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Quantum Graph Entanglement Implementation Plan

Created: 2025-05-30 Status: Planning Phase

Overview

Extend the QuantumGraph class to support genuine quantum entanglement by implementing composite state storage that spans multiple vertices. Current limitation: individual vertex storage cannot represent entangled states.

Current Bell Chain Example

The existing Bell chain example demonstrates the limitation:

Bell State Chain Examples

1. 3-vertex chain (aperiodic):
   q0 ---- bell_0_1 ---- q1 ---- bell_1_2 ---- q2
   |0⟩                   |0⟩                   |0⟩

2. 4-vertex ring (periodic):
        q0 ---- bell_0_1 ---- q1
        |                      |
   bell_3_0                bell_1_2
        |                      |
        q3 ---- bell_2_3 ---- q2
       |0⟩                   |0⟩

3. 6-vertex chain (aperiodic):
   q0 ---- bell_0_1 ---- q1 ---- bell_1_2 ---- q2 ---- bell_2_3 ---- q3 ---- bell_3_4 ---- q4 ---- bell_4_5 ---- q5
   |1⟩                   |1⟩                   |1⟩                   |1⟩                   |1⟩                   |1⟩

Legend:
- Vertices (qN): Quantum states (qubits)
- Edges (bell_i_j): Bell state entanglement operators
- |0⟩, |1⟩: Initial computational basis states

Issue: Each vertex stores independent states, edges have only metadata labels. Goal: Transform into genuine entangled Bell states |00⟩ + |11⟩ spanning vertex pairs.

Current Architecture Issues

Problem: One vertex = one quantum object Need: One quantum object across multiple vertices for entanglement

// Current (Limited)
vertex_q0: StateVector(|0⟩)  // 2D, independent
vertex_q1: StateVector(|0⟩)  // 2D, independent

// Required (Entangled) 
composite_system: StateVector(|00⟩ + |11⟩)  // 4D, spanning q0,q1

Implementation Plan

Phase 1: Core Composite State Storage ✅ COMPLETE

Status: Implemented and validated in entangledGraphPOC.ts

File: packages/quantum/examples/poc/entangledGraphPOC.ts (~100 lines)

class CompositeQuantumManager {
  private composites: Map<string, QuantumObject>;
  private elementToComposite: Map<string, string>;
  
  // Core operations implemented
  setComposite(elementIds: string[], obj: QuantumObject): void;
  getComposite(elementIds: string[]): QuantumObject | undefined;
  getCompositeForElement(elementId: string): QuantumObject | undefined;
}

Validation Results:

File: packages/quantum/src/qgraph/types.ts (~80 lines)

interface ICompositeSystem {
  id: string;
  vertexIds: Set<string>;
  state: StateVector;
  dimension: number;
}

interface EntanglementOperation {
  sourceVertices: string[];
  targetVertices: string[];
  operator: IOperator;
  resultingComposite: string;
}

Phase 2: QuantumGraph Integration ✅ COMPLETE

Status: Implemented and validated in entangledGraphPOC.ts

File: packages/quantum/examples/poc/entangledGraphPOC.ts (extended implementation)

class CompositeQuantumGraph {
  private compositeManager: CompositeQuantumManager;
  
  // Composite methods implemented
  setCompositeQuantumObject(elementIds: string[], obj: QuantumObject): void;
  getCompositeQuantumObject(elementIds: string[]): QuantumObject | undefined;
  
  // Enhanced backward compatible methods
  getVertexQuantumObject(nodeId: string): QuantumObject | undefined; // composite priority
  setVertexQuantumObject(nodeId: string, obj: QuantumObject): void; // prevents overwrites
}

Integration Status:

Next Phase Ready: Entanglement operations (CNOT, Bell state creation, measurement)

Phase 3: Graph-State Entanglement Operations ✅ COMPLETE - 3-4 days)

Current Status: Phase 3C Complete - QuantumGraph class wired to general operations module with test coverage.

Phase 3A: General Operation Infrastructure ✅ COMPLETE

Implementation Summary: The general operation infrastructure provides a foundation for applying arbitrary quantum operations to arbitrary subsets of graph elements. This addresses the core requirement to create entanglement ON existing graph states rather than attaching pre-entangled objects.

Files Implemented:

Key Features Implemented:

Current Capabilities:

Architecture Benefits:

Phase 3B: Enhanced Operation Implementation ✅ COMPLETE

Implementation Summary: Enhanced the general operations module to use existing quantum module functionality instead of placeholder implementations. All multi-element operations now work using proven quantum library components.

Key Enhancements Made:

Integration with Existing Quantum Module:

Phase 3C: QuantumGraph Integration ✅ COMPLETE

Implementation Summary: Wired up the QuantumGraph class to use the general operations module instead of placeholder implementations. Added comprehensive test coverage and example code.

Key Integration Made:

Test Coverage Added:

Current Working Functionality:

Composite System Step-by-Step Operation

The current POC composite system works as follows:

  1. Composite Manager Structure

  2. Setting Composite Relationships

  3. Retrieval with Composite Priority

  4. Composite Object Creation

Required Changes for Graph-State Entanglement

Problem: Current system attaches pre-entangled objects to graph elements instead of creating entanglement from existing graph states.

Solution: Transform to act ON graph states themselves:

  1. Initialize Graph with Individual States

  2. Graph-Level Gate Operations

  3. Entanglement Creation Process

  4. State Management During Operations

  5. Operation Interface

Current General Implementation: The implemented general operations framework provides flexible quantum operations without requiring specific pre-defined functions:

// Create Bell state between vertices using general operations
graph.applyOperation([vertex1], hadamardOperator);  // Apply H to first vertex
graph.applyOperation([vertex1, vertex2], cnotOperator);  // Apply CNOT to both

// Create multi-vertex entangled states using general operations
graph.applyOperation([vertex1, vertex2, vertex3], ghzOperator);  // Any multi-qubit operator

// Apply arbitrary operators to arbitrary element subsets
graph.applyVertexOperation(vertexIds, operator);  // Vertex-specific operations
graph.applyEdgeOperation(edgeIds, operator);     // Edge-specific operations
graph.applyOperation(elementIds, operator);      // Mixed vertex/edge operations

// Measurement operations using general framework
const result = graph.measureSubsystem(vertexIds, projector);

Design Philosophy: Instead of implementing specific entanglement functions, the general operations framework allows users to apply any quantum operator to any subset of graph elements, providing maximum flexibility while maintaining the same functionality.

Phase 4: Quantum Circuit Operations (2-3 days) NEXT

Proposed Enhancement: Higher-level convenience functions built on the general operations framework:

// Convenience wrappers using the general operations
function applyCNOT(graph: QuantumGraph, controlVertex: string, targetVertex: string): void {
  graph.applyOperation([controlVertex, targetVertex], cnotOperator);
}

function applyHadamard(graph: QuantumGraph, vertex: string): void {
  graph.applyVertexOperation([vertex], hadamardOperator);
}

function applyToffoli(graph: QuantumGraph, control1: string, control2: string, target: string): void {
  graph.applyOperation([control1, control2, target], toffoliOperator);
}

// Circuit sequence execution using general framework
interface CircuitOperation {
  elementIds: string[];
  operator: IOperator;
}

function executeCircuit(graph: QuantumGraph, operations: CircuitOperation[]): void {
  operations.forEach(op => graph.applyOperation(op.elementIds, op.operator));
}

Note: These are convenience functions that internally use the general applyOperation(), applyVertexOperation(), and applyEdgeOperation() methods already implemented.

Phase 5: Enhanced Bell Chain Example (1-2 days) NEXT

Proposed Enhancement: Update Bell chain example to use general operations framework:

export function createEntangledBellChain(config: BellChainConfig): QuantumGraph {
  const graph = new QuantumGraph();
  
  // Add individual vertices with |0⟩ states
  for (let i = 0; i < config.numVertices; i++) {
    graph.addNode({id: `q${i}`, type: 'qubit', properties: {...}});
    graph.setVertexQuantumObject(`q${i}`, StateVector.computationalBasis(2, 0));
  }
  
  // Create actual Bell state entanglement using general operations
  for (let i = 0; i < config.numVertices - 1; i++) {
    // Apply Hadamard using general vertex operation
    graph.applyVertexOperation([`q${i}`], hadamardOperator);
    
    // Apply CNOT using general multi-element operation
    graph.applyOperation([`q${i}`, `q${i+1}`], cnotOperator);
    
    console.log(`Created Bell pair between q${i} and q${i+1}`);
  }
  
  // Handle periodic boundary using general operations
  if (config.periodic && config.numVertices > 2) {
    graph.applyOperation([`q${config.numVertices-1}`, `q0`], cnotOperator);
  }
  
  return graph;
}

export function verifyEntanglement(graph: QuantumGraph): void {
  // Use existing quantum module functions for verification
  // Calculate entanglement entropy using existing information.ts functions
  // Verify Bell state correlations using composite state access
  // Show composite state dimensions using existing methods
}

Key Change: Uses the implemented general operations (applyVertexOperation, applyOperation) instead of specific gate functions.

Phase 6: Testing and Validation (2-3 days) REMAINING

Proposed Testing Strategy: Test the general operations framework functionality:

describe('Quantum Graph General Operations', () => {
  test('Multi-element operations', () => {
    // Test applyOperation() with CNOT on two vertices
    // Verify tensor product creation
    // Validate composite state storage
  });
  
  test('Vertex and edge operations', () => {
    // Test applyVertexOperation() with Hadamard
    // Test applyEdgeOperation() with operators on edges
    // Verify element type detection and routing
  });
  
  test('Composite state management', () => {
    // Test tensorProductStates() function
    // Test splitCompositeState() function
    // Verify integration with existing quantum module components
  });
  
  test('Subsystem operations', () => {
    // Test extractSubsystemState() and insertSubsystemState()
    // Test measureSubsystem() functionality
    // Verify partial trace operations
  });
  
  test('Integration with quantum module', () => {
    // Test StateVector.tensorProduct() integration
    // Test DensityMatrixOperator.fromPureState() integration
    // Verify partialTrace() integration
  });
});

Focus: Test the implemented general operations framework rather than specific quantum circuit functions.

Implementation Details

1. Composite State Identification

// Use deterministic IDs for composite systems
function generateCompositeId(vertexIds: string[]): string {
  return `composite_${[...vertexIds].sort().join('_')}`;
}

2. State Vector Tensor Operations

// Extend StateVector for tensor products
class StateVector {
  static tensorProduct(state1: StateVector, state2: StateVector): StateVector;
  static partialTrace(compositeState: StateVector, tracedVertices: number[]): StateVector;
}

3. Memory Management

// Handle large composite state spaces efficiently
class CompositeStateManager {
  private maxCompositeSize: number = 10; // Limit to 2^10 = 1024 dimensions
  
  private validateCompositeSize(vertexCount: number): void {
    if (Math.pow(2, vertexCount) > Math.pow(2, this.maxCompositeSize)) {
      throw new Error(`Composite system too large: ${vertexCount} qubits`);
    }
  }
}

4. Edge Operator Integration

// Apply edge operators to create entanglement
class QuantumGraph {
  applyEdgeOperator(edgeId: string): void {
    const edge = this.getEdge(edgeId);
    const operator = this.getEdgeQuantumObject(edgeId) as IOperator;
    
    if (operator) {
      this.applyCompositeOperation([edge.sourceId, edge.targetId], operator);
    }
  }
}

Success Criteria

  1. Genuine Bell State Creation: |00⟩ + |11⟩ stored as single composite state ✅ ACHIEVED
  2. Multi-element Operations: CNOT, Hadamard, and arbitrary operators work across vertices ✅ ACHIEVED
  3. Composite State Tracking: System knows which vertices are entangled together ✅ ACHIEVED
  4. Existing Library Integration: All operations use proven quantum module components ✅ ACHIEVED
  5. Tensor Product Composition: Individual states combine into entangled composites ✅ ACHIEVED
  6. State Splitting: Composite states can be decomposed via partial trace ✅ ACHIEVED

Remaining Criteria for Full Implementation: 7. Measurement Correlations: Measuring one qubit instantly affects its partner (Phase 4) 8. Circuit Operations: High-level quantum circuit operations (Phase 4) 9. Memory Efficiency: No redundant storage of individual states for entangled systems (Phase 5)

File Structure Summary

packages/quantum/src/qgraph/
├── CompositeQuantumManager.ts    # ✅ Core composite state storage
├── QuantumGraph.ts              # ✅ Enhanced with general operation support  
├── operations/
│   ├── general.ts               # ✅ General operation functions (implemented)
│   └── index.ts                 # ✅ Operations module exports
└── types.ts                     # ✅ Interface definitions

packages/quantum/examples/qgraph/
├── bellStateChain.ts            # ✅ Basic example (static structure)
├── basicOperations.ts           # ✅ Working examples with test coverage
└── entangledBellChain.ts        # 🔄 Enhanced example using general operations (next)

packages/quantum/__tests__/qgraph/
└── general-operations.test.ts   # ✅ Comprehensive test suite (complete)

packages/quantum/docs/
└── graph-entanglement-plan.md   # ✅ This document (updated)

Legend: ✅ Complete, 🔄 Next phase

Dependencies

Timeline

Total Estimated Time: 13-19 days

Notes

This implementation transforms the QuantumGraph from a static structure with quantum labels into a dynamic quantum computational framework capable of representing and manipulating genuine quantum entanglement across graph topologies.

Last Updated: 2025-07-06