Influence in Completely Bounded Block-multilinear Forms and Classical Simulation of Quantum Algorithms (2203.00212v1)

Abstract: The Aaronson-Ambainis conjecture (Theory of Computing '14) says that every low-degree bounded polynomial on the Boolean hypercube has an influential variable. This conjecture, if true, would imply that the acceptance probability of every $d$-query quantum algorithm can be well-approximated almost everywhere (i.e., on almost all inputs) by a $\mathrm{poly}(d)$-query classical algorithm. We prove a special case of the conjecture: in every completely bounded degree-$d$ block-multilinear form with constant variance, there always exists a variable with influence at least $1/\mathrm{poly}(d)$. In a certain sense, such polynomials characterize the acceptance probability of quantum query algorithms, as shown by Arunachalam, Bri\"et and Palazuelos (SICOMP '19). As a corollary we obtain efficient classical almost-everywhere simulation for a particular class of quantum algorithms that includes for instance $k$-fold Forrelation. Our main technical result relies on connections to free probability theory.

• The paper proves that completely bounded block-multilinear forms contain a variable with influence of at least 1/poly(d), confirming a special case of the Aaronson-Ambainis conjecture.
• It introduces a novel non-commutative root-influence inequality that leverages free probability theory to relate bounded norms with variable influences.
• The study establishes an efficient classical simulation method for k-fold Forrelation and similar quantum algorithms, achieving polynomial query complexity.

Overview of the Paper: Influence in Completely Bounded Block-multilinear Forms and Classical Simulation of Quantum Algorithms

This paper addresses a fundamental question in quantum complexity theory related to the Aaronson-Ambainis conjecture, which posits that every low-degree bounded polynomial defined on the Boolean hypercube possesses an influential variable. The implication of this conjecture, if proven, would be significant; it suggests that the acceptance probability of any quantum algorithm based on d queries can be closely approximated by a classical algorithm using polynomially more queries in almost all instances. This work successfully validates a special case of this conjecture concerning completely bounded degree-d block-multilinear forms that exhibit constant variance, demonstrating the existence of a variable with influence not less than 1/poly(d).

The authors leverage concepts from free probability theory to underpin their main results. The incorporation of free probability theory allows for innovative proofs and lends theoretical strength to the methods employed. This approach directly contributes to an enhanced understanding of influence in block-multilinear forms, which capture the essence of quantum query algorithms' acceptance probability. Subsequently, they establish an efficient classical simulation approach for a specific category of quantum algorithms, notably including k-fold Forrelation, which is known for its classical-quantum complexity gap.

Technical Contributions and Results

1. Theorem on Influence: A central result of this work is establishing that, for completely bounded block-multilinear forms, there always exists a variable with significant influence. This is quantified by Theorem 1.3, which corroborates the Aaronson-Ambainis conjecture in this restricted setting.
2. Novel Inequality: The development of a non-commutative root-influence inequality (Theorem 1.4) serves as a pivotal tool in achieving the main results. This inequality relates the completely bounded norm of these polynomials to the influences within them.
3. Connections to Free Probability: The application of results from free probability theory significantly strengthens the analytical framework. Theorems such as those found in the work of Kemp and Speicher on free Haar unitaries are creatively utilized to achieve bounding arguments fundamental to this research's success.
4. Classical Simulation: The corollary derived from the main results offers a method for simulating certain quantum algorithms classically with high efficiency on most inputs. Notably, this is achieved with a polynomial number of queries proportional to d in relation to the desired accuracy and tolerance.

Implications and Future Directions

The findings from this paper have important implications in both practical and theoretical dimensions of quantum computing. From a practical standpoint, they pave the way for classical algorithms that can simulate quantum processes with substantial accuracy across a broad set of inputs, thus contributing to the ongoing discourse on quantum advantage in computational tasks.

On the theoretical front, this work provides new insights into the structure of quantum algorithms and the role of influences within the polynomial frameworks representing them. It poses interesting questions regarding the generalization of these results to broader classes of quantum algorithms and highlights the potential of free probability theory to tackle complex problems within quantum information science.

Future research could explore deeper connections between free probability frameworks and quantum complexity measures, potentially unveiling more robust forms of the Aaronson-Ambainis conjecture and exploring its implications further. Additionally, the paper promotes the examination of influences in more generalized settings and their impact on classical simulation capabilities. The continued exploration of these avenues will likely yield new results that extend the boundaries of what is computationally feasible on classical versus quantum platforms.

In summary, this paper offers significant contributions to understanding the roles of influences in block-multilinear forms associated with quantum algorithms while demonstrating compelling advancements in classical simulation methods for these quantum processes.