[Paper Review] A Quantum Approximate Optimization Algorithm Applied to a Bounded Occurrence Constraint Problem

Edward Farhi, Jeffrey Goldstone+1 more|arXiv (Cornell University)|Dec 18, 2014

Quantum Computing Algorithms and ArchitectureComputer Science4 references128 citations

TL;DR

This paper applies the Quantum Approximate Optimization Algorithm (QAOA) at level $p=1$ to the bounded-occurrence Max E3LIN2 problem, where each variable appears in at most $D+1$ equations. It proves that the quantum algorithm satisfies $\left(\frac{1}{2} + \frac{1}{101D^{1/2}\ln D}\right)m$ equations on average, outperforming classical algorithms with $\left(\frac{1}{2} + \frac{\text{constant}}{D^{1/2}}\right)$, and achieves $\left(\frac{1}{2} + \frac{1}{2\sqrt{3e}\,D^{1/2}}\right)m$ on typical random instances.

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 $\left(\frac{1}{2} + \frac{1}{101 D^{1/2}\, l n\, D} ight)$ times the number of equations. A recent classical algorithm achieved $\left(\frac{1}{2} + \frac{constant}{D^{1/2}} ight)$. We also show that in the typical case the quantum computer will output a string that satisfies $\left(\frac{1}{2}+ \frac{1}{2\sqrt{3e}\, D^{1/2}} ight)$ times the number of equations.

Motivation & Objective

• To analyze the performance of the QAOA at level $p=1$ on the Max E3LIN2 problem with bounded variable occurrences.
• To establish a quantum advantage over classical algorithms in approximating solutions to Max E3LIN2 under bounded occurrence constraints.
• To derive a lower bound on the expected number of satisfied equations using quantum state preparation and measurement.
• To compare quantum performance with known classical approximation bounds, particularly those dependent on $D^{-1/2}$.

Proposed method

• The QAOA is applied using a fixed $\beta = \pi/4$ and optimized $\gamma$ to prepare a quantum state $|\gamma, \beta\rangle = e^{-i\beta B}e^{-i\gamma C}|s\rangle$, where $|s\rangle$ is the uniform superposition state.
• The objective function $C(z)$, counting satisfied equations, is encoded as a diagonal operator in the computational basis, with each clause contributing a term $\frac{1}{2}(1 \pm Z_a Z_b Z_c)$.
• The expected value $\langle -\gamma, \pi/4 | C | -\gamma, \pi/4 \rangle$ is computed to estimate the average number of satisfied equations.
• Variance analysis is performed by examining correlations between clauses, showing that only overlapping clauses contribute significantly to fluctuations.
• The analysis leverages the fact that $d_{abc}$ values are random $\pm1$ variables, and uses statistical independence to bound the variance of the expectation.
• The performance is evaluated both in expectation and in the typical case, where equations are assigned 0 or 1 with equal probability.

Experimental results

Research questions

• RQ1 Can the QAOA at $p=1$ achieve a better approximation ratio than classical algorithms for bounded-occurrence Max E3LIN2?
• RQ2 What is the expected fraction of equations satisfied by the QAOA when each variable appears in at most $D+1$ equations?
• RQ3 How does the quantum algorithm's performance compare to the classical $\left(\frac{1}{2} + \frac{\text{constant}}{D^{1/2}}\right)$ bound?
• RQ4 Does the QAOA outperform random guessing in the typical case where equation targets are assigned randomly?
• RQ5 Can the performance be improved by optimizing $\gamma$ and $\beta$ or increasing $p$?

Key findings

• The QAOA at $p=1$ satisfies at least $\left(\frac{1}{2} + \frac{1}{101D^{1/2}\ln D}\right)m$ equations in expectation, where $m$ is the total number of equations.
• For typical random instances with 50/50 assignment of equation targets, the algorithm satisfies $\left(\frac{1}{2} + \frac{1}{2\sqrt{3e}\,D^{1/2}}\right)m$ equations with high probability.
• The performance gain scales as $\sim D^{-1/2}\ln^{-1}D$, improving upon the classical $\sim D^{-1/2}$ bound in terms of the constant factor.
• The variance of the output is bounded by $\mathcal{O}(mD^2)$, ensuring concentration around the mean for large $m$.
• The analysis shows that only overlapping clauses (sharing at least one variable) contribute significantly to the variance, limiting the number of correlated terms.
• The result establishes a quantum advantage in approximation ratio for Max E3LIN2 under bounded occurrence constraints, even at $p=1$.