hybrid-tensor-network-qml

name: hybrid-tensor-network-qml description: > Hybrid tensor network architecture for quantum machine learning using post-selection as a trainable hyperparameter. Interpolates between classical and quantum tensor network edge cases by controlling quantum constraint enforcement via post-selection allocation. Use when designing hybrid quantum-classical ML models, tensor network quantum ML, or optimizing quantum resource allocation with limited post-selection budget. Activation: hybrid tensor network, quantum-classical interpolation, post-selection QML, trainable quantum constraints.

Hybrid Tensor Networks for QML

Overview

Hybrid tensor networks combine classical and quantum tensor networks in a unified framework, using post-selection as the key property controlling the interpolation between regimes. The amount of post-selection determines how strongly quantum constraints are enforced on the network.

Core Concept

Post-Selection as Hyperparameter

The framework introduces a new hyperparameter controlling the transition:

• 0 post-selection → Pure classical tensor network
• Full post-selection → Pure quantum tensor network
• Partial post-selection → Hybrid (practical regime for NISQ)

This hyperparameter complements bond dimension as a second axis for controlling model capacity.

Architecture

Step 1: Classical Tensor Network Backbone

Use classical tensor network (MPS, PEPS, TTN) as the base model:

• Efficient classical inference
• Well-understood training procedures
• Proven expressiveness for many tasks

Step 2: Quantum Edge Integration

Replace selected tensor network edges with quantum circuits:

• Each quantum edge requires post-selection to enforce quantum constraints
• Post-selection probability determines feasible quantum portion

Step 3: Trainable Post-Selection Allocation

Instead of fixed post-selection ratio:

Allocate post-selection budget to quantum model in a trainable manner
→ Optimize which edges get quantum treatment
→ Maximize quantum advantage within hardware constraints

Training Protocol

1. Initialize classical tensor network
2. Select subset of edges for quantum replacement
3. Define post-selection budget (hyperparameter)
4. Train with quantum inference on selected edges
5. Optimize post-selection allocation jointly with model parameters
6. Evaluate classical vs quantum vs hybrid performance

Comparison Framework

When comparing classical vs quantum tensor networks, report:

• Bond dimension (traditional hyperparameter)
• Post-selection ratio (new hyperparameter)
• Classical/quantum/hybrid accuracy
• Resource requirements (qubits, shots, post-selection success rate)

Key Insights

1. Post-selection is the bottleneck: Limited post-selection on real devices means pure quantum tensor networks may be impractical
2. Hybrid is the practical regime: Partial quantum constraints + classical backbone gives best tradeoff
3. Trainable allocation: Let the model learn where quantum matters most
4. Complementary to bond dimension: Two independent capacity controls

Design Patterns

Pattern 1: Budget-Constrained Hybrid Design

Fixed post-selection budget → Optimize allocation → Best hybrid architecture

Pattern 2: Progressive Quantum Integration

Start classical → Add quantum edges gradually → Monitor performance gain
→ Stop when budget exhausted or marginal gain negligible

Applications

• Quantum ML with limited qubits: Maximize advantage within hardware limits
• Tensor network compression: Use quantum edges for hard-to-classical-compress regions
• Benchmarking: Systematically compare classical vs quantum tensor networks

References

• Hybrid TN paper: arxiv:2605.02385 (Jäger, Bieniasz, Plenio, Rieser, 2026)
• Tensor Networks for ML: Stoudenmire & Schwab (2016)
• Post-selection in QML: Various works on post-selected quantum computing