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Register a new userOptimal Separation in Exact Query Complexities for Simons Problem
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Simons problem is one of the most important problems demonstrating the power of quantum computers, which achieves a large separation between quantum and classical query complexities. However, Simons discussion on his problem was limited to bounded-error setting, which means his algorithm can not always get the correct answer. Exact quantum algorithms for Simons problem have also been proposed, which deterministically solve the problem with O(n) queries. Also the quantum lower bound Omega(n) for Simons problem is known. Although these algorithms are either complicated or specialized, their results give an O(n) versus Omega(sqrt{2^{n}}) separation in exact query complexities for Simons problem (Omega(sqrt{2^{n}}) is the lower bound for classical probabilistic algorithms), but it has not been proved whether this separation is optimal. In this paper, we propose another exact quantum algorithm for solving Simons problem with O(n) queries, which is simple, concrete and does not rely on special query oracles. Our algorithm combines Simons algorithm with the quantum amplitude amplification technique to ensure its determinism. In particular, we show that Simons problem can be solved by a classical deterministic algorithm with O(sqrt{2^{n}}) queries (as we are aware, there were no classical deterministic algorithms for solving Simons problem with O(sqrt{2^{n}}) queries). Combining some previous results, we obtain the optimal separation in exact query complexities for Simons problem: Theta({n}) versus Theta({sqrt{2^{n}}}).
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Simons problem is an essential example demonstrating the faster speed of quantum computers than classical computers for solving some problems. The optimal separation between exact quantum and classical query complexities for Simons problem has been proved by Cai $&$ Qiu. Generalized Simons problem can be described as follows. Given a function $f:{{0, 1}}^n to {{0, 1}}^m$, with the property that there is some unknown hidden subgroup $S$ such that $f(x)=f(y)$ iff $x oplus yin S$, for any $x, yin {{0, 1}}^n$, where $|S|=2^k$ for some $0leq kleq n$. The goal is to find $S$. For the case of $k=1$, it is Simons problem. In this paper, we propose an exact quantum algorithm with $O(n-k)$ queries and an non-adaptive deterministic classical algorithm with $O(ksqrt{2^{n-k}})$ queries for solving the generalized Simons problem. Also, we prove that their lower bounds are $Omega(n-k)$ and $Omega(sqrt{k2^{n-k}})$, respectively. Therefore, we obtain a tight exact quantum query complexity $Theta(n-k)$ and an almost tight non-adaptive classical deterministic query complexities $Omega(sqrt{k2^{n-k}}) sim O(ksqrt{2^{n-k}})$ for this problem.
The query model offers a concrete setting where quantum algorithms are provably superior to randomized algorithms. Beautiful results by Bernstein-Vazirani, Simon, Aaronson, and others presented partial Boolean functions that can be computed by quantum algorithms making much fewer queries compared to their randomized analogs. To date, separations of $O(1)$ vs. $sqrt{N}$ between quantum and randomized query complexities remain the state-of-the-art (where $N$ is the input length), leaving open the question of whether $O(1)$ vs. $N^{1/2+Omega(1)}$ separations are possible? We answer this question in the affirmative. Our separating problem is a variant of the Aaronson-Ambainis $k$-fold Forrelation problem. We show that our variant: (1) Can be solved by a quantum algorithm making $2^{O(k)}$ queries to the inputs. (2) Requires at least $tilde{Omega}(N^{2(k-1)/(3k-1)})$ queries for any randomized algorithm. For any constant $varepsilon>0$, this gives a $O(1)$ vs. $N^{2/3-varepsilon}$ separation between the quantum and randomized query complexities of partial Boolean functions. Our proof is Fourier analytical and uses new bounds on the Fourier spectrum of classical decision trees, which could be of independent interest. Looking forward, we conjecture that the Fourier bounds could be further improved in a precise manner, and show that such conjectured bounds imply optimal $O(1)$ vs. $N^{1-varepsilon}$ separations between the quantum and randomized query complexities of partial Boolean functions.
We study the complexity of quantum query algorithms that make p queries in parallel in each timestep. This model is in part motivated by the fact that decoherence times of qubits are typically small, so it makes sense to parallelize quantum algorithms as much as possible. We show tight bounds for a number of problems, specifically Theta((n/p)^{2/3}) p-parallel queries for element distinctness and Theta((n/p)^{k/(k+1)} for k-sum. Our upper bounds are obtained by parallelized quantum walk algorithms, and our lower bounds are based on a relatively small modification of the adversary lower bound method, combined with recent results of Belovs et al. on learning graphs. We also prove some general bounds, in particular that quantum and classical p-parallel complexity are polynomially related for all total functions f when p is small compared to fs block sensitivity.
In the exact quantum query model a successful algorithm must always output the correct function value. We investigate the function that is true if exactly $k$ or $l$ of the $n$ input bits given by an oracle are 1. We find an optimal algorithm (for some cases), and a nontrivial general lower and upper bound on the minimum number of queries to the black box.
We study the query complexity of computing a function f:{0,1}^n-->R_+ in expectation. This requires the algorithm on input x to output a nonnegative random variable whose expectation equals f(x), using as few queries to the input x as possible. We exactly characterize both the randomized and the quantum query complexity by two polynomial degrees, the nonnegative literal degree and the sum-of-squares degree, respectively. We observe that the quantum complexity can be unboundedly smaller than the classical complexity for some functions, but can be at most polynomially smaller for functions with range {0,1}. These query complexities relate to (and are motivated by) the extension complexity of polytopes. The linear extension complexity of a polytope is characterized by the randomized communication complexity of computing its slack matrix in expectation, and the semidefinite (psd) extension complexity is characterized by the analogous quantum model. Since query complexity can be used to upper bound communication complexity of related functions, we can derive some upper bounds on psd extension complexity by constructing efficient quantum query algorithms. As an example we give an exponentially-close entrywise approximation of the slack matrix of the perfect matching polytope with psd-rank only 2^{n^{1/2+epsilon}}. Finally, we show there is a precise sense in which randomized/quantum query complexity in expectation corresponds to the Sherali-Adams and Lasserre hierarchies, respectively.
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