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Register a new userQuantum and Classical Query Complexities for Generalized Simons Problem
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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.
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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}}}).
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 introduce a minimal set of physically motivated postulates that the Hamiltonian H of a continuous-time quantum walk should satisfy in order to properly represent the quantum counterpart of the classical random walk on a given graph. We found that these conditions are satisfied by infinitely many quantum Hamiltonians, which provide novel degrees of freedom for quantum enhanced protocols, In particular, the on-site energies, i.e. the diagonal elements of H, and the phases of the off-diagonal elements are unconstrained on the quantum side. The diagonal elements represent a potential energy landscape for the quantum walk, and may be controlled by the interaction with a classical scalar field, whereas, for regular lattices in generic dimension, the off-diagonal phases of H may be tuned by the interaction with a classical gauge field residing on the edges, e.g., the electro-magnetic vector potential for a charged walker.
Let $f: X times Y rightarrow {0,1,bot }$ be a partial function and $mu$ be a distribution with support contained in $f^{-1}(0) cup f^{-1}(1)$. Let $mathsf{D}^{1,mu}_epsilon(f)$ be the classical one-way communication complexity of $f$ with average error under $mu$ at most $epsilon$, $mathsf{Q}^{1,mu}_epsilon(f)$ be the quantum one-way communication complexity of $f$ with average error under $mu$ at most $epsilon$ and $mathsf{Q}^{1,mu, *}_epsilon(f)$ be the entanglement assisted one-way communication complexity of $f$ with average error under $mu$ at most $epsilon$. We show: 1. If $mu$ is a product distribution, then $forall epsilon, eta > 0$, $$mathsf{D}^{1,mu}_{2epsilon + eta}(f) leq mathsf{Q}^{1,mu, *}_{epsilon}(f) /eta+Obigl(log(mathsf{Q}^{1,mu, *}_{epsilon}(f))/etabigr).$$ 2. If $mu$ is a non-product distribution, then $forall epsilon, eta > 0$ such that $epsilon/eta + eta < 0.5$, $$mathsf{D}^{1,mu}_{3eta}(f) = O(mathsf{Q}^{1,mu}_{{epsilon}}(f) cdot mathsf{CS}(f)/eta^4)enspace,$$ where [mathsf{CS}(f) = max_{y} min_{zin{0,1}} { vert {x~|~f(x,y)=z} vert} enspace.]
We provide lower and upper bounds on the information transmission capacity of one single use of a classical-quantum channel. The lower bound is expressed in terms of the Hoeffding capacity, that we define similarly to the Holevo capacity, but replacing the relative entropy with the Hoeffding distance. Similarly, our upper bound is in terms of a quantity obtained by replacing the relative entropy with the recently introduced max-relative entropy in the definition of the divergence radius of a channel.
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