
Can graph properties have exponential quantum speedup?
Quantum computers can sometimes exponentially outperform classical ones,...
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How symmetric is too symmetric for large quantum speedups?
Suppose a Boolean function f is symmetric under a group action G acting ...
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Quantum Implications of Huang's Sensitivity Theorem
Based on the recent breakthrough of Huang (2019), we show that for any t...
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Degree vs. Approximate Degree and Quantum Implications of Huang's Sensitivity Theorem
Based on the recent breakthrough of Huang (2019), we show that for any t...
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Towards Efficient Normalizers of Primitive Groups
We present the ideas behind an algorithm to compute normalizers of primi...
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Normalizers and permutational isomorphisms in simplyexponential time
We show that normalizers and permutational isomorphisms of permutation g...
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Symmetry and Quantum QuerytoCommunication Simulation
Buhrman, Cleve and Wigderson (STOC'98) showed that for every Boolean fun...
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Symmetries, graph properties, and quantum speedups
Aaronson and Ambainis (2009) and Chailloux (2018) showed that fully symmetric (partial) functions do not admit exponential quantum query speedups. This raises a natural question: how symmetric must a function be before it cannot exhibit a large quantum speedup? In this work, we prove that hypergraph symmetries in the adjacency matrix model allow at most a polynomial separation between randomized and quantum query complexities. We also show that, remarkably, permutation groups constructed out of these symmetries are essentially the only permutation groups that prevent superpolynomial quantum speedups. We prove this by fully characterizing the primitive permutation groups that allow superpolynomial quantum speedups. In contrast, in the adjacency list model for boundeddegree graphs (where graph symmetry is manifested differently), we exhibit a property testing problem that shows an exponential quantum speedup. These results resolve open questions posed by Ambainis, Childs, and Liu (2010) and Montanaro and de Wolf (2013).
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