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arXiv:1409.3323

The Structure of Promises in Quantum Speedups

Shalev Ben-David

Published Sep 11, 2014• 2 citations
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Abstract

It has long been known that in the usual black-box model, one cannot get super-polynomial quantum speedups without some promise on the inputs. In this paper, we examine certain types of symmetric promises, and show that they also cannot give rise to super-polynomial quantum speedups. We conclude that exponential quantum speedups only occur given "structured" promises on the input. Specifically, we show that there is a polynomial relationship of degree $12$ between $D(f)$ and $Q(f)$ for any function $f$ defined on permutations (elements of $\{0,1,\dots, M-1\}^n$ in which each alphabet element occurs exactly once). We generalize this result to all functions $f$ defined on orbits of the symmetric group action $S_n$ (which acts on an element of $\{0,1,\dots, M-1\}^n$ by permuting its entries). We also show that when $M$ is constant, any function $f$ defined on a "symmetric set" - one invariant under $S_n$ - satisfies $R(f)=O(Q(f)^{12(M-1)})$.

DOI
10.48550/arxiv.1409.3323

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It has long been known that in the usual black-box model, one cannot get super-polynomial quantum speedups without some promise on the inputs. In this paper, we examine certain types of symmetric promises, and show that they also cannot give rise to super-polynomial quantum speedups. We conclude that exponential quantum speedups only occur given "structured" promises on the input. Specifically, we show that there is a polynomial relationship of degree 121212 between D(f)D(f)D(f) and Q(f)Q(f)Q(f) for any function fff defined on permutations (elements of {0,1,…,M−1}n\{0,1,\dots, M-1\}^n{0,1,…,M−1}n in which each alphabet element occurs exactly once). We generalize this result to all functions fff defined on orbits of the symmetric group action SnS_nSn​ (which acts on an element of {0,1,…,M−1}n\{0,1,\dots, M-1\}^n{0,1,…,M−1}n by permuting its entries). We also show that when MMM is constant, any function fff defined on a "symmetric set" - one invariant under SnS_nSn​ - satisfies R(f)=O(Q(f)12(M−1))R(f)=O(Q(f)^{12(M-1)})R(f)=O(Q(f)12(M−1)).
In 1998, Beals, Buhrman, Cleve, Mosca, and de Wolf showed that no super-polynomial quantum speedup is possible in the query complexity setting unless there is a promise on the input. We examine several types of "unstructured" promises, and show that they also are not compatible with super-polynomial quantum speedups. We conclude that such speedups are only possible when the input is known to have some structure. Specifically, we show that there is a polynomial relationship of degree 18 between D(f) and Q(f) for any Boolean function f defined on permutations (elements of [n]^n in which each alphabet element occurs exactly once). More generally, this holds for all f defined on orbits of the symmetric group action (which acts on an element of [M]^n by permuting its entries). We also show that any Boolean function f defined on a "symmetric" subset of the Boolean hypercube has a polynomial relationship between R(f) and Q(f) - although in that setting, D(f) may be exponentially larger.
Suppose a Boolean function fff is symmetric under a group action GGG acting on the nnn bits of the input. For which GGG does this mean fff does not have an exponential quantum speedup? Is there a characterization of how rich GGG must be before the function fff cannot have enough structure for quantum algorithms to exploit? In this work, we make several steps towards understanding the group actions GGG which are "quantum intolerant" in this way. We show that sufficiently transitive group actions do not allow a quantum speedup, and that a "well-shuffling" property of group actions -- which happens to be preserved by several natural transformations -- implies a lack of super-polynomial speedups for functions symmetric under the group action. Our techniques are motivated by a recent paper by Chailloux (2018), which deals with the case where G=SnG=S_nG=Sn​. Our main application is for graph symmetries: we show that any Boolean function fff defined on the adjacency matrix of a graph (and symmetric under relabeling the vertices of the graph) has a power 666 relationship between its randomized and quantum query complexities, even if fff is a partial function. In particular, this means no graph property testing problems can have super-polynomial quantum speedups, settling an open problem of Ambainis, Childs, and Liu (2011).
Is there a general theorem that tells us when we can hope for exponential speedups from quantum algorithms, and when we cannot? In this paper, we make two advances toward such a theorem, in the black-box model where most quantum algorithms operate. First, we show that for any problem that is invariant under permuting inputs and outputs (like the collision or the element distinctness problems), the quantum query complexity is at least the 7th root of the classical randomized query complexity. (An earlier version of this paper gave the 9th root.) This resolves a conjecture of Watrous from 2002. Second, inspired by recent work of O'Donnell et al. (2005) and Dinur et al. (2006), we conjecture that every bounded low-degree polynomial has a "highly influential" variable. Assuming this conjecture, we show that every T-query quantum algorithm can be simulated on most inputs by a poly(T)-query classical algorithm, and that one essentially cannot hope to prove P!=BQP relative to a random oracle.
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).
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))D(f) = \mathrm{poly}(Q(f))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 kkk-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 ppp-biased hypercubes and functions with efficiently identifiable domains, and then show some hardness results for general completion techniques.
Given a problem which is intractable for both quantum and classical algorithms, can we find a sub-problem for which quantum algorithms provide an exponential advantage? We refer to this problem as the "sculpting problem." In this work, we give a full characterization of sculptable functions in the query complexity setting. We show that a total function f can be restricted to a promise P such that Q(f|_P)=O(polylog(N)) and R(f|_P)=N^{Omega(1)}, if and only if f has a large number of inputs with large certificate complexity. The proof uses some interesting techniques: for one direction, we introduce new relationships between randomized and quantum query complexity in various settings, and for the other direction, we use a recent result from communication complexity due to Klartag and Regev. We also characterize sculpting for other query complexity measures, such as R(f) vs. R_0(f) and R_0(f) vs. D(f). Along the way, we prove some new relationships for quantum query complexity: for example, a nearly quadratic relationship between Q(f) and D(f) whenever the promise of f is small. This contrasts with the recent super-quadratic query complexity separations, showing that the maximum gap between classical and quantum query complexities is indeed quadratic in various settings - just not for total functions! Lastly, we investigate sculpting in the Turing machine model. We show that if there is any BPP-bi-immune language in BQP, then every language outside BPP can be restricted to a promise which places it in PromiseBQP but not in PromiseBPP. Under a weaker assumption, that some problem in BQP is hard on average for P/poly, we show that every paddable language outside BPP is sculptable in this way.