Advanced Quantum Builders

Target Audience: Researchers, algorithm developers, quantum computing enthusiasts

This guide covers advanced quantum computation builders designed for specialized research and educational applications. These tools demonstrate cutting-edge quantum algorithms that provide theoretical advantages but require fault-tolerant quantum hardware for practical use.

⚠️ Current Limitations: Most features require 100-4000+ qubits with low error rates. Current NISQ (Noisy Intermediate-Scale Quantum) hardware is limited to ~100 qubits with high error rates, so only toy examples work on real quantum computers today.

Table of Contents

  1. Quantum Tree Search - Game AI with Grover’s algorithm
  2. Quantum Constraint Solver - CSP solving (Sudoku, N-Queens)
  3. Quantum Pattern Matcher - Configuration optimization, hyperparameter tuning
  4. Quantum Arithmetic - Modular arithmetic for cryptography
  5. Period Finder (Shor’s Algorithm) - RSA factorization
  6. Phase Estimator - Eigenvalue extraction for quantum chemistry
  7. When to Use These Builders
  8. Performance & Cost Comparison

Quantum Tree Search uses Grover’s algorithm to explore game trees and decision trees quadratically faster than classical minimax search. It’s ideal for scenarios where position evaluation is expensive (e.g., neural network evaluation in chess engines).

When to Use

✅ Good Fits:

❌ Not Suitable For:

API Reference

Basic Usage:

open FSharp.Azure.Quantum
open FSharp.Azure.Quantum.QuantumTreeSearch

let problem = quantumTreeSearch {
    initialState startingGameState
    maxDepth 4                    // Search 4 moves ahead
    branchingFactor 16            // Average moves per position
    evaluateWith evaluatePosition // Evaluation function
    generateMovesWith generateMoves
    topPercentile 0.2             // Consider top 20% of moves
    backend localBackend
    shots 100
}

match solve problem with
| Ok result ->
    printfn "Best move: %d" result.BestMove
    printfn "Evaluation score: %.4f" result.Score
    printfn "Paths explored: %d" result.PathsExplored
| Error err ->
    printfn "Error: %s" err.Message

Configuration Options:

Option Type Description Default
initialState 'State Starting game/decision state Required
maxDepth int Maximum search depth (plies) Required
branchingFactor int Average moves per position Required
evaluateWith 'State -> float Position evaluation function Required
generateMovesWith 'State -> 'State list Generate successor states Required
topPercentile float Consider top X% of moves (0.0-1.0) 0.2
backend IQuantumBackend Quantum backend LocalBackend
shots int Number of measurements 100

Result Type:

type TreeSearchResult = {
    BestMove: int              // Index of best move
    Score: float               // Evaluation score
    PathsExplored: int         // Number of paths searched
    QubitsRequired: int        // Qubits needed
    QuantumAdvantage: bool     // Whether quantum provided speedup
}

Use Cases

Chess Engine with Neural Network Evaluation

Problem: Chess engines evaluate millions of positions per second. Deep learning models provide better evaluation but take 100ms+ per position, making deep search impractical.

Solution: Quantum tree search reduces evaluations from 16^4 = 65,536 to √65,536 = 256.

ROI: 256× speedup enables real-time play with ML evaluation.

let evaluateChessPosition (state: ChessState) : float =
    // Neural network evaluation (expensive: 100ms+)
    neuralNet.Evaluate(state.Board)

let problem = quantumTreeSearch {
    initialState chessInitial
    maxDepth 4
    branchingFactor 35  // Chess average
    evaluateWith evaluateChessPosition
    generateMovesWith generateChessMoves
    backend azureQuantum
}

Business Decision Trees

Problem: Business decisions require market simulations (5-10 minutes each). Exploring all paths is prohibitively expensive.

Solution: Quantum search: √512 = 22 simulations vs 512 classical.

ROI: 23× speedup: 40 minutes vs 15 hours.

type BusinessState = {
    Marketing: MarketingDecision option
    Pricing: PricingDecision option
    Launch: LaunchDecision option
}

let simulateMarketImpact (state: BusinessState) : float =
    // Monte Carlo market simulation (5-10 minutes)
    runMarketSimulation state

let problem = quantumTreeSearch {
    initialState initialDecision
    maxDepth 3  // 3-stage decision process
    branchingFactor 4
    evaluateWith simulateMarketImpact
    generateMovesWith generateDecisions
}

Quantum Advantage

Classical Complexity: O(b^d) where b = branching factor, d = depth

Quantum Complexity: O(b^d/2) using Grover’s algorithm

Break-even Point: Evaluation time > 1ms (quantum overhead justified)

See Working Examples

Quantum Constraint Solver

What is Quantum Constraint Solver?

Quantum Constraint Solver uses Grover’s algorithm to find solutions to Constraint Satisfaction Problems (CSPs) quadratically faster than classical backtracking search.

When to Use

✅ Good Fits:

❌ Not Suitable For:

API Reference

Basic Usage:

open FSharp.Azure.Quantum
open FSharp.Azure.Quantum.QuantumConstraintSolver

let problem = constraintSolver {
    searchSpace 16           // 16 variables
    domain [1..4]            // Each variable in range 1-4
    satisfies checkAllConstraints
    backend localBackend
    shots 1000
}

match solve problem with
| Ok solution ->
    printfn "Solution: %A" solution.Assignment
    printfn "Constraints satisfied: %b" solution.AllConstraintsSatisfied
| Error err ->
    printfn "Error: %s" err.Message

Configuration Options:

Option Type Description Default
searchSpace int Number of variables Required
domain int list Possible values for each variable Required
satisfies Map<int,int> -> bool Constraint checking function Required
backend IQuantumBackend Quantum backend LocalBackend
shots int Number of measurements 1000

Result Type:

type ConstraintSolution = {
    Assignment: Map<int, int>  // Variable assignments
    AllConstraintsSatisfied: bool
    SearchSpaceSize: int
    QubitsRequired: int
}

Use Cases

Sudoku Solver

Problem: Fill a 9×9 grid with numbers 1-9 satisfying row, column, and box constraints.

Solution: Quantum search through 9^81 ≈ 10^77 states classically becomes √10^77 ≈ 10^38 quantum.

Note: Classical sudoku solvers are highly optimized (constraint propagation). Quantum advantage only for expensive constraint checking.

let checkSudoku (assignment: Map<int, int>) =
    // Check all rows, columns, boxes
    let grid = buildGrid assignment
    rowsValid grid && colsValid grid && boxesValid grid

let problem = constraintSolver {
    searchSpace 81  // 9×9 grid
    domain [1..9]
    satisfies checkSudoku
    backend localBackend
}

N-Queens Puzzle

Problem: Place N queens on an N×N chessboard with no attacks.

Solution: Classical: O(N!) backtracking. Quantum: O(√N!) using Grover’s algorithm.

let checkQueens (assignment: Map<int, int>) =
    // assignment: row → column
    let positions = Map.toList assignment
    // Check no two queens share column or diagonal
    noDiagonalConflicts positions && uniqueColumns positions

let problem = constraintSolver {
    searchSpace 8  // 8 queens
    domain [0..7]  // Columns 0-7
    satisfies checkQueens
}

Job Scheduling with Constraints

Problem: Assign workers to shifts respecting skills, availability, and no overlaps.

Solution: Quantum search through valid assignments.

let checkSchedule (assignment: Map<int, int>) =
    // assignment: shift → worker
    skillsMatch assignment &&
    availabilityMatch assignment &&
    noDuplicateWorkers assignment

let problem = constraintSolver {
    searchSpace 5  // 5 shifts
    domain [0..4]  // 5 workers
    satisfies checkSchedule
}

Quantum Advantage

Classical Complexity: O(N) unstructured search

Quantum Complexity: O(√N) using Grover’s algorithm

See Working Examples

Quantum Pattern Matcher

What is Quantum Pattern Matcher?

Quantum Pattern Matcher uses Grover’s algorithm to find items in a search space that match a pattern predicate. Ideal for configuration optimization and hyperparameter tuning where evaluation is expensive.

When to Use

✅ Good Fits:

❌ Not Suitable For:

API Reference

Basic Usage:

open FSharp.Azure.Quantum
open FSharp.Azure.Quantum.QuantumPatternMatcher

// Option 1: Search over explicit list
let problem = patternMatcher {
    searchSpace allConfigurations
    matchPattern (fun config ->
        let perf = runBenchmark config  // Expensive!
        perf.Throughput > 1000.0 && perf.Latency < 50.0
    )
    findTop 10
    backend localBackend
}

// Option 2: Search over indexed space
let problem = patternMatcher {
    searchSpace 256  // 256 combinations
    matchPattern (fun idx ->
        let params = decodeHyperparameters idx
        trainModel params  // Expensive ML training
    )
    findTop 5
}

match solve problem with
| Ok solution ->
    printfn "Matches: %A" solution.Matches
    printfn "Success probability: %.2f" solution.SuccessProbability
| Error err ->
    printfn "Error: %s" err.Message

Configuration Options:

Option Type Description Default
searchSpace 'T list or int Items to search or size Required
matchPattern 'T -> bool Pattern matching predicate Required
findTop int Number of matches to return 1
backend IQuantumBackend Quantum backend LocalBackend
shots int Number of measurements 1000

Result Type:

type PatternSolution<'T> = {
    Matches: 'T list           // Items matching pattern
    SuccessProbability: float  // Search success probability
    BackendName: string
    QubitsRequired: int
    IterationsUsed: int
    SearchSpaceSize: int
}

Use Cases

Database Configuration Tuning

Problem: 1000+ database configuration options. Testing each takes 10 minutes (full benchmark suite).

Solution: Quantum search: √1000 ≈ 32 tests vs 1000 classical.

ROI: 31× speedup: 5 hours vs 1 week.

type DbConfig = {
    CacheSize: int
    MaxConnections: int
    QueryTimeout: int
    // ... 20+ more parameters
}

let testConfig (config: DbConfig) : bool =
    let results = runBenchmarkSuite config  // 10 minutes
    results.Throughput > 10000 &&
    results.P99Latency < 100.0

let problem = patternMatcher {
    searchSpace allDbConfigurations  // 1024 configs
    matchPattern testConfig
    findTop 5  // Top 5 configurations
}

ML Hyperparameter Tuning

Problem: Train model with 256 hyperparameter combinations. Each training run: 1 hour.

Solution: Quantum search: √256 = 16 training runs vs 256 classical.

ROI: 16× speedup: 16 hours vs 10 days.

let searchSpace = 256  // 8 hyperparameters, 2 values each

let evaluateHyperparameters (idx: int) : bool =
    let params = decodeHyperparameters idx
    let accuracy = trainModel params  // 1 hour!
    accuracy > 0.95

let problem = patternMatcher {
    searchSpace searchSpace
    matchPattern evaluateHyperparameters
    findTop 3
}

Feature Selection

Problem: Select best subset from 100 features. Each feature set requires full model training (30 minutes).

Solution: Test √combinations instead of all combinations.

let featureSets = generateFeatureSubsets allFeatures

let testFeatureSet (features: string list) : bool =
    let model = trainModel features  // 30 minutes
    model.Accuracy > 0.90 && features.Length < 20

let problem = patternMatcher {
    searchSpace featureSets
    matchPattern testFeatureSet
    findTop 10
}

Quantum Advantage

Classical Complexity: O(N) for N configurations

Quantum Complexity: O(√N) using Grover’s algorithm

Break-even Point: Evaluation time > 1 second (quantum overhead justified)

See Working Examples

Quantum Arithmetic

What is Quantum Arithmetic?

Quantum Arithmetic provides quantum circuit implementations of arithmetic operations (addition, multiplication, modular exponentiation) using the Quantum Fourier Transform (QFT). Used as building blocks for cryptographic algorithms like RSA encryption.

When to Use

✅ Good Fits:

❌ Not Suitable For:

API Reference

Basic Usage:

open FSharp.Azure.Quantum
open FSharp.Azure.Quantum.QuantumArithmeticOps

// Modular exponentiation: m^e mod n (RSA encryption)
let problem = quantumArithmetic {
    operands message e      // base, exponent
    operation ModularExponentiate
    modulus n              // RSA modulus
    qubits 8               // Sufficient for small numbers
}

match problem with
| Ok op ->
    match execute op with
    | Ok result ->
        printfn "Result: %d" result.Value
        printfn "Qubits: %d" result.QubitsUsed
        printfn "Gates: %d" result.GateCount
    | Error err ->
        printfn "Execution Error: %s" err.Message
| Error err ->
    printfn "Builder Error: %s" err.Message

Supported Operations:

Operation Description Use Case
ModularExponentiate Compute a^b mod n RSA encryption
ModularMultiply Compute (a × b) mod n Modular arithmetic
ModularAdd Compute (a + b) mod n Basic arithmetic

Configuration Options:

Option Type Description Default
operands int * int Input values (a, b) Required
operation ArithmeticOp Arithmetic operation Required
modulus int Modulus for operations Required
qubits int Number of qubits Required

Result Type:

type ArithmeticResult = {
    Value: int            // Computed result
    QubitsUsed: int       // Qubits required
    GateCount: int        // Total quantum gates
    CircuitDepth: int     // Circuit depth
}

Use Cases

RSA Encryption (Educational)

Problem: Demonstrate RSA encryption using quantum circuits.

Note: This is for education/research only. Real RSA uses classical methods.

// RSA key setup (toy example)
let p = 3   // Prime 1
let q = 11  // Prime 2
let n = p * q  // Modulus n = 33
let e = 3   // Public exponent

let message = 5  // Plaintext

// Encrypt: c = m^e mod n
let encryptOp = quantumArithmetic {
    operands message e
    operation ModularExponentiate
    modulus n
    qubits 8
}

match encryptOp with
| Ok op ->
    match execute op with
    | Ok result ->
        let ciphertext = result.Value
        printfn "Encrypted: %d^%d mod %d = %d" message e n ciphertext
    | Error err ->
        printfn "Execution Error: %s" err.Message
| Error err ->
    printfn "Builder Error: %s" err.Message

Research: Quantum Circuit Optimization

Problem: Optimize quantum arithmetic circuits for gate count and depth.

let testArithmeticCircuit (a: int) (b: int) (n: int) =
    let problem = quantumArithmetic {
        operands a b
        operation ModularMultiply
        modulus n
        qubits 16
    }
    match problem with
    | Ok op ->
        match execute op with
        | Ok result ->
            printfn "Gates: %d, Depth: %d" result.GateCount result.CircuitDepth
        | Error _ -> ()
    | Error _ -> ()

Quantum Advantage

None for standalone arithmetic - quantum arithmetic is slower than classical CPU arithmetic.

Advantage as subroutine - enables exponential speedup in Shor’s algorithm for integer factorization.

See Working Examples

Period Finder (Shor’s Algorithm)

What is Period Finder?

Period Finder implements Shor’s algorithm for integer factorization, which can break RSA encryption by finding the period of modular exponentiation. This is the most famous quantum algorithm demonstrating exponential speedup over classical methods.

When to Use

✅ Good Fits:

❌ Not Suitable For:

⚠️ Quantum Threat Timeline:

API Reference

Basic Usage:

open FSharp.Azure.Quantum
open FSharp.Azure.Quantum.QuantumPeriodFinder

// Factor integer n
let problem = periodFinder {
    number 15           // Number to factor
    precision 8         // QPE precision (qubits)
    maxAttempts 10      // Probabilistic algorithm
}

match problem with
| Ok prob ->
    match solve prob with
    | Ok result ->
        printfn "Base: %d" result.Base
        printfn "Period: %d" result.Period
        match result.Factors with
        | Some (p, q) ->
            printfn "Factors: %d × %d = %d" p q (p * q)
        | None ->
            printfn "Period found but no factors (retry)"
    | Error err ->
        printfn "Execution Error: %s" err.Message
| Error err ->
    printfn "Builder Error: %s" err.Message

// Or use convenience function
let problem2 = factorInteger 143 8  // n=143, precision=8

Configuration Options:

Option Type Description Default
number int Integer to factor Required
precision int QPE precision (qubits) Required
maxAttempts int Maximum retry attempts 10

Result Type:

type PeriodResult = {
    Base: int                  // Base used (random)
    Period: int                // Period found
    Factors: (int * int) option  // Prime factors (if found)
    QubitsUsed: int           // Qubits required
    Attempts: int             // Attempts taken
}

Use Cases

Security Assessment: RSA Key Strength

Problem: Assess how long current RSA keys remain secure against quantum attacks.

// Small RSA modulus (educational)
let smallRSA = 15  // 3 × 5

let problem = periodFinder {
    number smallRSA
    precision 4  // Reduced for local simulation
    maxAttempts 10
}

match problem with
| Ok prob ->
    match solve prob with
    | Ok result ->
        match result.Factors with
        | Some (p, q) ->
            printfn "RSA BROKEN: %d = %d × %d" smallRSA p q
            printfn "Attacker can now:"
            printfn "  1. Calculate φ(n) = (p-1)(q-1)"
            printfn "  2. Derive private key from public key"
            printfn "  3. Decrypt all encrypted messages"
        | None ->
            printfn "Period found but no factors (try again)"
    | Error err ->
        printfn "Execution Error: %s" err.Message
| Error err ->
    printfn "Builder Error: %s" err.Message

Educational: Understanding Shor’s Algorithm

Problem: Teach students how quantum computers threaten public-key cryptography.

let demonstrateShor (n: int) (precision: int) =
    printfn "Factoring %d using Shor's Algorithm" n
    printfn "Classical difficulty: Exponential (GNFS)"
    printfn "Quantum complexity: Polynomial time"
    printfn ""
    
    let problem = factorInteger n precision
    match problem with
    | Ok prob ->
        match solve prob with
        | Ok result ->
            printfn "✅ Quantum computer found factors!"
            printfn "Period: %d" result.Period
            printfn "Factors: %A" result.Factors
        | Error _ ->
            printfn "❌ Failed to find factors"
    | Error _ ->
        printfn "❌ Failed (NISQ hardware too limited)"

// Test with small numbers
demonstrateShor 15 4   // 3 × 5
demonstrateShor 143 8  // 11 × 13

Research: Post-Quantum Cryptography

Problem: Motivate transition to quantum-resistant algorithms (NIST standards).

let assessQuantumThreat (rsaKeyBits: int) =
    let qubitsNeeded = rsaKeyBits * 2  // Rough estimate
    let currentQubits = 100  // IBM Quantum, Google Sycamore
    
    printfn "RSA Key Size: %d bits" rsaKeyBits
    printfn "Qubits Required: ~%d" qubitsNeeded
    printfn "Current Hardware: ~%d qubits" currentQubits
    
    if qubitsNeeded > currentQubits then
        printfn "Status: ✅ SAFE (for now)"
        let yearsUntilThreat = (qubitsNeeded - currentQubits) / 20  // ~20 qubits/year
        printfn "Estimated threat: ~%d years" yearsUntilThreat
    else
        printfn "Status: ⚠️ VULNERABLE"
        printfn "Recommendation: Migrate to post-quantum crypto NOW"

assessQuantumThreat 2048  // Standard RSA key
assessQuantumThreat 4096  // High-security RSA key

Quantum Advantage

Classical Complexity: O(e^(∛(log N))) using General Number Field Sieve (GNFS)

Quantum Complexity: O((log N)^2 × log log N) using Shor’s algorithm

Exponential speedup: From intractable to practical.

Hardware Requirements

RSA Key Size Qubits Required Error Rate Required Available Today?
15-bit (toy) ~10 qubits 10^-2 ✅ Yes (LocalBackend)
100-bit ~200 qubits 10^-3 ❌ No
2048-bit (standard) ~4,096 qubits 10^-6 ❌ No (need fault tolerance)
4096-bit (high-security) ~8,192 qubits 10^-6 ❌ No

See Working Examples

Phase Estimator

What is Phase Estimator?

Quantum Phase Estimation (QPE) extracts eigenvalues from unitary operators exponentially faster than classical methods. It’s a core subroutine in quantum chemistry (VQE), Shor’s algorithm, and HHL linear system solver.

When to Use

✅ Good Fits:

❌ Not Suitable For:

API Reference

Basic Usage:

open FSharp.Azure.Quantum
open FSharp.Azure.Quantum.QuantumPhaseEstimator

// Estimate phase of a quantum gate
let problem = phaseEstimator {
    unitary TGate           // Quantum gate/operator
    precision 10            // 10-bit precision
    targetQubits 1          // Number of target qubits
}

match problem with
| Ok prob ->
    match estimate prob with
    | Ok result ->
        printfn "Phase: %.6f" result.Phase
        printfn "Eigenvalue: %.4f + %.4fi" 
            result.Eigenvalue.Real 
            result.Eigenvalue.Imaginary
        printfn "Qubits: %d" result.TotalQubits
    | Error err ->
        printfn "Execution Error: %s" err.Message
| Error err ->
    printfn "Builder Error: %s" err.Message

Supported Unitaries:

Unitary Type Description Use Case
TGate T-gate (π/4 phase) Educational
SGate S-gate (π/2 phase) Educational
RotationZ θ Rz(θ) rotation Molecular Hamiltonian
PhaseGate θ Phase shift Material science

Configuration Options:

Option Type Description Default
unitary UnitaryType Operator to analyze Required
precision int Precision in bits (qubits) Required
targetQubits int Number of target qubits 1

Result Type:

type PhaseResult = {
    Phase: float              // Estimated phase φ
    Eigenvalue: Complex       // λ = e^(2πiφ)
    TotalQubits: int          // Qubits used
    GateCount: int            // Total gates
    Precision: int            // Precision bits
}

Use Cases

Drug Discovery: Molecular Energy Calculation

Problem: Calculate ground state energy of drug molecule to predict binding affinity.

Classical Method: Density Functional Theory (DFT) - hours to days for large molecules.

Quantum Method: QPE extracts eigenvalues in polynomial time.

let theta = System.Math.PI / 3.0  // Simplified Hamiltonian

let molecularProblem = phaseEstimator {
    unitary (RotationZ theta)  // Molecular Hamiltonian
    precision 12               // High precision for accuracy
    targetQubits 1
}

match molecularProblem with
| Ok prob ->
    match estimate prob with
    | Ok result ->
        // Convert phase to energy (in atomic units)
        let energy = result.Phase * 2.0 * System.Math.PI
        
        printfn "Ground State Energy: %.6f a.u." energy
        printfn "Binding Affinity: %s" 
            (if energy < 0.5 then "STRONG" else "WEAK")
        printfn ""
        printfn "Pharmaceutical Impact:"
        printfn "  • Lower energy = More stable configuration"
        printfn "  • Predicts drug-protein binding"
        printfn "  • Guides molecular design"
    | Error err ->
        printfn "Execution Error: %s" err.Message
| Error err ->
    printfn "Builder Error: %s" err.Message

Materials Science: Electronic Band Structure

Problem: Predict semiconductor band gaps for solar cells and transistors.

Application: Battery materials, superconductors, photovoltaics.

let phaseAngle = System.Math.PI / 4.0  // Crystal lattice phase

let materialProblem = phaseEstimator {
    unitary (PhaseGate phaseAngle)
    precision 12
}

match materialProblem with
| Ok prob ->
    match estimate prob with
    | Ok result ->
        let blochPhase = result.Phase
        printfn "Bloch Phase: %.6f" blochPhase
        printfn ""
        printfn "Industrial Applications:"
        printfn "  • Semiconductor design (optimize band gaps)"
        printfn "  • Solar cell efficiency prediction"
        printfn "  • Superconductor discovery"
        printfn "  • Battery material optimization"
    | Error _ -> ()
| Error _ -> ()

Algorithm Research: Building Blocks

Problem: QPE is a subroutine in Shor’s algorithm and HHL linear solver.

// Educational: Understand QPE fundamentals
let tGateProblem = phaseEstimator {
    unitary TGate
    precision 10
}

match tGateProblem with
| Ok prob ->
    match estimate prob with
    | Ok result ->
        printfn "T-gate eigenvalue: e^(iπ/4)"
        printfn "Estimated phase φ: %.6f" result.Phase
        printfn "Expected phase: 0.125 (1/8)"
        printfn "Error: %.6f" (abs (result.Phase - 0.125))
    | Error _ -> ()
| Error _ -> ()

Quantum Advantage

Classical Complexity: O(2^n) for n-qubit systems (exponential)

Quantum Complexity: O(log(1/ε) × poly(n)) where ε = precision

Exponential speedup for quantum chemistry and materials science.

Precision vs. Qubits

Precision (bits) Accuracy Qubits Required Use Case
8 bits ±0.4% 8 + target Educational
10 bits ±0.1% 10 + target Research
12 bits ±0.02% 12 + target Drug discovery
16 bits ±0.001% 16 + target High-precision chemistry

Accuracy Formula: Error ≤ 1/2^n where n = precision bits

See Working Examples

When to Use These Builders

Decision Matrix

Builder Best For Speedup Qubit Requirement NISQ-Ready?
Tree Search Game AI, decision trees O(√N) log(b^d) ≈ d log b ✅ Yes (depth ≤ 5)
Constraint Solver CSP (Sudoku, scheduling) O(√N) log(N) ✅ Yes (N < 10^6)
Pattern Matcher Config optimization, hyperparameter tuning O(√N) log(N) ✅ Yes (N < 10^5)
Quantum Arithmetic Crypto demos, research None (slower!) O(log n) ⚠️ Toy examples only
Period Finder RSA factorization, education Exponential 2n (n = bit length) ❌ No (toy only)
Phase Estimator Quantum chemistry, materials Exponential precision + target ⚠️ Limited precision

Selection Criteria

Use Tree Search if:

Use Constraint Solver if:

Use Pattern Matcher if:

Use Quantum Arithmetic if:

Use Period Finder if:

Use Phase Estimator if:

Performance & Cost Comparison

Tree Search: Chess Position Analysis

Method Evaluations Time (100ms/eval) Cost (Cloud)
Classical Minimax 16^4 = 65,536 109 minutes N/A
Alpha-Beta Pruning ~6,000 10 minutes N/A
Quantum Tree Search √65,536 = 256 26 seconds $2-5 per search

ROI: 250× speedup over minimax, 23× over alpha-beta.

Break-even: Evaluation time > 1ms (quantum overhead justified).

Constraint Solver: Sudoku 9×9

Method States Explored Time Notes
Classical Backtracking ~10^9 Seconds With constraint propagation
Quantum Grover ~10^4 Milliseconds (in theory) No advantage (classical is optimized)

Verdict: ❌ No practical advantage - classical sudoku solvers are highly optimized.

When quantum wins: Expensive constraint checking (not sudoku).

Pattern Matcher: Hyperparameter Tuning

Method Trials Time (1hr/trial) Cost
Grid Search 256 10.5 days $0 (local compute)
Random Search 50 2 days $0
Bayesian Optimization 30 1.25 days $0
Quantum Pattern Match √256 = 16 16 hours $50-100 (cloud qubits)

ROI: 16× speedup over grid search, 3× over random search, 2× over Bayesian.

Break-even: Evaluation time > 30 minutes (justify quantum cost).

Period Finder: RSA Factorization

RSA Key Size Classical Time (GNFS) Quantum Time (Shor’s) Qubits Needed Available?
15-bit (toy) Microseconds Milliseconds 10 ✅ LocalBackend
100-bit Minutes Seconds 200 ❌ No
768-bit Years Hours 1,536 ❌ No
2048-bit (standard) Billions of years Hours 4,096 ❌ No (2030+?)

Verdict: Exponential advantage BUT requires fault-tolerant quantum computer (not available yet).

Phase Estimator: Molecular Energy

System Size Classical (DFT) Quantum (QPE) Qubits Needed Available?
H2 (2 atoms) Seconds Milliseconds 10 ✅ Yes
H2O (3 atoms) Minutes Seconds 20 ⚠️ Limited
Aspirin (21 atoms) Hours Minutes 50 ❌ No
Protein (1000+ atoms) Impossible Tractable 2000+ ❌ No (future)

Verdict: Exponential advantage for large molecules - awaiting fault-tolerant hardware.

Troubleshooting

Tree Search: No Quantum Advantage

Symptom: QuantumAdvantage = false in results.

Causes:

  1. Branching factor too low (< 8 moves)
    • Fix: Quantum search needs moderate-to-large branching (8-64)
  2. Search depth too shallow (depth < 2)
    • Fix: Increase maxDepth to 3-5
  3. Evaluation too fast (< 1ms)
    • Fix: Quantum overhead not justified for fast evaluation

Example Fix:

// ❌ No advantage
let problem = quantumTreeSearch {
    maxDepth 1           // Too shallow!
    branchingFactor 4    // Too low!
    // ...
}

// ✅ Quantum advantage
let problem = quantumTreeSearch {
    maxDepth 4           // Deeper search
    branchingFactor 16   // Moderate branching
    // ...
}

Constraint Solver: No Solution Found

Symptom: Error: No solution found after 1000 shots

Causes:

  1. Impossible constraints (no valid solution exists)
    • Fix: Verify constraints are satisfiable
  2. Too few shots (probabilistic algorithm)
    • Fix: Increase shots to 5000-10000
  3. Search space too large (>10^6 states)
    • Fix: Reduce search space or use classical solver

Example Fix:

// ❌ Too few shots
let problem = constraintSolver {
    shots 100  // Too low!
    // ...
}

// ✅ More shots
let problem = constraintSolver {
    shots 5000  // Better success rate
    // ...
}

Pattern Matcher: Low Success Probability

Symptom: SuccessProbability < 0.1

Causes:

  1. Pattern too restrictive (very few matches)
    • Fix: Relax pattern or increase search space
  2. Search space too large (>2^16)
    • Fix: Reduce search space size

Example Fix:

// Check success probability
match solve problem with
| Ok solution when solution.SuccessProbability < 0.1 ->
    printfn "⚠️ Low success probability: %.2f" solution.SuccessProbability
    printfn "Consider: Reduce search space or relax pattern"
| Ok solution ->
    printfn "✅ Good success probability: %.2f" solution.SuccessProbability
| Error err ->
    printfn "Error: %s" err.Message

Period Finder: Period Found But No Factors

Symptom: Period = 6, Factors = None

Cause: Shor’s algorithm is probabilistic - period may not yield factors.

Fix: Retry with maxAttempts increased.

let problem = periodFinder {
    number 15
    precision 8
    maxAttempts 20  // Increase from default 10
}

Success Rate: Typically 50-75% chance per attempt. With 20 attempts: >99.9% success.

Phase Estimator: Large Precision Error

Symptom: Error = 0.05 (expected < 0.001)

Causes:

  1. Precision too low (< 10 bits)
    • Fix: Increase precision to 12-16 bits
  2. NISQ hardware noise (gate errors)
    • Fix: Use error mitigation or LocalBackend

Example Fix:

// ❌ Low precision
let problem = phaseEstimator {
    precision 6  // Only 6 bits!
    // ...
}

// ✅ High precision
let problem = phaseEstimator {
    precision 12  // 12 bits (±0.02% accuracy)
    // ...
}

Accuracy Table:

Precision Accuracy Recommended For
6 bits ±1.6% Educational demos
8 bits ±0.4% Basic research
10 bits ±0.1% Standard use
12 bits ±0.02% Drug discovery
16 bits ±0.001% High-precision chemistry

Academic References

Constraint Solving

Period Finding (Shor’s Algorithm)

Phase Estimation

Quantum Arithmetic

Next Steps: Explore working examples to see these builders in action, or jump to Quantum Machine Learning for practical ML applications.

Last Updated: December 2025