Pacal Koiran, Jürgen Landes, Natacha Portier, Penghui Yao
We present general methods for proving lower bounds on the query complexity of nonadaptive quantum algorithms. Our results are based on the adversary method of Ambainis.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
This paper employs a powerful argument, called an algorithmic argument, to prove lower bounds of the quantum query complexity of a multiple-block ordered search problem in which, given a block number i, we are to find a location of a target keyword in an ordered list of the i-th block. Apart from much studied polynomial and adversary methods for quantum query complexity lower bounds, our argument shows that the multiple-block ordered search needs a large number of nonadaptive oracle queries on a black-box model of quantum computation that is also supplemented with advice. Our argument is also …
A Boolean function $f:\{0,1\}^d \mapsto \{0,1\}$ is unate if, along each coordinate, the function is either nondecreasing or nonincreasing. In this note, we prove that any nonadaptive, one-sided error unateness tester must make $Ω(\frac{d}{\log d})$ queries. This result improves upon the $Ω(\frac{d}{\log^2 d})$ lower bound for the same class of testers due to Chen et al. (STOC, 2017).
We say a turnstile streaming algorithm is "non-adaptive" if, during updates, the memory cells written and read depend only on the index being updated and random coins tossed at the beginning of the stream (and not on the memory contents of the algorithm). Memory cells read during queries may be decided upon adaptively. All known turnstile streaming algorithms in the literature are non-adaptive. We prove the first non-trivial update time lower bounds for both randomized and deterministic turnstile streaming algorithms, which hold when the algorithms are non-adaptive. While there has been abunda…
In this note, we show that quantum lower bounds obtained using the adversary method hold in the Hamiltonian oracle model.
In this note we construct an explicit optimal (negative-weight) adversary matrix for the element distinctness problem, given that the size of the alphabet is sufficiently large.
We design a nonadaptive algorithm that, given oracle access to a function $f: \{0,1\}^n \to \{0,1\}$ which is $α$-far from monotone, makes poly$(n, 1/α)$ queries and returns an estimate that, with high probability, is an $\widetilde{O}(\sqrt{n})$-approximation to the distance of $f$ to monotonicity. The analysis of our algorithm relies on an improvement to the directed isoperimetric inequality of Khot, Minzer, and Safra (SIAM J. Comput., 2018). Furthermore, we rule out a poly$(n, 1/α)$-query nonadaptive algorithm that approximates the distance to monotonicity significantly better by showing th…
We prove a tight quantum query lower bound $Ω(n^{k/(k+1)})$ for the problem of deciding whether there exist $k$ numbers among $n$ that sum up to a prescribed number, provided that the alphabet size is sufficiently large. This is an extended and simplified version of an earlier preprint of one of the authors arXiv:1204.5074.
We investigate query-to-communication lifting theorems for models related to the quantum adversary bounds. Our results are as follows: 1. We show that the classical adversary bound lifts to a lower bound on randomized communication complexity with a constant-sized gadget. We also show that the classical adversary bound is a strictly stronger lower bound technique than the previously-lifted measure known as critical block sensitivity, making our lifting theorem one of the strongest lifting theorems for randomized communication complexity using a constant-sized gadget. 2. Turning to quantum mode…
The polynomial method and the adversary method are the two main techniques to prove lower bounds on quantum query complexity, and they have so far been considered as unrelated approaches. Here, we show an explicit reduction from the polynomial method to the multiplicative adversary method. The proof goes by extending the polynomial method from Boolean functions to quantum state generation problems. In the process, the bound is even strengthened. We then show that this extended polynomial method is a special case of the multiplicative adversary method with an adversary matrix that is independen…
We prove a quantum query lower bound Ω(n^{(d+1)/(d+2)}) for the problem of deciding whether an input string of size n contains a k-tuple which belongs to a fixed orthogonal array on k factors of strength d<=k-1 and index 1, provided that the alphabet size is sufficiently large. Our lower bound is tight when d=k-1. The orthogonal array problem includes the following problems as special cases: k-sum problem with d=k-1, k-distinctness problem with d=1, k-pattern problem with d=0, (d-1)-degree problem with 1<=d<=k-1, unordered search with d=0 and k=1, and graph collision with d=0 and k=2.
We present a new method for proving lower bounds on quantum query algorithms. The new method is an extension of adversary method, by analyzing the eigenspace structure of the problem. Using the new method, we prove a strong direct product theorem for quantum search. This result was previously proven by Klauck, Spalek and de Wolf (quant-ph/0402123) using polynomials method. No proof using adversary method was known before.
We show that all known classical adversary lower bounds on randomized query complexity are equivalent for total functions, and are equal to the fractional block sensitivity $\text{fbs}(f)$. That includes the Kolmogorov complexity bound of Laplante and Magniez and the earlier relational adversary bound of Aaronson. This equivalence also implies that for total functions, the relational adversary is equivalent to a simpler lower bound, which we call rank-1 relational adversary. For partial functions, we show unbounded separations between $\text{fbs}(f)$ and other adversary bounds, as well as betw…
The general adversary bound is a semi-definite program (SDP) that lower-bounds the quantum query complexity of a function. We turn this lower bound into an upper bound, by giving a quantum walk algorithm based on the dual SDP that has query complexity at most the general adversary bound, up to a logarithmic factor. In more detail, the proof has two steps, each based on "span programs," a certain linear-algebraic model of computation. First, we give an SDP that outputs for any boolean function a span program computing it that has optimal "witness size." The optimal witness size is shown to coin…
We propose a new method for proving lower bounds on quantum query algorithms. Instead of a classical adversary that runs the algorithm with one input and then modifies the input, we use a quantum adversary that runs the algorithm with a superposition of inputs. If the algorithm works correctly, its state becomes entangled with the superposition over inputs. We bound the number of queries needed to achieve a sufficient entanglement and this implies a lower bound on the number of queries for the computation. Using this method, we prove two new $Ω(\sqrt{N})$ lower bounds on computing AND of ORs a…
We prove a generalization of the parallel adversary method to multi-valued functions, and apply it to prove that there is no parallel quantum advantage for approximate counting.
We introduce a new quantum adversary method to prove lower bounds on the query complexity of the quantum state generation problem. This problem encompasses both, the computation of partial or total functions and the preparation of target quantum states. There has been hope for quite some time that quantum state generation might be a route to tackle the {\sc Graph Isomorphism} problem. We show that for the related problem of {\sc Index Erasure} our method leads to a lower bound of $Ω(\sqrt N)$ which matches an upper bound obtained via reduction to quantum search on $N$ elements. This closes an …
Certificate Games and Consequences for the Classical Adversary Bound
本文提出证书博弈复杂度，通过零通信博弈刻画经典对手界，并证明量子证书博弈复杂度可比量子查询复杂度大平方倍，最终揭示除私有随机性外所有证书博弈模型均坍缩到经典对手界。
Quantum Query Lower Bounds for Key Recovery Attacks on the Even-Mansour Cipher
本文研究 Even-Mansour 密码密钥恢复的量子查询复杂度，证明即使允许自适应查询，任意量子算法仍需 Ω(n) 次查询，从而表明 Kuwakado-Morii 攻击在常数因子内最优。
Query Complexity of Testing Structured Parenthesis Languages
本研究针对结构化括号语言（如Dyck语言）的成员资格测试，证明了测试Dyck语言需Ω(n^{2/5})查询下界，并发现一类常数可测的excursion语言，同时构造了需要多项式查询的Hidden String语言，揭示了查询复杂度的边界。
Quantum Query Complexity of Finding a Tarski Fixed Point on a High-Dimensional Grid
该研究针对高维网格上寻找塔斯基不动点问题，提出树过滤敌手方法，证明了量子查询下界，在部分参数范围优于经典下界，为建立量子复杂度下界提供了新途径。
Generalizing earlier work characterizing the quantum query complexity of computing a function of an unknown classical ``black box'' function drawn from some set of such black box functions, we investigate a more general quantum query model in which the goal is to compute functions of N by N ``black box'' unitary matrices drawn from a set of such matrices, a problem with applications to determining properties of quantum physical systems. We characterize the existence of an algorithm for such a query problem, with given error and number of queries, as equivalent to the feasibility of a certain s…
The problem of distinguishing between a random function and a random permutation on a domain of size $N$ is important in theoretical cryptography, where the security of many primitives depend on the problem's hardness. We study the quantum query complexity of this problem, and show that any quantum algorithm that solves this problem with bounded error must make $Ω(N^{1/5}/\log N)$ queries to the input function. Our lower bound proof uses a combination of the Collision Problem lower bound and Ambainis's adversary theorem.
Small-Bias Quantum Approximate Counting via the Multiplicative Adversary Method
通过乘法对手方法，我们证明了量子近似计数两权重判定版本的下界，并给出基于单查询进展的推导，同时将唯一OR归约到该问题。
On query complexity measures and their relations for symmetric functions
本文探讨了对称函数类中量子查询复杂性下界与其他基于决策树的复杂性度量之间的关系，为正对手方法等给出了显式构造，并刻画了Gap Majority的量子查询复杂性及紧分离。
Lower Bounds for Learning Quantum States with Single-Copy Measurements
本文研究单副本测量下的量子态层析与影子层析，证明非自适应及自适应测量的样本复杂度和测量数下界，确立Pauli层析与样本均值策略的最优性。
A Brief Introduction to Quantum Query Complexity
本文介绍量子查询复杂度，重点阐述四种求解下界的主要技术及对手方法推导上界的双重作用，为初学者和研究人员提供自成体系的入门指南。
The problem of finding a local minimum of a black-box function is central for understanding local search as well as quantum adiabatic algorithms. For functions on the Boolean hypercube {0,1}^n, we show a lower bound of Omega(2^{n/4}/n) on the number of queries needed by a quantum computer to solve this problem. More surprisingly, our approach, based on Ambainis' quantum adversary method, also yields a lower bound of Omega(2^{n/2}/n^2) on the problem's classical randomized query complexity. This improves and simplifies a 1983 result of Aldous. Finally, in both the randomized and quantum cases, …
This note complements the paper "One-Way Ticket to Las Vegas and the Quantum Adversary" (arxiv:2301.02003). I develop the ideas behind the adversary bound - universal algorithm duality therein in a different form, using the same perspective as Barnum-Saks-Szegedy in which query algorithms are defined as sequences of feasible reduced density matrices rather than sequences of unitaries. This form may be faster to understand for a general quantum information audience: It avoids defining the "unidirectional relative $γ_{2}$-bound" and relating it to query algorithms explicitly. This proof is also …
The degree of a polynomial representing (or approximating) a function f is a lower bound for the number of quantum queries needed to compute f. This observation has been a source of many lower bounds on quantum algorithms. It has been an open problem whether this lower bound is tight. We exhibit a function with polynomial degree M and quantum query complexity Ω(M^{1.321...}). This is the first superlinear separation between polynomial degree and quantum query complexity. The lower bound is shown by a new, more general version of quantum adversary method.
We describe new lower bounds for randomized communication complexity and query complexity which we call the partition bounds. They are expressed as the optimum value of linear programs. For communication complexity we show that the partition bound is stronger than both the rectangle/corruption bound and the γ_2/generalized discrepancy bounds. In the model of query complexity we show that the partition bound is stronger than the approximate polynomial degree and classical adversary bounds. We also exhibit an example where the partition bound is quadratically larger than polynomial degree and cl…
We introduce a new technique proving formula size lower bounds based on the linear programming bound originally introduced by Karchmer, Kushilevitz and Nisan [11] and the theory of stable set polytope. We apply it to majority functions and prove their formula size lower bounds improved from the classical result of Khrapchenko [13]. Moreover, we introduce a notion of unbalanced recursive ternary majority functions motivated by a decomposition theory of monotone self-dual functions and give integrally matching upper and lower bounds of their formula size. We also show monotone formula size lower…
A one-query lower bound for unitary synthesis and breaking quantum cryptography
本文证明酉合成存在单查询下界，即某些酉算子无法被仅查询一次布尔函数的神谕算法近似实现，并由此推出相对随机神谕存在抗单查询敌手的量子密码原语，表明实现随机酉算子和破解量子密码比经典搜索或态制备更难。
In this work, we initiate a formal study of probably approximately correct (PAC) learning under evasion attacks, where the adversary's goal is to \emph{misclassify} the adversarially perturbed sample point $\widetilde{x}$, i.e., $h(\widetilde{x})\neq c(\widetilde{x})$, where $c$ is the ground truth concept and $h$ is the learned hypothesis. Previous work on PAC learning of adversarial examples have all modeled adversarial examples as corrupted inputs in which the goal of the adversary is to achieve $h(\widetilde{x}) \neq c(x)$, where $x$ is the original untampered instance. These two definitio…
Given an implicit $n\times n$ matrix $A$ with oracle access $x^TA x$ for any $x\in \mathbb{R}^n$, we study the query complexity of randomized algorithms for estimating the trace of the matrix. This problem has many applications in quantum physics, machine learning, and pattern matching. Two metrics are commonly used for evaluating the estimators: i) variance; ii) a high probability multiplicative-approximation guarantee. Almost all the known estimators are of the form $\frac{1}{k}\sum_{i=1}^k x_i^T A x_i$ for $x_i\in \mathbb{R}^n$ being i.i.d. for some special distribution. Our main results ar…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。