Abstract. Aaronson and Ambainis [ Theory Comput., 10 (2014), pp. 133–166] and Chailloux [ Proceedings of the 10 th Innovations in Theoretical Computer Science Conference, 2018, pp. 19:1–19:7] 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 bounded-degree 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 [ Lecture Notes in Comput. Sci. 6845, Springer, 2011, pp. 365–376] and Montanaro and de Wolf [ Theory Comput., 7 (2016)].
Abstract. Aaronson and Ambainis [ Theory Comput., 10 (2014), pp. 133–166] and Chailloux [ Proceedings of the 10 th Innovations in Theoretical Computer Science Conference, 2018, pp. 19:1–19:7] 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 bounded-degree 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 [ Lecture Notes in Comput. Sci. 6845, Springer, 2011, pp. 365–376] and Montanaro and de Wolf [ Theory Comput., 7 (2016)].
We study when partial Boolean functions can (and cannot) exhibit superpolynomial quantum query speedups, and develop a general framework for ruling out such speedups via two complementary lenses: promise-aware complexity measures and function completions. First, we introduce promise versions of standard combinatorial measures (including block sensitivity and related variants) and prove that if the relevant promise and completion measures ``collapse'', then deterministic and quantum query complexities are necessarily polynomially related, i.e. D(f)=poly(Q(f)). We then analyze structured families of promises, including symmetric partial functions and promises supported on Hamming slices, obtaining sharp (up to polynomial factors) characterizations in terms of a single gap parameter for the symmetric case and refined slice-dependent bounds for k-slice domains. Next, we formalize completion complexity as the minimum of a measure over total completions of a partial function, and show that completability of a measure captures the possibility of superpolynomial quantum speedups. Finally, we apply this viewpoint to derive broad non-speedup criteria for some classes of functions admitting well-behaved completions, such as functions defined on a single slice or a union of slices, functions with low maximum influence on both the standard and p-biased hypercubes and functions with efficiently identifiable domains, and then show some hardness results for general completion techniques.
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 super-polynomial quantum speedups. We prove this by fully characterizing the primitive permutation groups that allow super-polynomial quantum speedups. In contrast, in the adjacency list model for bounded-degree 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).
In this work, we study bipartiteness and expansion testing, two canonical problems in graph property testing in the bounded-degree model through the lens of quantum query complexity. In the classical setting, it is known that Θ(N) queries are necessary and sufficient for both these testing problems (Goldreich and Ron, 1999, 2000 & 2002), where N denotes the number of vertices of the input graph. Due to their significance, (Ambainis, Childs, and Liu, 2011) initiated the study of these problems in the quantum setting and designed quantum algorithms for bipartiteness and expansion testing that perform O(N1/3) queries, showing a polynomial speedup. They also proved that Ω(N1/4) queries are necessary for expansion testing, but the possibility of an exponential quantum advantage for bipartiteness testing remained open. Despite significant effort, there has been no improvement in these results in the last decade and a half. In this work, we prove essentially tight Ω(N1/3) quantum query lower bounds for both bipartiteness and expansion testing, thereby completely characterizing the quantum query complexity of these problems up to polylogarithmic factors. While our proofs use the polynomial method similarly to Ambainis, Childs, and Liu, we use intermediate problems that we relate to the main problems via reductions, and perform a more precise analysis of the resulting polynomials, leading to the near-optimal lower bounds.
Aaronson and Ambainis (Theory of Computing, 2014) conjectured that quantum query algorithms admit efficient almost-everywhere classical simulation: for any T-query quantum algorithm, its acceptance probability can be approximated on a (1−δ) fraction of inputs, up to ε additive error, using poly(T,1/ε,1/δ) classical queries. At a high level, the conjecture suggests that exponential quantum speedups are possible only on sufficiently structured inputs.
In this work, we make progress on this conjecture by proving it for quantum algorithms that make massively parallel quantum queries. In contrast, Yamakawa and Zhandry (Journal of the ACM, 2024) showed that quantum algorithms restricted to parallel queries can still achieve exponential speedups over classical algorithms for sampling problems. We establish our simulation theorem by proving the stronger statement that parallel-query quantum algorithms cannot distinguish the uniform distribution over oracles from oracles drawn from so-called "dense distributions". Our main technical contribution is a coupling theorem that relates the uniform distribution over oracles to oracles drawn from dense distributions.
We further extend this approach beyond the purely parallel setting. We prove simulation theorems for quantum algorithms with constant rounds of adaptivity, and for quantum algorithms with a bounded quantum-query prefix followed by a massively parallel quantum-query stage.
Smoothed analysis is a central framework in classical algorithms for explaining the performance of algorithms beyond the worst case, often explaining why algorithms perform well in practice. We initiate a systematic study of its quantum counterpart and show the following results. (1) We show that there is a total function whose smoothed quantum query complexity is exponentially smaller than its classical query complexity. (2) We give near-tight characterizations of smoothed randomized and quantum query complexities for symmetric Boolean functions, unifying the worst-case complexity results of [Beals et al, FOCS'98] and average-case complexity results of [Ambainis and de Wolf, STACS'00]. (3) We study string problems such as pattern matching and edit distance and, in various regimes, give polynomial to superpolynomial quantum speedups. Our main technical ingredients include a near-tight quantum algorithm for ε-approximating the number of collisions between two non-repetitive strings, improving the result of Le Gall and Ng [QIC'22]. Together, our results show that smoothing can reveal larger quantum speedups than worst-case analysis suggests, opening a path towards quantum advantage on more realistic inputs.
Abstract There are few known exponential speedups for quantum algorithms and these tend to fall into even fewer families. One speedup that has mostly resisted generalization is the use of quantum walks to traverse the welded-tree graph, due to Childs, Cleve, Deotto, Farhi, Gutmann, and Spielman. We show how to generalize this to a large class of hierarchical graphs in which the vertices are grouped into “supervertices” which are arranged according to a d-dimensional lattice. Supervertices can have different sizes, and edges between supervertices correspond to random connections between their constituent vertices. The hitting times of quantum walks on these graphs are related to the localization properties of zero modes in certain disordered tight binding Hamiltonians. The speedups range from superpolynomial to exponential, depending on the underlying dimension and the random graph model. We also provide concrete realizations of these hierarchical graphs, and introduce a general method for constructing graphs with efficient quantum traversal times using graph sparsification.