[Paper Review] A QAOA-inspired circuit for Grover's unstructured search using a transverse field

Jiang Zhang, Eleanor Rieffel+1 more|arXiv (Cornell University)|Feb 8, 2017

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper presents a QAOA-inspired quantum circuit that achieves Grover's quadratic speedup for unstructured search by alternately applying a problem Hamiltonian (oracle) and a transverse field. Using a spin-coherent-state analysis, it demonstrates a transition rate of order $\Theta(1/\sqrt{N})$ and overlaps of order $\Theta(1)$, yielding a query complexity of $T \simeq \sqrt{N} \cdot \pi/(2\sqrt{2})$, marking the first non-Trotterized QAOA with quantum advantage.

ABSTRACT

Inspired by a class of algorithms proposed by Farhi et al., namely the quantum approximate optimization algorithm (QAOA), we present a circuit-based quantum algorithm to search for a needle in a haystack, obtaining the same quadratic speedup achieved by Grover's original algorithm. In our algorithm, the problem Hamiltonian (oracle) and a transverse field are applied alternately to the system in a periodic manner. We introduced a novel technique, a spin-coherent-state approach, to analyze the composite unitary in a single period. This composite unitary drives a closed transition between two states that have high degrees of overlap with the initial state and the target state, respectively. The transition rate in our algorithm is of order $\Theta(1/\sqrt N)$, and the overlaps are of order $\Theta(1)$, yielding a nearly optimal query complexity of $T\simeq \sqrt N (\pi/2\sqrt 2\,)$. Our algorithm is the first example of a QAOA circuit that demonstrates a quantum advantage with large number of iterations that is not derived from Trotterization of an adiabatic quantum optimization (AQO) algorithm. It also suggests that the analysis required to understand QAOA circuits involves a very different process from estimating the energy gap of a Hamiltonian in AQO.

Motivation & Objective

• To develop a QAOA-inspired quantum algorithm that achieves the quadratic speedup of Grover's algorithm without relying on adiabatic evolution or Trotterization.
• To analyze the dynamics of a periodically driven quantum system composed of a problem Hamiltonian and a transverse field using a novel spin-coherent-state approach.
• To demonstrate that a non-adiabatic, non-Trotterized QAOA circuit can still yield quantum advantage in search problems.
• To establish a connection between QAOA dynamics and the structure of coherent state transitions, distinct from energy gap analysis in adiabatic quantum optimization.
• To achieve nearly optimal query complexity $T \simeq \sqrt{N} \cdot \pi/(2\sqrt{2})$ through high overlap with initial and target states.

Proposed method

• The algorithm applies the problem Hamiltonian (oracle) and a transverse field alternately in a periodic sequence, forming a composite unitary evolution.
• A spin-coherent-state approach is used to analyze the effective evolution over a single period, enabling the identification of a closed transition between two states.
• The method identifies two states—one with high overlap to the initial state and one with high overlap to the target state—through the composite unitary.
• The transition rate between these states is derived as $\Theta(1/\sqrt{N})$, scaling with the inverse of the square root of the search space size.
• The analysis shows that overlaps with initial and target states are both $\Theta(1)$, ensuring efficient population transfer.
• The query complexity is estimated as $T \simeq \sqrt{N} \cdot \pi/(2\sqrt{2})$, approaching the theoretical optimum of $\pi\sqrt{N}/2$.

Experimental results

Research questions

• RQ1 Can a QAOA-inspired circuit achieve Grover's quadratic speedup without relying on adiabatic evolution or Trotterization?
• RQ2 What is the role of the transverse field in enabling coherent population transfer between initial and target states in a non-adiabatic setting?
• RQ3 How does the spin-coherent-state formalism enable the analysis of composite unitary evolution in a QAOA circuit?
• RQ4 What is the scaling of the transition rate and state overlaps in the proposed algorithm, and how do they affect query complexity?
• RQ5 Does this approach demonstrate quantum advantage in search with a large number of iterations, despite not being derived from adiabatic quantum optimization?

Key findings

• The algorithm achieves a transition rate of order $\Theta(1/\sqrt{N})$, which is optimal for unstructured search.
• The overlaps of the evolved state with the initial and target states are both of order $\Theta(1)$, indicating strong population transfer.
• The query complexity is $T \simeq \sqrt{N} \cdot \pi/(2\sqrt{2})$, approaching the theoretical minimum for Grover's algorithm.
• The method provides a new analytical framework using spin-coherent states to understand QAOA dynamics beyond energy gap estimation.
• This is the first QAOA circuit with quantum advantage derived from periodic driving, not from adiabatic evolution or Trotterization.
• The results suggest that QAOA analysis requires fundamentally different tools than those used in adiabatic quantum optimization.