Explanations

Understanding-oriented documentation that clarifies concepts, design decisions, and theoretical foundations of ATLAS-Q.

• Tensor Networks
  • Overview
  • Tensor Network Fundamentals
  • Matrix Product States (MPS)
  • Canonical Forms
  • Entanglement and Bond Dimensions
  • Matrix Product Operators (MPO)
  • Tensor Network Contraction
  • Higher-Dimensional Tensor Networks
  • Computational Complexity
  • Connection to Other Formalisms
  • Advantages and Limitations
  • Summary
  • See Also
• Adaptive Truncation
  • Overview
  • SVD and Optimal Truncation
  • Truncation Criteria
  • Error Accumulation and Bounds
  • Adaptive Truncation Strategies
  • Memory Budget Management
  • Special Cases and Optimizations
  • Truncation-Free Methods
  • Implementation in ATLAS-Q
  • Comparison with Other Methods
  • Summary
  • See Also
• Algorithms
  • Overview
  • Time-Dependent Variational Principle (TDVP)
  • Density Matrix Renormalization Group (DMRG)
  • Variational Quantum Eigensolver (VQE)
  • Quantum Approximate Optimization Algorithm (QAOA)
  • Stabilizer Formalism
  • Period-Finding Algorithm
  • Algorithm Selection Guide
  • Summary
• GPU Acceleration
  • Overview
  • GPU Architecture Fundamentals
  • PyTorch CUDA Backend
  • Custom Triton Kernels
  • cuQuantum Integration
  • GPU Memory Management
  • Tensor Core Acceleration
  • Performance Profiling
  • Multi-GPU Scaling
  • Best Practices
  • Summary
• Numerical Stability
  • Overview
  • Floating-Point Arithmetic
  • Sources of Numerical Error
  • Condition Numbers
  • Canonical Forms and Orthogonality
  • Truncation Error Analysis
  • Robust Linear Algebra
  • Mixed Precision Strategies
  • Diagnostics and Monitoring
  • Best Practices
  • Common Issues and Solutions
  • Summary
• Performance Model
  • Overview
  • Computational Complexity
  • Memory Scaling
  • Time Complexity
  • Scaling Laws
  • Performance Predictions
  • Bottleneck Analysis
  • Optimization Strategies
  • Case Studies
  • Best Practices
  • Summary
• Design Decisions
  • Overview
  • Core Architecture
  • API Design
  • Backend Selection
  • Numerical Stability
  • Memory Management
  • Modularity and Extension
  • Error Handling
  • Future-Proofing
  • Lessons Learned
  • Summary
• Comparisons
  • Overview
  • ATLAS-Q vs Qiskit Aer
  • ATLAS-Q vs Cirq
  • ATLAS-Q vs PennyLane
  • ATLAS-Q vs ITensor
  • ATLAS-Q vs TeNPy
  • ATLAS-Q vs Full Statevector
  • Unique ATLAS-Q Features
  • Performance Summary
  • Ecosystem Integration
  • Summary and Recommendations

Explanation Topics

Tensor Networks

Matrix Product States (MPS), Matrix Product Operators (MPO), and Projected Entangled Pair States (PEPS). Explains tensor network representation, contraction schemes, and canonical forms.

Adaptive Truncation

Bond dimension selection strategies, energy-based truncation, entropy criteria, and global error bounds. Discusses trade-offs between accuracy and efficiency.

Algorithms

Mathematical foundations of TDVP, VQE, QAOA, stabilizer formalism, and period-finding. Includes complexity analysis and convergence properties.

GPU Acceleration

GPU optimization strategies, custom Triton kernels, memory management, and cuQuantum integration. Explains performance characteristics and bottlenecks.

Numerical Stability

Sources of numerical error, ill-conditioned matrices, mixed-precision strategies, and robust linear algebra. Discusses condition number monitoring and automatic promotion.

Performance Model

Computational complexity, memory scaling, and performance predictions for different simulation regimes. Helps choose appropriate methods for specific problem sizes.

Design Decisions

Architectural choices, API design rationale, and implementation trade-offs. Explains why ATLAS-Q works the way it does.

Comparisons

Comparison with other quantum simulation frameworks (Qiskit Aer, Cirq, ITensor, TeNPy). Discusses when to use each tool.

Theoretical Background

Mathematical notation used throughout ATLAS-Q documentation:

• \(|\psi\rangle\): Quantum state vector

• \(\chi\): Bond dimension

• \(\epsilon\): Truncation tolerance

• \(S\): Entanglement entropy

• \(\sigma_i\): Singular values

• \(\hat{H}\): Hamiltonian operator

Matrix Product State (MPS) representation:

\[|\psi\rangle = \sum_{s_1,\ldots,s_n} A^{[1]}_{s_1} A^{[2]}_{s_2} \cdots A^{[n]}_{s_n} |s_1 s_2 \ldots s_n\rangle\]

where \(A^{[i]}_{s_i}\) are rank-3 tensors with shape \([\chi_{i-1}, 2, \chi_i]\).

Truncation criterion:

\[\text{keep } k \text{ such that } \sum_{i=1}^k \sigma_i^2 \geq (1 - \epsilon^2) \sum_{i=1}^{r} \sigma_i^2\]

References

Key papers and resources:

• Verstraete et al., “Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems,” Advances in Physics (2008)

• Haegeman et al., “Time-Dependent Variational Principle for Quantum Lattices,” Physical Review Letters (2011)

• Peruzzo et al., “A variational eigenvalue solver on a photonic quantum processor,” Nature Communications (2014)

• Farhi & Goldstone, “A Quantum Approximate Optimization Algorithm,” arXiv:1411.4028 (2014)

• Aaronson & Gottesman, “Improved Simulation of Stabilizer Circuits,” Physical Review A (2004)

See Citing ATLAS-Q for how to cite ATLAS-Q in academic work.