Andris Ambainis
We show that, for almost all N-variable Boolean functions f, at least N/4-O(\sqrt{N} log N) queries are required to compute f in quantum black-box model with bounded error.
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Given a Boolean function $f$ provided as a black-box with $n$ variables, this paper will propose a quantum algorithm for testing if a certain variable is junta or $ε$-far from being junta. The proposed algorithm constructs another black-box using two copies of the provided black-box. The constructed black-box is used with the partial diffusion operator in an amplitude amplification technique to test whether the variable being tested is junta or not, using $\mathcal{O}(\sqrt{2^n})$ queries to the constructed black-box. The proposed algorithm considers a Boolean function of general form, contrar…
In this paper, we present a fast quantum algorithm to measure the Hamming distance between two or more Boolean functions provided as black-boxes. The proposed algorithm constructs a new black-box with a certain property which is utilized to solve this problem. The introduced algorithm converts measuring the Hamming distance between Boolean functions to measuring entanglement between qubits, using concurrence entanglement measure.
We prove that, to compute a Boolean function $f$ on $N$ variables with error probability $ε$, any quantum black-box algorithm has to query at least $\frac{1 - 2\sqrtε}{2} ρ_f N = \frac{1 - 2\sqrtε}{2} \bar{S}_f$ times, where $ρ_f$ is the average influence of variables in $f$, and $\bar{S}_f$ is the average sensitivity. It's interesting to contrast this result with the known lower bound of $Ω(\sqrt{S_f})$, where $S_f$ is the sensitivity of $f$. This lower bound is tight for some functions. We also show for any polynomial $\tilde{f}$ that approximates $f$ with error probability $ε$, $deg(\tilde{…
Given a black-box representing an unknown Boolean function $f$ of $n$ variables, in this paper we propose a fast quantum algorithm to test whether or not a certain variable in the function $f$ is a junta variable. The proposed algorithm creates entanglement between the variable under test and an auxiliary qubit, where the entanglement is measured using concurrence measure to decide if the variable is junta. The paper shows applications to the proposed algorithm in learning and categorization of Boolean functions.
The goal in function property testing is to determine whether a black-box Boolean function has a certain property or is epsilon-far from having that property. The performance of the algorithm is judged by how many calls need to be made to the black box in order to determine, with high probability, which of the two alternatives is the case. Here we present two quantum algorithms, the first to determine whether the function is linear and the second to determine whether it is symmetric (invariant under permutations of the arguments). Both require O(epsilon^{-2/3}) calls to the oracle, which is be…
In view of the remarkable interest raised by the introduction of the so called PR-box we analyze its nonlocal features from the point of view of Parameter and Outcome Dependence.
The paper has been withdrawn by the author due to unhappy mistake in the initial scope of the work.
We examine the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}^N in the black-box model. We show that, in the black-box model, the exponential quantum speed-up obtained for partial functions (i.e. problems involving a promise on the input) by Deutsch and Jozsa and by Simon cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with bounded-error using T black-box queries then there is a classical deterministic algorithm that computes f exactly with O(T^6) queries. We also give asymptotically tig…
Boolean and quantum circuits have commonalities and differences. To formalize the syntactical commonality we introduce syntactic circuits where the gates are black boxes. Syntactic circuits support various semantics. One semantics is provided by Boolean circuits, another by quantum circuits. Quantum semantics is a generalization of Boolean but, because of entanglement, the generalization is not straightforward. We consider only unitary quantum circuits here.
Let a Boolean function be available as a black-box (oracle) and one likes to devise an algorithm to test whether it has certain property or it is $ε$-far from having that property. The efficiency of the algorithm is judged by the number of calls to the oracle so that one can decide, with high probability, between these two alternatives. The best known quantum algorithm for testing whether a function is linear or $ε$-far $(0 < ε< \frac{1}{2})$ from linear functions requires $O(ε^{-\frac{2}{3}})$ many calls [Hillery and Andersson, Physical Review A 84, 062329 (2011)]. We show that this can be im…
The oracle identification problem (OIP) is, given a set $S$ of $M$ Boolean oracles out of $2^{N}$ ones, to determine which oracle in $S$ is the current black-box oracle. We can exploit the information that candidates of the current oracle is restricted to $S$. The OIP contains several concrete problems such as the original Grover search and the Bernstein-Vazirani problem. Our interest is in the quantum query complexity, for which we present several upper and lower bounds. They are quite general and mostly optimal: (i) The query complexity of OIP is $O(\sqrt{N\log M \log N}\log\log M)$ for {\it…
Using the Naimark dilation theory we investigate the question under what conditions an observable which is a coarse graining of another observable is a function of it. To this end, conditions for the separability and for the Boolean structure of an observable are given.
An $n\overset{p}{\mapsto}m$ random access code (RAC) is an encoding of $n$ bits into $m$ bits such that any initial bit can be recovered with probability at least $p$, while in a quantum RAC (QRAC), the $n$ bits are encoded into $m$ qubits. Since its proposal, the idea of RACs was generalized in many different ways, e.g. allowing the use of shared entanglement (called entanglement-assisted random access code, or simply EARAC) or recovering multiple bits instead of one. In this paper we generalize the idea of RACs to recovering the value of a given Boolean function $f$ on any subset of fixed si…
In this survey, we describe two recent developments in quantum algorithms. The first new development is a quantum algorithm for evaluating a Boolean formula consisting of AND and OR gates of size N in time O(\sqrt{N}). This provides quantum speedups for any problem that can be expressed via Boolean formulas. This result can be also extended to span problems, a generalization of Boolean formulas. This provides an optimal quantum algorithm for any Boolean function in the black-box query model. The second new development is a quantum algorithm for solving systems of linear equations. In contrast …
Quantum algorithms can be analyzed in a query model to compute Boolean functions where input is given in a black box and the aim is to compute function value for arbitrary input using as few queries as possible. We concentrate on quantum query algorithm designing tasks in this paper. The main aim of the research was to find new efficient algorithms and develop general algorithm designing techniques. First, we present several exact quantum query algorithms for certain problems that are better than classical counterparts. Next, we introduce algorithm transformation methods that allow significant…
Analysis of Boolean Functions Related to Binary Input Binary Output Two-party Nonlocal Games
本文穷举四变量布尔函数以表达二元输入二元输出双方非定域博弈，发现除CHSH外部分博弈量子成功率更高，但CHSH博弈在量子和经典技术的分离效率上表现最佳。
In Weighted Model Counting (WMC) we assign weights to Boolean literals and we want to compute the sum of the weights of the models of a Boolean function where the weight of a model is the product of the weights of its literals. WMC was shown to be particularly effective for performing inference in graphical models, with a complexity of $O(n2^w)$ where $n$ is the number of variables and $w$ is the treewidth. In this paper, we propose a quantum algorithm for performing WMC, Quantum WMC (QWMC), that modifies the quantum model counting algorithm to take into account the weights. In turn, the model…
Boolean Matching Reversible Circuits: Algorithm and Complexity
本文首次研究可逆电路的布尔匹配问题，证明在输入取反与置换下的等价性可量子多项式时间求解，实现设计自动化的量子指数加速，并给出输入输出同时取反下的不可能性结果。
We consider orthomodular posets endowed with a symmetric difference. We call them ODPs. Expressed in the quantum logic language, we consider quantum logics with an XOR-type connective. We study three classes of "almost Boolean" ODPs, two of them defined by requiring rather specific behaviour of infima and the third by a Boolean-like behaviour of Frink ideals. We establish a (rather surprising) inclusion between the three classes, shadding thus light on their intrinsic properties. (More details can be found in the Introduction that follows.) Let us only note that the orthomodular posets pursued…
A recent paper on quantum walks by Childs et al. [STOC'03] provides an example of a black-box problem for which there is a quantum algorithm with exponential speedup over the best classical randomized algorithm for the problem, but where the quantum algorithm does not involve any use of the quantum Fourier transform. They give an exponential lower bound for a classical randomized algorithm solving the black-box graph traversal problem defined in their paper. In this note we give an improved lower bound for this problem via a straightforward and more complete analysis.
The degrees of polynomials representing or approximating Boolean functions are a prominent tool in various branches of complexity theory. Sherstov recently characterized the minimal degree deg_{\eps}(f) among all polynomials (over the reals) that approximate a symmetric function f:{0,1}^n-->{0,1} up to worst-case error \eps: deg_{\eps}(f) = ~Θ(deg_{1/3}(f) + \sqrt{n\log(1/\eps)}). In this note we show how a tighter version (without the log-factors hidden in the ~Θ-notation), can be derived quite easily using the close connection between polynomials and quantum algorithms.
A Quantum Algorithm for the Classification of Patterns of Boolean Functions
本文提出一种量子算法，通过单次预言机查询以概率1分类不平衡布尔函数的层级类别，并详述了可扩展的设计方法论。
Optimal T depth quantum circuits for implementing arbitrary Boolean functions
本文提出一种通用构造方法，通过分析布尔函数的代数标准型，为任意布尔函数实现精确为 $\lceil \log_2 k \rceil$ 的最优 Toffoli 及 T 深度量子线路，促进近期量子器件的高效实现。
This work studies the quantum query complexity of Boolean functions in a scenario where it is only required that the query algorithm succeeds with a probability strictly greater than 1/2. We show that, just as in the communication complexity model, the unbounded error quantum query complexity is exactly half of its classical counterpart for any (partial or total) Boolean function. Moreover, we show that the "black-box" approach to convert quantum query algorithms into communication protocols by Buhrman-Cleve-Wigderson [STOC'98] is optimal even in the unbounded error setting. We also study a se…
In this article we give several new results on the complexity of algorithms that learn Boolean functions from quantum queries and quantum examples. Hunziker et al. conjectured that for any class C of Boolean functions, the number of quantum black-box queries which are required to exactly identify an unknown function from C is $O(\frac{\log |C|}{\sqrt{{\hatγ}^{C}}})$, where $\hatγ^{C}$ is a combinatorial parameter of the class C. We essentially resolve this conjecture in the affirmative by giving a quantum algorithm that, for any class C, identifies any unknown function from C using $O(\frac{\l…
In his 2003 paper "Towards an algebraic theory of Boolean circuits", Lafont notes that the class of reversible circuits over a set of k truth values is finitely generated when k is odd. He cites a private communication for the proof. The purpose of this short note is to make the content of that communication available.
A non-local box is an abstract device into which Alice and Bob input bits x and y respectively and receive outputs a and b respectively, where a, b are uniformly distributed and the parity of a+b equals xy. Such boxes have been central to the study of quantum or generalized non-locality as well as the simulation of non-signaling distributions. In this paper, we start by studying how many non-local boxes Alice and Bob need in order to compute a Boolean function f. We provide tight upper and lower bounds in terms of the communication complexity of the function both in the deterministic and rando…
Many quantum algorithms can be analyzed in a query model to compute Boolean functions where input is given by a black box. As in the classical version of decision trees, different kinds of quantum query algorithms are possible: exact, zero-error, bounded-error and even nondeterministic. In this paper, we study the latter class of algorithms. We introduce a fresh notion in addition to already studied nondeterministic algorithms and introduce dual nondeterministic quantum query algorithms. We examine properties of such algorithms and prove relations with exact and nondeterministic quantum query …
The premise of this note is the following observation: the formalism of Bohrification is a natural place for the interpretation of general non-signalling theories. We demonstrate it through an analysis of so-called box-worlds, a popular framework for the discussion of systems exhibiting super-quantum correlations. In particular, we show that non-signalling box-world states are precisely the internal probability valuations on an internal frame in a Kripke topos naturally associated with a given box world.
Algebra of Nonlocal Boxes and the Collapse of Communication Complexity
本文通过代数研究非局域盒的线路结构，定义盒的乘积与轨道，证明了通信复杂度的坍缩现象，并恢复噪声PR盒，为理解非局域资源提供了新框架。
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