Quantum approximate optimization algorithm

Background

The Quantum Approximate Optimization Algorithm (QAOA) is a hybrid iterative method for solving combinatorial optimization problems. In this tutorial, we demonstrate how to implement the QAOA algorithm using Qiskit Runtime for solving a simple max-cut problem.

In a max-cut problem, we want to partition nodes of a graph in a way that maximizes the number of edges between nodes in differing groups. The desired max-cut partition for the following graph is clear: the 0th-node on the left should be separated from the rest of the nodes on the right by a cut. We will find this answer by applying QAOA by using Qiskit Runtime primitives and sessions.

Setup

Step 1: Map classical inputs to a quantum problem

To demonstrate max-cut, we'll create a graph using the rustworkx library, and create Pauli Hamiltonian that encodes the cost in a manner such that the minimum expectation value of the operator corresponds to the maximum number of edges between the nodes in two different groups.

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Output:

For this simple example, the operator is a linear combination of terms with Z operators on nodes connected by an edge (recall that the 0th qubit is farthest right): $IIIZZ + IIZIZ + IZIIZ + ZIIIZ$. Once the operator is constructed, the ansatz for the QAOA algorithm can easily be built by using the QAOAAnsatz circuit from the Qiskit circuit library.

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Output:

The previous image illustrates the ansatz in basic gates for clarity. However, it can be expressed in multiple levels of decomposition by changing the reps argument or by drawing the circuit without the decompose method. For example, the following representation directly shows the QAOA structure with the default reps value, which is reps=1.

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Output:

Step 2: Optimize problem for quantum execution.

We can schedule a series of qiskit.transpiler passes to optimize our circuit for a selected backend. This includes a few components:

• optimization_level: The lowest optimization level does the bare minimum needed to get the circuit running on the device; it maps the circuit qubits to the device qubits and adds swap gates to allow all 2-qubit operations. The highest optimization level is much smarter and uses lots of tricks to reduce the overall gate count. Since multi-qubit gates have high error rates and qubits decohere over time, the shorter circuits should give better results.
• Dynamical Decoupling: We can apply a sequence of gates to idling qubits. This cancels out some unwanted interactions with the environment.