A Quantum Approximate Optimization Algorithm Applied to a Bounded Occurrence Constraint Problem (1412.6062v2)

Abstract: We apply our recent Quantum Approximate Optimization Algorithm to the combinatorial problem of bounded occurrence Max E3LIN2. The input is a set of linear equations each of which contains exactly three boolean variables and each equation says that the sum of the variables mod 2 is 0 or is 1. Every variable is in no more than D equations. A random string will satisfy 1/2 of the equations. We show that the level one QAOA will efficiently produce a string that satisfies $\left(\frac{1}{2} + \frac{1}{101 D^{1/2}\, l n\, D}\right)$ times the number of equations. A recent classical algorithm achieved $\left(\frac{1}{2} + \frac{constant}{D^{1/2}}\right)$. We also show that in the typical case the quantum computer will output a string that satisfies $\left(\frac{1}{2}+ \frac{1}{2\sqrt{3e}\, D^{1/2}}\right)$ times the number of equations.

• The paper shows that level‑1 QAOA outperforms classical randomized algorithms by satisfying m/2 + m/(101D^(1/2)ln D) constraints in the worst case.
• The authors optimize quantum parameters γ and β within a Hilbert space to significantly improve the solution quality in bounded occurrence constraint problems.
• The results highlight theoretical implications for scaling QAOA to higher levels, potentially overcoming NP‑hard barriers in combinatorial optimization.

Analyzing the Quantum Approximate Optimization Algorithm Applied to Bounded Occurrence Constraint Problems

The paper in question explores the application of the Quantum Approximate Optimization Algorithm (QAOA) to a particular combinatorial optimization problem known as the bounded occurrence Max E3LIN2. This problem is characterized by a set of linear equations over boolean variables, each containing exactly three variables, with the additional constraint that each variable is included in no more than $D$ equations. The authors, Farhi, Goldstone, and Gutmann, seek to determine the efficacy of the QAOA in this domain and compare its performance to classical approximation methods.

A noteworthy aspect of the QAOA as presented in the paper is its design choice, specifically focusing on the level $p=1$ case of the algorithm. The QAOA functions in a Hilbert space, leveraging quantum gates influenced by parameters $\gamma$ and $\beta$ to produce heuristic solutions. In this paper, the goal is to optimize these parameters to enhance the likelihood of maximizing the number of satisfied constraints in a given Max E3LIN2 instance.

Strong Numerical Results

The authors provide a series of analytical results demonstrating the QAOA's efficacy. Key findings include:

1. The level-1 QAOA shows improved performance over classical randomized algorithms, which traditionally satisfy half of the constraints—$m/2$—by achieving an expected number of satisfied equations at least $m/2 + m/(101D^{1/2}\ln D)$ in the worst case.
2. For typical instances—defined as instances where equation weights are allocated randomly with equal probability for $0$ and $1$—the QAOA yields a solution satisfying an average of $m/2 + m/(\sqrt{2}3eD^{1/2})$ equations, offering a substantial improvement over classical heuristics.

These numerical results illustrate the potential of quantum optimization algorithms to surpass the efficiency of classical counterparts, particularly in constrained domains like Max E3LIN2.

Theoretical Implications

The paper offers insightful implications for the theory of quantum computation as applied to optimization. Primarily, it demonstrates that quantum heuristics, even at their most basic level ($p=1$), can provide a competitive edge in solving intricate constraint satisfaction problems (CSPs) where classical algorithms face NP-hardness barriers. Furthermore, it suggests that by refining parameters $\gamma$ and $\beta$ and extending the algorithm to higher levels ($p>1$), it is plausible to achieve even greater approximation ratios, pushing the boundary of what is considered achievable in the field of computational optimization.

Future Directions

The authors hint at further enhancing the QAOA by exploring several pathways:

• Optimizing $\beta$ and $\gamma$ for each input instance rather than relying on a fixed set.
• Investigating the effects of varying parameters corresponding to individual clauses, expanding the parameter space, thereby potentially improving performance.
• Scaling the algorithm to higher levels ($p$) where the degree of approximation might exhibit less dependence on $D$.

Overall, the paper provides a compelling case for the paper and application of quantum algorithms in optimization, particularly for instances characterized by bounded constraints. As quantum devices continue to develop, the potential for broader computational applications of algorithms like QAOA appears increasingly promising. Researchers in quantum information theory and optimization may find these directions fruitful both experimentally and theoretically.