension of ACK is the incorporation of hardware-aware strategies into the cutting algorithm (Ren et al., 5 Sep 2024). Rather than minimizing sampling overhead alone, the optimal partitioning process is co-optimized alongside physical hardware constraints. The partitioning algorithm dynamically considers the target hardware topology through graph similarity heuristics, minimizing SWAP gate insertion and circuit depth.

The overall cost function combines the number of cuts (directly related to sampling overhead) and the hardware routing cost (quantified via graph edit distance), tuned through a parameter $\alpha$ that scales with two-qubit gate density. Recursive edge contraction algorithms (inspired by Karger-Stein methods) enable polynomial-time exploration of partitioning solutions, with performance demonstrated as up to 64% subcircuit depth reduction and up to 2.7$\times$ fidelity enhancement compared to non-hardware-aware baseline approaches.

6. Circuit Folding, Modular Management, and Resource Efficiency

ACK frameworks such as CiFold introduce modular and qubit-level workload management by identifying repeated structures across quantum circuits and folding them into parallel modules (Kan et al., 24 Dec 2024). This folding is realized through dynamic programming (e.g., LCCS algorithms), meta-graph construction, and edge-growth partitioning. Resource overhead—quantified via the Quantum Resource Overhead (QRO) metric—is substantially reduced by consolidating repeated modules, improving both classical reconstruction efficiency and quantum resource utilization.

Extensive benchmarking shows reductions in quantum resource usage of up to 799.2% against previous approaches. The method maintains stable QRO as circuit width increases and enhances observable value accuracy under noise models, making the modular ACK approach scalable for circuits up to 190 qubits in practical scenarios.

7. Extension to Continuous Variables and Hybrid Architectures

ACK has been further generalized to infinite-dimensional, continuous-variable quantum systems. The QPD-based simulation framework accommodates non-Gaussian resource state generation, such as GKP and cat states, by expanding the target state in terms of available Gaussian or cat states (Hu et al., 9 Sep 2025). However, a no-go theorem demonstrates that nonlocal Gaussian operations cannot be exactly knitted using only separable operations—any classical communication protocol in CV settings leads to infinite sampling overhead unless the unitary is strictly local.

Advanced techniques such as unitary-transformed projective squeezing extend ACK to hybrid architectures (qubits-coupled-to-bosonic-modes), enabling noise-resilient projection onto highly squeezed or non-Gaussian resource states. Algorithms like linear-combination-of-unitaries (LCU) and virtual quantum error detection (VQED) leverage these architectures for efficient state distillation and error mitigation (Anai et al., 29 Nov 2024).

The QPD framework in CV scenarios provides mechanisms for approximate simulation, with explicit formulas governing sampling overhead and error scaling, and suggests adaptability for measurement-informed strategies, though with heightened classical cost.

---

Adaptive Circuit Knitting synthesizes circuit partitioning, quasiprobability simulation, entanglement-resource engineering, hardware-awareness, error mitigation, and classical communication into a cohesive framework for scalable, high-fidelity quantum algorithm execution across near-term and distributed quantum architectures. While ACK offers substantial overhead reduction, improved resilience against errors, and practical resource management, its performance is fundamentally bounded by entanglement-theoretic limits and sampling overhead scaling, particularly in high-entanglement or infinite-dimensional regimes. Its continued development integrates advanced optimization, modular compression, and hybrid-system algorithms, positioning ACK as a critical component in the roadmap for practical quantum computing.