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arXiv:1907.09531

Between the deterministic and non-deterministic query complexity

Dániel Gerbner

Published Jul 22, 2019• 0 citations
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Abstract

We consider problems that can be solved by asking certain queries. The deterministic query complexity $D(P,n)$ of a problem $P$ is the smallest number of queries needed to ask in order to find the solution with an input of size $n$ (in the worst case), while the non-deterministic query complexity $D_0(P,n)$ is the smallest number of queries needed to ask, in case we know the solution, to prove that it is indeed the solution (in the worst case). Equivalently, $D(P,n)$ is the largest number of queries needed to find the solution in case an Adversary is answering the queries, while $D_0(P)$ is the largest number of queries needed to find the solution in case an Adversary chooses the input. We define a series of quantities between these two values, $D_k(P,n)$ is the largest number of queries needed to find the solution in case an Adversary chooses the input, and answers the queries, but he can change the input at most $k$ times. We give bounds on $D_k(P,n)$ for various problems $P$.

DOI
10.48550/arxiv.1907.09531

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We consider problems that can be solved by asking certain queries. The deterministic query complexity D(P,n)D(P,n)D(P,n) of a problem PPP is the smallest number of queries needed to ask in order to find the solution with an input of size nnn (in the worst case), while the non-deterministic query complexity D0(P,n)D_0(P,n)D0​(P,n) is the smallest number of queries needed to ask, in case we know the solution, to prove that it is indeed the solution (in the worst case). Equivalently, D(P,n)D(P,n)D(P,n) is the largest number of queries needed to find the solution in case an Adversary is answering the queries, while D0(P)D_0(P)D0​(P) is the largest number of queries needed to find the solution in case an Adversary chooses the input. We define a series of quantities between these two values, Dk(P,n)D_k(P,n)Dk​(P,n) is the largest number of queries needed to find the solution in case an Adversary chooses the input, and answers the queries, but he can change the input at most kkk times. We give bounds on Dk(P,n)D_k(P,n)Dk​(P,n) for various problems PPP.
The randomized query complexity R(f)R(f)R(f) of a boolean function f ⁣:{0,1}n→{0,1}f\colon\{0,1\}^n\to\{0,1\}f:{0,1}n→{0,1} is famously characterized (via Yao's minimax) by the least number of queries needed to distinguish a distribution D0D_0D0​ over 000-inputs from a distribution D1D_1D1​ over 111-inputs, maximized over all pairs (D0,D1)(D_0,D_1)(D0​,D1​). We ask: Does this task become easier if we allow query access to infinitely many samples from either D0D_0D0​ or D1D_1D1​? We show the answer is no: There exists a hard pair (D0,D1)(D_0,D_1)(D0​,D1​) such that distinguishing D0∞D_0^\inftyD0∞​ from D1∞D_1^\inftyD1∞​ requires Θ(R(f))Θ(R(f))Θ(R(f)) many queries. As an application, we show that for any composed function f∘gf\circ gf∘g we have R(f∘g)≥Ω(fbs(f)R(g))R(f\circ g) \geq Ω(\mathrm{fbs}(f)R(g))R(f∘g)≥Ω(fbs(f)R(g)) where fbs\mathrm{fbs}fbs denotes fractional block sensitivity.
We study pseudo-deterministic query complexity - randomized query algorithms that are required to output the same answer with high probability on all inputs. We prove Ω(√n) lower bounds on the pseudo-deterministic complexity of a large family of search problems based on unsatisfiable random CNF instances, and also for the promise problem (FIND1) of finding a 1 in a vector populated with at least half one’s. This gives an exponential separation between randomized query complexity and pseudo-deterministic complexity, which is tight in the quantum setting. As applications we partially solve a related combinatorial coloring problem, and we separate random tree-like Resolution from its pseudo-deterministic version. In contrast to our lower bound, we show, surprisingly, that in the zero-error, average case setting, the three notions (deterministic, randomized, pseudo-deterministic) collapse.
Quantum query complexity enjoys a variety of pleasing joint computation properties: for example, a composition theorem asserting Q(f∘g)=Θ(Q(f)Q(g))Q(f\circ g)=Θ(Q(f)Q(g))Q(f∘g)=Θ(Q(f)Q(g)) for all Boolean functions fff and ggg; a direct sum theorem asserting that computing kkk copies of a function (or search problem) costs Ω(k)Ω(k)Ω(k) times as much as the cost of computing one copy; and a direct product theorem asserting that for Boolean functions, even succeeding at the direct sum problem with exponentially small probability still requires Ω(k)Ω(k)Ω(k) times the cost of computing one copy to bounded error. However, all of these results are strictly for worst-case quantum query complexity. For example, if we have a fixed distribution μμμ over inputs, the direct sum theorem says nothing about the quantum query complexity of computing kkk copies of fff when the input comes from the product distribution μkμ^kμk instead of being worst-case. (Note that while a standard Yao-type minimax theorem guarantees a hard distribution for the direct sum problem, there's no guarantee that this hard distribution is a product distribution.) A similar problem occurs for the direct product theorem and the composition theorem: none of these results respect distributions. In this work, we give distributional joint computation lower bounds for the composition, direct sum, and direct product problems. Along the way, we introduce some new tools for handling quantum query lower bounds, including (a) a new ``multiplicative'' variant of the γ2γ_2γ2​ norm (which we use in place of the multiplicative adversary method for proving the direct product theorem), and (b) a new ``Shaltiel-free'' measure of quantum query complexity, which we show characterizes the composition behavior of distributional quantum query complexity and satisfies pleasing properties.
The randomized query complexity 𝖱(f) of a boolean function f: {0,1}ⁿ → {0,1} is famously characterized (via Yao’s minimax) by the least number of queries needed to distinguish a distribution 𝒟₀ over 0-inputs from a distribution 𝒟₁ over 1-inputs, maximized over all pairs (𝒟₀,𝒟₁). We ask: Does this task become easier if we allow query access to infinitely many samples from either 𝒟₀ or 𝒟₁? We show the answer is no: There exists a hard pair (𝒟₀,𝒟₁) such that distinguishing 𝒟₀^∞ from 𝒟₁^∞ requires Θ(𝖱(f)) many queries. As an application, we show that for any composed function f∘g we have 𝖱(f∘g) ≥ Ω(fbs(f)𝖱(g)) where fbs denotes fractional block sensitivity.
The randomized query complexity 𝖱(f) of a boolean function f: {0,1}ⁿ → {0,1} is famously characterized (via Yao’s minimax) by the least number of queries needed to distinguish a distribution 𝒟₀ over 0-inputs from a distribution 𝒟₁ over 1-inputs, maximized over all pairs (𝒟₀,𝒟₁). We ask: Does this task become easier if we allow query access to infinitely many samples from either 𝒟₀ or 𝒟₁? We show the answer is no: There exists a hard pair (𝒟₀,𝒟₁) such that distinguishing 𝒟₀^∞ from 𝒟₁^∞ requires Θ(𝖱(f)) many queries. As an application, we show that for any composed function f∘g we have 𝖱(f∘g) ≥ Ω(fbs(f)𝖱(g)) where fbs denotes fractional block sensitivity.
In this note we investigate the relationship between worst-case quantum query complexity and average-case classical query complexity. Specifically, we show that if a quantum computer can evaluate a total Boolean function f with bounded error using T queries in the worst case, then a deterministic classical computer can evaluate f using O(T^5) queries in the average case, under a uniform distribution of inputs. If f is monotone, we show furthermore that only O(T^3) queries are needed. Previously, Beals et al. (1998) showed that if a quantum computer can evaluate f with bounded error using T queries in the worst case, then a deterministic classical computer can evaluate f using O(T^6) queries in the worst case, or O(T^4) if f is monotone. The optimal bound is conjectured to be O(T^2), but improving on O(T^6) remains an open problem. Relating worst-case quantum complexity to average-case classical complexity may suggest new ways to reduce the polynomial gap in the ordinary worst-case versus worst-case setting.