Optimizing Quantum Circuits

Introduction

In the previous article, we tackled the intricacies of implementing Shor’s algorithm for solving the Elliptic Curve Discrete Logarithm Problem (ECDLP)—a complex and challenging topic. To give you a pause, this article is a lighter read, focusing on the practical art of optimizing quantum circuits. Think of it as a chance to step back, relax, and fine-tune the circuits we’ve already built.

Optimization might not sound as flashy as breaking cryptographic protocols, but it’s a crucial skill in quantum computing. By the end of this article, you’ll have a clear understanding of Qiskit’s optimization passes and how they can help streamline your quantum circuits for better efficiency and performance.

Optimization Passes in Qiskit

Here’s a breakdown of the optimization passes available in Qiskit and how they transform quantum circuits. The source code for these passes is openly available in the Qiskit repository (link).

1. Collecting Clifford and Linear Functions

What are Clifford gates?

Clifford gates include gates like Hadamard ($H$) and CNOT gates, along with their combinations. They form a group called the Clifford group, which maps to themselves under conjugation by Pauli operators ($X$, $Y$, $Z$).

What it does:

Replaces a block of consecutive Clifford gates with a single equivalent gate.

Example:

• Before optimization:

OPENQASM 2.0;
qreg q[1];

h q[0];
s q[0];
h q[0];

• After optimization:

OPENQASM 2.0;
qreg q[1];

sdg q[0]; // Equivalent single Clifford gate

2. Commutative Cancellation

What it does:

Identifies and cancels self-inverse gates (i.e., gates that are their own inverse) that appear in pairs.

Example:

• Before optimization:

OPENQASM 2.0;
qreg q[1];

h q[0];
h q[0];

• After optimization:

OPENQASM 2.0;
qreg q[1];

// No gates left after optimization

3. Commutative Inverse Cancellation

What it does:

Removes pairs of inverse gates that may not be adjacent by leveraging commutation properties.

Example:

• Before optimization:

OPENQASM 2.0;
qreg q[1];

h q[0];
x q[0];
h q[0];
x q[0];

• After optimization: Exploiting commutativity of $X$ and $H$ gates, the two $X$ gates cancel out.

OPENQASM 2.0;
qreg q[1];

h q[0];
h q[0];

(Then the two $H$ gates are optimized by commutative cancellation.)

4. CNOT Cancellation

What it does:

Removes adjacent pairs of CNOT gates with the same control and target qubits.

Example:

• Before optimization:

OPENQASM 2.0;
qreg q[2];

cx q[0], q[1];
cx q[0], q[1];

• After optimization: The two consecutive CNOT gates cancel.

OPENQASM 2.0;
qreg q[2];

// No gates remain

5. Hoare Logic Optimization

What is Hoare logic?

Hoare logic is a formal system for reasoning about the correctness of programs, applied here to quantum circuits.

What it does:

Simplifies blocks of gates by analyzing their logic using Hoare rules, powered by Z3 solver.

Example:

• Before optimization:

OPENQASM 2.0;
qreg q[2];

cx q[0], q[1];

• After optimization: The CNOT gate is simplified using Hoare logic.

OPENQASM 2.0;
qreg q[2];

// No gates remain

Since the quantum register q is initialized as $\ket{00}$, the CNOT gate becomes a no-op.

6. Optimizing Commutation

What it does:

Exploits the commutation of single-qubit gates with two-qubit gates to reorder operations for better efficiency.

Example:

• Before optimization:

OPENQASM 2.0;
qreg q[2];

rz(0.5) q[0];
cx q[0], q[1];
rz(0.5) q[0];

• After optimization: The two $R_z(0.5)$ gates are commuted past the CNOT gate.

OPENQASM 2.0;
qreg q[2];

cx q[0], q[1];
rz(1.0) q[0];

Note: $R_z(0.5)$ and $R_z(0.5)$ combine to $R_z(1.0)$; $R_z$ (rotation around the $Z$ axis) is parameterized by the sum of the angles.

7. Optimizing SWAP Before Measurement

What it does:

Removes unnecessary SWAP gates (that swap two qubits) followed immediately by measurement gates since the swap can be easily done in post-processing.

Example:

• Before optimization:

OPENQASM 2.0;
qreg q[2];
creg c[1];

swap q[0], q[1];
measure q[1] -> c[0];

• After optimization: The SWAP gate is removed.

OPENQASM 2.0;
qreg q[2];
creg c[1];

measure q[0] -> c[0];

8. Removing Diagonal Gates Before Measurement

Why don’t diagonal gates affect measurement outcomes?

Diagonal gates only alter the phase of a quantum state. Not affecting the amplitude, they don’t change the probabilities of measurement outcomes. For instance, $Z$ gate is diagonal:

\[\begin{array}{c} Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \\ Z \prths{a \ket{0} + b \ket{1}} = a \ket{0} - b \ket{1}. \end{array}\]

$Z$ gate doesn’t change the probabilities of measuring $\ket{0}$ or $\ket{1}$, only the phase.

What it does:

Removes diagonal gates that precede measurements.

Example:

• Before optimization:

OPENQASM 2.0;
qreg q[1];
creg c[1];

rz(0.5) q[0];
measure q[0] -> c[0];

• After optimization: The $R_z(0.5)$ gate is removed.

OPENQASM 2.0;
qreg q[1];
creg c[1];

measure q[0] -> c[0];

Using Qiskit’s Transpiler for Optimization

To apply these optimization passes, Qiskit’s transpile function is used. The optimization level determines how aggressive the optimization is:

• 0: No optimization
• 1: Light optimization (e.g., gate cancellation)
• 2: Medium optimization (e.g., commutation and consolidation)
• 3: Aggressive optimization (e.g., Hoare logic)

Here’s how we transpiled a quantum circuit implementation in the previous article using optimization level 3:

from qiskit import transpile

optimized_circuit = transpile(original_circuit, optimization_level=3)