Circuit Cutting in Quantum Circuits

• Circuit cutting is a hybrid quantum–classical technique that partitions a large circuit into smaller fragments for independent execution and classical recombination.
• It uses spatial gate cuts to reduce circuit width and temporal wire cuts to reduce circuit depth, addressing NISQ constraints such as limited qubit count and coherence.
• Advanced methods integrate optimization, classical shadows, and error mitigation to lower overhead and improve reconstruction accuracy.

Circuit cutting is a hybrid quantum–classical technique in which a circuit that is too large, too deep, or too nonlocal for the available hardware is partitioned into smaller fragments, the fragments are executed independently, and the target quantity for the original circuit is recovered by classical post-processing (Yang et al., 2024). It was originally designed for expectation-value estimation, but the literature now spans spatial gate cuts for distributed execution, temporal wire cuts for sequential execution, full-output reconstruction, classical-shadow prediction, error mitigation, near-Clifford simulation, and direct sampling from trained variational circuits (Cambiucci et al., 12 Apr 2025, Wagner et al., 9 Jul 2025).

1. Foundations and cut models

Circuit cutting is motivated by the standard NISQ constraints: limited qubit count, limited coherence and depth, imperfect gate fidelity, and, in distributed settings, expensive inter-device communication (Cambiucci et al., 12 Apr 2025). In the most common taxonomy, spatial circuit cutting is the gate-cut approach that reduces circuit width by partitioning qubits across multiple QPUs and cutting inter-partition gates, whereas temporal circuit cutting is the qubit-wire-cut approach that reduces circuit depth by partitioning operations into sequential segments while keeping the same width (Cambiucci et al., 12 Apr 2025). The literature also distinguishes ancilla-free and ancilla-assisted wire-cutting models, and with or without classical communication between fragments (Li et al., 2024, Bechtold et al., 2023).

The object being reconstructed depends on the application. In the canonical setting, cutting estimates an observable of the original circuit. In more recent work, the target may instead be the full output distribution, a family of observables reused through classical shadows, or the bitstring distribution of a trained QAOA ansatz used for optimization (Perlin et al., 2020, Chen et al., 2022, Wagner et al., 9 Jul 2025). That distinction is substantive: recovering an expectation value is a linear task, whereas recovering a usable sample distribution is sensitive to distributional distortion, tail behavior, and finite-shot regularization.

The cut itself may be imposed on a wire, on a gate, or on a structured interface derived from a graph or hypergraph model of the circuit. Hypergraph-based formulations make this explicit by using a primal hypergraph for spatial cutting and a dual hypergraph for temporal cutting, so that the same partitioning machinery targets different physical costs: communication across QPUs in the spatial case, and qubit initialization or reset overhead in the temporal case (Cambiucci et al., 12 Apr 2025).

2. Mathematical formalisms and reconstruction overhead

A standard quasiprobabilistic formulation expresses a nonlocal channel as a signed combination of local or communication-assisted channels,

$\mathcal U_{AB} = \sum_{i=1}^m a_i \mathcal F_i,$

and measures cost through the quasiprobability extent

$$\gamma_S(\mathcal E) \coloneqq \inf\Big\{ \sum_{i=1}^m |a_i| : \mathcal E = \sum_{i=1}^m a_i \mathcal F_i,\; \mathcal F_i\in S,\; a_i\in\mathbb R \Big\},$$

with single-cut sampling overhead factor $\gamma_S(\mathcal E)^2$ (Piveteau et al., 28 Mar 2025). In the corresponding Monte Carlo view, the number of shots scales as

$$\sim \frac{2 a_1^2}{\epsilon^2}\log\!\left(\frac{2}{\delta}\right),$$

so the $\ell_1$-norm of the reconstruction coefficients is the central cost parameter (Piveteau et al., 28 Mar 2025).

For wire cutting, a widely used Pauli-basis identity is

$$\rho \simeq \frac{1}{2}\sum_{M\in\{I,X,Y,Z\}} M \otimes \operatorname{tr}_n(M_{(n)}\rho),$$

which underlies the separation of upstream measurement and downstream preparation across a cut wire (Liu et al., 2022). CutQC-style exact wire cutting generalizes this to $k$ cuts through

$$\rho=\frac{1}{2^k}\sum_{M\in\mathcal B^k}\bigotimes_{j=1}^{N_f}\rho_{f_j}(M(f_j)),$$

but the number of fragment configurations grows exponentially with the number of cut interfaces (Li et al., 2024).

Several later papers sharpen the overhead theory. Randomized-measurement wire cutting introduces the identity-channel decomposition

$$\mathrm{id} = (2d+1)\,\mathbb E_z\!\left[(-1)^z \Psi_z\right]$$

for a $d=2^k$ dimensional cut subsystem, yielding $$\gamma_S(\mathcal E) \coloneqq \inf\Big\{ \sum_{i=1}^m |a_i| : \mathcal E = \sum_{i=1}^m a_i \mathcal F_i,\; \mathcal F_i\in S,\; a_i\in\mathbb R \Big\},$$0 sample complexity and an information-theoretic lower bound of $$\gamma_S(\mathcal E) \coloneqq \inf\Big\{ \sum_{i=1}^m |a_i| : \mathcal E = \sum_{i=1}^m a_i \mathcal F_i,\; \mathcal F_i\in S,\; a_i\in\mathbb R \Big\},$$1 for comparable wire-cutting procedures (2207.14734). For gate cutting, the optimal overhead for an arbitrary two-qubit unitary with KAK coefficients $$\gamma_S(\mathcal E) \coloneqq \inf\Big\{ \sum_{i=1}^m |a_i| : \mathcal E = \sum_{i=1}^m a_i \mathcal F_i,\; \mathcal F_i\in S,\; a_i\in\mathbb R \Big\},$$2 is

$$\gamma_S(\mathcal E) \coloneqq \inf\Big\{ \sum_{i=1}^m |a_i| : \mathcal E = \sum_{i=1}^m a_i \mathcal F_i,\; \mathcal F_i\in S,\; a_i\in\mathbb R \Big\},$$3

and for $$\gamma_S(\mathcal E) \coloneqq \inf\Big\{ \sum_{i=1}^m |a_i| : \mathcal E = \sum_{i=1}^m a_i \mathcal F_i,\; \mathcal F_i\in S,\; a_i\in\mathbb R \Big\},$$4 identical copies,

$$\gamma_S(\mathcal E) \coloneqq \inf\Big\{ \sum_{i=1}^m |a_i| : \mathcal E = \sum_{i=1}^m a_i \mathcal F_i,\; \mathcal F_i\in S,\; a_i\in\mathbb R \Big\},$$5

which is strictly cheaper than cutting the gates independently; the same work shows no advantage from classical communication for arbitrary two-qubit unitary cuts in this setting (Schmitt et al., 2023).

The role of side information has also been formalized through quantum instruments. In that framework, side information enlarges the operational decomposition set from channels to signed combinations of instrument branches, but its value depends strongly on the free-operation class: it is essential for feasibility in the no-communication LO setting, gives no extent reduction for states, and gives no advantage in SEP or PPT for Hermitian-preserving trace-preserving targets; whether $$\gamma_S(\mathcal E) \coloneqq \inf\Big\{ \sum_{i=1}^m |a_i| : \mathcal E = \sum_{i=1}^m a_i \mathcal F_i,\; \mathcal F_i\in S,\; a_i\in\mathbb R \Big\},$$6 can occur remains open (Piveteau et al., 28 Mar 2025).

3. Partitioning, placement, and decomposition-aware optimization

Because overhead grows rapidly with the cut interface, cut placement is itself an optimization problem. A general architecture-aware strategy represents the circuit as a hypergraph $$\gamma_S(\mathcal E) \coloneqq \inf\Big\{ \sum_{i=1}^m |a_i| : \mathcal E = \sum_{i=1}^m a_i \mathcal F_i,\; \mathcal F_i\in S,\; a_i\in\mathbb R \Big\},$$7, with qubits as vertices and multi-qubit gates as hyperedges in the primal representation, then uses the dual hypergraph $$\gamma_S(\mathcal E) \coloneqq \inf\Big\{ \sum_{i=1}^m |a_i| : \mathcal E = \sum_{i=1}^m a_i \mathcal F_i,\; \mathcal F_i\in S,\; a_i\in\mathbb R \Big\},$$8 when the goal is temporal wire cutting (Cambiucci et al., 12 Apr 2025). On top of this representation, Stoer–Wagner, Kernighan–Lin, and especially Fiduccia–Mattheyses heuristics are used to seek balanced bipartitions with few crossed hyperedges; the same work introduces the coupling base

$$\gamma_S(\mathcal E) \coloneqq \inf\Big\{ \sum_{i=1}^m |a_i| : \mathcal E = \sum_{i=1}^m a_i \mathcal F_i,\; \mathcal F_i\in S,\; a_i\in\mathbb R \Big\},$$9

and coupling ratio

$\gamma_S(\mathcal E)^2$0

as structural indicators of cut difficulty (Cambiucci et al., 12 Apr 2025).

Problem-graph-aware placement is particularly prominent for QAOA. In separator-based QAOA cutting for MaxCut, the minimum number of wire cuts per layer is tied to the cardinality of a minimum vertex separator of the problem graph, so the preprocessing problem becomes finding a low-cardinality balanced vertex separator, then shrinking it before any quantum decomposition is performed (Wagner et al., 9 Jul 2025). For $\gamma_S(\mathcal E)^2$1 QAOA, the reported workflow shrinks the separator to a single vertex, producing exactly two wire cuts, one per layer, and this cut minimization is crucial because reconstruction overhead “increases exponentially with the number of cuts and quickly becomes prohibitive if circuit cutting is performed naively” (Wagner et al., 9 Jul 2025). In a related randomized-measurement analysis, a balanced vertex separator of size $\gamma_S(\mathcal E)^2$2 yields QAOA simulation overhead roughly $\gamma_S(\mathcal E)^2$3 (2207.14734).

Distributed systems place additional pressure on placement quality. FitCut converts the circuit DAG into a weighted graph whose vertices are two-qubit gates and whose edge weights count shared qubits, then applies constrained community detection and a capacity-aware merge heuristic (Kan et al., 2024). Its primary objective proxies cut count through

$\gamma_S(\mathcal E)^2$4

while a secondary objective maximizes worker utilization through idle-qubit penalties. In the reported experiments, FitCut reduces cut-search time by factors ranging from $\gamma_S(\mathcal E)^2$5 to $\gamma_S(\mathcal E)^2$6 relative to Qiskit CKT and improves worker-side utilization by up to $\gamma_S(\mathcal E)^2$7 times, with a system-wide improvement of $\gamma_S(\mathcal E)^2$8 times (Kan et al., 2024).

A separate line of work observes that decomposition and cut placement should not be treated independently. For MCX and CCCX gates, partition-aware decompositions using a small number of ancillas can compress control information crossing the partition so that a multi-control gate is replaced by one or two cut CNOTs rather than many cross-boundary two-qubit gates (Tamura et al., 27 Mar 2026). The strongest reported variant, dec2Ad, lowers the asymptotic sampling overhead from $\gamma_S(\mathcal E)^2$9 in prior direct multi-control cutting to $$\sim \frac{2 a_1^2}{\epsilon^2}\log\!\left(\frac{2}{\delta}\right),$$0 without classical communication and $$\sim \frac{2 a_1^2}{\epsilon^2}\log\!\left(\frac{2}{\delta}\right),$$1 with classical communication, at the price of two ancillas and dirty-ancilla semantics (Tamura et al., 27 Mar 2026).

4. Execution-level optimization and measurement reduction

Once a cut plan is fixed, a second layer of optimization targets the fragment execution burden itself. One example is ancilla-assisted, commuting-group wire cutting: quantum input initializations are converted into output measurements using ancillas, and the resulting Pauli observables are grouped into commuting families via MUBs-based grouping (Li et al., 2024). For a fragment with $$\sim \frac{2 a_1^2}{\epsilon^2}\log\!\left(\frac{2}{\delta}\right),$$2 quantum inputs and $$\sim \frac{2 a_1^2}{\epsilon^2}\log\!\left(\frac{2}{\delta}\right),$$3 quantum outputs, this reduces the number of required fragment executions from

$$\sim \frac{2 a_1^2}{\epsilon^2}\log\!\left(\frac{2}{\delta}\right),$$4

to

$$\sim \frac{2 a_1^2}{\epsilon^2}\log\!\left(\frac{2}{\delta}\right),$$5

when $$\sim \frac{2 a_1^2}{\epsilon^2}\log\!\left(\frac{2}{\delta}\right),$$6, while preserving exact reconstruction in the CutQC-style wire-cutting model (Li et al., 2024).

Another strategy is to optimize the basis in which the cut state is represented. RICCO introduces a parameterized unitary at the cut so that, after optimization, many boundary observables vanish and the upstream fragment can be reconstructed from fewer measurement settings (Uchehara et al., 2022). In the reported benchmarks, one cut required $$\sim \frac{2 a_1^2}{\epsilon^2}\log\!\left(\frac{2}{\delta}\right),$$7 circuit executions for QCUT and $$\sim \frac{2 a_1^2}{\epsilon^2}\log\!\left(\frac{2}{\delta}\right),$$8 for RICCO, while two cuts required $$\sim \frac{2 a_1^2}{\epsilon^2}\log\!\left(\frac{2}{\delta}\right),$$9 for QCUT and $\ell_1$0 for RICCO; however, the optimization overhead itself was large, about $\ell_1$1 circuit executions on average for the two-qubit-cut random-circuit case, and at least $\ell_1$2 the QCUT execution count in the VQE experiment (Uchehara et al., 2022).

A simpler exact pruning rule is to omit basis elements whose total contribution vanishes. “Golden cutting points” are cuts at which one Pauli basis element passes “no information” through the cut, so its downstream fragment executions can be skipped without changing the reconstructed observable (Chen et al., 2023). For $\ell_1$3 golden cuts and $\ell_1$4 regular cuts, the paper reports reconstruction complexity $\ell_1$5 and circuit-evaluation cost $\ell_1$6, with up to $\ell_1$7 wall-time reduction on IBM hardware and no loss of accuracy in the tested instances (Chen et al., 2023).

State-dependent optimization extends this logic further. Because downstream cut fragments begin in known states and upstream fragments are measured in selected observables, state-dependent simplifications become broadly applicable after cutting (Li et al., 6 Jun 2025). The proposed SDO framework combines initial-state dependent optimization, measure-state dependent optimization, and biased observable selection; in noisy simulations of QAOA, QFT, and Bernstein–Vazirani, the method consistently mitigates noise and improves overall circuit performance (Li et al., 6 Jun 2025).

5. Applications and specialized variants

A major branch of the literature uses circuit cutting to reconstruct full output distributions rather than single expectation values. Maximum-likelihood fragment tomography recasts each fragment as a Choi-like tomographic object, projects noisy fragment estimates back to the physical positive semidefinite cone, and then contracts the fragment models into a valid global probability distribution (Perlin et al., 2020). In numerical experiments on clustered random unitary circuits, MLFT consistently improves over direct fragment recombination and can estimate the output of a fragmented circuit with higher fidelity than full execution under the same shot budget (Perlin et al., 2020).

Classical-shadow variants replace full fragment tomography by reusable randomized measurements on fragment Choi states. For observables that factorize across fragments, the divide-and-conquer shadow method gives per-fragment sample complexity

$\ell_1$8

and the numerical results show an advantage over uncut shadow tomography for high-weight observables (Chen et al., 2022). The underlying mechanism is that the exponential dependence on observable weight can be traded for smaller per-fragment operator size, provided the cut overhead remains manageable.

Circuit cutting has also been repurposed for error mitigation. Simulated Quantum Error Mitigation cuts away the Pauli Check Sandwiching mitigation gadget, executes the main ansatz on hardware, simulates the small correction fragment classically, and recombines qubitwise-mitigated distributions into a full output distribution (Liu et al., 2022). In hardware experiments on IBM Auckland with a two-local SU(2) VQE ansatz, the Bayesian-recombined SQEM energies were $\ell_1$9 vs $$\rho \simeq \frac{1}{2}\sum_{M\in\{I,X,Y,Z\}} M \otimes \operatorname{tr}_n(M_{(n)}\rho),$$0 for LiH, $$\rho \simeq \frac{1}{2}\sum_{M\in\{I,X,Y,Z\}} M \otimes \operatorname{tr}_n(M_{(n)}\rho),$$1 vs $$\rho \simeq \frac{1}{2}\sum_{M\in\{I,X,Y,Z\}} M \otimes \operatorname{tr}_n(M_{(n)}\rho),$$2 for HF, and $$\rho \simeq \frac{1}{2}\sum_{M\in\{I,X,Y,Z\}} M \otimes \operatorname{tr}_n(M_{(n)}\rho),$$3 vs $$\rho \simeq \frac{1}{2}\sum_{M\in\{I,X,Y,Z\}} M \otimes \operatorname{tr}_n(M_{(n)}\rho),$$4 for BeH$$\rho \simeq \frac{1}{2}\sum_{M\in\{I,X,Y,Z\}} M \otimes \operatorname{tr}_n(M_{(n)}\rho),$$5, all relative to the corresponding noiseless ansatz values (Liu et al., 2022).

For classical simulation, circuit cutting can be combined with other structure. SuperSim isolates non-Clifford operations using cuts so that the large resulting fragments are mostly Clifford and can be simulated with Stim, while the residual non-Clifford fragments are handled exactly (Smith et al., 2023). In the reported HWEA benchmarks, a runtime crossover appears around the mid-20-qubit range, and the framework scales to hundreds of qubits with modest runtimes on suitable near-Clifford instances (Smith et al., 2023).

The most conceptually distinct application is pure sampling from trained variational circuits. For depth-$$\rho \simeq \frac{1}{2}\sum_{M\in\{I,X,Y,Z\}} M \otimes \operatorname{tr}_n(M_{(n)}\rho),$$6 QAOA on MaxCut, circuit cutting was applied not to expectation-value evaluation but to final bitstring sampling (Wagner et al., 9 Jul 2025). In noiseless simulation, the reconstructed distribution is broadened and shifted toward suboptimal objective values; the paper gives a per-bitstring suppression guarantee of at most $$\rho \simeq \frac{1}{2}\sum_{M\in\{I,X,Y,Z\}} M \otimes \operatorname{tr}_n(M_{(n)}\rho),$$7, so $$\rho \simeq \frac{1}{2}\sum_{M\in\{I,X,Y,Z\}} M \otimes \operatorname{tr}_n(M_{(n)}\rho),$$8 additional samples compensate the worst-case probability loss (Wagner et al., 9 Jul 2025). On IBM hardware, however, the 25-node instance shows the opposite effect: the reduced fragment width and depth lower hardware noise enough that the cut distribution is shifted closer to the optimum than the uncut distribution, making the relevant trade-off “distribution distortion versus noise mitigation” rather than the older “exactness versus overhead” framing (Wagner et al., 9 Jul 2025).

6. Scalability, systems perspective, and open questions

Resource-estimation studies consistently show that circuit cutting is not a universally scalable substitute for larger hardware. A broad fault-tolerant benchmarking study using Qiskit cutting and Azure Quantum Resource Estimator reports that cutting can reduce physical-qubit requirements by about $$\rho \simeq \frac{1}{2}\sum_{M\in\{I,X,Y,Z\}} M \otimes \operatorname{tr}_n(M_{(n)}\rho),$$9 on average, but for realistic Hamiltonian simulation, QFT, QPE, QAOA, block encoding, and random circuits, the exponential growth of quantum runtime and classical recombination usually dominates (Yang et al., 2024). Representative figures include QFT on 10 qubits, where the baseline runtime is $k$0 s and the cut runtime is $k$1 s, and Ising $k$2, where the baseline is $k$3 s and the cut runtime is $k$4 s, despite substantial qubit reductions (Yang et al., 2024).

A systems view sharpens that conclusion. In distributed QNN training, a cut-aware estimator call decomposes into partitioning, subexperiment generation, execution, and reconstruction,

$k$5

and the observed bottleneck is often reconstruction rather than fragment execution (Singh et al., 18 Feb 2026). The reported median reconstruction shares are $k$6, $k$7, and $k$8 for one, two, and three cuts, respectively, with poor parallel scaling and speed-up often near $k$9 or below it; nevertheless, test accuracy and robustness are preserved in the measured regimes (Singh et al., 18 Feb 2026). This suggests that asymptotic cut-count analysis alone is insufficient for estimator-driven workloads.

MAESTROCUT pushes this systems interpretation further by treating circuit cutting as an online control problem over partitioning, shot allocation, estimator choice, and secure dispatch (Punch et al., 31 Aug 2025). Its partitioner optimizes

$$\rho=\frac{1}{2^k}\sum_{M\in\mathcal B^k}\bigotimes_{j=1}^{N_f}\rho_{f_j}(M(f_j)),$$0

re-partitions under a CUSUM drift trigger, and allocates shots by the closed-form rule

$$\rho=\frac{1}{2^k}\sum_{M\in\mathcal B^k}\bigotimes_{j=1}^{N_f}\rho_{f_j}(M(f_j)),$$1

derived from a topology-aware variance surrogate (Punch et al., 31 Aug 2025). Tier-2 emulation reports stable jitter, high reliability, and about $$\rho=\frac{1}{2^k}\sum_{M\in\mathcal B^k}\bigotimes_{j=1}^{N_f}\rho_{f_j}(M(f_j)),$$2 software overhead for the confidentiality layer under stress scenarios (Punch et al., 31 Aug 2025). The underlying implication is that practical cutting on cloud-era NISQ hardware may depend as much on runtime orchestration as on the reconstruction identity itself.

Several open questions remain structural rather than implementational. The power of side information is still unresolved in the physically important LOCC setting (Piveteau et al., 28 Mar 2025). For QAOA, simple “classical cutting” by enumerating separator assignments can be cheaper than layer-by-layer quantum cutting, especially as $$\rho=\frac{1}{2^k}\sum_{M\in\mathcal B^k}\bigotimes_{j=1}^{N_f}\rho_{f_j}(M(f_j)),$$3 grows (Wagner et al., 9 Jul 2025). More generally, the literature repeatedly returns to the same barrier: the number of cuts, the size of the cut interface, and the reconstruction layer jointly determine whether cutting functions as a practical scaling tool, a noise-mitigation tool, a classical-simulation aid, or merely an expensive formal reduction. This suggests that the modern subject is no longer just circuit decomposition, but the joint optimization of cut placement, fragment structure, estimator family, shot allocation, and runtime policy.