Qubit Routing and Compilation: Optimizing Quantum Circuits for Real Hardware

Table of Contents

1. Introduction
2. What Is Qubit Routing?
3. The Need for Compilation in Quantum Computing
4. Logical vs Physical Qubit Mapping
5. Coupling Constraints in Hardware
6. Overview of Routing Algorithms
7. SWAP Insertion Strategies
8. Routing Cost Metrics
9. Compilation Workflow in Qiskit
10. Layout Selection Techniques
11. SABRE: Swap-Based Adaptive Routing
12. Lookahead Routing and Heuristics
13. Commutativity and Gate Reordering
14. Circuit Rewriting for Optimization
15. Hardware-Aware Compilation Tools
16. Mapping and Routing in t|ket>
17. Compilation for Trapped Ions vs Superconducting Qubits
18. Impact of Routing on Fidelity and Execution Time
19. Visualization and Debugging of Routing Paths
20. Conclusion

1. Introduction

Qubit routing is the process of adapting an ideal quantum circuit to the specific physical constraints of a quantum device, ensuring valid gate execution paths. It’s a crucial step in the compilation process for real hardware.

2. What Is Qubit Routing?

Routing finds a mapping from logical qubits to physical qubits while satisfying coupling constraints, often involving inserting SWAP operations to move qubit states.

3. The Need for Compilation in Quantum Computing

• Logical circuits assume full connectivity
• Physical hardware is constrained
• Compilation ensures valid and optimized execution

4. Logical vs Physical Qubit Mapping

• Logical qubits: defined by algorithm
• Physical qubits: actual device layout
  Routing establishes the best mapping between the two.

5. Coupling Constraints in Hardware

Qubits are not fully connected. Only certain pairs can perform two-qubit gates. Devices expose these constraints via a coupling map.

6. Overview of Routing Algorithms

• Exact (search-based): optimal but slow
• Heuristic: scalable and fast
• Examples: SABRE, Greedy, Beam search

7. SWAP Insertion Strategies

When qubits are non-adjacent:

• Insert SWAP gates to move states closer
• Prioritize gates with early deadlines or high weight

8. Routing Cost Metrics

• Circuit depth
• Number of SWAPs
• Fidelity impact
• Total gate count

9. Compilation Workflow in Qiskit

from qiskit import transpile
transpiled = transpile(circuit, backend, optimization_level=3)

10. Layout Selection Techniques

• Trivial layout: assign qubits in order
• Dense layout: place connected logical qubits close
• Noise-aware layout: prefer higher-fidelity qubits

11. SABRE: Swap-Based Adaptive Routing

Qiskit’s default heuristic for routing:

• Balances SWAP cost vs lookahead
• Adapts dynamically to gate queue

12. Lookahead Routing and Heuristics

Evaluates future gate needs to plan optimal current SWAPs.

13. Commutativity and Gate Reordering

Reorders gates that commute to expose better parallelism and reduce SWAP overhead.

14. Circuit Rewriting for Optimization

• Gate merging
• Cancellation (e.g., CX followed by CX = I)
• Rebase to native gates

15. Hardware-Aware Compilation Tools

• Qiskit: PassManager, transpiler stages
• t|ket>: RoutingPass, MappingPass
• Q#: ResourceEstimator

16. Mapping and Routing in t|ket>

• Uses advanced cost models and placement strategies
• Provides visual feedback on routing

17. Compilation for Trapped Ions vs Superconducting Qubits

• Trapped ions: all-to-all but slow gates
• Superconducting: fast gates but strict topology

18. Impact of Routing on Fidelity and Execution Time

Poor routing = more SWAPs = more errors
Optimized routing = shorter time and higher success

19. Visualization and Debugging of Routing Paths

Use:

circuit.draw('mpl')

To compare pre- and post-routing layouts and gate placement.

20. Conclusion

Qubit routing and compilation bridge the gap between abstract quantum algorithms and real hardware execution. Understanding the routing process helps developers create efficient, hardware-compatible quantum circuits and minimize execution errors.