Conditional Operations and Classical Registers in Quantum Circuits

Table of Contents

1. Introduction
2. Classical Registers in Qiskit
3. Classical vs Quantum Information Flow
4. Measurement and Classical Register Mapping
5. Mid-Circuit Measurement
6. Conditional Gates Based on Classical Bits
7. Syntax for Conditional Operations in Qiskit
8. Example: Conditional X Gate
9. Multi-Qubit Classical Conditioning
10. Circuit Execution and Timing Considerations
11. Supported Gates and Conditional Constraints
12. Conditional Execution and Feedback Loops
13. Simulating Circuits with Classical Conditions
14. Classical Control Flow with Qiskit Extensions
15. Using c_if() for Conditional Logic
16. Classical Expressions in Qiskit (OpenQASM 3)
17. Classical Registers vs Classical Memory
18. Device Support for Conditional Execution
19. Debugging and Best Practices
20. Conclusion

1. Introduction

Quantum programs often require interaction between quantum and classical domains. Conditional operations allow quantum gates to be applied only if a specific measurement outcome is observed.

2. Classical Registers in Qiskit

Classical registers store the results of quantum measurements. Defined during circuit initialization:

qc = QuantumCircuit(2, 2)  # 2 qubits, 2 classical bits

3. Classical vs Quantum Information Flow

Quantum information evolves through unitary gates. Classical bits store irreversible measurement outcomes. Conditional operations link these two layers.

4. Measurement and Classical Register Mapping

qc.measure(0, 0)  # Maps qubit 0 to classical bit 0

5. Mid-Circuit Measurement

Measurements can occur mid-execution:

qc.measure(0, 0)

This enables real-time decision making using classical control.

6. Conditional Gates Based on Classical Bits

Apply a gate only if a classical bit has a certain value:

qc.x(1).c_if(qc.cregs[0], 1)

7. Syntax for Conditional Operations in Qiskit

Use .c_if(classical_register, value) to gate execution.

8. Example: Conditional X Gate

from qiskit import QuantumCircuit
qc = QuantumCircuit(2, 1)
qc.h(0)
qc.measure(0, 0)
qc.x(1).c_if(qc.cregs[0], 1)

This applies an X gate to qubit 1 if qubit 0 was measured as 1.

9. Multi-Qubit Classical Conditioning

qc.measure([0, 1], [0, 1])
qc.h(2).c_if(qc.cregs[0], 3)  # Only if bits 0 and 1 are both 1 (binary 11)

10. Circuit Execution and Timing Considerations

Conditional operations depend on:

• Latency of measurement readout
• Device ability to pause/resume based on classical bits
• Supported by most simulators, but limited on some hardware

11. Supported Gates and Conditional Constraints

Not all gates are conditionally executable on real hardware. Typically supported:

• X, CX
• Phase gates (Z, S, T)
• Delay (on hardware)
  Gate support varies by backend.

12. Conditional Execution and Feedback Loops

Used in:

• Quantum teleportation
• Quantum error correction
• Adaptive quantum metrology

13. Simulating Circuits with Classical Conditions

Aer simulator supports conditional operations:

backend = Aer.get_backend('qasm_simulator')

14. Classical Control Flow with Qiskit Extensions

Qiskit extensions allow loops and conditional branching:

from qiskit.circuit import IfElseOp
# Available in OpenQASM 3.0 through future support

15. Using c_if() for Conditional Logic

The .c_if() method binds an instruction to a classical bitmask and target value.

16. Classical Expressions in Qiskit (OpenQASM 3)

OpenQASM 3 supports full classical expressions:

if (c == 0b11) {
    x q[2];
}

Still under development in Qiskit full support.

17. Classical Registers vs Classical Memory

• Classical Registers: named bit collections (creg[0], creg[1])
• Classical Memory: used in OpenQASM 3 for dynamic access

18. Device Support for Conditional Execution

• Most IBM Q devices support basic conditionals
• Real-time feedback limited to latency ~µs
• Long-term goal: autonomous feedback and error correction

19. Debugging and Best Practices

• Always map qubits to classical bits explicitly
• Simulate circuits before hardware execution
• Check backend support for conditional gates

20. Conclusion

Conditional operations connect the quantum and classical worlds. Mastering them allows implementation of complex quantum logic, adaptive protocols, and interactive quantum algorithms.