Quantum Supremacy through the Quantum Approximate Optimization Algorithm (1602.07674v2)

Abstract: The Quantum Approximate Optimization Algorithm (QAOA) is designed to run on a gate model quantum computer and has shallow depth. It takes as input a combinatorial optimization problem and outputs a string that satisfies a high fraction of the maximum number of clauses that can be satisfied. For certain problems the lowest depth version of the QAOA has provable performance guarantees although there exist classical algorithms that have better guarantees. Here we argue 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. We contrast this with the case of sampling from the output of a quantum computer running the Quantum Adiabatic Algorithm (QADI) with the restriction that the Hamiltonian that governs the evolution is gapped and stoquastic. Here we show that there is an oracle that would allow sampling from the QADI but even with this oracle, if one could efficiently classically sample from the output of the QAOA, the Polynomial Hierarchy would collapse. This suggests that the QAOA is an excellent candidate to run on near term quantum computers not only because it may be of use for optimization but also because of its potential as a route to establishing quantum supremacy.

• The paper demonstrates that even low-depth QAOA produces output distributions that resist classical simulation, implying quantum supremacy under key complexity assumptions.
• It uses complexity theory to argue that efficient classical sampling of QAOA results would collapse the Polynomial Hierarchy, underscoring its computational challenge.
• The research highlights QAOA's promise for near-term quantum devices, providing strategic insights for tackling combinatorial optimization problems.

Analyzing Quantum Supremacy via the Quantum Approximate Optimization Algorithm

The paper "Quantum Supremacy through the Quantum Approximate Optimization Algorithm" by Edward Farhi and Aram W. Harrow presents an in-depth paper of the Quantum Approximate Optimization Algorithm (QAOA) and posits its potential for showcasing Quantum Supremacy. The QAOA is a quantum algorithm designed to leverage the capabilities of gate-model quantum computers, specifically aimed at solving combinatorial optimization problems. The key assertion of the paper is that QAOA can reach a form of Quantum Supremacy by generating output distributions that cannot be efficiently simulated by any classical device with bounded polynomial resources, assuming the validity of certain complexity-theoretic assumptions.

Key Contributions and Results

1. Quantum Approximate Optimization Algorithm (QAOA): The QAOA is tailored for finding approximate solutions to combinatorial optimization problems with a shallow quantum circuit. Even the lowest depth version of QAOA is argued to be computationally challenging to simulate classically. The paper suggests that if classical algorithms were capable of efficiently sampling the QAOA’s output distribution, the Polynomial Hierarchy (PH) would collapse, corroborating QAOA's computational prowess.
2. Connection with Quantum Supremacy: The authors draw on complexity theory to argue that QAOA cannot be efficiently simulated classically under the supposition that such an accomplishment would lead to the collapse of the PH. This collapse is a widely believed unlikely phenomenon in complexity theory, which serves to illustrate the nontriviality of simulating quantum processes on classical systems.
3. Comparison with Quantum Adiabatic Algorithm: The Quantum Adiabatic Algorithm (QADI), another contending quantum algorithm, is also explored for optimization tasks. However, the paper contends that under certain assumptions—pertaining to the Hamiltonian's stoquastic properties—QADI may be more amenable to classical simulation compared with QAOA.
4. Complexity Theoretic Implications: The paper explores the collapse of the PH, revealing that efficient classical computation of specific matrix elements associated with quantum circuits and successful sampling from such circuits would result in severe implications for established complexity boundaries.
5. Empirical Prospects: From an empirical perspective, the paper positions QAOA as a promising candidate for implementation on near-term quantum computers. Its low-depth nature and optimization capability make it an attractive algorithm to exploit quantum hardware's computational capacity.

Implications and Future Directions

The QAOA's potential for Quantum Supremacy holds significant implications for the quantum computing landscape, particularly regarding the algorithm's ability to challenge classical computing paradigms. The arguments presented in this paper underpin a fundamental shift in how computational problems could be approached with quantum resources, advocating for more exploration of quantum algorithms in broader computational contexts.

Future research might explore extending QAOA to higher depths or new problem domains to further its potential supremacy. Additionally, empirical studies on quantum devices could provide insights into practical performance gaps between QAOA and classical algorithms. Exploring ways to relax assumptions around problem conditions or constraints could also broaden the applicability of QAOA.

Overall, this paper contributes to the growing literature on quantum computing by providing a rigorous analysis of QAOA and its role in demonstrating quantum advantages over classical computation. Its findings underscore the depth and potential of quantum algorithms, reinforcing the need for continued research into quantum computational supremacy.