ShotQC: Enhanced Quantum Circuit Cutting

• ShotQC is an enhanced quantum circuit cutting framework that decomposes large circuits into subcircuits to enable simulation on noisy NISQ devices.
• It employs a two-phase shot allocation strategy with adaptive, variance-driven distribution and flexible cut parameter optimization to minimize quantum resource requirements.
• Benchmarks show up to 19× variance reduction and significant shot savings while maintaining classical postprocessing efficiency despite exponential scaling in the number of cuts.

ShotQC is an enhanced quantum circuit cutting framework developed to address the severe sampling overhead associated with simulating large quantum circuits on distributed noisy intermediate-scale quantum (NISQ) devices. It introduces variance-driven shot allocation and flexible cut parameterization to minimize the number of quantum resources required, while retaining the classical postprocessing structure of prior circuit-cutting methodologies (Chen et al., 2024).

1. Problem Context and Circuit Cutting Formalism

NISQ processors, limited to tens of qubits and significant noise, cannot natively accommodate circuits of practical scale. Circuit cutting enables simulation beyond hardware size by decomposing a large circuit into independent subcircuits: wires are "cut," identity channels are represented as sums over measure-and-prepare channels, subcircuits are executed on hardware, and results are recombined classically. The dominant bottleneck is sampling overhead: with $K$ cuts and Pauli-basis wire decomposition, the number of subcircuit configurations scales as $\ell^K$ ($\ell=4$ or $8$), and the total number of shots required to keep the statistical error fixed increases as $O(16^K)$ in the no-communication setting. This exponential scaling rapidly makes naive circuit cutting impractical (Chen et al., 2024).

2. ShotQC: Core Optimizations

ShotQC achieves sampling overhead reduction through two major algorithmic advances:

2.1 Variance-Driven Shot Distribution

Let $\{e\}$ denote all measure–prepare subcircuit configurations, and $N_e$ the number of shots allocated to configuration $e$. For each output bitstring $i$, the final probability is reconstructed as $p_i = \sum_e c_{i,e} P_e$, with $\ell^K$0 the empirical probability from subcircuit $\ell^K$1.

The total variance (summed over all outcomes) is $\ell^K$2, which can be propagated as

$\ell^K$3

where $\ell^K$4 are variance-propagation coefficients dependent on partial estimates $\ell^K$5 and cut parameterization $\ell^K$6.

The optimal shot allocation to minimize error is achieved by the Cauchy–Schwarz bound:

$\ell^K$7

with equality when $\ell^K$8. This leads to the allocation rule:

$\ell^K$9

Practical implementation is two-phase: an initial fraction $\ell=4$0 of shots is distributed equally to gather coarse $\ell=4$1, then $\ell=4$2 is computed and the remaining shots are adaptively assigned, iteratively refining both $\ell=4$3 and $\ell=4$4 in subsequent segments to mitigate bias from poor initial estimation.

2.2 Cut Parameterization Optimization

Standard Pauli decomposition fixes the identity channel as a sum over measure-and-prepare operations, typically with $\ell=4$5 or $\ell=4$6 terms per cut. ShotQC generalizes this by introducing free parameters $\ell=4$7 within the cut decomposition, subject to completeness constraints (e.g., $\ell=4$8 and similar for other observables and preparations), yielding a $\ell=4$9-dimensional parameter space for each cut.

Optimizing over $8$0 directly targets minimization of the "Loss" function,

$8$1

thereby reducing the minimum achievable sampling overhead at fixed error. Gradient-based and stochastic optimization (Adam, SGD, simulated annealing) are employed, using an initial sample to update $8$2 and feed into subsequent variance estimates and shot distribution.

The flexible choice between $8$3 versus $8$4 for the number of basis elements per cut allows tuning computational trade-offs: higher $8$5 yields larger potential variance reduction but exponentially more classical terms.

3. Complexity and Scaling Analysis

The standard circuit cutting sampling overhead is $8$6. When ShotQC achieves a reduction factor $8$7 in $8$8 (and consequently cumulative variance), the total shot requirement becomes $8$9. In benchmarks, $O(16^K)$0 ranged from $O(16^K)$1 to $O(16^K)$2 ($O(16^K)$3 lower variance on average).

Crucially, ShotQC does not worsen the scaling of classical postprocessing: all additional computations—evaluating $O(16^K)$4, updating $O(16^K)$5, and optimally redistributing shots—require no more Kronecker-product terms than the base method (either $O(16^K)$6 or $O(16^K)$7), and all other steps are polynomial in $O(16^K)$8 and $O(16^K)$9.

4. Stepwise Algorithmic Workflow

The ShotQC procedure is as follows:

1. Partition target circuit with $\{e\}$0 cuts, generating disjoint subcircuits.
2. Enumerate all $\{e\}$1 configurations ($\{e\}$2 being $\{e\}$3 or $\{e\}$4).
3. Assign $\{e\}$5 prior shots equally, obtain coarse probability estimates $\{e\}$6.
4. Optionally, optimize cut parameters $\{e\}$7 to minimize $\{e\}$8.
5. Distribute remaining shots iteratively, allocating per the variance-driven optimal $\{e\}$9 rule, recalculating as new empirical data arrives.
6. Reconstruct global output probabilities by combining all $N_e$0 through classical postprocessing, possibly renormalizing.
7. Return final estimated distribution.

5. Empirical Results and Benchmarks

ShotQC was tested on a suite of established quantum benchmarks, including QAOA circuits on Erdős–Rényi and regular graphs, Google supremacy circuits, ripple-carry adders, and the approximate quantum Fourier transform (AQFT). Multiple ShotQC regimes were examined, varying in whether cut parameter optimization is used (dubbed "economical" versus full) and in chosen $N_e$1.

Findings included:

• Maximum observed variance reduction was $N_e$2 (AQFT, $N_e$3 qubits); the mean across $N_e$4 benchmarks was $N_e$5.
• With $N_e$6 and no cut-parameter optimization, average variance reduction was $N_e$7 with negligible added classical cost.
• $N_e$8 parameterization produced lower minimum variance but higher risk of classical timeouts in large cases.
• Overhead-reduction quantification: for the erdos_22 circuit, baseline needed $N_e$9 shots; with ShotQC Setting B, only $e$0 (2.8$e$1 fewer).

Representative benchmark results (variance normalized; Setting B uses full ShotQC, Setting A is more economical):

Circuit	Baseline	Setting A	Setting B
erdos_22	1.00	0.45	0.42
regular_18	1.00	0.54	0.50
supremacy_14	1.00	0.40	0.38
adder_20	1.00	0.50	0.48
aqft_15	1.00	0.47	0.45

Trade-off studies reveal that the optimal prior-shot ratio is near $e$2–$e$3; segmentation achieves diminishing returns beyond 3–5 iterations; and $e$4 parameter optimization converges more slowly but achieves lower variance.

6. Limitations and Potential Extensions

Although ShotQC achieves constant-factor reductions in required quantum resources, the underlying scaling with the number of cuts $e$5 remains exponential, and the effectiveness of its variance-driven allocation hinges on accurate priors, which may suffer under severe hardware noise. Full cut-parameter optimization for large $e$6 is computationally demanding and may time out on large circuits.

Proposed extensions include: combining ShotQC with LOCC-based randomized-measurement schemes to further reduce overhead; variance-driven automated cut point selection; adoption of non-Pauli or entangled-state basis decompositions; ML-assisted tuning of prior-shot and segmentation parameters; and validation via real-hardware implementation (Chen et al., 2024).

7. Significance and Context

ShotQC augments the standard circuit-cutting paradigm by combining adaptive, variance-driven shot distribution with parameter-optimized cut decompositions. This combination achieves multi-fold reductions (up to $e$7 in sampling overhead for representative benchmark circuits) in quantum resources required for simulating large circuits on NISQ processors, yet maintains classical postprocessing costs in the same exponential class as the underlying circuit cutting. While scaling in the number of cuts remains the limiting factor, ShotQC represents a significant improvement in practicality and efficiency for distributed quantum simulation strategies (Chen et al., 2024).