Quantum Compilers

Quantum Compilation Pipeline

A quantum compiler transforms high-level quantum circuits into hardware-executable instructions:

1. Frontend parsing: accept quantum program (OpenQASM, QIR)
2. Optimization: reduce circuit depth and gate count
3. Transpilation: map to native gate set
4. Layout/routing: map logical qubits to physical qubits
5. Scheduling: assign timing and parallelism
6. Backend code generation: produce pulse-level instructions

Gate Synthesis

Gate synthesis decomposes arbitrary unitaries into native gate sets:

For a single qubit, any unitary can be decomposed as:

For two qubits, the KAK decomposition (Cartan decomposition) writes any as:

with at most 3 CNOT gates.

Circuit Optimization

Circuit optimization reduces gate count and depth:

• Gate cancellation: remove adjacent inverse gates
• Gate merging: combine multiple gates into one
• Template matching: replace subcircuits with equivalent shorter ones
• Algebraic simplification: use gate identities

For example, and .

Qubit Routing

Qubit routing maps logical qubits to physical qubits with limited connectivity:

1. Initial placement: assign logical to physical qubits
2. SWAP insertion: add SWAP gates to enable non-adjacent interactions

The routing problem is NP-hard. Heuristics include:

• SABRE: look-ahead SWAP routing
• Graph-based: use hardware connectivity graph
• Noise-aware: consider error rates when routing

Transpilation Levels

Qiskit provides transpilation levels 0-3:

• Level 0: no optimization, default layout
• Level 1: light optimization, trivial layout
• Level 2: medium optimization, layout by noise
• Level 3: heavy optimization, full routing and scheduling

Higher levels produce shorter circuits but take longer to compile.

Resource Estimation

Resource estimation counts the resources needed for a quantum algorithm:

• Qubit count: number of physical qubits
• Gate count: number of gates (by type)
• Circuit depth: depth of the circuit
• T-gate count: number of T gates (costly in fault-tolerant QC)

Resource estimation guides hardware requirements and algorithm design.

Python: Circuit Optimization

import numpy as np

def naive_hadamard_circuit(n):
    # n Hadamards -> can be simplified.
    return ['H'] * n

def optimized_hadamard_circuit(n):
    # n Hadamards = I if n is even, H if n is odd.
    return ['H'] if n % 2 else []

def count_gates(circuit):
    return len(circuit)

for n in [10, 100, 1000]:
    naive = naive_hadamard_circuit(n)
    optimized = optimized_hadamard_circuit(n)
    print(f"n={n}: naive={count_gates(naive)}, optimized={count_gates(optimized)}")

Transpilation Algorithms

The SABRE algorithm for qubit routing:

1. Initial placement: map logical to physical qubits using graph matching
2. Forward pass: for each gate, find shortest path and insert SWAPs
3. Backward pass: optimize SWAP insertion
4. Iterate: repeat until no improvement

The algorithm runs in time where is the gate count and is the qubit count.

Native Gate Sets

Different hardware has different native gate sets:

The transpiler must convert arbitrary gates to these native sets.