Pure Magic Scheduling

• Pure Magic Scheduling is a paradigm characterized by idealized resource allocation methods that optimize efficiency in both classical and quantum architectures.
• It employs interference-aware online schedulers and qubit allocation techniques to minimize resource wastage and meet stringent performance targets.
• Empirical evaluations show significant improvements, such as reduced makespan in classical systems and lowered qubit-time volume in quantum applications.

Pure Magic Scheduling denotes a class of idealized scheduling paradigms that pursue near-optimal resource efficiency, rapid interference learning, and dynamic adaptation in both classical and quantum computational architectures. Across contexts ranging from cloud datacenter task allocation to quantum state distillation and fault-tolerant logical circuit routing, Pure Magic Scheduling embodies methodologies that minimize wasted resources, maximize throughput, and transparently meet stringent performance or fidelity targets. This concept is best illustrated by interference-aware online schedulers in heterogeneous systems (Romero et al., 2018) and demand-driven qubit scheduling architectures in quantum computation (Hofmeyr et al., 6 Dec 2025, Wang et al., 29 Sep 2025, Ding et al., 2018), as well as combinatorial NP-verification algorithms for processor scheduling (Dwibedy et al., 2022).

1. Formulation in Heterogeneous Systems and Quantum Architectures

Pure Magic Scheduling formalizes resource allocation as a multi-dimensional optimization problem. In cloud-scale multi-core systems, resources span inter-server (CPU, memory, accelerator) and intra-server (NUMA domain, socket, core type) heterogeneity. Applications $\mathcal{A} = \{a_1, ..., a_N\}$ with resource demand vectors $r_i$ and QoS targets $\delta_i$ must be assigned to servers $\mathcal{S} = \{s_1, ..., s_M\}$, minimizing cumulative interference cost and migration overhead, subject to slowdown constraints:

$$\text{minimize}_{x,\, \mu(t)} \quad \sum_{i,\, k,\, j} x_{ij} x_{kj} \mathrm{Cost}_{i,k}(s_j) + \sum_t \mathrm{Overhead}(\mu(t))$$

where $x_{ij}$ is the binary decision variable for placement, $\mathrm{Cost}_{i,k}(s_j)$ encodes pairwise interference cost, and $\mathrm{Overhead}(\mu(t))$ represents migration penalties (Romero et al., 2018). In quantum architectures, resources include logical data qubits, ancilla patches for magic-state cultivation, and routing infrastructure. Pure Magic schedules optimize the spacetime volume $V = N \times T$ where $N$ is the qubit count and $T$ is circuit duration, subject to connectivity and exclusivity constraints for routing trees and cultivation progress (Hofmeyr et al., 6 Dec 2025, Ding et al., 2018).

2. Architectural Foundations and Algorithmic Engines

The Mage online scheduler exemplifies Pure Magic in classical systems, employing multi-level learning engines:

• Node-local agents rapidly probe application footprints via hardware counters and synthetic microbenchmarks.
• Global coordinators aggregate probe data, learn interference models involving hardware fingerprints and co-resident feature vectors, and drive placement/migration decisions.
• Placement APIs abstract all mapping decisions behind a “magic box” that outputs the server/core assignment minimizing predicted interference and satisfying all QoS constraints (Romero et al., 2018).

In quantum fault-tolerant design, Pure Magic architecture eliminates static bus-routing by dynamically repurposing all ancilla qubits as routers or cultivators. State variables include $r_i(t)$ (routing flag), $c_i(t)$ (cultivation progress), and $u_i(t)$ (cultivation status). Scheduling proceeds via greedy Steiner-tree packing, favoring shortest routing paths and continual cultivation (Hofmeyr et al., 6 Dec 2025).

For NP-complete processor scheduling, the Magic Scheduling (MS) algorithm operates as a non-deterministic verifier traversing the Scheduling Solution Space Tree (SSST) or Weighted SSST, encoding all possible job-to-processor assignments along root-to-leaf paths, and verifying optimal makespan via certificate checking (Dwibedy et al., 2022).

3. Online Data Mining, Feature Extraction, and Interference Modeling

Pure Magic’s operational core is fast, low-overhead online learning. Probing stages extract concise feature vectors (IPC, bandwidth, cache sensitivity) on candidate servers in milliseconds. Interference models (decision-tree ensembles or latent-factor regressors) use this data plus residency fingerprints to predict pairwise costs and slowdown. Rapid convergence is achieved via Bayesian update from prior probe histories, yielding high-accuracy cost estimation within $\sim$10 ms (Romero et al., 2018).

In quantum settings, cultivation progress for magic-state qubits follows exponential distributions (e.g., $E[\tau_i] = 1/\lambda$, $\lambda \simeq 0.00227$ for distance-17 codes). Routing and cultivation decisions are interleaved using task graphs and dependency sets, minimizing schedule volume and latency via greedy packing and dynamic interruption of slow cultivations (Hofmeyr et al., 6 Dec 2025).

Resource allocation in multi-level distillation pipelines utilizes dynamic integer-linear programming (ILP) decompositions, balancing buffer sizes against burst-then-steady consumption patterns to minimize overall qubit-time volume (Wang et al., 29 Sep 2025). Scheduling metrics include projective volume, scheduling efficiency (ideal/measured volume), and cultivation time.

4. Dynamic Monitoring, Migration, and Resource Repurposing

Continuous monitoring underpins Pure Magic scheduling. Local agents and schedulers observe fine-grained metrics (QoS, tail latency, instruction rates). If performance deviates ($\mathrm{Slowdown}_i(t) > \delta_i + \varepsilon_i$ for $\tau_{\mathrm{violation}}$), migration queries identify alternate placements with minimum net cost, factoring live migration and cache warm-up overheads (Romero et al., 2018). Migration thresholds and hysteresis prevent oscillatory placement.

Quantum pipelines apply dynamic qubit allocation and ancilla reuse. On stall (e.g., buffer underrun), ancilla patches from stalled consumers are temporarily reallocated to lower-level factories, resuming production after buffers are replenished. This yields as much as 70% peak qubit savings and 26–37% average reductions in qubit-time volume compared to static scheduling, as demonstrated on application benchmarks spanning quantum chemistry, RSA factoring, Ising, Heisenberg, and Hubbard models (Wang et al., 29 Sep 2025).

Hierarchical mapping approaches for multi-level distillation circuits combine gate reordering, qubit renaming, braid repulsion, dipole moment braid rotation, and recursive graph partitioning. Layer-aware embedding and waypoint-based inter-level routing further cut permutation latency and area usage (Ding et al., 2018).

5. Complexity Analysis, Hardness Results, and Theoretical Limits

Pure Magic Scheduling exposes fundamental complexity frontiers underpinning practical and theoretical scheduling. For multi-processor assignments, the SSST size is $\frac{m^{n+1}-1}{m-1}$, encoding $m^n$ schedules for $n$ jobs and $m$ machines. Verification (MS) is polynomial time, but deterministic solution is NP-hard, linked directly to Partition and MPSP (Dwibedy et al., 2022). Multi-user scheduling extends to MUMPSP, retaining NP-completeness.

In quantum architectures, mapping and routing utilize force-directed annealing (O($m^2$) per iteration for repulsion), graph partitioning (O($(n+m)\log n)$), and hierarchical stitching combining community detection and Valiant-style randomized routing. Scheduling is tractable for embedding and partitioning, but global co-design of multi-level logic, circuit timing, and resource allocation remains a combinatorial challenge (Ding et al., 2018).

6. Empirical Outcomes and Performance Metrics

Empirical evaluation validates substantial resource and performance improvements under Pure Magic Scheduling:

• Mage scheduler offers $\sim$38% makespan and completion-time improvements over greedy scheduling, with $\sim$15% gains over batch interference-aware baselines like Paragon; tail-latency speedups up to 2–3$\times$ and near-zero QoS violations (Romero et al., 2018).
• Pure Magic quantum scheduling increases efficiency by 19–223%, reduces average magic-state cultivation time by 2.6–9.7$\times$, and decreases required qubits by up to 80% for small layouts (Hofmeyr et al., 6 Dec 2025).
• Dynamic distillation pipelines achieve 16–70% reduction in qubit cost and 26–37% reduction in total qubit-time volume over static sequential/parallel architectures (Wang et al., 29 Sep 2025).
• Hierarchical scheduling for distillation factories demonstrates 5.64$\times$ reduction in space-time volume compared to prior best designs, with near-optimal area/latency results (Ding et al., 2018).

Table: Representative Qubit-Time Volume Improvements (Wang et al., 29 Sep 2025)

Application	d-levels	Static Seq (M)	Static Par (M)	Dynamic (M, ↓%)
Ising	(3,9)	7.25	7.05	5.25 (–28%)
Heisenberg	(5,15)	67.5	67.0	48.5 (–28%)
Hubbard	(5,17)	87.6	74.4	62.1 (–29%)
Chemistry	(5,17)	3.40	2.89	2.93 (–16%)
Factoring	(5,17)	2.08	2.49	2.09 (–16%)

7. Limitations, Extensions, and Open Problems

Limitations of Pure Magic Scheduling include assumptions of stable application/qubit workloads post-probing, exclusion of network/storage interference in current architectural models, and potential underprediction of migration costs under large in-memory footprints (Romero et al., 2018). Dynamic pipelines require careful buffer tuning to avoid qubit starvation or excessive idling (Wang et al., 29 Sep 2025).

Potential extensions identified include GPU/FPGA co-scheduling (extending interference models to accelerators), power-aware scheduling, storage I/O modeling via micro-benchmarks, fault-tolerance integration, and predictive failure handling. Theoretical open problems span extension of SSST concepts to broader NP-complete domains and refinement of lower-bound makespan estimates under multi-user and multi-resource constraints (Dwibedy et al., 2022).

A plausible implication is that continued advances in Pure Magic Scheduling—spanning data mining, interference modeling, resource repurposing, and scalable mapping—will further compress scheduling overheads and resource footprints within both classical and quantum computing systems, driving progress toward practical large-scale architectures with robust fault-tolerance and resource efficiency.