Andrew M. Childs, Richard Cleve, Stephen P. Jordan, David Yonge-Mallo
Recently, Farhi, Goldstone, and Gutmann gave a quantum algorithm for evaluating NAND trees that runs in time O(sqrt(N log N)) in the Hamiltonian query model. In this note, we point out that their algorithm can be converted into an algorithm using O(N^{1/2 + epsilon}) queries in the conventional quantum query model, for any fixed epsilon > 0.
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We present an O(\sqrt{N}) discrete query quantum algorithm for evaluating balanced binary NAND formulas and an O(N^{{1/2}+O(\frac{1}{\sqrt{\log N}})}) discrete query quantum algorithm for evaluating arbitrary binary NAND formulas.
We give a quantum algorithm for the binary NAND tree problem in the Hamiltonian oracle model. The algorithm uses a continuous time quantum walk with a run time proportional to sqrt N. We also show a lower bound of sqrt N for the NAND tree problem in the Hamiltonian oracle model.
The negative-AND (NAND) gate is universal for classical computation making it an important target for development. A seminal quantum computing algorithm by Farhi, Goldstone and Gutmann has demonstrated its realization by means of quantum scattering yielding a quantum algorithm that evaluates the output faster than any classical algorithm. Here, we derive the NAND outputs analytically from scattering theory using a tight-binding (TB) model and show the restrictions on the TB parameters in order to still maintain the NAND gate function. We map the quantum NAND tree onto a conjugated molecular sy…
For every NAND formula of size N, there is a bounded-error N^{1/2+o(1)}-time quantum algorithm, based on a coined quantum walk, that evaluates this formula on a black-box input. Balanced, or ``approximately balanced,'' NAND formulas can be evaluated in O(sqrt{N}) queries, which is optimal. It follows that the (2-o(1))-th power of the quantum query complexity is a lower bound on the formula size, almost solving in the positive an open problem posed by Laplante, Lee and Szegedy.
In the age of post-Moore era, the next-generation computing model would be a hybrid architecture consisting of different physical components such as photonic chips. In 2008, it has been proposed that the solving of NAND-tree problem can be sped up by quantum walk. Such scheme is groundbreaking due to the universality of NAND gate. However, experimental demonstration has never been achieved so far, mostly due to the challenge in preparing the propagating initial state. Here we propose an alternative solution by including a structure called "quantum slide", where a propagating Gaussian wave-pack…
We say a unitary operator acting on a set of qubits has been compiled if it has been expressed as a SEO (sequence of elementary operations, like CNOTs and single-qubit operations). SEO's are often represented as quantum circuits. arXiv:quant-ph/0702144 by Farhi-Goldstone-Gutmann has inspired a recent flurry of papers, that propose quantum algorithms for evaluating NAND formulas via quantum walks over tree graphs. These algorithms use two types of unitary evolution: oracle and non-oracle. Non-oracle evolutions are independent of the NAND formula input, whereas oracle evolutions depend on this i…
We propose a physical implementation of the quantum NAND tree evaluation algorithm. Our approach, based on continuous time quantum walks, uses the wave interference of a single electron in a heirarchical set of tunnel coupled quantum dots. We find that the query complexity of the NAND tree evaluation does not suffer strongly from disorder and dephasing, nor is it directly limited by temperature or restricted dimensionality for 2-d structures. Finally, we suggest a potential application of this algorithm to the efficient determination of high-order correlation functions of complex quantum syste…
We show that the quantum query complexity of evaluating NAND-tree instances with average choice complexity at most $W$ is $O(W)$, where average choice complexity is a measure of the difficulty of winning the associated two-player game. This generalizes a superpolynomial speedup over classical query complexity due to Zhan et al. [Zhan et al., ITCS 2012, 249-265]. We further show that the player with a winning strategy for the two-player game associated with the NAND-tree can win the game with an expected $\widetilde{O}(N^{1/4}\sqrt{{\cal C}(x)})$ quantum queries against a random opponent, where…
The continuous-time query model is a variant of the discrete query model in which queries can be interleaved with known operations (called "driving operations") continuously in time. Interesting algorithms have been discovered in this model, such as an algorithm for evaluating nand trees more efficiently than any classical algorithm. Subsequent work has shown that there also exists an efficient algorithm for nand trees in the discrete query model; however, there is no efficient conversion known for continuous-time query algorithms for arbitrary problems. We show that any quantum algorithm in t…
We give a quantum algorithm for evaluating a class of boolean formulas (such as NAND trees and 3-majority trees) on a restricted set of inputs. Due to the structure of the allowed inputs, our algorithm can evaluate a depth $n$ tree using $O(n^{2+\logω})$ queries, where $ω$ is independent of $n$ and depends only on the type of subformulas within the tree. We also prove a classical lower bound of $n^{Ω(\log\log n)}$ queries, thus showing a (small) super-polynomial speed-up.
We study the quantum query complexity of finding a certificate for a d-regular, k-level balanced NAND formula. Up to logarithmic factors, we show that the query complexity is Theta(d^{(k+1)/2}) for 0-certificates, and Theta(d^{k/2}) for 1-certificates. In particular, this shows that the zero-error quantum query complexity of evaluating such formulas is O(d^{(k+1)/2}) (again neglecting a logarithmic factor). Our lower bound relies on the fact that the quantum adversary method obeys a direct sum theorem.
We study randomized and quantum query (a.k.a. decision tree) complexity for all total Boolean functions, with emphasis to derandomization and dequantization (removing quantumness from algorithms). Firstly, we show that $D(f) = O(Q_1(f)^3)$ for any total function $f$, where $D(f)$ is the minimal number of queries made by a deterministic query algorithm and $Q_1(f)$ is the number of queries made by any quantum query algorithm (decision tree analog in quantum case) with one-sided constant error; both algorithms compute function $f$. Secondly, we show that for all total Boolean functions $f$ holds…
Quantum algorithms for graph problems are considered, both in the adjacency matrix model and in an adjacency list-like array model. We give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree, and Single Source Shortest Paths. For example we show that the query complexity of Minimum Spanning Tree is in Theta(n^{3/2}) in the matrix model and in Theta(sqrt{nm}) in the array model, while the complexity of Connectivity is also in Theta(n^{3/2}) in the matrix model, but in Theta(n) in the array model. The upp…
Quantum walks on binary trees are used in many quantum algorithms to achieve important speedup over classical algorithms. The formulation of this kind of algorithms as quantum circuit presents the advantage of being easily readable, executable on circuit based quantum computers and simulators and optimal on the usage of resources. We propose a strategy to compose quantum circuit that performs quantum walk on binary trees following universal gate model quantum computation principles. We give a particular attention to NAND formula evaluation algorithm as it could have many applications in game t…
It has recently been shown that starting with a classical query algorithm (decision tree) and a guessing algorithm that tries to predict the query answers, we can design a quantum algorithm with query complexity $O(\sqrt{GT})$ where $T$ is the query complexity of the classical algorithm (depth of the decision tree) and $G$ is the maximum number of wrong answers by the guessing algorithm [arXiv:1410.0932, arXiv:1905.13095]. In this paper we show that, given some constraints on the classical algorithms, this quantum algorithm can be implemented in time $\tilde O(\sqrt{GT})$. Our algorithm is bas…
Improved Quantum Query Upper Bounds Based on Classical Decision Trees
本文基于经典决策树结构给出量子查询算法的通用构造，实现最高二次加速，并揭示猜测复杂度等于秩，且通过优化权重方案获得更强上界。
Lin and Lin have recently shown how starting with a classical query algorithm (decision tree) for a function, we may find upper bounds on its quantum query complexity. More precisely, they have shown that given a decision tree for a function $f:\{0,1\}^n\to[m]$ whose input can be accessed via queries to its bits, and a guessing algorithm that predicts answers to the queries, there is a quantum query algorithm for $f$ which makes at most $O(\sqrt{GT})$ quantum queries where $T$ is the depth of the decision tree and $G$ is the maximum number of mistakes of the guessing algorithm. In this paper w…
In this paper we study the separation between the deterministic (classical) query complexity ($D$) and the exact quantum query complexity ($Q_E$) of several Boolean function classes using the parity decision tree method. We first define the Query Friendly (QF) functions on $n$ variables as the ones with minimum deterministic query complexity $(D(f))$. We observe that for each $n$, there exists a non-separable class of QF functions such that $D(f)=Q_E(f)$. Further, we show that for some values of $n$, all the QF functions are non-separable. Then we present QF functions for certain other values …
In Exact Quantum Query model, almost all of the Boolean functions for which non-trivial query algorithms exist are symmetric in nature. The most well known techniques in this domain exploit parity decision trees, in which the parity of two bits can be obtained by a single query. Thus, exact quantum query algorithms outperforming parity decision trees are rare. In this paper we first obtain optimal exact quantum query algorithms ($Q_{algo}(f)$) for a direct sum based class of $Ω\left( 2^{\frac{\sqrt{n}}{2}} \right)$ non-symmetric functions. We construct these algorithms by analyzing the algebra…
Fat-Tree QRAM: A High-Bandwidth Shared Quantum Random Access Memory for Parallel Queries
提出 Fat-Tree QRAM 架构，通过使用 $O(N)$ 个量子比特在 $O(\log (N))$ 时间内并行执行 $O(\log (N))$ 次查询，在保证保真度的同时显著提升了并行查询能力，并提供了超导电路实现方案与查询调度协议。
Quantum Advantage in Decision Trees: A Weighted Graph and $L_1$ Norm Approach
本文提出利用加权图与L1范数方法分析单次查询量子决策树，以揭示量子优势的必要条件并实现指数级优势。
Span program is a linear-algebraic model of computation originally proposed for studying the complexity theory. Recently, it has become a useful tool for designing quantum algorithms. In this paper, we present a time-efficient span-program-based quantum algorithm for the following problem. Let $T$ be an arbitrary tree. Given query access to the adjacency matrix of a graph $G$ with $n$ vertices, we need to determine whether $G$ contains $T$ as a subgraph, or $G$ does not contain $T$ as a minor, under the promise that one of these cases holds. We call this problem the subgraph/not-a-minor proble…
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