From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz (1709.03489v2)

Abstract: The next few years will be exciting as prototype universal quantum processors emerge, enabling implementation of a wider variety of algorithms. Of particular interest are quantum heuristics, which require experimentation on quantum hardware for their evaluation, and which have the potential to significantly expand the breadth of quantum computing applications. A leading candidate is Farhi et al.'s Quantum Approximate Optimization Algorithm, which alternates between applying a cost-function-based Hamiltonian and a mixing Hamiltonian. Here, we extend this framework to allow alternation between more general families of operators. The essence of this extension, the Quantum Alternating Operator Ansatz, is the consideration of general parametrized families of unitaries rather than only those corresponding to the time-evolution under a fixed local Hamiltonian for a time specified by the parameter. This ansatz supports the representation of a larger, and potentially more useful, set of states than the original formulation, with potential long-term impact on a broad array of application areas. For cases that call for mixing only within a desired subspace, refocusing on unitaries rather than Hamiltonians enables more efficiently implementable mixers than was possible in the original framework. Such mixers are particularly useful for optimization problems with hard constraints that must always be satisfied, defining a feasible subspace, and soft constraints whose violation we wish to minimize. More efficient implementation enables earlier experimental exploration of an alternating operator approach to a wide variety of approximate optimization, exact optimization, and sampling problems. Here, we introduce the Quantum Alternating Operator Ansatz, lay out design criteria for mixing operators, detail mappings for eight problems, and provide brief descriptions of mappings for diverse problems.

• The paper introduces a generalized framework that extends QAOA by incorporating a wider range of unitary operations to navigate complex optimization landscapes.
• The paper details design criteria for mixing operators and encoding strategies, enabling efficient exploration of state spaces in combinatorial problems.
• The paper provides resource assessments—covering qubits, gate depth, and noise robustness—to support future empirical and theoretical quantum heuristic experiments.

Quantum Alternating Operator Ansatz for Optimization Problems

The paper extends the Quantum Approximate Optimization Algorithm (QAOA), originally proposed by Farhi et al., to a more generalized framework known as the Quantum Alternating Operator Ansatz (QAOA). This extension broadens the scope of QAOA by allowing more varied unitary operations, rather than being limited to those corresponding to the time evolution of a fixed local Hamiltonian. The paper meticulously explores how this broadened framework can be applied to various optimization problems, especially those with hard constraints, thus potentially supporting a wider array of quantum heuristic methods.

The exploration starts with a comprehensive review of the original QAOA, where algorithms alternate between applying a cost-function-based Hamiltonian and a mixing Hamiltonian. The authors propose a new formulation, which involves a wider selection of unitary operations, dubbed as mixing operators. The utility of these operators lies in their ability to explore larger and more useful state spaces by enabling more efficient transitions within the computational landscape.

The researchers provide methodological details and design criteria for the mixing operators and initial states, integral components of their proposed ansatz. They ensure these elements can be efficiently implemented, which is crucial for early experimentation on emerging quantum hardware. Consequently, this paper proposes several innovative mapping strategies, particularly for problems represented by strings, subsets, and permutations. For instance, the one-hot encoding and binary encoding techniques are employed to handle different aspects of these problems.

The paper delivers careful analyses of key problems, such as variations of the Max-$k$-ColorableSubgraph, MaxIndependentSet, MinGraphColoring, and various scheduling problems like the Traveling Salesperson Problem (TSP). Each problem is examined with its specific constraints and objectives, expanding the QAOA framework to incorporate problem-specific operators and configurations efficiently. The authors delve into the specifics of building partitioned mixing Hamiltonians, controlled operators, and different encoding strategies for these problems.

For practitioners, the paper provides resource assessments in terms of qubits, gate depth, and complexity, examining how different mixers and strategies affect resource requirements. The implementation frameworks incorporate considerations for existing quantum computational limitations like qubit connectivity and noise robustness.

Critically, the paper proposes an empirical exploration into the effectiveness of these newly proposed circuits as quantum heuristics. The authors call for further research into parameter optimization, error correction, and more effective mixers to fully realize the potential of QAOA and its extension. Such work would involve improved simulation techniques and experimental trials on quantum hardware.

Overall, the introduction of QAOA presents a substantial leap in extending quantum optimization capabilities beyond what was previously possible with the original QAOA. The new framework offers promising insights into enhancing approximation approaches for complex combinatorial problems, setting the stage for future empirical and theoretical investigations in quantum algorithm development.