• Corpus ID: 119071769

A Quantum Approximate Optimization Algorithm Applied to a Bounded Occurrence Constraint Problem

@article{Farhi2014AQA,
  title={A Quantum Approximate Optimization Algorithm Applied to a Bounded Occurrence Constraint Problem},
  author={Edward Farhi and Jeffrey Goldstone and Sam Gutmann},
  journal={arXiv: Quantum Physics},
  year={2014}
}
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… 
The Quantum Approximate Optimization Algorithm Needs to See the Whole Graph: Worst Case Examples
TLDR
The Quantum Approximate Optimization Algorithm can be applied to search problems on graphs with a cost function that is a sum of terms corresponding to the edges, and it is shown that the QAOA with $(d-1)^{2p} < n^A$ for any $A<1$, can only achieve an approximation ratio of 1/2 on bipartite random d-regular graphs for d large.
Bounds on approximating Max kXOR with quantum and classical local algorithms
TLDR
A tight upper bound on the maximum satisfying fraction of nearly all large random regular Max kXOR instances is computed by numerically calculating the ground state energy density P(k) of a mean-field k-spin glass and grows with k much faster than the performance of both one-local algorithms.
Quantum Supremacy through the Quantum Approximate Optimization Algorithm
TLDR
It is argued that beyond its possible computational value the QAOA can exhibit a form of Quantum Supremacy in that, based on reasonable complexity theoretic assumptions, the output distribution of even the lowest depth version cannot be efficiently simulated on any classical device.
The Quantum Approximate Optimization Algorithm Needs to See the Whole Graph: A Typical Case
TLDR
This work focuses on finding big independent sets in random graphs with dn/2 edges keeping d fixed and n large, and shows that if p is less than a d-dependent constant times log n, the QAOA cannot do better than finding an independent set of size .854 times the optimal for d large.
Near-optimal quantum circuit for Grover's unstructured search using a transverse field
TLDR
This paper presents a circuit-based quantum algorithm to search for a needle in a haystack, obtaining the same quadratic speedup achieved by Grover's original algorithm, and introduces a technique, based on spin-coherent states, to analyze the composite unitary in a single period.
Improving the Quantum Approximate Optimization Algorithm with postselection
TLDR
This work considers a well-studied quantum algorithm for combinatorial optimization: the Quantum Approximate Optimization Algorithm (QAOA) applied to the MaxCut problem on 3-regular graphs and derives theoretical upper and lower bounds showing that a constant (though small) increase of the fraction of satisfied edges is indeed achievable.
Beating the random assignment on constraint satisfaction problems of bounded degree
We show that for any odd k and any instance I of the max-kXOR constraint satisfaction problem, there is an efficient algorithm that finds an assignment satisfying at least a 1/2 + Omega(1/sqrt(D))
Applying the Quantum Alternating Operator Ansatz to the Graph Matching Problem
TLDR
This algorithm uses a W -state as input and it is proved that this input state is better compared to using the empty matching as the input state and the expected size of the matchings corresponding to the output states of the QAOA+ algorithm when ran on a 2-regular graph is greater than the expected matching size obtained from a uniform distribution over all matchings.
Performance of QAOA on Typical Instances of Constraint Satisfaction Problems with Bounded Degree
TLDR
Using the quantum approximate optimization algorithm (QAOA), it is shown that a fraction of the constraints can be satisfied for typical instances, with the assignment efficiently produced by QAOA.
Quantum Algorithms, Architecture, and Error Correction
Quantum algorithms have the potential to provide exponential speedups over some of the best known classical algorithms. These speedups may enable quantum devices to solve currently intractable
...
...

References

SHOWING 1-4 OF 4 REFERENCES
Non-approximability results for optimization problems on bounded degree instances
TLDR
Some non-approximability results for restrictions of basic combinatorial optimization problems to instances of bounded “degree” or bounded ”width” are proved.
Some optimal inapproximability results
We prove optimal, up to an arbitrary ε > 0, inapproximability results for Max-E k-Sat for k ≥ 3, maximizing the number of satisfied linear equations in an over-determined system of linear equations
On bounded occurrence constraint satisfaction
On bounded occurrence constraint satisfaction , 2000
  • Journal of ACM