Universal Quantum Computation with Ising Anyons

This guide demonstrates how to achieve universal quantum computation using Ising anyons (Majorana zero modes), which naturally support only Clifford operations.

The Challenge

Ising anyons can only perform Clifford operations natively through braiding:

But Clifford gates alone are NOT universal! You cannot implement arbitrary quantum algorithms with just Clifford operations.

The Solution: Magic State Distillation

To achieve universality, we add non-Clifford gates (specifically T-gates) via magic state distillation:

  1. Prepare noisy magic states T⟩ = ( 0⟩ + e^(iπ/4) 1⟩) / √2
  2. Distill to high fidelity using error detection codes
  3. Inject purified magic states to implement T-gates via gate teleportation
  4. Combine with native Clifford ops → Universal computation!

Gate Set: Clifford + T = Universal quantum computation ✓

Quick Start Example

open FSharp.Azure.Quantum.Topological

// 1. Prepare noisy magic states (15 needed for one distillation round)
let random = System.Random()
let noisyErrorRate = 0.05  // 5% error

let noisyStates = 
    [1..15]
    |> List.map (fun _ -> 
        MagicStateDistillation.prepareNoisyMagicState noisyErrorRate AnyonSpecies.AnyonType.Ising
    )
    |> List.choose Result.toOption

// 2. Distill to high-fidelity magic state
let distillResult = 
    MagicStateDistillation.distill15to1 random noisyStates
    |> Result.defaultWith (fun err -> failwith err.Message)

let purifiedState = distillResult.PurifiedState

printfn "Input fidelity:  %.4f" (noisyStates |> List.averageBy (fun s -> s.Fidelity))
printfn "Output fidelity: %.6f" purifiedState.Fidelity
printfn "Error suppression: %.1fx" 
    ((1.0 - (List.averageBy (fun s -> s.Fidelity) noisyStates)) / (1.0 - purifiedState.Fidelity))

// 3. Create a topological qubit (|0⟩ state)
let sigma = AnyonSpecies.Particle.Sigma
let vacuum = AnyonSpecies.Particle.Vacuum

let dataQubit = 
    let left = FusionTree.leaf sigma
    let right = FusionTree.leaf sigma
    let tree = FusionTree.fuse left right vacuum
    FusionTree.create tree AnyonSpecies.AnyonType.Ising

// 4. Apply T-gate using magic state injection
let tGateResult = 
    MagicStateDistillation.applyTGate random dataQubit purifiedState
    |> Result.defaultWith (fun err -> failwith err.Message)

printfn "\nT-gate applied!"
printfn "Gate fidelity: %.6f" tGateResult.GateFidelity
printfn "Output: T|0⟩"

Output:

Input fidelity:  0.9500
Output fidelity: 0.995625
Error suppression: 11.4x

T-gate applied!
Gate fidelity: 0.995625
Output: T|0⟩

How Magic State Distillation Works

The 15-to-1 Protocol (Bravyi-Kitaev 2005)

Input: 15 noisy magic states with error rate p
Output: 1 purified magic state with error rate p_out ≈ 35p³

Key property: Cubic error suppression

Iterative Distillation

Apply 15-to-1 multiple times for exponential error suppression:

// Prepare 15^2 = 225 noisy states
let round1States = [1..225] |> List.map (fun _ -> 
    MagicStateDistillation.prepareNoisyMagicState 0.10 AnyonSpecies.AnyonType.Ising
) |> List.choose Result.toOption

// 2 rounds of distillation: p → 35p³ → 35(35p³)³ = 35⁴p⁹
let finalState = 
    MagicStateDistillation.distillIterative random 2 round1States
    |> Result.defaultWith (fun err -> failwith err.Message)

printfn "Input error:  %.4f" 0.10
printfn "Output error: %.8f" (1.0 - finalState.Fidelity)
printfn "Suppression:  %.1fx" (0.10 / (1.0 - finalState.Fidelity))

Output:

Input error:  0.1000
Output error: 0.00000043
Suppression:  232558.1x

Resource Estimation

Estimate how many noisy states you need for a target fidelity:

let targetFidelity = 0.9999  // 99.99% fidelity
let noisyFidelity = 0.95     // Start with 95% fidelity

let estimate = 
    MagicStateDistillation.estimateResources targetFidelity noisyFidelity

printfn "%s" (MagicStateDistillation.displayResourceEstimate estimate)

Output:

Resource Estimate for 99.99% fidelity:
  Distillation Rounds: 2
  Noisy States Required: 225
  Output Fidelity: 99.9956%
  Overhead Factor: 225x

Example: Implementing Toffoli Gate

The Toffoli gate (CCNOT) can be decomposed into Clifford + T gates:

Toffoli = H·CNOT·T†·CNOT·T·CNOT·T†·CNOT·T·H

This requires:

// For 99.99% fidelity Toffoli gate
let tGatesNeeded = 4
let estimate = MagicStateDistillation.estimateResources 0.9999 0.95

let totalNoisyStates = tGatesNeeded * estimate.NoisyStatesRequired

printfn "Toffoli Gate Resource Requirements:"
printfn "  T-gates needed: %d" tGatesNeeded
printfn "  Rounds per T-gate: %d" estimate.DistillationRounds
printfn "  Noisy states per T-gate: %d" estimate.NoisyStatesRequired
printfn "  Total noisy states: %d" totalNoisyStates
printfn "  Gate fidelity: %.4f%%" (estimate.OutputFidelity * 100.0)

Output:

Toffoli Gate Resource Requirements:
  T-gates needed: 4
  Rounds per T-gate: 2
  Noisy states per T-gate: 225
  Total noisy states: 900
  Gate fidelity: 99.9956%

Complete Algorithm Workflow

Here’s a complete example implementing a simple quantum algorithm:

open FSharp.Azure.Quantum.Topological

let runQuantumAlgorithm () =
    let random = System.Random()
    
    // Step 1: Resource planning
    printfn "=== Step 1: Resource Planning ==="
    let targetFidelity = 0.999
    let noisyErrorRate = 0.05
    
    let estimate = 
        MagicStateDistillation.estimateResources targetFidelity (1.0 - noisyErrorRate)
    
    printfn "Algorithm requires: %d noisy magic states" estimate.NoisyStatesRequired
    
    // Step 2: Prepare and distill magic states
    printfn "\n=== Step 2: Magic State Preparation ==="
    let noisyStates = 
        [1..estimate.NoisyStatesRequired]
        |> List.map (fun _ -> 
            MagicStateDistillation.prepareNoisyMagicState noisyErrorRate AnyonSpecies.AnyonType.Ising
        )
        |> List.choose Result.toOption
    
    let purifiedState = 
        MagicStateDistillation.distillIterative random estimate.DistillationRounds noisyStates
        |> Result.defaultWith (fun err -> failwith err.Message)
    
    printfn "Distilled magic state fidelity: %.6f" purifiedState.Fidelity
    
    // Step 3: Build quantum circuit
    printfn "\n=== Step 3: Build Quantum Circuit ==="
    
    // Initialize qubit to |0⟩
    let sigma = AnyonSpecies.Particle.Sigma
    let vacuum = AnyonSpecies.Particle.Vacuum
    let qubit = 
        let tree = FusionTree.fuse (FusionTree.leaf sigma) (FusionTree.leaf sigma) vacuum
        FusionTree.create tree AnyonSpecies.AnyonType.Ising
    
    printfn "Initial state: |0⟩"
    
    // Apply Hadamard (Clifford - native via braiding)
    printfn "Apply H (Hadamard) → |+⟩"
    
    // Apply T-gate (non-Clifford - via magic state)
    let afterT = 
        MagicStateDistillation.applyTGate random qubit purifiedState
        |> Result.defaultWith (fun err -> failwith err.Message)
    
    printfn "Apply T (via magic state) → T|+⟩"
    printfn "T-gate fidelity: %.6f" afterT.GateFidelity
    
    // Apply another Hadamard
    printfn "Apply H (Hadamard)"
    
    // Step 4: Measure
    printfn "\n=== Step 4: Measurement ==="
    printfn "Circuit complete with fidelity: %.4f%%" (afterT.GateFidelity * 100.0)
    
    printfn "\n✓ Universal quantum computation achieved!"
    printfn "✓ Combined Clifford (braiding) + T-gates (magic states)"
    printfn "✓ Can implement any quantum algorithm"

// Run it!
runQuantumAlgorithm()

Performance Characteristics

Error Suppression

Input Fidelity Output Fidelity (1 round) Improvement
90% (10% error) 96.5% (3.5% error) 2.9×
95% (5% error) 99.56% (0.44% error) 11.4×
99% (1% error) 99.9965% (0.0035% error) 286×

Resource Overhead

Target Fidelity Rounds Noisy States Overhead
99% 1 15 15×
99.9% 1-2 15-225 15-225×
99.99% 2 225 225×
99.999% 2-3 225-3,375 225-3,375×

Rule of thumb: Each additional “9” in fidelity requires ~15× more resources.

Integration with Topological Error Correction

Magic state distillation works synergistically with topological protection:

  1. Topological protection (from braiding):
    • Protects against local perturbations
    • Exponentially suppressed errors with system size
    • Handles Clifford operations
  2. Magic state distillation (for T-gates):
    • Error detection on encoded states
    • Polynomial (cubic) error suppression per round
    • Handles non-Clifford operations

Combined: Full fault-tolerant universal quantum computation!

API Reference

Core Functions

// Prepare noisy magic state
val prepareNoisyMagicState : 
    errorRate:float -> 
    anyonType:AnyonSpecies.AnyonType -> 
    TopologicalResult<MagicState>

// Single round of 15-to-1 distillation
val distill15to1 : 
    random:Random -> 
    inputStates:MagicState list -> 
    TopologicalResult<DistillationResult>

// Iterative distillation (multiple rounds)
val distillIterative : 
    random:Random -> 
    rounds:int -> 
    initialStates:MagicState list -> 
    TopologicalResult<MagicState>

// Apply T-gate via magic state injection
val applyTGate : 
    random:Random -> 
    dataQubit:FusionTree.State -> 
    magicState:MagicState -> 
    TopologicalResult<TGateResult>

// Estimate resources for target fidelity
val estimateResources : 
    targetFidelity:float -> 
    noisyStateFidelity:float -> 
    ResourceEstimate

Types

type MagicState = {
    QubitState: FusionTree.State
    Fidelity: float
    ErrorRate: float
}

type DistillationResult = {
    PurifiedState: MagicState
    AcceptanceProbability: float
    InputStatesConsumed: int
    Syndromes: bool list
}

type TGateResult = {
    OutputState: FusionTree.State
    CorrectionApplied: bool
    GateFidelity: float
}

type ResourceEstimate = {
    TargetFidelity: float
    DistillationRounds: int
    NoisyStatesRequired: int
    OutputFidelity: float
    OverheadFactor: int
}

Further Reading

See Also