quantum circuit learning, quantum ai
Train parameterized quantum circuits.
102 technical terms and definitions
Train parameterized quantum circuits.
Use quantum effects for computation.
Energy quantization in small structures.
Add quantum effects to classical simulation.
Protect quantum information from noise.
Encode classical data in quantum states.
Use quantum computers for generation.
Use quantum states as feature spaces.
Apply QML to chemistry problems.
Apply quantum computing to ML problems.
Design of quantum neural networks.
Neural networks running on quantum hardware.
Estimate eigenvalues of operators.
Use quantum computers for sampling.
Quantum algorithms based on walks.
Fraction of absorbed photons that cause reaction.
Use quantum for better sampling.
Quantum ML explores quantum computers for AI. Very early stage. Potential for certain optimization problems.
Partially ballistic regime.
Separate Fermi levels under non-equilibrium.
Measure carrier lifetime.
Quaternion embeddings for KGs.
Query classification categorizes requests to select optimal processing strategies.
Break complex query into sub-queries for better retrieval.
Add terms to improve retrieval.
Query expansion adds related terms to improve retrieval coverage.
Generate multiple related queries to improve coverage.
Query expansion rewrites or augments user query for better retrieval. LLM can generate alternative phrasings.
Store complete results.
Reformulate query for better results.
Query rewriting rephrases queries to improve retrieval effectiveness.
Transform user query to better match document format before retrieval.
Test examples in few-shot evaluation.
Interpret user query intent.
Query understanding interprets user intent reformulating or expanding queries.
Answer questions about text. Reading comprehension.
QA as auxiliary task.
Break complex questions into sub-questions.
Break complex questions into simpler sub-questions.
Generate questions from text.
Queue management controls waiting lots at equipment minimizing cycle time.
Minimize waiting between steps.
Queue time is duration work waits before next operation begins.
Time wafers wait between process steps.
Use queue of negatives (MoCo).
Message queues (RabbitMQ, SQS) decouple request intake from processing. Handle bursts, enable async workflows.
# Semiconductor Manufacturing & Queueing Theory: A Mathematical Deep Dive ## 1. Introduction Semiconductor fabrication presents one of the most mathematically rich queueing environments in existence. Key characteristics include: - **Reentrant flow**: Wafers visit the same machine groups multiple times (e.g., photolithography 20–30 times) - **Process complexity**: 400–800 processing steps over 2–3 months - **Batch processing**: Furnaces, wet benches process multiple wafers simultaneously - **Sequence-dependent setups**: Recipe changes require significant time - **Tool dedication**: Some products can only run on specific tools - **High variability**: Equipment failures, rework, yield issues - **Multiple product mix**: Hundreds of different products simultaneously ## 2. Foundational Queueing Mathematics ### 2.1 The M/M/1 Queue The foundational single-server queue with: - **Arrival rate**: $\lambda$ (Poisson process) - **Service rate**: $\mu$ (exponential service times) - **Utilization**: $\rho = \frac{\lambda}{\mu}$ **Key metrics**: $$ W = \frac{\rho}{\mu(1-\rho)} $$ $$ L = \frac{\rho^2}{1-\rho} $$ Where: - $W$ = Average waiting time - $L$ = Average queue length ### 2.2 Kingman's Formula (G/G/1 Approximation) The **core insight** for semiconductor manufacturing—the G/G/1 approximation: $$ W_q \approx \left(\frac{\rho}{1-\rho}\right) \cdot \left(\frac{C_a^2 + C_s^2}{2}\right) \cdot \bar{s} $$ **Variable definitions**: | Symbol | Definition | |--------|------------| | $\rho$ | Utilization (arrival rate / service rate) | | $C_a^2$ | Squared coefficient of variation of interarrival times | | $C_s^2$ | Squared coefficient of variation of service times | | $\bar{s}$ | Mean service time | **Critical insight**: The term $\frac{\rho}{1-\rho}$ is **explosively nonlinear**: | Utilization ($\rho$) | Queueing Multiplier $\frac{\rho}{1-\rho}$ | |---------------------|-------------------------------------------| | 50% | 1.0× | | 70% | 2.3× | | 80% | 4.0× | | 90% | 9.0× | | 95% | 19.0× | | 99% | 99.0× | ### 2.3 Pollaczek-Khinchine Formula (M/G/1) For Poisson arrivals with general service distribution: $$ W_q = \frac{\lambda \mathbb{E}[S^2]}{2(1-\rho)} = \frac{\rho}{1-\rho} \cdot \frac{1+C_s^2}{2} \cdot \frac{1}{\mu} $$ ### 2.4 Little's Law The **universal connector** in queueing theory: $$ L = \lambda W $$ Where: - $L$ = Average number in system (WIP) - $\lambda$ = Throughput (arrival rate) - $W$ = Average time in system (cycle time) **Properties**: - Exact (not an approximation) - Distribution-free - Universally applicable - Foundational for fab metrics ## 3. The VUT Equation (Factory Physics) The practical "working equation" for semiconductor cycle time: $$ CT = T_0 \cdot \left[1 + \left(\frac{C_a^2 + C_s^2}{2}\right) \cdot \left(\frac{\rho}{1-\rho}\right)\right] $$ ### 3.1 Component Breakdown | Factor | Symbol | Meaning | |--------|--------|---------| | **V** (Variability) | $\frac{C_a^2 + C_s^2}{2}$ | Process and arrival randomness | | **U** (Utilization) | $\frac{\rho}{1-\rho}$ | Congestion penalty | | **T** (Time) | $T_0$ | Raw (irreducible) processing time | ### 3.2 Cycle Time Bounds **Best Case Cycle Time**: $$ CT_{best} = T_0 + \frac{(W_0 - 1)}{r_{bottleneck}} \cdot \mathbf{1}_{W_0 > 1} $$ **Practical Worst Case (PWC)**: $$ CT_{PWC} = T_0 + \frac{(n-1) \cdot W_0}{r_{bottleneck}} $$ Where: - $T_0$ = Raw processing time - $W_0$ = WIP level - $n$ = Number of stations - $r_{bottleneck}$ = Bottleneck rate ## 4. Reentrant Line Theory ### 4.1 Mathematical Formulation A reentrant line has: - $K$ stations (machine groups) - $J$ steps (operations) - Each step $j$ is processed at station $s(j)$ - Products visit the same station multiple times **State descriptor**: $$ \mathbf{n} = (n_1, n_2, \ldots, n_J) $$ where $n_j$ = number of jobs at step $j$. ### 4.2 Stability Conditions For a reentrant line to be stable: $$ \rho_k = \sum_{j:\, s(j)=k} \frac{\lambda}{\mu_j} < 1 \quad \forall k \in \{1, \ldots, K\} $$ > **Critical Result**: This condition is **necessary but NOT sufficient**! > > The **Lu-Kumar network** demonstrated that even with all $\rho_k < 1$, certain scheduling policies (including FIFO) can make the system **unstable**—queues grow unboundedly. ### 4.3 Fluid Models Deterministic approximation treating jobs as continuous flow: $$ \frac{dq_j(t)}{dt} = \lambda_j(t) - \mu_j(t) $$ **Applications**: - Capacity planning - Stability analysis - Bottleneck identification - Long-run behavior prediction ### 4.4 Diffusion Limits (Heavy Traffic) In heavy traffic ($\rho \to 1$), the queue length process converges to **Reflected Brownian Motion (RBM)**: $$ Z(t) = X(t) + L(t) $$ Where: - $Z(t)$ = Queue length process - $X(t)$ = Net input process (Brownian motion) - $L(t)$ = Regulator process (reflection at zero) **Brownian motion parameters**: - Drift: $\theta = \lambda - \mu$ - Variance: $\sigma^2 = \lambda \cdot C_a^2 + \mu \cdot C_s^2$ ## 5. Variability Propagation ### 5.1 Sources of Variability 1. **Arrival variability** ($C_a^2$): Order patterns, lot releases 2. **Process variability** ($C_s^2$): Equipment, recipes, operators 3. **Flow variability**: Propagation through network 4. **Failure variability**: Random equipment downs ### 5.2 The Linking Equations For departures from a queue: $$ C_d^2 = \rho^2 C_s^2 + (1-\rho^2) C_a^2 $$ **Interpretation**: - High-utilization stations ($\rho \to 1$): Export **service variability** - Low-utilization stations ($\rho \to 0$): Export **arrival variability** ### 5.3 Equipment Failures and Effective Variability When tools fail randomly: $$ C_{s,eff}^2 = C_{s,0}^2 + 2 \cdot \frac{(1-A)}{A} \cdot \frac{MTTR}{t_0} $$ Where: - $C_{s,0}^2$ = Inherent process variability - $A = \frac{MTBF}{MTBF + MTTR}$ = Availability - $MTBF$ = Mean Time Between Failures - $MTTR$ = Mean Time To Repair - $t_0$ = Processing time **Example calculation**: For $A = 0.95$, $MTTR = t_0$: $$ \Delta C_s^2 = 2 \cdot \frac{0.05}{0.95} \cdot 1 \approx 0.105 $$ ## 6. Batch Processing Mathematics ### 6.1 Bulk Service Queues (M/G^b/1) Characteristics: - Customers arrive singly (Poisson) - Server processes up to $b$ customers simultaneously - Service time same regardless of batch size **Analysis tools**: - Probability generating functions - Embedded Markov chains at departure epochs ### 6.2 Minimum Batch Trigger (MBT) Policies Wait until at least $b$ items accumulate before processing. **Effects**: - Creates artificial correlation between arrivals - Dramatically increases effective $C_a^2$ - Higher cycle times despite efficient tool usage **Effective arrival variability** can increase by factors of **2–5×**. ### 6.3 Optimal Batch Size Balancing setup efficiency against queue time: $$ B^* = \sqrt{\frac{2DS}{ph}} $$ Where: - $D$ = Demand rate - $S$ = Setup cost/time - $p$ = Processing cost per item - $h$ = Holding cost **Trade-off**: - Smaller batches → More setups, less waiting - Larger batches → Fewer setups, longer queues ## 7. Queueing Network Analysis ### 7.1 Jackson Networks **Assumptions**: - Poisson external arrivals - Exponential service times - Probabilistic routing **Product-form solution**: $$ \pi(\mathbf{n}) = \prod_{i=1}^{K} \pi_i(n_i) $$ Each queue behaves independently in steady state. ### 7.2 BCMP Networks Extensions to Jackson networks: - Multiple job classes - Various service disciplines (FCFS, PS, LCFS-PR, IS) - General service time distributions (with constraints) **Product-form maintained**: $$ \pi(n_1, n_2, \ldots, n_K) = C \prod_{i=1}^{K} f_i(n_i) $$ ### 7.3 Mean Value Analysis (MVA) For closed networks (fixed WIP): $$ W_k(n) = \frac{1}{\mu_k}\left(1 + Q_k(n-1)\right) $$ **Iterative algorithm**: 1. Compute wait times given queue lengths at $n-1$ jobs 2. Calculate queue lengths at $n$ jobs 3. Determine throughput 4. Repeat ### 7.4 Decomposition Approximations (QNA) For realistic fabs, use **decomposition methods**: 1. **Traffic equations**: Solve for effective arrival rates $\lambda_i$ $$ \lambda_i = \gamma_i + \sum_{j=1}^{K} \lambda_j p_{ji} $$ 2. **Linking equations**: Track $C_a^2$ propagation 3. **G/G/m formulas**: Apply at each station independently 4. **Aggregation**: Combine results for system metrics ## 8. Scheduling Theory for Fabs ### 8.1 Basic Priority Rules | Rule | Description | Optimal For | |------|-------------|-------------| | FIFO | First In, First Out | Fairness | | SRPT | Shortest Remaining Processing Time | Mean flow time | | EDD | Earliest Due Date | On-time delivery | | SPT | Shortest Processing Time | Mean waiting time | ### 8.2 Fluctuation Smoothing Policies Developed specifically for semiconductor manufacturing: - **FSMCT** (Fluctuation Smoothing for Mean Cycle Time): - Prioritizes jobs that smooth the output stream - Reduces mean cycle time - **FSVCT** (Fluctuation Smoothing for Variance of Cycle Time): - Reduces cycle time variability - Improves delivery predictability ### 8.3 Heavy Traffic Scheduling In the limit as $\rho \to 1$, optimal policies often take forms: - **cμ-rule**: Prioritize class with highest $c_i \mu_i$ $$ \text{Priority index} = c_i \cdot \mu_i $$ where $c_i$ = holding cost, $\mu_i$ = service rate - **Threshold policies**: Switch based on queue length thresholds - **State-dependent priorities**: Dynamic adjustment based on system state ### 8.4 Computational Complexity **State space dimension** = Number of (step × product) combinations For realistic fabs: **thousands of dimensions** Dynamic programming approaches suffer the **curse of dimensionality**: $$ |\mathcal{S}| = \prod_{j=1}^{J} (N_{max} + 1) $$ Where $J$ = number of steps, $N_{max}$ = maximum queue size per step. ## 9. Key Mathematical Insights ### 9.1 Summary Table | Insight | Mathematical Expression | Practical Implication | |---------|------------------------|----------------------| | Nonlinear congestion | $\frac{\rho}{1-\rho}$ | Small utilization increases near capacity cause huge cycle time jumps | | Variability multiplies | $\frac{C_a^2 + C_s^2}{2}$ | Reducing variability is as powerful as reducing utilization | | Variability propagates | $C_d^2 = \rho^2 C_s^2 + (1-\rho^2) C_a^2$ | Upstream problems cascade downstream | | Batching costs | MBT inflates $C_a^2$ | "Efficient" batching often increases total cycle time | | Reentrant instability | Lu-Kumar example | Simple policies can destabilize feasible systems | | Universal law | $L = \lambda W$ | Connects WIP, throughput, and cycle time | ### 9.2 The Central Trade-off $$ \text{Cycle Time} \propto \frac{1}{1-\rho} \times \text{Variability} $$ **The fundamental tension**: Pushing utilization higher improves asset ROI but triggers explosive cycle time growth through the $\frac{\rho}{1-\rho}$ nonlinearity—amplified by every source of variability. ## 10. Modern Developments ### 10.1 Stochastic Processing Networks Generalizations of classical queueing: - Simultaneous resource possession - Complex synchronization constraints - Non-idling constraints ### 10.2 Robust Queueing Theory Optimize for **worst-case performance** over uncertainty sets: $$ \min_{\pi} \max_{\theta \in \Theta} J(\pi, \theta) $$ Rather than assuming specific stochastic distributions. ### 10.3 Machine Learning Integration - **Reinforcement Learning**: Train dispatch policies from simulation $$ Q(s, a) \leftarrow Q(s, a) + \alpha \left[ r + \gamma \max_{a'} Q(s', a') - Q(s, a) \right] $$ - **Neural Networks**: Approximate complex distributions - **Data-driven estimation**: Real-time parameter learning ### 10.4 Digital Twin Technology Combines: - Analytical queueing models (fast, interpretable) - High-fidelity simulation (detailed, accurate) - Real-time sensor data (current state) For predictive control and optimization. ## Common Notation Reference | Symbol | Meaning | |--------|---------| | $\lambda$ | Arrival rate | | $\mu$ | Service rate | | $\rho$ | Utilization ($\lambda/\mu$) | | $C_a^2$ | Squared CV of interarrival times | | $C_s^2$ | Squared CV of service times | | $W$ | Waiting time | | $W_q$ | Waiting time in queue | | $L$ | Number in system | | $L_q$ | Number in queue | | $CT$ | Cycle time | | $T_0$ | Raw processing time | | $WIP$ | Work in process | ## Key Formulas Quick Reference ### B.1 Single Server Queues ``` M/M/1: W = 1/(μ - λ) M/G/1: W_q = λE[S²]/(2(1-ρ)) G/G/1 (Kingman): W_q ≈ (ρ/(1-ρ)) × ((C_a² + C_s²)/2) × (1/μ) ``` ### B.2 Factory Physics ``` VUT Equation: CT = T₀ × [1 + ((C_a² + C_s²)/2) × (ρ/(1-ρ))] Little's Law: L = λW Departure CV: C_d² = ρ²C_s² + (1-ρ²)C_a² ``` ### B.3 Availability ``` Availability: A = MTBF/(MTBF + MTTR) Effective C_s²: C_s² = C_s0² + 2((1-A)/A)(MTTR/t₀) ```
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