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Sparse Operations Implementation Plan

Created: 2025-05-30 Related Task: T74 - Optimize Quantum Operator Performance

Overview

Implementation plan for sparse storage and operations to optimize quantum operator performance. Target: extend practical limits from 6-7 qubits to 10-12 qubits while maintaining API compatibility.

Current Performance Bottlenecks

State vector operations scale well (16 qubits in 75ms), indicating dense matrix operations in operators are the primary bottleneck.

Simple Transparent Implementation Strategy

Core Principle: Keep It Really Simple, Stupid (KIRSS)

Key Insight: Keep exact same classes and APIs, only change internal storage.

StateVector Sparse Storage

Current Dense Implementation:

class StateVector {
  amplitudes: Complex[] = [amp1, amp2, amp3, ...]; // Dense array
}

Proposed Sparse Implementation:

class StateVector {
  private dense: Complex[] | null = null;
  private sparse: Map<number, Complex> = new Map(); // Only non-zero values
  private sparsityThreshold: number = 0.8; // Use sparse when >80% zeros
  
  getState(index: number): Complex {
    return this.sparse.get(index) || math.complex(0, 0);
  }
  
  get amplitudes(): Complex[] {
    if (!this.dense) {
      // Convert sparse to dense when needed
      this.dense = Array(this.dimension).fill(math.complex(0, 0));
      this.sparse.forEach((val, idx) => this.dense[idx] = val);
    }
    return this.dense;
  }
}

MatrixOperator Optimizations

Specialized Operator Classes (same interface):

  1. IdentityOperator - No matrix storage
class IdentityOperator implements IOperator {
  apply(state: IStateVector): StateVector {
    return state.clone(); // No matrix multiplication!
  }
  
  compose(other: IOperator): IOperator {
    return other; // I ∘ A = A
  }
}
  1. DiagonalOperator - Store only diagonal elements
class DiagonalOperator implements IOperator {
  private diagonal: Complex[];
  
  apply(state: IStateVector): StateVector {
    // Element-wise multiplication, no full matrix
    const newAmps = state.amplitudes.map((amp, i) => 
      math.multiply(amp, this.diagonal[i])
    );
    return new StateVector(state.dimension, newAmps);
  }
}
  1. SparseMatrixOperator - Compressed storage for sparse matrices
class SparseMatrixOperator implements IOperator {
  private sparseData: Map<string, Complex> = new Map(); // "i,j" -> value
  
  apply(state: IStateVector): StateVector {
    // Sparse matrix-vector multiplication
    return sparseMatrixVectorMultiply(this.sparseData, state);
  }
}

Performance Trade-offs

Small Objects (2-8 qubits)

Large Objects (10+ qubits)

Implementation Status

Phase 1: Infrastructure - ✅ COMPLETE

Phase 2: Core Optimizations - ✅ COMPLETE

Phase 2: Core Optimizations - ✅ COMPLETE

Phase 3: Testing and Validation - 🔄 NEXT

Phase 4: Advanced Features - ⬜ FUTURE

API Compatibility

Zero Breaking Changes:

Transparency Mechanism:

Files Modified

Created Files

Modified Files

Future Files (Phase 4)

Success Metrics

Quantum Random Walk Application Analysis

Updated: 2025-06-03 - T76 Quantum Random Walk Implementation

Memory Requirements for 2D Quantum Random Walks

The T74 sparse infrastructure enables practical quantum random walk simulations on large lattices. Analysis shows dramatic memory savings for the specific structure of quantum walk operators.

State Space Structure

Memory Analysis Summary

Dense Implementation (Current Default):

Size     | Total Dim | State(KB) | Dense Op(MB) | Practical Limit
---------|-----------|-----------|--------------|----------------
10x10    |       400 |         6 |            2 | ✅ Acceptable
15x15    |       900 |        14 |           12 | ⚠️ Borderline  
20x20    |      1600 |        25 |           39 | ❌ Impractical
25x25    |      2500 |        39 |           95 | ❌ Prohibitive
50x50    |     10000 |       156 |         1526 | ❌ Impossible

Sparse Implementation (T74 Infrastructure):

Size     | Total Dim | Sparse(KB) | Dense(MB) | Memory Reduction
---------|-----------|------------|-----------|------------------
10x10    |       400 |         83 |         2 |             30x
25x25    |      2500 |        523 |        95 |            187x
50x50    |     10000 |       2102 |      1526 |            743x
100x100  |     40000 |       8422 |     24414 |          2,968x
200x200  |    160000 |      33719 |    390625 |         11,863x

Component-Wise Memory Breakdown

State Vector (always dense):

Coin Operator (block diagonal structure):

Shift Operator (extremely sparse):

Sparse Structure Optimizations

Coin Operator Optimization:

// Block diagonal structure: I_position ⊗ C_coin
// Each 4x4 block is the same coin matrix C
const coinBlocks = positionDim;          // Number of blocks
const nonZerosPerBlock = 16;             // 4x4 dense blocks
const totalCoinNonZeros = coinBlocks * nonZerosPerBlock;
const coinOperatorBytes = totalCoinNonZeros * (16 + 24); // Value + overhead

Shift Operator Optimization:

// Highly sparse: each row has at most 1 non-zero entry
// Movement: up/down/left/right based on coin state
let shiftNonZeros = 0;
for (position in lattice) {
  for (coinState in [up, down, left, right]) {
    if (canMoveInDirection(position, coinState)) {
      shiftNonZeros++; // Exactly one non-zero per valid move
    }
  }
}
// Result: ~90% of total dimension for interior points

Performance Implications

Evolution Step Operations:

Memory Scaling:

Implementation Strategy for T76

Phase 1: Leverage T74 Infrastructure

  1. Use IdentityOperator for position identity terms
  2. Use sparse matrix utilities for shift operators
  3. Implement coin operators as sparse block diagonal
  4. Target 50×50 lattices as primary use case

Phase 2: Quantum Walk Specific Optimizations

  1. Custom shift operator class with movement logic
  2. Block diagonal coin operator implementation
  3. Composite state management with sparse representations
  4. Memory-efficient evolution algorithms

Target Performance (T76):

Conclusion

The T74 sparse infrastructure makes quantum random walks practically feasible for research-scale lattices:

  1. 50×50 lattices are comfortably achievable (743x memory reduction)
  2. 100×100 lattices are feasible for research applications (2,968x reduction)
  3. Linear memory scaling O(n²) instead of quartic O(n⁴)
  4. Dramatic performance improvements enable practical quantum walk simulations

Risk Mitigation

This plan prioritizes simplicity and backward compatibility while achieving the performance goals of T74 and enabling the ambitious quantum random walk simulations of T76.

Last Updated: 2025-07-06