FSharp.Azure.Quantum v1.3.1 — nugetz

pkg

Thorium/FSharp.Azure.Quantumv1.3.1

.NET 10.0

F# library for Azure Quantum with quantum-first optimization (Graph Coloring, TSP, MaxCut, Knapsack, Network Flow, Portfolio), Azure Quantum backend integration (IonQ, Rigetti, Quantinuum, Atom Computing), extensible circuit validation, local quantum simulation, QAOA with automatic parameter optimization (Nelder-Mead), OpenQASM 2.0 support, error mitigation techniques (ZNE, PEC, REM), Quantum Random Number Generation (QRNG), and educational quantum algorithms (Grover, QFT, Amplitude Amplification) with cloud backend execution. Multi-provider architecture supports IonQ, Rigetti, Quantinuum, Atom Computing, IBM, Amazon Braket, and Google Cirq.

github.com ↗

License

Unlicense

Deps

Install Size

—

Vulns

✓ 0

Published

Feb 22, 2026

Get Started

$ dotnet add package FSharp.Azure.Quantum

Readme

FSharp.Azure.Quantum

Quantum-First F# Library - Solve combinatorial optimization problems using quantum algorithms (QAOA - Quantum Approximate Optimization Algorithm) with automatic cloud/local backend selection.

flowchart TD

    %% Top layer
    A[.NET API High-level algorithms familiar to normal developers]

    %% Quantum algorithms
    B[Quantum Algorithms, 
       Ph.D stuff]

    %% Backend interfaces
    C[Backend Interfaces: IQuantumBackend / ITopologicalBackend]

    %% Quantum Gate Backends
    D[Quantum Gate Backends: 
        LocalSimulator: Ph.D stuff, 
        Azure Quantum as a service]


    %% Topological Backends
    E[Topological Backend simulator, 
       more Ph.D stuff]

    %% Flow
    A --> B --> C
    C --> D
    C --> E

Status
Quick Start
Problem Builders

dotnet add package FSharp.Azure.Quantum

open FSharp.Azure.Quantum

// Graph Coloring: Register Allocation
let problem = graphColoring {
    node "R1" ["R2"; "R3"]
    node "R2" ["R1"; "R4"]
    node "R3" ["R1"; "R4"]
    node "R4" ["R2"; "R3"]
    colors ["EAX"; "EBX"; "ECX"; "EDX"]
}

// Solve using quantum optimization (QAOA)
match GraphColoring.solve problem 4 None with
| Ok solution ->
    printfn "Colors used: %d" solution.ColorsUsed
    solution.Assignments 
    |> Map.iter (fun node color -> printfn "%s → %s" node color)
| Error msg -> 
    printfn "Error: %s" msg

using FSharp.Azure.Quantum;
using static FSharp.Azure.Quantum.CSharpBuilders;

// MaxCut: Circuit Partitioning
var vertices = new[] { "A", "B", "C", "D" };
var edges = new[] {
    (source: "A", target: "B", weight: 1.0),
    (source: "B", target: "C", weight: 2.0),
    (source: "C", target: "D", weight: 1.0),
    (source: "D", target: "A", weight: 1.0)
};

var problem = MaxCutProblem(vertices, edges);
var result = MaxCut.solve(problem, null);

if (result.IsOk) {
    var solution = result.ResultValue;
    Console.WriteLine($"Cut Value: {solution.CutValue}");
    Console.WriteLine($"Partition S: {string.Join(", ", solution.PartitionS)}");
    Console.WriteLine($"Partition T: {string.Join(", ", solution.PartitionT)}");
}

open FSharp.Azure.Quantum

let problem = graphColoring {
    node "Task1" ["Task2"; "Task3"]
    node "Task2" ["Task1"; "Task4"]
    node "Task3" ["Task1"; "Task4"]
    node "Task4" ["Task2"; "Task3"]
    colors ["Slot A"; "Slot B"; "Slot C"]
    objective MinimizeColors
}

match GraphColoring.solve problem 3 None with
| Ok solution ->
    printfn "Valid coloring: %b" solution.IsValid
    printfn "Colors used: %d/%d" solution.ColorsUsed 3
    printfn "Conflicts: %d" solution.ConflictCount
| Error msg -> printfn "Error: %s" msg

let vertices = ["A"; "B"; "C"; "D"]
let edges = [
    ("A", "B", 1.0)
    ("B", "C", 2.0)
    ("C", "D", 1.0)
    ("D", "A", 1.0)
]

let problem = MaxCut.createProblem vertices edges

match MaxCut.solve problem None with
| Ok solution ->
    printfn "Partition S: %A" solution.PartitionS
    printfn "Partition T: %A" solution.PartitionT
    printfn "Cut value: %.2f" solution.CutValue
| Error msg -> printfn "Error: %s" msg

let items = [
    ("laptop", 3.0, 1000.0)   // (id, weight, value)
    ("phone", 0.5, 500.0)
    ("tablet", 1.5, 700.0)
    ("monitor", 2.0, 600.0)
]

let problem = Knapsack.createProblem items 5.0  // capacity = 5.0

match Knapsack.solve problem None with
| Ok solution ->
    printfn "Total value: $%.2f" solution.TotalValue
    printfn "Total weight: %.2f/%.2f" solution.TotalWeight problem.Capacity
    printfn "Items: %A" (solution.SelectedItems |> List.map (fun i -> i.Id))
| Error msg -> printfn "Error: %s" msg

let cities = [
    ("Seattle", 0.0, 0.0)
    ("Portland", 1.0, 0.5)
    ("San Francisco", 2.0, 1.5)
    ("Los Angeles", 3.0, 3.0)
]

let problem = TSP.createProblem cities

match TSP.solve problem None with
| Ok tour ->
    printfn "Optimal route: %s" (String.concat " → " tour.Cities)
    printfn "Total distance: %.2f" tour.TotalDistance
| Error msg -> printfn "Error: %s" msg

let assets = [
    ("AAPL", 0.12, 0.15, 150.0)  // (symbol, return, risk, price)
    ("GOOGL", 0.10, 0.12, 2800.0)
    ("MSFT", 0.11, 0.14, 350.0)
]

let problem = Portfolio.createProblem assets 10000.0  // budget

match Portfolio.solve problem None with
| Ok allocation ->
    printfn "Portfolio value: $%.2f" allocation.TotalValue
    printfn "Expected return: %.2f%%" (allocation.ExpectedReturn * 100.0)
    printfn "Risk: %.2f" allocation.Risk
    
    allocation.Allocations 
    |> List.iter (fun (symbol, shares, value) ->
        printfn "  %s: %.2f shares ($%.2f)" symbol shares value)
| Error msg -> printfn "Error: %s" msg

let nodes = [
    NetworkFlow.SourceNode("Factory", 100)
    NetworkFlow.IntermediateNode("Warehouse", 80)
    NetworkFlow.SinkNode("Store1", 40)
    NetworkFlow.SinkNode("Store2", 60)
]

let routes = [
    NetworkFlow.Route("Factory", "Warehouse", 5.0)
    NetworkFlow.Route("Warehouse", "Store1", 3.0)
    NetworkFlow.Route("Warehouse", "Store2", 4.0)
]

let problem = { NetworkFlow.Nodes = nodes; Routes = routes }

match NetworkFlow.solve problem None with
| Ok flow ->
    printfn "Total cost: $%.2f" flow.TotalCost
    printfn "Fill rate: %.1f%%" (flow.FillRate * 100.0)
| Error msg -> printfn "Error: %s" msg

open FSharp.Azure.Quantum

// Define tasks with dependencies
let taskA = scheduledTask {
    taskId "TaskA"
    duration (hours 2.0)
    priority 10.0
}

let taskB = scheduledTask {
    taskId "TaskB"
    duration (hours 1.5)
    after "TaskA"  // Dependency
    requires "Worker" 2.0
    deadline 180.0
}

let taskC = scheduledTask {
    taskId "TaskC"
    duration (minutes 30.0)
    after "TaskA"
    requires "Machine" 1.0
}

// Define resources
let worker = resource {
    resourceId "Worker"
    capacity 3.0
}

let machine = resource {
    resourceId "Machine"
    capacity 2.0
}

// Build scheduling problem
let problem = scheduling {
    tasks [taskA; taskB; taskC]
    resources [worker; machine]
    objective MinimizeMakespan
    timeHorizon 500.0
}

// Solve with quantum backend for resource constraints
let backend = BackendAbstraction.createLocalBackend()
match solveQuantum backend problem with
| Ok solution ->
    printfn "Makespan: %.2f hours" solution.Makespan
    solution.Schedule 
    |> List.iter (fun assignment ->
        printfn "%s: starts %.2f, ends %.2f" 
            assignment.TaskId assignment.StartTime assignment.EndTime)
| Error msg -> printfn "Error: %s" msg

open FSharp.Azure.Quantum.QuantumTreeSearchBuilder

// Search decision tree with quantum parallelism
let searchProblem = QuantumTreeSearch.quantumTreeSearch {
    depth 5                    // Tree depth
    branchingFactor 3          // Children per node
    target 42                  // Target node value
    qubits 10
}

match search searchProblem with
| Ok result ->
    printfn "Found path: %A" result.Path
    printfn "Depth explored: %d" result.DepthReached
    printfn "Success probability: %.2f%%" (result.Probability * 100.0)
| Error msg ->
    printfn "Error: %s" msg

open FSharp.Azure.Quantum.QuantumConstraintSolverBuilder

// Solve boolean satisfiability (SAT) problem
let satProblem = QuantumConstraintSolver.constraintSolver {
    variables ["x"; "y"; "z"]
    clause ["x"; "!y"]         // x OR NOT y
    clause ["!x"; "z"]         // NOT x OR z
    clause ["y"; "!z"]         // y OR NOT z
    qubits 8
}

match solve satProblem with
| Ok solution ->
    printfn "Satisfying assignment:"
    solution.Assignment 
    |> Map.iter (fun var value -> printfn "  %s = %b" var value)
    printfn "Clauses satisfied: %d/%d" solution.SatisfiedClauses solution.TotalClauses
| Error msg ->
    printfn "Error: %s" msg

open FSharp.Azure.Quantum.QuantumPatternMatcherBuilder

// Find pattern in data stream with quantum speedup
let matchProblem = QuantumPatternMatcher.patternMatcher {
    haystack [1; 2; 3; 4; 5; 6; 7; 8]
    needle [3; 4; 5]
    tolerance 0                 // Exact match
    qubits 12
}

match find matchProblem with
| Ok result ->
    printfn "Pattern found at positions: %A" result.Positions
    printfn "Total matches: %d" result.MatchCount
    printfn "Search probability: %.2f%%" (result.Probability * 100.0)
| Error msg ->
    printfn "Error: %s" msg

open FSharp.Azure.Quantum.QuantumArithmeticBuilder

// Quantum integer addition
let addProblem = QuantumArithmeticOps.quantumArithmetic {
    operands 15 27              // 15 + 27
    operation Add
    qubits 10
}

match compute addProblem with
| Ok result ->
    printfn "Result: %d" result.Value
    printfn "Carry: %b" result.Carry
    printfn "Gates used: %d" result.GateCount
    printfn "Circuit depth: %d" result.CircuitDepth
| Error msg ->
    printfn "Error: %s" msg

// Modular exponentiation for RSA
let rsaProblem = QuantumArithmeticOps.quantumArithmetic {
    operands 5 3                // base=5, exponent=3
    operation ModularExponentiate
    modulus 33                  // RSA modulus
    qubits 12
}

open FSharp.Azure.Quantum.QuantumPeriodFinderBuilder

// Find period of modular function (Shor's algorithm)
let shorsProblem = QuantumPeriodFinder.periodFinder {
    number 15                   // Composite to factor
    base_ 7                     // Coprime base
    precision 12                // QPE precision bits
    maxAttempts 10              // Probabilistic retries
}

match findPeriod shorsProblem with
| Ok result ->
    printfn "Period found: %d" result.Period
    match result.Factors with
    | Some (p, q) ->
        printfn "Factors: %d × %d = %d" p q (p * q)
        printfn "⚠️  RSA modulus factored!"
    | None ->
        printfn "Retry with different base (probabilistic)"
| Error msg ->
    printfn "Error: %s" msg

open FSharp.Azure.Quantum.QuantumPhaseEstimatorBuilder
open FSharp.Azure.Quantum.Algorithms.QuantumPhaseEstimation

// Estimate eigenphase of unitary operator
let qpeProblem = QuantumPhaseEstimator.phaseEstimator {
    unitary (RotationZ (Math.PI / 4.0))  // Operator U
    eigenstate 0                         // |ψ⟩ = |0⟩
    precision 16                         // 16-bit phase precision
    qubits 20
}

match estimate qpeProblem with
| Ok result ->
    printfn "Estimated phase: %.6f" result.Phase
    printfn "Eigenvalue: e^(2πi × %.6f)" result.Phase
    printfn "Confidence: %.2f%%" (result.Confidence * 100.0)
    printfn "Application: Molecular energy = %.4f Hartree" (result.Phase * 2.0 * Math.PI)
| Error msg ->
    printfn "Error: %s" msg

Builder	Primary Application	Quantum Advantage	Hardware Readiness
Tree Search	Game AI, Route Planning	O(√N) speedup	2028+ (requires 50+ qubits)
Constraint Solver	SAT, Logic Puzzles	O(√2^n) speedup	2030+ (requires 100+ qubits)
Pattern Matcher	DNA Alignment, Data Mining	O(√N) speedup	2027+ (requires 30+ qubits)
Arithmetic	Cryptography, RSA	Polynomial vs exponential	2026+ (requires 4096+ qubits for RSA-2048)
Period Finder	Shor's Algorithm, Cryptanalysis	Exponential speedup	2029+ (requires 4096+ qubits)
Phase Estimator	Quantum Chemistry, VQE	Polynomial speedup	2025+ (10-20 qubits sufficient for small molecules)

using FSharp.Azure.Quantum;
using static FSharp.Azure.Quantum.CSharpBuilders;

// Tree Search
var treeSearch = TreeSearch(depth: 5, branchingFactor: 3, target: 42);
var searchResult = ExecuteTreeSearch(treeSearch);

// Constraint Solver
var satProblem = ConstraintProblem(
    variables: new[] { "x", "y", "z" },
    clauses: new[] { 
        new[] { "x", "!y" },
        new[] { "!x", "z" }
    }
);
var satResult = SolveConstraints(satProblem);

// Pattern Matcher
var pattern = PatternMatch(
    haystack: new[] { 1, 2, 3, 4, 5, 6, 7, 8 },
    needle: new[] { 3, 4, 5 },
    tolerance: 0
);
var matchResult = FindPattern(pattern);

// Arithmetic
var add = ArithmeticAdd(15, 27);
var addResult = ComputeArithmetic(add);

var rsa = ModularExponentiate(baseValue: 5, exponent: 3, modulus: 33);
var rsaResult = ComputeArithmetic(rsa);

// Period Finder (Shor's)
var shors = FactorInteger(15, precision: 12);
var factorResult = FindPeriod(shors);

// Phase Estimator
var qpe = EstimateRotationZ(Math.PI / 4.0, precision: 16);
var phaseResult = EstimatePhase(qpe);

open FSharp.Azure.Quantum
open FSharp.Azure.Quantum.MachineLearning

// Setup backend and architecture
let backend = LocalBackend() :> IQuantumBackend
let featureMap = AngleEncoding
let variationalForm = RealAmplitudes 2  // 2 layers

// Prepare training data (features and labels as separate arrays)
let trainFeatures = [|
    [| 0.1; 0.2 |]
    [| 0.9; 0.8 |]
    [| 0.3; 0.1 |]
    [| 0.8; 0.9 |]
|]
let trainLabels = [| 0; 1; 0; 1 |]

// Configure training
let config = {
    LearningRate = 0.1
    MaxEpochs = 100
    ConvergenceThreshold = 0.001
    Shots = 1000
    Verbose = false
    Optimizer = VQC.SGD
}

// Initialize parameters
let numQubits = trainFeatures.[0].Length
let numParams = VariationalForms.parameterCount variationalForm numQubits
let initialParams = Array.init numParams (fun _ -> Random().NextDouble() * 2.0 * Math.PI)

// Train the classifier
match VQC.train backend featureMap variationalForm initialParams trainFeatures trainLabels config with
| Ok result ->
    // Make predictions
    let testPoint = [| 0.5; 0.5 |]
    match VQC.predict backend featureMap variationalForm result.Parameters testPoint 1000 with
    | Ok prediction ->
        printfn "Prediction: %d (probability: %.2f%%)" 
            prediction.Label (prediction.Probability * 100.0)
    | Error msg -> printfn "Error: %s" msg
| Error msg -> printfn "Training failed: %s" msg

open FSharp.Azure.Quantum
open FSharp.Azure.Quantum.MachineLearning

// Setup backend and quantum feature map
let backend = LocalBackend() :> IQuantumBackend
let featureMap = ZZFeatureMap  // Quantum feature map with entanglement

// Training data
let trainData = [| [| 0.1; 0.2 |]; [| 0.9; 0.8 |]; [| 0.3; 0.1 |]; [| 0.8; 0.9 |] |]
let trainLabels = [| 0; 1; 0; 1 |]

// SVM configuration
let config = {
    C = 1.0
    Tolerance = 1e-3
    MaxIterations = 100
    Verbose = false
}

// Train SVM with quantum kernel
match QuantumKernelSVM.train backend featureMap trainData trainLabels config 1000 with
| Ok model ->
    // Evaluate on test data
    let testData = [| [| 0.5; 0.5 |]; [| 0.2; 0.8 |] |]
    let testLabels = [| 0; 1 |]
    
    match QuantumKernelSVM.evaluate backend model testData testLabels 1000 with
    | Ok accuracy -> printfn "Test accuracy: %.2f%%" (accuracy * 100.0)
    | Error msg -> printfn "Evaluation error: %s" msg
| Error msg -> printfn "Training error: %s" msg

open FSharp.Azure.Quantum.Business

// Automated hyperparameter tuning and model selection
let automl = autoML {
    dataset trainData
    target "label_column"
    features ["feature1"; "feature2"; "feature3"]
    maxTrials 20
    timeout (minutes 30.0)
}

match AutoML.run automl with
| Ok result ->
    printfn "Best model accuracy: %.2f%%" (result.BestAccuracy * 100.0)
    printfn "Best hyperparameters: %A" result.BestHyperparameters
    
    // Use best model for predictions
    let predictions = AutoML.predict result.BestModel newData
    predictions |> List.iter (printfn "Prediction: %A")
| Error msg -> printfn "Error: %s" msg

open FSharp.Azure.Quantum.Business

// Detect outliers in network traffic for security monitoring
let detector = anomalyDetection {
    data securityLogs
    features ["packet_size"; "connection_duration"; "failed_logins"]
    threshold 0.95
    sensitivity High
}

match AnomalyDetection.detect detector with
| Ok anomalies ->
    printfn "Found %d anomalies" anomalies.Length
    anomalies |> List.iter (fun a ->
        printfn "⚠️  Anomaly score: %.3f - %A" a.Score a.DataPoint)
| Error msg -> printfn "Error: %s" msg

open FSharp.Azure.Quantum.Business

// Classify transactions as fraudulent or legitimate
let classifier = binaryClassification {
    trainingData transactions
    positiveClass "fraud"
    negativeClass "legitimate"
    features ["amount"; "location"; "time"; "merchant_type"]
    algorithm QuantumSVM
}

match BinaryClassification.train classifier with
| Ok model ->
    // Evaluate performance
    let metrics = BinaryClassification.evaluate model testTransactions
    printfn "Precision: %.2f%%" (metrics.Precision * 100.0)
    printfn "Recall: %.2f%%" (metrics.Recall * 100.0)
    printfn "F1 Score: %.2f" metrics.F1Score
    
    // Classify new transactions
    let newTransaction = ["amount", 1500.0; "location", "foreign"]
    match BinaryClassification.predict model newTransaction with
    | Ok prediction ->
        printfn "Classification: %s (%.2f%% confidence)" 
            prediction.Class (prediction.Probability * 100.0)
    | Error msg -> printfn "Error: %s" msg
| Error msg -> printfn "Training failed: %s" msg

open FSharp.Azure.Quantum.Business

// Predict which customers are likely to cancel service
let model = predictiveModel {
    historicalData customerData
    targetVariable "churned"
    predictors ["tenure"; "monthly_charges"; "total_charges"; "contract_type"]
    horizon (days 30.0)
}

match PredictiveModel.build model with
| Ok trained ->
    // Identify high-risk customers
    let predictions = PredictiveModel.predict trained activeCustomers
    let highRisk = predictions |> List.filter (fun p -> p.ChurnProbability > 0.7)
    
    printfn "High-risk customers: %d" highRisk.Length
    highRisk |> List.iter (fun customer ->
        printfn "Customer %s: %.1f%% churn risk" 
            customer.Id (customer.ChurnProbability * 100.0))
| Error msg -> printfn "Error: %s" msg

open FSharp.Azure.Quantum.Business

// Find similar products using quantum-enhanced search
let search = similaritySearch {
    catalog productCatalog
    features ["category"; "price"; "rating"; "features"]
    similarityMetric QuantumKernel
    topK 10
}

let targetProduct = "laptop-xyz-2024"

match SimilaritySearch.findSimilar search targetProduct with
| Ok recommendations ->
    printfn "Customers who viewed %s also liked:" targetProduct
    recommendations |> List.iter (fun (product, similarity) ->
        printfn "  %s (%.1f%% similar)" product.Name (similarity * 100.0))
| Error msg -> printfn "Error: %s" msg

open FSharp.Azure.Quantum.Classical.HybridSolver

// TSP with automatic routing
let distances = array2D [[0.0; 10.0; 15.0]; 
                          [10.0; 0.0; 20.0]; 
                          [15.0; 20.0; 0.0]]

match solveTsp distances None None None with
| Ok solution ->
    printfn "Method used: %A" solution.Method           // Classical or Quantum
    printfn "Reasoning: %s" solution.Reasoning          // Why this method?
    printfn "Time: %.2f ms" solution.ElapsedMs
    printfn "Route: %A" solution.Result.Route
    printfn "Distance: %.2f" solution.Result.TotalDistance
| Error msg -> printfn "Error: %s" msg

// MaxCut with quantum backend config
let vertices = ["A"; "B"; "C"; "D"]
let edges = [("A", "B", 1.0); ("B", "C", 2.0); ("C", "D", 1.0)]
let problem = MaxCut.createProblem vertices edges

let quantumConfig = {
    Backend = IonQ "ionq.simulator"
    WorkspaceId = "your-workspace-id"
    Location = "eastus"
    ResourceGroup = "quantum-rg"
    SubscriptionId = "sub-id"
    MaxCostUSD = Some 50.0          // Cost guard
    EnableComparison = true         // Compare with classical
}

match solveMaxCut problem (Some quantumConfig) None None with
| Ok solution ->
    printfn "Method: %A" solution.Method
    printfn "Cut Value: %.2f" solution.Result.CutValue
    match solution.Recommendation with
    | Some recommendation -> printfn "Advisor: %s" recommendation.Reasoning
    | None -> ()
| Error msg -> printfn "Error: %s" msg

// Knapsack
match solveKnapsack knapsackProblem None None None with
| Ok solution ->
    printfn "Total Value: %.2f" solution.Result.TotalValue
    printfn "Items: %A" solution.Result.SelectedItems
| Error msg -> printfn "Error: %s" msg

// Graph Coloring
match solveGraphColoring graphProblem 3 None None None with
| Ok solution ->
    printfn "Colors Used: %d/3" solution.Result.ColorsUsed
    printfn "Valid: %b" solution.Result.IsValid
| Error msg -> printfn "Error: %s" msg

// Portfolio Optimization
match solvePortfolio portfolioProblem None None None with
| Ok solution ->
    printfn "Portfolio Value: $%.2f" solution.Result.TotalValue
    printfn "Expected Return: %.2f%%" (solution.Result.ExpectedReturn * 100.0)
| Error msg -> printfn "Error: %s" msg

graph TB
    subgraph "Layer 1: High-Level Builders"
        GC["GraphColoring Builder<br/>graphColoring { }"]
        MC["MaxCut Builder<br/>MaxCut.createProblem"]
        KS["Knapsack Builder<br/>Knapsack.createProblem"]
        TS["TSP Builder<br/>TSP.createProblem"]
        PO["Portfolio Builder<br/>Portfolio.createProblem"]
        NF["NetworkFlow Builder<br/>NetworkFlow module"]
        SCHED["TaskScheduling Builder<br/>scheduledTask { }"]
    end
    
    subgraph "Layer 2: Quantum Solvers"
        QGC["QuantumGraphColoringSolver<br/>(QAOA)"]
        QMC["QuantumMaxCutSolver<br/>(QAOA)"]
        QKS["QuantumKnapsackSolver<br/>(QAOA)"]
        QTS["QuantumTspSolver<br/>(QAOA)"]
        QPO["QuantumPortfolioSolver<br/>(QAOA)"]
        QNF["QuantumNetworkFlowSolver<br/>(QAOA)"]
        QSCHED["QuantumSchedulingSolver<br/>(QAOA)"]
    end
    
    subgraph "Layer 3: Quantum Backends"
        LOCAL["LocalBackend<br/>(≤20 qubits)"]
        IONQ["IonQBackend<br/>(Azure Quantum)"]
        RIGETTI["RigettiBackend<br/>(Azure Quantum)"]
        ATOM["AtomComputingBackend<br/>(Azure Quantum, 100+ qubits)"]
        QUANTINUUM["QuantinuumBackend<br/>(Azure Quantum, 99.9%+ fidelity)"]
    end
    
    GC --> QGC
    MC --> QMC
    KS --> QKS
    TS --> QTS
    PO --> QPO
    NF --> QNF
    SCHED --> QSCHED
    
    QGC --> LOCAL
    QMC --> LOCAL
    QKS --> LOCAL
    QTS --> LOCAL
    QPO --> LOCAL
    QNF --> LOCAL
    QSCHED --> LOCAL
    
    QGC -.-> IONQ
    QMC -.-> IONQ
    QKS -.-> IONQ
    QTS -.-> IONQ
    QPO -.-> IONQ
    QNF -.-> IONQ
    QSCHED -.-> IONQ
    
    QGC -.-> RIGETTI
    QMC -.-> RIGETTI
    QKS -.-> RIGETTI
    QTS -.-> RIGETTI
    QPO -.-> RIGETTI
    QNF -.-> RIGETTI
    QSCHED -.-> RIGETTI
    
    QGC -.-> ATOM
    QMC -.-> ATOM
    QKS -.-> ATOM
    QTS -.-> ATOM
    QPO -.-> ATOM
    QNF -.-> ATOM
    QSCHED -.-> ATOM
    
    QGC -.-> QUANTINUUM
    QMC -.-> QUANTINUUM
    QKS -.-> QUANTINUUM
    QTS -.-> QUANTINUUM
    QPO -.-> QUANTINUUM
    QNF -.-> QUANTINUUM
    QSCHED -.-> QUANTINUUM
    
    style GC fill:#90EE90
    style MC fill:#90EE90
    style KS fill:#90EE90
    style TS fill:#90EE90
    style PO fill:#90EE90
    style NF fill:#90EE90
    style SCHED fill:#90EE90
    style QGC fill:#FFA500
    style QMC fill:#FFA500
    style QKS fill:#FFA500
    style QTS fill:#FFA500
    style QPO fill:#FFA500
    style QNF fill:#FFA500
    style QSCHED fill:#FFA500
    style LOCAL fill:#5B9BD5
    style IONQ fill:#70AD47
    style RIGETTI fill:#ED7D31
    style ATOM fill:#FFC000
    style QUANTINUUM fill:#A85CC8

// F# computation expression
let problem = graphColoring {
    node "R1" ["R2"]
    colors ["Red"; "Blue"]
}

// Delegates to Layer 2
GraphColoring.solve problem 2 None

// Called internally by GraphColoring.solve
QuantumGraphColoringSolver.solve 
    backend          // IQuantumBackend
    problem          // GraphColoringProblem
    quantumConfig    // QAOA parameters

Backend	Qubits	Speed	Cost	Use Case
LocalBackend	≤20	Fast (ms)	Free	Development, testing, small problems
IonQBackend	29+ (sim), 11 (QPU)	Moderate (seconds)	Paid	Production, large problems
RigettiBackend	40+ (sim), 80 (QPU)	Moderate (seconds)	Paid	Production, large problems
AtomComputingBackend	100+ (sim/QPU Phoenix)	Moderate (seconds)	Paid	Large-scale problems, all-to-all connectivity
QuantinuumBackend	20-32 (sim), 20-32 (QPU)	Moderate (seconds)	Paid	High-fidelity (99.9%+), trapped-ion

// Local simulation (default)
let backend = BackendAbstraction.createLocalBackend()

// Azure Quantum (cloud)
let connectionString = "InstrumentationKey=..."

// Using Azure Quantum Workspace (Recommended for cloud backends)
open FSharp.Azure.Quantum.Backends.AzureQuantumWorkspace

let workspace = createDefault "subscription-id" "resource-group" "workspace-name" "eastus"

// Using Azure Quantum Workspace (Recommended)
open FSharp.Azure.Quantum.Backends.AzureQuantumWorkspace

let workspace = createDefault "subscription-id" "resource-group" "workspace-name" "eastus"

// IonQ Backend
let backend_ionq = BackendAbstraction.createFromWorkspace workspace "ionq.simulator"

// Rigetti Backend  
let backend_rigetti = BackendAbstraction.createFromWorkspace workspace "rigetti.sim.qvm"

// Quantinuum Backend (trapped-ion, highest fidelity)
let backend_quantinuum = BackendAbstraction.createFromWorkspace workspace "quantinuum.sim.h1-1sc"

// Atom Computing Backend (neutral atoms, 100+ qubits, all-to-all connectivity)
let backend_atom = BackendAbstraction.createFromWorkspace workspace "atom-computing.sim"

// Rigetti Backend (superconducting, fast gates)
let backend_rigetti = BackendAbstraction.createRigettiBackend(
    connectionString = "YOUR_CONNECTION_STRING",
    targetId = "rigetti.sim.qvm"  // or "rigetti.qpu.*" for hardware
)

// Atom Computing Backend (neutral atoms, 100+ qubits, all-to-all connectivity)
let backend_atom = BackendAbstraction.createAtomComputingBackend(
    connectionString = "YOUR_CONNECTION_STRING",
    targetId = "atom-computing.sim"  // or "atom-computing.qpu.phoenix" for hardware
)

// Quantinuum Backend (trapped-ion, highest fidelity)
let backend_quantinuum = BackendAbstraction.createQuantinuumBackend(
    connectionString = "YOUR_CONNECTION_STRING",
    targetId = "quantinuum.sim.h1-1sc"  // See available targets below
)

// Pass to solver
match GraphColoring.solve problem 3 (Some backend_quantinuum) with
| Ok solution -> 
    printfn "Backend used: %s" solution.BackendName

open FSharp.Azure.Quantum.Backends.DWaveBackend
open FSharp.Azure.Quantum.Backends.RealDWaveBackend
open FSharp.Azure.Quantum.Backends.DWaveTypes

// Option 1: Mock D-Wave backend (testing, no credentials needed)
let mockBackend = createMockDWaveBackend Advantage_System6_1 (Some 42)

// Option 2: Real D-Wave backend (production, requires API token)
let dwaveConfig = {
    ApiToken = "YOUR_DWAVE_TOKEN"  // Get from https://cloud.dwavesys.com/leap/
    Endpoint = "https://cloud.dwavesys.com/sapi/v2/"
    Solver = "Advantage_system6.1"  // 5640 qubits, Pegasus topology
    TimeoutMs = Some 300000  // 5 minutes
}
let dwaveBackend = RealDWaveBackend.create dwaveConfig

// Option 3: From environment variables (DWAVE_API_TOKEN, DWAVE_SOLVER)
match RealDWaveBackend.createFromEnv() with
| Ok backend -> printfn "D-Wave backend ready: %s" backend.Name
| Error msg -> printfn "No D-Wave credentials: %s" msg

// Build QAOA circuit for MaxCut (automatically converted to QUBO/Ising)
let vertices = ["A"; "B"; "C"; "D"; "E"]
let edges = [
    ("A", "B", 1.0); ("B", "C", 2.0); ("C", "D", 1.0)
    ("D", "E", 1.5); ("E", "A", 1.2)
]

let problem = MaxCut.createProblem vertices edges

// Solve using D-Wave backend (implements IQuantumBackend)
// D-Wave automatically extracts QUBO from QAOA circuit and uses quantum annealing
match MaxCut.solve problem (Some dwaveBackend) with
| Ok solution ->
    printfn "Cut value: %.2f" solution.CutValue
    printfn "Partition S: %A" solution.PartitionS
    printfn "Partition T: %A" solution.PartitionT
| Error msg -> printfn "Error: %s" msg

// Small problem: Use local simulation
let smallProblem = MaxCut.createProblem ["A"; "B"; "C"] [("A","B",1.0)]
let result1 = MaxCut.solve smallProblem None  // Fast, free

// Medium problem: Use Azure Quantum
let mediumProblem = 
    MaxCut.createProblem 
        [for i in 1..20 -> sprintf "V%d" i]
        [for i in 1..19 -> (sprintf "V%d" i, sprintf "V%d" (i+1), 1.0)]

let azureBackend = BackendAbstraction.createIonQBackend(conn, "ionq.simulator")
let result2 = MaxCut.solve mediumProblem (Some azureBackend)  // 20-29 qubits

// Large problem: Use D-Wave quantum annealer
let largeProblem =
    MaxCut.createProblem
        [for i in 1..100 -> sprintf "V%d" i]  // 100 vertices!
        [for i in 1..99 -> (sprintf "V%d" i, sprintf "V%d" (i+1), 1.0)]

// Create D-Wave backend (mock for testing or real for production)
let dwaveBackend = 
    // Option 1: Mock backend (no credentials)
    DWaveBackend.createMockDWaveBackend Advantage_System6_1 None
    
    // Option 2: Real backend (requires DWAVE_API_TOKEN env var)
    // match RealDWaveBackend.createFromEnv() with
    // | Ok backend -> backend
    // | Error _ -> DWaveBackend.createDefaultMockBackend()

let result3 = MaxCut.solve largeProblem (Some dwaveBackend)  // 2000+ qubits

Problem Size	Backend	Qubits	Speed	Cost	Best For
Small (≤20 variables)	LocalBackend	≤20	Milliseconds	Free	Development, testing, prototyping
Medium (20-29 variables)	IonQ/Rigetti	29-80	Seconds	~$10-50/run	Gate-based quantum algorithms (QAOA, VQE)
Medium-High Fidelity (20-32 variables)	Quantinuum	20-32	Seconds	~$50-100/run	High-precision quantum chemistry, error-sensitive algorithms
Large (30-100+ variables)	Atom Computing	100+	Seconds	~$20-80/run	Large-scale optimization, all-to-all connectivity benefits
Very Large (100+ variables)	D-Wave	2000+	Seconds	~$1-10/run	Optimization problems (MaxCut, TSP, scheduling)

type IQuantumBackend =
    /// Execute circuit and return quantum state (not just measurements)
    abstract member ExecuteToState: ICircuit -> Result<QuantumState, QuantumError>
    
    /// Apply single operation to existing state
    abstract member ApplyOperation: QuantumOperation -> QuantumState -> Result<QuantumState, QuantumError>
    
    /// Check if backend supports a specific operation
    abstract member SupportsOperation: QuantumOperation -> bool
    
    /// Initialize quantum state for n qubits
    abstract member InitializeState: int -> Result<QuantumState, QuantumError>
    
    /// Backend's native state representation
    abstract member NativeStateType: QuantumStateType
    
    /// Backend identifier
    abstract member Name: string

open FSharp.Azure.Quantum
open FSharp.Azure.Quantum.Core.BackendAbstraction

// Step 1: Develop locally
let localBackend = createLocalBackend()
let circuit = CircuitBuilder.create 3
              |> addGate (H 0)
              |> addGate (CNOT (0,1))
              |> addGate (CNOT (1,2))

match localBackend.ExecuteToState circuit with
| Ok state -> 
    printfn "Local test passed: %d amplitudes" (state.GetAmplitudes().Length)
| Error e -> 
    printfn "Error: %A" e

// Step 2: Same code on cloud backend (no changes needed!)
let workspace = AzureQuantumWorkspace.createDefault "sub-id" "rg" "workspace" "eastus"
let cloudBackend = createFromWorkspace workspace "ionq.simulator"

match cloudBackend.ExecuteToState circuit with  // Identical call!
| Ok state -> 
    printfn "Cloud execution passed: Backend=%s" cloudBackend.Name
| Error e -> 
    printfn "Error: %A" e

open FSharp.Azure.Quantum.Backends.AzureQuantumWorkspace
open FSharp.Azure.Quantum.Core.BackendAbstraction
open System.Net.Http

// Step 1: Check quota with workspace
use workspace = 
    createDefault 
        "your-subscription-id"
        "your-resource-group"
        "your-workspace-name"
        "eastus"

async {
    // Check remaining quota before execution
    let! quota = workspace.GetTotalQuotaAsync()
    
    match quota.Remaining with
    | Some remaining when remaining < 10.0 ->
        printfn "⚠️  Low quota - stopping"
    | Some remaining ->
        printfn "✅ Sufficient quota: %.2f credits" remaining
        
        // Step 2: Use HTTP backend for proven execution
        use httpClient = new HttpClient()
        let backend = createIonQBackend
            httpClient
            "https://your-workspace.quantum.azure.com"
            "ionq.simulator"
        
        // Step 3: Convert circuit and execute
        let circuit = quantumCircuit { H 0; CNOT 0 1 }
        let wrapper = CircuitWrapper(circuit) :> ICircuit
        
        match convertCircuitToProviderFormat wrapper "ionq.simulator" with
        | Ok json ->
            match backend.Execute wrapper 1000 with
            | Ok result -> printfn "Success!"
            | Error msg -> printfn "Error: %s" msg
        | Error msg -> 
            printfn "Circuit conversion failed: %s" msg
    | None -> 
        printfn "✅ Unlimited quota"
} |> Async.RunSynchronously

// Set environment variables:
// export AZURE_QUANTUM_SUBSCRIPTION_ID="..."
// export AZURE_QUANTUM_RESOURCE_GROUP="..."
// export AZURE_QUANTUM_WORKSPACE_NAME="..."
// export AZURE_QUANTUM_LOCATION="eastus"

match createFromEnvironment() with
| Ok workspace -> 
    printfn "✅ Workspace loaded: %s" workspace.Config.WorkspaceName
| Error msg -> 
    printfn "⚠️  Environment not configured: %s" msg

// Convert circuits to provider-specific formats
let circuit = quantumCircuit { H 0; CNOT 0 1; RX (0, Math.PI / 4.0) }
let wrapper = CircuitWrapper(circuit) :> ICircuit

// To IonQ JSON
match convertCircuitToProviderFormat wrapper "ionq.simulator" with
| Ok ionqJson -> printfn "IonQ: %s" ionqJson
| Error msg -> printfn "Error: %s" msg

// To Rigetti Quil
match convertCircuitToProviderFormat wrapper "rigetti.sim.qvm" with
| Ok quilProgram -> printfn "Quil: %s" quilProgram
| Error msg -> printfn "Error: %s" msg

open FSharp.Azure.Quantum.Backends.AzureQuantumWorkspace
open FSharp.Azure.Quantum.Core.BackendAbstraction

// Step 1: Create workspace
use workspace = 
    createDefault 
        "your-subscription-id"
        "your-resource-group"
        "your-workspace-name"
        "eastus"

// Step 2: Create SDK backend (NEW!)
let backend = createFromWorkspace workspace "ionq.simulator"

// Step 3: Build circuit
let circuit = quantumCircuit {
    H 0
    CNOT 0 1
    MEASURE_ALL
}

let wrapper = CircuitWrapper(circuit) :> ICircuit

// Step 4: Execute on Azure Quantum
match backend.Execute wrapper 1000 with
| Ok result ->
    printfn "✅ Job completed!"
    printfn "   Backend: %s" result.BackendName
    printfn "   Shots: %d" result.NumShots
    printfn "   Job ID: %s" (result.Metadata.["job_id"] :?> string)
    
    // Analyze measurements
    let counts = result.Measurements |> Array.countBy id
    counts |> Array.iter (fun (bitstring, count) ->
        printfn "   %A: %d times" bitstring count)
| Error msg ->
    printfn "❌ Error: %s" msg

async {
    // Check quota before execution
    let! quota = workspace.GetTotalQuotaAsync()
    
    match quota.Remaining with
    | Some remaining when remaining < 10.0 ->
        printfn "⚠️  Low quota: %.2f credits - stopping" remaining
    | Some remaining ->
        printfn "✅ Quota available: %.2f credits" remaining
        
        // Create backend and execute
        let backend = createFromWorkspace workspace "ionq.simulator"
        match backend.Execute circuit 1000 with
        | Ok result -> printfn "Success!"
        | Error msg -> printfn "Error: %s" msg
    | None ->
        printfn "✅ Unlimited quota"
        // Execute...
} |> Async.RunSynchronously

Feature	LocalBackend	HTTP Backend	SDK Backend
Setup	None	HttpClient + URL	Workspace object
Quota Checking	❌	❌	✅
Provider Discovery	❌	❌	✅
Job Polling	❌ (instant)	Manual	✅ Automatic
Resource Cleanup	❌	Manual	✅ IDisposable
Circuit Conversion	❌	Manual	✅ Automatic
Max Qubits	20	29 (IonQ) / 40 (Rigetti)	29 (IonQ) / 40 (Rigetti)
Cost	Free	Paid	Paid
Production Ready	✅	✅	✅
Best For	Testing	Manual control	Full integration

// open FSharp.Azure.Quantum
// open FSharp.Azure.Quantum.CircuitBuilder

// Build circuit using F# circuit builder
let circuit = 
    CircuitBuilder.empty 2
    |> CircuitBuilder.addGate (H 0)
    |> CircuitBuilder.addGate (CNOT (0, 1))
    |> CircuitBuilder.addGate (RZ (0, System.Math.PI / 4.0))

// Export to OpenQASM 2.0 string
let qasmCode = OpenQasm.export circuit
printfn "%s" qasmCode

// Export to .qasm file
OpenQasm.exportToFile circuit "bell_state.qasm"

OPENQASM 2.0;
include "qelib1.inc";
qreg q[2];
h q[0];
cx q[0],q[1];
rz(0.7853981634) q[0];

// open FSharp.Azure.Quantum
// open System.IO

// Parse OpenQASM string
let qasmCode = """
OPENQASM 2.0;
include "qelib1.inc";
qreg q[3];
h q[0];
cx q[0],q[1];
cx q[1],q[2];
"""

match OpenQasmImport.parse qasmCode with
| Ok circuit ->
    printfn "Loaded %d-qubit circuit with %d gates" 
        circuit.QubitCount circuit.Gates.Length
    // Use circuit with LocalBackend or export to another format
| Error msg -> 
    printfn "Parse error: %s" msg

// Import from file
match OpenQasmImport.parseFromFile "grover.qasm" with
| Ok circuit -> printfn "Loaded %d-qubit circuit" circuit.QubitCount
| Error msg -> printfn "Error: %s" msg

using FSharp.Azure.Quantum;
using FSharp.Azure.Quantum.CircuitBuilder;

// Export circuit to OpenQASM
var circuit = CircuitBuilder.empty(2)
    .AddGate(Gate.NewH(0))
    .AddGate(Gate.NewCNOT(0, 1));

var qasmCode = OpenQasm.export(circuit);
File.WriteAllText("circuit.qasm", qasmCode);

// Import from OpenQASM
var qasmInput = File.ReadAllText("qiskit_circuit.qasm");
var result = OpenQasmImport.parse(qasmInput);

if (result.IsOk) {
    var imported = result.ResultValue;
    Console.WriteLine($"Loaded {imported.QubitCount}-qubit circuit");
}

// 1. Load circuit from Qiskit
let qiskitCircuit = OpenQasmImport.parseFromFile "qiskit_algorithm.qasm"

match qiskitCircuit with
| Ok circuit ->
    // 2. Run on LocalBackend for testing
    let localBackend = BackendAbstraction.createLocalBackend()
    let testResult = LocalSimulator.QaoaSimulator.simulate circuit 1000
    
    printfn "Local test: %d samples" testResult.Shots
    
    // 3. Transpile for IonQ hardware
    let transpiled = GateTranspiler.transpileForBackend "ionq.qpu" circuit
    
    // 4. Execute on IonQ
    let ionqBackend = BackendAbstraction.createIonQBackend(
        connectionString,
        "ionq.qpu"
    )
    
    // 5. Export results back to Qiskit format
    OpenQasm.exportToFile transpiled "results_ionq.qasm"
| Error msg -> 
    printfn "Import failed: %s" msg

// Original circuit
let original = { QubitCount = 3; Gates = [H 0; CNOT (0, 1); RZ (1, 1.5708)] }

// Export → Import → Compare
let qasm = OpenQasm.export original
let imported = OpenQasmImport.parse qasm

match imported with
| Ok circuit ->
    assert (circuit.QubitCount = original.QubitCount)
    assert (circuit.Gates.Length = original.Gates.Length)
    printfn "✅ Round-trip successful"
| Error msg -> 
    printfn "❌ Round-trip failed: %s" msg

open FSharp.Azure.Quantum.ErrorMitigation

// Configure ZNE
let zneConfig = {
    NoiseScalings = [
        ZeroNoiseExtrapolation.IdentityInsertion 0.0    // baseline (1.0x)
        ZeroNoiseExtrapolation.IdentityInsertion 0.5    // 1.5x noise
        ZeroNoiseExtrapolation.IdentityInsertion 1.0    // 2.0x noise
    ]
    PolynomialDegree = 2
    MinSamples = 1000
}

// Apply ZNE to circuit expectation value
// Mock circuit and observable for demonstration
let circuit = QuantumCircuit.empty()  // Mock circuit
let observable = PauliOperator.Z(0)   // Mock observable

async {
    let! result = ZeroNoiseExtrapolation.mitigate circuit observable zneConfig backend
    
    match result with
    | Ok zneResult ->
        printfn "Zero-noise value: %f" zneResult.ZeroNoiseValue
        printfn "R² goodness of fit: %f" zneResult.GoodnessOfFit
        printfn "Measured values:"
        zneResult.MeasuredValues 
        |> List.iter (fun (noise, value) -> printfn "  λ=%.1f: %f" noise value)
    | Error msg -> 
        printfn "ZNE failed: %s" msg
}

open FSharp.Azure.Quantum.ErrorMitigation

// Configure PEC with noise model
let pecConfig = {
    NoiseModel = {
        SingleQubitDepolarizing = 0.001  // 0.1% per single-qubit gate
        TwoQubitDepolarizing = 0.01      // 1% per two-qubit gate
        ReadoutError = 0.02              // 2% readout error
    }
    Samples = 1000
    Seed = Some 42
}

// Apply PEC to circuit
async {
    let! result = ProbabilisticErrorCancellation.mitigate circuit observable pecConfig backend
    
    match result with
    | Ok pecResult ->
        printfn "Corrected expectation: %f" pecResult.CorrectedExpectation
        printfn "Uncorrected (noisy): %f" pecResult.UncorrectedExpectation
        printfn "Error reduction: %.1f%%" (pecResult.ErrorReduction * 100.0)
        printfn "Overhead: %.1fx" pecResult.Overhead
    | Error msg -> 
        printfn "PEC failed: %s" msg
}

open FSharp.Azure.Quantum.ErrorMitigation

// Step 1: Calibrate (one-time cost per backend)
let remConfig = 
    ReadoutErrorMitigation.defaultConfig
    |> ReadoutErrorMitigation.withCalibrationShots 10000
    |> ReadoutErrorMitigation.withConfidenceLevel 0.95

async {
    // Calibrate confusion matrix (run once, cache result)
    let! calibrationResult = ReadoutErrorMitigation.calibrate remConfig backend
    
    match calibrationResult with
    | Ok calibMatrix ->
        printfn "Calibration complete:"
        printfn "  Qubits: %d" calibMatrix.Qubits
        printfn "  Shots: %d" calibMatrix.CalibrationShots
        printfn "  Backend: %s" calibMatrix.Backend
        
        // Step 2: Correct measurement histogram (zero overhead!)
        let noisyHistogram = Map.ofList [("00", 0.9); ("01", 0.05); ("10", 0.03); ("11", 0.02)]  // Mock noisy results
        let correctedResult = ReadoutErrorMitigation.correctHistogram noisyHistogram calibMatrix remConfig
        
        match correctedResult with
        | Ok corrected ->
            printfn "\nCorrected histogram:"
            corrected.Histogram 
            |> Map.iter (fun state prob -> printfn "  |%s⟩: %.4f" state prob)
            
            printfn "\nConfidence intervals (95%%):"
            corrected.ConfidenceIntervals
            |> Map.iter (fun state (lower, upper) -> 
                printfn "  |%s⟩: [%.4f, %.4f]" state lower upper)
        | Error msg -> 
            printfn "Correction failed: %s" msg
    | Error msg -> 
        printfn "Calibration failed: %s" msg
}

open FSharp.Azure.Quantum.ErrorMitigation

// Define selection criteria
let criteria = {
    CircuitDepth = 25
    QubitCount = 6
    Backend = Types.Backend.IonQBackend
    MaxCostUSD = Some 10.0
    RequiredAccuracy = Some 0.95
}

// Get recommended strategy
let recommendation = ErrorMitigationStrategy.selectStrategy criteria

printfn "Recommended: %s" (
    match recommendation.Primary with
    | ZeroNoiseExtrapolation _ -> "Zero-Noise Extrapolation (ZNE)"
    | ProbabilisticErrorCancellation _ -> "Probabilistic Error Cancellation (PEC)"
    | ReadoutErrorMitigation _ -> "Readout Error Mitigation (REM)"
    | Combined _ -> "Combined Techniques"
)
printfn "Reasoning: %s" recommendation.Reasoning
printfn "Estimated cost multiplier: %.1fx" recommendation.EstimatedCostMultiplier
printfn "Estimated accuracy: %.1f%%" (recommendation.EstimatedAccuracy * 100.0)

// Apply recommended strategy
let noisyHistogram = Map.ofList [("00", 0.9); ("01", 0.05); ("10", 0.03); ("11", 0.02)]  // Mock noisy results
let mitigatedResult = ErrorMitigationStrategy.applyStrategy noisyHistogram recommendation.Primary

match mitigatedResult with
| Ok result ->
    printfn "\nMitigation successful:"
    printfn "  Technique: %A" result.AppliedTechnique
    printfn "  Used fallback: %b" result.UsedFallback
    printfn "  Actual cost: %.1fx" result.ActualCostMultiplier
    result.Histogram |> Map.iter (fun k v -> printfn "    %s: %f" k v)
| Error msg ->
    printfn "Mitigation failed: %s" msg

Circuit Type	Best Technique	Error Reduction	Cost	Why
Shallow (< 20 gates)	REM	50-90%	~0x	Readout dominates
Medium (20-50 gates)	ZNE	30-50%	3x	Balanced gate/readout
Deep (> 50 gates)	PEC or ZNE+REM	40-70%	10-100x	High gate errors
Cost-constrained	REM	50-90%	~0x	Free after calibration
High accuracy	PEC	2-3x	10-100x	Research/benchmarking

open FSharp.Azure.Quantum
open FSharp.Azure.Quantum.ErrorMitigation

// Define MaxCut problem
let vertices = ["A"; "B"; "C"; "D"]
let edges = [
    ("A", "B", 1.0)
    ("B", "C", 2.0)
    ("C", "D", 1.0)
    ("D", "A", 1.0)
]

let problem = MaxCut.problem vertices edges

// Solve with ZNE error mitigation
let zneConfig = {
    NoiseScalings = [
        ZeroNoiseExtrapolation.IdentityInsertion 0.0
        ZeroNoiseExtrapolation.IdentityInsertion 0.5
        ZeroNoiseExtrapolation.IdentityInsertion 1.0
    ]
    PolynomialDegree = 2
    MinSamples = 1000
}

async {
    // Standard solve (noisy)
    let! noisyResult = MaxCut.solve problem None
    
    // Solve with ZNE (error-mitigated)
    let! mitigatedResult = MaxCut.solveWithErrorMitigation problem (Some zneConfig) None
    
    match noisyResult, mitigatedResult with
    | Ok noisy, Ok mitigated ->
        printfn "Noisy cut value: %.2f" noisy.CutValue
        printfn "ZNE-mitigated cut value: %.2f" mitigated.CutValue
        printfn "Improvement: %.1f%%" ((mitigated.CutValue - noisy.CutValue) / noisy.CutValue * 100.0)
    | _ -> 
        printfn "Error occurred"
}

// Custom QAOA parameters
let quantumConfig : QuantumGraphColoringSolver.QuantumGraphColoringConfig = {
    OptimizationShots = 100        // Shots per optimization step
    FinalShots = 1000              // Shots for final measurement
    EnableOptimization = true      // Enable parameter optimization
    InitialParameters = (0.5, 0.5) // Starting (gamma, beta)
}

// Use custom config
let backend = BackendAbstraction.createLocalBackend()
match QuantumGraphColoringSolver.solve backend problem quantumConfig with
| Ok result -> printfn "Colors used: %d" result.ColorsUsed
| Error msg -> printfn "Error: %s" msg

Problem Type	Small (LocalBackend)	Medium	Large (Cloud Required)
Graph Coloring	≤20 nodes	20-30 nodes	30+ nodes
MaxCut	≤20 vertices	20-30 vertices	30+ vertices
Knapsack	≤20 items	20-30 items	30+ items
TSP	≤8 cities	8-12 cities	12+ cities
Portfolio	≤20 assets	20-30 assets	30+ assets
Network Flow	≤15 nodes	15-25 nodes	25+ nodes
Task Scheduling	≤15 tasks	15-25 tasks	25+ tasks

// ✅ QUANTUM: QAOA-based optimization
GraphColoring.solve problem 3 None

// ❌ NO CLASSICAL FALLBACK: If quantum fails, returns Error
// Users should use dedicated classical libraries for that use case

open FSharp.Azure.Quantum.GroverSearch

// Search for item satisfying predicate
let searchConfig = {
    MaxIterations = Some 10
    SuccessThreshold = 0.9
    OptimizeIterations = true
    Shots = 1000
    RandomSeed = Some 42
}

// Search 8-item space for items where f(x) = true
let predicate x = x = 3 || x = 5  // Looking for indices 3 or 5

match Search.searchWithPredicate 3 predicate searchConfig with
| Ok result ->
    printfn "Found solutions: %A" result.Solutions
    printfn "Success probability: %.2f%%" (result.SuccessProbability * 100.0)
    printfn "Iterations: %d" result.IterationsApplied
| Error msg -> 
    printfn "Search failed: %s" msg

open FSharp.Azure.Quantum.GroverSearch.AmplitudeAmplificationAdapter
open FSharp.Azure.Quantum.Core.BackendAbstraction

// Execute amplitude amplification on quantum hardware backend
let backend = createIonQBackend(connectionString, "ionq.simulator")

// Define oracle (marks solution states)
let oracle = StateOracle(fun state -> state = 3 || state = 5)

// Configure amplitude amplification
let config = {
    NumQubits = 3
    StatePreparation = UniformSuperposition  // or BasisState(n), PartialSuperposition(states)
    Oracle = oracle
    Iterations = 2  // Optimal iterations for 2 solutions in 8-element space
}

match executeWithBackend config backend 1000 with
| Ok measurementCounts ->
    printfn "Amplitude amplification executed on quantum hardware"
    measurementCounts 
    |> Map.iter (fun state count -> printfn "  |%d⟩: %d counts" state count)
| Error msg ->
    printfn "Backend execution failed: %s" msg

// Convenience functions
executeUniformAmplification 3 oracle 2 backend 1000      // Uniform initial state
executeBasisStateAmplification 3 5 oracle 2 backend 1000 // Start from |5⟩

QFT: |j⟩ → (1/√N) Σₖ e^(2πijk/N) |k⟩

open FSharp.Azure.Quantum.GroverSearch.QFTBackendAdapter
open FSharp.Azure.Quantum.Core.BackendAbstraction

// Execute QFT on quantum hardware backend
let backend = createIonQBackend(connectionString, "ionq.simulator")
let config = { NumQubits = 5; ApplySwaps = true; Inverse = false }

match executeQFTWithBackend config backend 1000 None with
| Ok measurementCounts ->
    printfn "QFT executed on quantum hardware"
    measurementCounts 
    |> Map.iter (fun state count -> printfn "  |%d⟩: %d counts" state count)
| Error msg ->
    printfn "Backend execution failed: %s" msg

// Convenience functions
executeStandardQFT 5 backend 1000      // Standard QFT with swaps
executeInverseQFT 5 backend 1000       // Inverse QFT (QFT†)
executeQFTOnState 5 7 backend 1000     // QFT on specific input state |7⟩

open FSharp.Azure.Quantum.QuantumArithmeticOps

// RSA encryption: m^e mod n
let encryptOp = quantumArithmetic {
    operands 5 3           // message=5, exponent=3
    operation ModularExponentiate
    modulus 33             // RSA modulus
    qubits 8
}

match execute encryptOp with
| Ok result -> 
    printfn "Encrypted: %d" result.Value
    printfn "Gates: %d, Depth: %d" result.GateCount result.CircuitDepth
| Error msg -> 
    printfn "Error: %s" msg

using static FSharp.Azure.Quantum.CSharpBuilders;

var encrypt = ModularExponentiate(baseValue: 5, exponent: 3, modulus: 33);
var result = ExecuteArithmetic(encrypt);

open FSharp.Azure.Quantum.QuantumPeriodFinder

// Factor RSA modulus (security analysis)
let problem = periodFinder {
    number 15              // Composite to factor
    precision 8            // QPE precision
    maxAttempts 10         // Retries (probabilistic)
}

match solve problem with
| Ok result ->
    match result.Factors with
    | Some (p, q) -> 
        printfn "Factors: %d × %d" p q
        printfn "⚠️  RSA Security Broken!"
    | None -> 
        printfn "Try again (probabilistic)"
| Error msg -> 
    printfn "Error: %s" msg

var problem = FactorInteger(15, precision: 8);
var result = ExecutePeriodFinder(problem);

open FSharp.Azure.Quantum.QuantumPhaseEstimator

// Estimate molecular ground state energy
let problem = phaseEstimator {
    unitary (RotationZ (Math.PI / 3.0))  // Molecular Hamiltonian
    precision 12                          // 12-bit energy precision
}

match estimate problem with
| Ok result ->
    printfn "Phase: %.6f" result.Phase
    printfn "Energy: %.4f a.u." (result.Phase * 2.0 * Math.PI)
    printfn "Application: Drug binding affinity prediction"
| Error msg ->
    printfn "Error: %s" msg

var problem = EstimateRotationZ(Math.PI / 3.0, precision: 12);
var result = ExecutePhaseEstimator(problem);

Feature Category	FSharp.Azure.Quantum	IBM Qiskit	Microsoft Azure Quantum SDK	Google Cirq	Amazon Braket SDK
Primary Language	F# (with C# interop)	Python	Python, C#, Q#	Python	Python
License	Unlicense (Public Domain)	Apache 2.0	MIT	Apache 2.0	Apache 2.0
Target Audience	.NET developers, optimization problems	General quantum computing	Enterprise quantum developers	Google hardware users	AWS cloud users

🎯 OPTIMIZATION SOLVERS
MaxCut	✅ Built-in (QAOA)	✅ Qiskit Optimization	❌ Manual	❌ Manual	❌ Manual
Knapsack	✅ Built-in (QAOA)	✅ Qiskit Optimization	❌ Manual	❌ Manual	❌ Manual
TSP	✅ Built-in (QAOA)	✅ Qiskit Optimization	❌ Manual	❌ Manual	❌ Manual
Portfolio Optimization	✅ Built-in (QAOA)	✅ Qiskit Finance	❌ Manual	❌ Manual	❌ Manual
Task Scheduling	✅ Built-in (QAOA)	❌ Manual	❌ Manual	❌ Manual	❌ Manual
Network Flow	✅ Built-in (QAOA)	❌ Manual	❌ Manual	❌ Manual	❌ Manual
Graph Coloring	✅ Built-in (QAOA)	❌ Manual implementation	❌ Manual	❌ Manual	❌ Manual

🤖 QUANTUM MACHINE LEARNING
VQC (Variational Classifier)	✅ Built-in	✅ Qiskit Machine Learning	❌ Manual	✅ TFQ integration	✅ Built-in
Quantum Kernel SVM	✅ Built-in	✅ Qiskit Machine Learning	❌ Manual	❌ Limited	✅ Built-in
Feature Maps	✅ ZZ, Pauli, Angle	✅ Extensive library	❌ Manual	✅ Via TFQ	✅ Built-in
Variational Forms	✅ RealAmplitudes, EfficientSU2	✅ Extensive ansatz library	❌ Manual	✅ Via Cirq	✅ Built-in
AutoML Integration	✅ Built-in	❌ External tools	❌ No	❌ No	❌ No

📊 BUSINESS PROBLEM BUILDERS
Anomaly Detection	✅ Built-in	❌ Manual	❌ No	❌ No	❌ No
Binary Classification	✅ Built-in	✅ Qiskit ML	❌ No	❌ No	❌ No
Predictive Modeling	✅ Built-in	❌ Manual	❌ No	❌ No	❌ No
Similarity Search	✅ Built-in	❌ Manual	❌ No	❌ No	❌ No

🔬 QUANTUM ALGORITHMS
QAOA	✅ Production-ready, auto-optimized	✅ Qiskit Optimization	✅ Q# samples	✅ Manual	✅ Built-in
VQE	✅ Built-in (chemistry)	✅ Qiskit Nature	✅ Q# samples	✅ Built-in	✅ Built-in
Grover's Algorithm	✅ Educational	✅ Built-in	✅ Q# samples	✅ Built-in	✅ Built-in
Shor's Algorithm	✅ Educational (period finder)	✅ Built-in	✅ Q# samples	✅ Built-in	✅ Built-in
QFT	✅ Built-in	✅ Built-in	✅ Q# built-in	✅ Built-in	✅ Built-in
HHL (Linear Systems)	✅ Built-in	✅ Qiskit Aqua	❌ Manual	❌ Manual	❌ Manual
Amplitude Amplification	✅ Built-in	✅ Built-in	✅ Q# built-in	✅ Built-in	❌ Manual

🖥️ LOCAL SIMULATION
Local Simulator	✅ Built-in (≤20 qubits)	✅ Aer (≤30 qubits)	✅ Full-state (≤30 qubits)	✅ Built-in (≤20 qubits)	✅ Local simulator
Noise Simulation	❌ Some	✅ AerSimulator noise models	✅ Open/Closed systems	✅ Built-in	✅ Built-in
GPU Acceleration	❌ No	✅ Aer GPU	✅ Yes	✅ Yes	✅ Yes
State Vector	✅ Pure F# implementation	✅ C++ backend	✅ C++ backend	✅ C++ backend	✅ C++ backend

☁️ CLOUD BACKENDS
Azure Quantum (IonQ)	✅ Native	✅ Via Qiskit Runtime	✅ Native	❌ No	❌ No
Azure Quantum (Rigetti)	✅ Native	✅ Via Qiskit Runtime	✅ Native	❌ No	❌ No
IBM Quantum	❌ Via OpenQASM export	✅ Native	❌ No	❌ No	❌ No
D-Wave Quantum Annealer	✅ Native	✅ Via Ocean SDK	✅ Native	❌ No	✅ Native
AWS Braket	❌ Via OpenQASM export	✅ Via plugin	❌ No	❌ No	✅ Native
Google Quantum	❌ Via OpenQASM export	✅ Via plugin	❌ No	✅ Native	❌ No

🔄 INTEROPERABILITY
OpenQASM 2.0 Import	✅ Full support	✅ Native	✅ Via conversion	✅ Full support	✅ Full support
OpenQASM 2.0 Export	✅ Full support	✅ Native	✅ Via conversion	✅ Full support	✅ Full support
QUIL	❌ No	❌ Via plugin	✅ Rigetti native	❌ No	✅ Rigetti support

🛡️ ERROR MITIGATION
Zero-Noise Extrapolation	✅ Built-in (30-50% reduction)	✅ Qiskit Experiments	❌ Manual	✅ Via Mitiq integration	❌ Manual
Probabilistic Error Cancellation	✅ Built-in (2-3x accuracy)	✅ Via Mitiq	❌ Manual	✅ Via Mitiq integration	❌ Manual
Readout Error Mitigation	✅ Built-in (50-90% reduction)	✅ Qiskit Experiments	❌ Manual	✅ Via Mitiq integration	❌ Manual
Automatic Strategy Selection	✅ Built-in	❌ Manual	❌ No	❌ Manual	❌ No

💻 API DESIGN
Computation Expressions	✅ F# native pattern	❌ N/A (Python)	❌ No	❌ N/A (Python)	❌ N/A (Python)
Type Safety	✅ F# compile-time checks	⚠️ Python dynamic typing	⚠️ Python/C# mixed	⚠️ Python dynamic typing	⚠️ Python dynamic typing
Fluent API (C#)	✅ Built-in	❌ N/A	✅ Native C#	❌ N/A	❌ N/A
Functional Programming	✅ F# first-class	❌ Object-oriented	⚠️ Mixed	⚠️ Mixed	❌ Object-oriented
Result Type Error Handling	✅ F# Result<T,E>	❌ Exceptions	❌ Exceptions	❌ Exceptions	❌ Exceptions

🤖 HYBRID CLASSICAL-QUANTUM
Automatic Problem Routing	✅ HybridSolver (optional)	❌ Manual	❌ Manual	❌ Manual	❌ Manual
Classical Fallback	✅ Built-in (small problems)	❌ Manual	❌ No	❌ No	❌ No
Cost Guards	✅ MaxCostUSD limits	❌ Manual	❌ Manual	❌ Manual	❌ Manual
Quantum Advantage Analysis	✅ Built-in reasoning	❌ Manual	❌ No	❌ No	❌ No

🧪 QUANTUM CHEMISTRY
VQE for Molecules	✅ Built-in (H₂, H₂O)	✅ Qiskit Nature	✅ Q# Chemistry	✅ OpenFermion integration	✅ Built-in
Hamiltonian Construction	✅ Built-in	✅ Qiskit Nature	✅ Broombridge format	✅ OpenFermion	✅ OpenFermion
UCC Ansatz	✅ Built-in	✅ Qiskit Nature	✅ Q# Chemistry	✅ OpenFermion	✅ Built-in

📚 ECOSYSTEM
Circuit Visualization	❌ Mermaid, or Export to OpenQASM → Qiskit	✅ Native (matplotlib)	✅ Q# visualizer	✅ Native (matplotlib)	✅ Native (matplotlib)
Documentation Quality	✅ Comprehensive (MD docs)	✅ Extensive (tutorials)	✅ Microsoft Docs	✅ Google Docs	✅ AWS Docs
Example Projects	✅ 30+ working examples	✅ 100+ tutorials	⚠️ Limited examples	✅ 50+ tutorials	✅ 40+ examples
Community Size	⚠️ Small (new library)	✅ Large (6.7k stars)	⚠️ Medium	✅ Medium (Google)	⚠️ Medium
GitHub Stars	⚠️ New project	✅ 6,700+	⚠️ Not standalone repo	✅ 4,000+	✅ 800+
Stack Overflow Support	⚠️ Limited (F# quantum niche)	✅ Extensive	⚠️ Medium	⚠️ Medium	⚠️ Limited

🔧 DEVELOPMENT EXPERIENCE
IDE Support	✅ Visual Studio, VS Code	✅ Jupyter, VS Code	✅ Visual Studio, VS Code	✅ Jupyter, VS Code	✅ Jupyter, VS Code
REPL/Interactive	✅ F# Interactive (FSI)	✅ Jupyter Notebooks	✅ Q# Jupyter	✅ Jupyter Notebooks	✅ Jupyter Notebooks
Package Manager	✅ NuGet	✅ pip	✅ NuGet, pip	✅ pip	✅ pip
Installation	✅ dotnet add package	✅ pip install qiskit	✅ pip install azure-quantum	✅ pip install cirq	✅ pip install amazon-braket-sdk

⚡ PERFORMANCE
Small Problems (<10 qubits)	✅ LocalBackend (ms)	✅ Aer (ms)	✅ Full-state (ms)	✅ Cirq (ms)	✅ Local (ms)
Medium Problems (10-20 qubits)	✅ LocalBackend (<1s)	✅ Aer (<1s)	✅ Full-state (<1s)	✅ Cirq (<10s)	✅ Local (<1s)
Large Problems (20+ qubits)	⚠️ Cloud required	✅ Aer GPU (30 qubits)	✅ Cloud	⚠️ Cloud required	✅ Cloud

💰 COST
Local Development	✅ Free	✅ Free	✅ Free	✅ Free	✅ Free
Cloud QPU Access	💰 Azure Quantum pricing	💰 IBM Quantum pricing	💰 Azure Quantum pricing	💰 Google Quantum pricing	💰 AWS Braket pricing
D-Wave Quantum	💰 ~-10/run	💰 Via Ocean SDK	💰 Azure marketplace	❌ N/A	💰 AWS Braket

open FSharp.Azure.Quantum.Topological

// Create backend for Ising anyons (Microsoft's approach)
let backend = TopologicalBackend.createSimulator AnyonSpecies.AnyonType.Ising 10

// Create entangled state via braiding
let! result = topological backend {
    // Initialize 4 sigma anyons
    do! TopologicalBuilder.initialize AnyonSpecies.AnyonType.Ising 4
    
    // Braiding creates entanglement geometrically
    do! TopologicalBuilder.braid 0  // Braid anyons 0-1
    do! TopologicalBuilder.braid 2  // Braid anyons 2-3
    
    // Measure fusion outcome
    let! (outcome, _) = TopologicalBuilder.measure 0
    return outcome
}

match result with
| Ok particle ->
    printfn "Fusion outcome: %A" particle  // Vacuum or Psi
| Error err ->
    printfn "Error: %s" err.Message

Aspect	Gate-Based QC	Topological QC
State	Qubit amplitudes	Fusion trees
Operations	H, CNOT, RZ gates	Braid, Measure
Error Correction	Active (surface codes)	Passive (topology)
Hardware	IonQ, Rigetti, IBM	Microsoft Majorana (experimental)

Category	Gates
Pauli	X, Y, Z, H
Phase	S, S†, T, T†
Rotation	RX(θ), RY(θ), RZ(θ)
Two-qubit	CNOT (CX), CZ, SWAP
Three-qubit	CCX (Toffoli)

Thorium/FSharp.Azure.Quantumv1.3.1

Get Started

Readme

FSharp.Azure.Quantum

Table of Contents

Status

Quick Start

Installation

F# Computation Expressions

C# Fluent API

Problem Builders

Graph Coloring

MaxCut

Knapsack (0/1)

Traveling Salesperson Problem (TSP)

Portfolio Optimization

Network Flow

Task Scheduling

Advanced Quantum Builders

Quantum Tree Search Builder

Quantum Constraint Solver Builder

Quantum Pattern Matcher Builder

Quantum Arithmetic Builder

Quantum Period Finder Builder

Quantum Phase Estimator Builder

Advanced Builder Features

C# API for Advanced Builders

Library Scope & Focus

Quantum Machine Learning (QML)

Variational Quantum Classifier (VQC)

Quantum Kernel SVM

Business Problem Builders

AutoML - Automated Machine Learning

Anomaly Detection - Security & Fraud Detection

Binary Classification - Fraud Detection

Predictive Modeling - Customer Churn Prediction

Similarity Search - Product Recommendations

HybridSolver - Automatic Classical/Quantum Routing

Supported Problems

Features

When to Use HybridSolver vs Direct Builders

Architecture

3-Layer Quantum-Only Architecture

Layer Responsibilities

Layer 1: High-Level Builders 🟢

Layer 2: Quantum Solvers 🟠

Layer 3: Quantum Backends 🔵

D-Wave Quantum Annealer

Backend Comparison

Unified Backend Architecture

Azure Quantum Workspace Management

SDK Backend - Full Azure Quantum Integration

OpenQASM 2.0 Support

Why OpenQASM?

Export Circuits to OpenQASM

Import Circuits from OpenQASM

C# API

Supported Gates

Workflow: Qiskit → F# → IonQ

Round-Trip Compatibility

Use Cases

Error Mitigation

Why Error Mitigation?

Available Techniques

1️⃣ Zero-Noise Extrapolation (ZNE)

2️⃣ Probabilistic Error Cancellation (PEC)

3️⃣ Readout Error Mitigation (REM)

4️⃣ Automatic Strategy Selection

Decision Matrix

Real-World Example: MaxCut with ZNE

Testing & Validation

Further Reading

QAOA Algorithm Internals

How Quantum Optimization Works

QAOA Configuration

Documentation

Problem Size Guidelines

Design Philosophy

Rule 1: Quantum-Only Library

Clean API Layers