Forrelation: A Problem that Optimally Separates Quantum from Classical Computing (1411.5729v1)

Abstract: We achieve essentially the largest possible separation between quantum and classical query complexities. We do so using a property-testing problem called Forrelation, where one needs to decide whether one Boolean function is highly correlated with the Fourier transform of a second function. This problem can be solved using 1 quantum query, yet we show that any randomized algorithm needs ~sqrt(N)/log(N) queries (improving an ~N^{1/4} lower bound of Aaronson). Conversely, we show that this 1 versus ~sqrt(N) separation is optimal: indeed, any t-query quantum algorithm whatsoever can be simulated by an O(N^{1-1/2t})-query randomized algorithm. Thus, resolving an open question of Buhrman et al. from 2002, there is no partial Boolean function whose quantum query complexity is constant and whose randomized query complexity is linear. We conjecture that a natural generalization of Forrelation achieves the optimal t versus ~N^{1-1/2t} separation for all t. As a bonus, we show that this generalization is BQP-complete. This yields what's arguably the simplest BQP-complete problem yet known, and gives a second sense in which Forrelation "captures the maximum power of quantum computation."

• The paper defines the Forrelation problem, demonstrating a remarkable 1 quantum query solution compared to \(\Omega(\sqrt{N}/\log N)\) for classical algorithms, proving the largest known query complexity separation.
• It proves this 1 vs. \(\widetilde{\Omega}(\sqrt{N})\) gap is optimal for any black-box function, settling a long-standing open question regarding constant quantum versus linear classical query complexity.
• The research provides insights into classical simulation of quantum algorithms and the limits of quantum speedup, with implications for BQP-completeness, sampling, and property testing.

An Analysis of "Forrelation: A Problem that Optimally Separates Quantum from Classical Computing"

The paper "Forrelation: A Problem that Optimally Separates Quantum from Classical Computing" by Scott Aaronson and Andris Ambainis explores a fundamental question about the potential advantages of quantum computing over classical computing in the black-box query complexity model. It presents the Forrelation problem, which achieves an optimal separation between quantum and classical query complexity, and addresses whether a quantum speedup of constant to linear is possible within any partial Boolean function. This paper extends previous work, refines lower bounds, and provides a comprehensive analysis of the power and limitations of quantum computing when assessed through the lens of query complexity.

Key Contributions and Results

1. Definition and Analysis of Forrelation Problem:
   • The Forrelation problem involves determining whether a Boolean function is correlated with the Fourier transform of another function. It can be solved with 1 quantum query, but the paper demonstrates that any classical randomized algorithm would require $\Omega(\sqrt{N}/\log N)$ queries. This constitutes the largest proven gap in quantum and classical query complexity setups.
2. Optimality of Forrelation Gap:
   • The authors establish the optimality of the $1$ vs. $\widetilde{\Omega}(\sqrt{N})$ separation using any black-box function, addressing and resolving an open question posed by Buhrman et al. in 2002, by showing that no partial Boolean function exists with constant quantum and linear randomized query complexities. This is underlined by a general result showing that any $t$-query quantum algorithm can be efficiently simulated by an $O(N^{1-1/2t})$-query randomized algorithm.
3. Extension to $k$-fold Forrelation and Implications for $\mathsf{BQP}$-Completeness:
   • A generalization to $k$-fold Forrelation shows that even a small fraction of queries can exhibit significant quantum advantages. Moreover, it is proven that a variant of this problem is $\mathsf{BQP}$-complete, thereby offering the simplest $\mathsf{BQP}$-complete problem and providing a fresh perspective on the capabilities of quantum computing.
4. Classical Simulation of Quantum Algorithms:
   • The paper provides a methodology for simulating quantum algorithms with bounded degree polynomials by producing constructively balanced polynomials for efficient sampling. This offers a fundamental insight into the borderline between classical and quantum computational strengths.
5. Lower Bounds for Sampling and Relational Problems:
   • The authors explore extensions related to sampling problems, showing that the potential quantum vs. classical complexity gap could be even larger in these settings. Heuristics suggest that sampling problems maintain the $\Omega(N/\log N)$ classical lower bound, hinting that quantum algorithms retain their advantages over classical methods even in broader settings.
6. Property Testing in Quantum Settings:
   • Forrelation as a property-testing problem is analyzed, marking a milestone in quantum property testing by showing considerable separations between the efficiency of quantum and classical testers. This is substantiated through robust theoretical arguments occupying the spheres of concentration bounds and multilinearity.

Implications and Future Directions

The implications of these findings are far-reaching. They suggest that the theoretical speedup offered by quantum algorithms has tangible limits, which are delineated by the Forrelation problem. Practically, this informs the community about the fundamental advantages and boundaries of quantum systems. The enriched understanding of complexity separations and completeness also deeply influences how future quantum algorithms will be designed and analyzed, especially in decision, relational, and sampling problems beyond mere theoretical curiosity.

In future developments, verifying these quantum-classical separations experimentally might provide insights into how quantum processing units could be implemented to surpass classical computations. Furthermore, there is room for exploring whether similarly intuitive and easy-to-construct problems could exhibit such complexities and separations, as this paper suggests. The potential for this research to enhance encryption, optimization, and beyond remains vast, where quantum computing may find its most compelling applications yet.