Peter Hoyer, Robert Spalek
Shor's and Grover's famous quantum algorithms for factoring and searching show that quantum computers can solve certain computational problems significantly faster than any classical computer. We discuss here what quantum computers_cannot_ do, and specifically how to prove limits on their computational power. We cover the main known techniques for proving lower bounds, and exemplify and compare the methods.
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
We present several families of total boolean functions which have exact quantum query complexity which is a constant multiple (between 1/2 and 2/3) of their classical query complexity, and show that optimal quantum algorithms for these functions cannot be obtained by simply computing parities of pairs of bits. We also characterise the model of nonadaptive exact quantum query complexity in terms of coding theory and completely characterise the query complexity of symmetric boolean functions in this context. These results were originally inspired by numerically solving the semidefinite programs …
We show that in the quantum query model the complexity of detecting a triangle in an undirected graph on $n$ nodes can be done using $O(n^{1+{3\over 7}}\log^{2}n)$ quantum queries. The same complexity bound applies for outputting the triangle if there is any. This improves upon the earlier bound of $O(n^{1+{1\over 2}})$.
Near-Optimal Separations of Certificate Complexity from Randomized and Quantum Query Complexity
构造全布尔函数，证明量子查询复杂度与证书复杂度存在近四次分离，解决长期开放问题，结果在对数因子内最优。
An Exponential Separation Between Quantum Query Complexity and the Polynomial Degree
本文证明部分布尔函数的精确多项式次数与近似量子查询复杂度之间存在指数级分离，并在无界字母表下实现常数与多项式分离。
It is known since the work of [AA14] that for any permutation symmetric function $f$, the quantum query complexity is at most polynomially smaller than the classical randomized query complexity, more precisely that $R(f) = \widetilde{O}\left(Q^7(f)\right)$. In this paper, we improve this result and show that $R(f) = {O}\left(Q^3(f)\right)$ for a more general class of symmetric functions. Our proof is constructive and relies largely on the quantum hardness of distinguishing a random permutation from a random function with small range from Zhandry [Zha15].
In the SEARCH WITH ADVICE problem, a single entry of interest within a database of N entries is to be found assuming that an ordering of the entries, from that with the highest probability of being the entry of interest (as determined by a so-called advice distribution) to that with the lowest, is provided. We present a quantum algorithm that, in the presence of significant levels of quantum noise, solves SEARCH WITH ADVICE for a power law advice distribution with average-case query complexity O(1) as N tends to infinity. Since as we also show the best classical algorithms for this problem exh…
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…
We will show that if there exists a quantum query algorithm that exactly computes some total Boolean function f by making T queries, then there is a classical deterministic algorithm A that exactly computes f making O(T^3) queries. The best know bound previously was O(T^4) due to Beals et al.
Classical and Quantum Query Complexity of Boolean Functions under Indefinite Causal Order
本文研究布尔函数在不确定因果顺序下的经典与量子查询复杂度，证明经典广义模型中原有下界仍成立，构造出可通过因果不确定性降低查询复杂度的布尔函数并实现多项式分离，还给出量子查询复杂度可降低的布尔函数。
We study the quantum query complexity of constant-sized subgraph containment. Such problems include determining whether an $ n $-vertex graph contains a triangle, clique or star of some size. For a general subgraph $ H $ with $ k $ vertices, we show that $ H $ containment can be solved with quantum query complexity $ O(n^{2-\frac{2}{k}-g(H)}) $, with $ g(H) $ a strictly positive function of $ H $. This is better than $ \tilde{O}\s{n^{2-2/k}} $ by Magniez et al. These results are obtained in the learning graph model of Belovs.
We prove a very general lower bound technique for quantum and randomized query complexity, that is easy to prove as well as to apply. To achieve this, we introduce the use of Kolmogorov complexity to query complexity. Our technique generalizes the weighted, unweighted methods of Ambainis, and the spectral method of Barnum, Saks and Szegedy. As an immediate consequence of our main theorem, adversary methods can only prove lower bounds for boolean functions $f$ in $O(\min(\sqrt{n C_0(f)},\sqrt{n C_1(f)}))$, where $C_0, C_1$ is the certificate complexity, and $n$ is the size of the input. We also…
We are concerned with the Hidden Subgroup Problem for finite groups. We present a simplified analysis of a quantum algorithm proposed by Hallgren, Russell and Ta-Shma as well as a detailed proof of a lower bound on the probability of success of the algorithm.
It has long been known that any Boolean function that depends on n input variables has both degree and exact quantum query complexity of Omega(log n), and that this bound is achieved for some functions. In this paper we study the case of approximate degree and bounded-error quantum query complexity. We show that for these measures the correct lower bound is Omega(log n / loglog n), and we exhibit quantum algorithms for two functions where this bound is achieved.
An important and usual problem is to search all states we want from a database with a large number of states. In such, recall is vital. Grover's original quantum search algorithm has been generalized to the case of multiple solutions, but no one has calculated the query complexity in this case. We will use a generalized algorithm with higher precision to solve such a search problem that we should find all marked states and show that the practical query complexity increases with the number of marked states. In the end we will introduce an algorithm for the problem on a ``duality computer'' and …
Harry Buhrman et al gave an Omega(sqrt n) lower bound for monotone graph properties in the adjacency matrix query model. Their proof is based on the polynomial method. However for some properties stronger lower bounds exist. We give an Omega(n^{3/2}) bound for Graph Connectivity using Andris Ambainis' method, and an O(n^{3/2} log n) upper bound based on Grover's search algorithm. In addition we study the adjacency list query model, where we have almost matching lower and upper bounds for Strong Connectivity of directed graphs.
We present quantum query complexity bounds for testing algebraic properties. For a set S and a binary operation on S, we consider the decision problem whether $S$ is a semigroup or has an identity element. If S is a monoid, we want to decide whether S is a group. We present quantum algorithms for these problems that improve the best known classical complexity bounds. In particular, we give the first application of the new quantum random walk technique by Magniez, Nayak, Roland, and Santha that improves the previous bounds by Ambainis and Szegedy. We also present several lower bounds for testin…
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…
State conversion generalizes query complexity to the problem of converting between two input-dependent quantum states by making queries to the input. We characterize the complexity of this problem by introducing a natural information-theoretic norm that extends the Schur product operator norm. The complexity of converting between two systems of states is given by the distance between them, as measured by this norm. In the special case of function evaluation, the norm is closely related to the general adversary bound, a semi-definite program that lower-bounds the number of input queries needed …
In this paper we study the complexity of quantum query algorithms computing the value of Boolean function and its relation to the degree of algebraic polynomial representing this function. We pay special attention to Boolean functions with quantum query algorithm complexity lower than the deterministic one. Relation between the degree of representing polynomial and potentially possible quantum algorithm complexity has been already described; unfortunately, there are few examples of quantum algorithms to illustrate theoretical evaluation of the complexity. Work in this direction was aimed (1) t…
Answer Partitions and Oracle Access Determine Quantum Query Complexity
本文探讨量子加速的物理和上下文限制，指出量子优势源于将问题结构与高效干涉模式匹配，并受测量上下文、热力学不可逆性和上下文性约束。
We obtain the strongest separation between quantum and classical query complexity known to date -- specifically, we define a black-box problem that requires exponentially many queries in the classical bounded-error case, but can be solved exactly in the quantum case with a single query (and a polynomial number of auxiliary operations). The problem is simple to define and the quantum algorithm solving it is also simple when described in terms of certain quantum Fourier transforms (QFTs) that have natural properties with respect to the algebraic structures of finite fields. These QFTs may be of …
We show a nearly quadratic separation between deterministic communication complexity and the logarithm of the partition number, which is essentially optimal. This improves upon a recent power 1.5 separation of Göös, Pitassi, and Watson (FOCS 2015). In query complexity, we establish a nearly quadratic separation between deterministic (and even randomized) query complexity and subcube partition complexity, which is also essentially optimal. We also establish a nearly power 1.5 separation between quantum query complexity and subcube partition complexity, the first superlinear separation between t…
Variational learning algorithms for quantum query complexity
本文提出变分学习算法，通过参数化量子电路和基于错误概率的损失函数研究量子查询复杂性，以4次查询解决5位模5的Hamming模问题，并可在NISQ设备上实现。
We show that any boolean function can be evaluated optimally by a quantum query algorithm that alternates a certain fixed, input-independent reflection with a second reflection that coherently queries the input string. Originally introduced for solving the unstructured search problem, this two-reflections structure is therefore a universal feature of quantum algorithms. Our proof goes via the general adversary bound, a semi-definite program (SDP) that lower-bounds the quantum query complexity of a function. By a quantum algorithm for evaluating span programs, this lower bound is known to be ti…
A 1.5-Query Lower Bound for the Unitary Synthesis Problem
本文将酉合成问题的单查询下界推广到1.5查询的分数查询领域，通过保守链式方法证明任意n量子比特酉变换需要超出分数查询阈值的资源，并由此证明伪随机量子态对1.5查询敌手仍然安全。
I offer a case that quantum query complexity still has loads of enticing and fundamental open problems -- from relativized QMA versus QCMA and BQP versus IP, to time/space tradeoffs for collision and element distinctness, to polynomial degree versus quantum query complexity for partial functions, to the Unitary Synthesis Problem and more.
We present a quantum algorithm which identifies with certainty a hidden subgroup of an arbitrary finite group G in only a polynomial (in log |G|) number of calls to the oracle. This is exponentially better than the best classical algorithm. However our quantum algorithm requires exponential time, as in the classical case. Our algorithm utilizes a new technique for constructing error-free algorithms for non-decision problems on quantum computers.
Quantum Query Complexity of Boolean Functions under Indefinite Causal Order
该研究将量子查询复杂度框架推广至量子超映射，证明带因果序量子控制的量子电路无法降低查询复杂度，但部分函数用两次查询时因果不定超映射的计算误差更低。
On the query complexity of unitary channel certification
研究揭示了单位ary通道认证的查询复杂性，证明了在不同算法下存在指数级的最坏情况与平均情况之间的差异。
The number of qubits used by a quantum algorithm will be a crucial computational resource for the foreseeable future. We show how to obtain the classical query complexity for continuous problems. We then establish a simple formula for a lower bound on the qubit complexity in terms of the classical query complexity
Randomized query complexity can beat certificate complexity
本文构造了一个全布尔函数，证明其随机化查询复杂度可远低于证书复杂度，达到平方根级别的最优，同时量子查询复杂度也接近最优。
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 …
In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized query complexity for a total boolean function is given by the function $f$ on $n=2^k$ bits defined by a complete binary tree of NAND gates of depth $k$, which achieves $R_0(f) = O(D(f)^{0.7537\ldots})$. We show this is false by giving an example of a total boolean function $f$ on $n$ bits whose deterministic query complexity is $Ω(n/\log(n))$ while its zero-error randomized query complexity is $\tilde O(\sqrt{n})$. We further show that the quantum query complexity of the same fun…
State complexity of quantum finite automata is one of the interesting topics in studying the power of quantum finite automata. It is therefore of importance to develop general methods how to show state succinctness results for quantum finite automata. One such method is presented and demonstrated in this paper. In particular, we show that state succinctness results can be derived out of query complexity results.
Query-optimal quantum simulation of Lindblad evolution
本文提出一种在块编码模型中具有最优加性查询复杂度的Lindblad演化模拟算法，利用换能器框架和重用电路线性组合，解决了乘性依赖是否必要的问题。
The set equality problem is to tell whether two sets $A$ and $B$ are equal or disjoint under the promise that one of these is the case. This problem is related to the Graph Isomorphism problem. It was an open problem to find any $ω(1)$ query lower bound when sets $A$ and $B$ are given by quantum oracles. We will show that any error-bounded quantum query algorithm that solves the set equality problem must evaluate oracles $Ω(\sqrt[5]{\frac{n}{\ln n}})$ times, where $n=|A|=|B|$.
In this note we study the number of quantum queries required to identify an unknown multilinear polynomial of degree d in n variables over a finite field F_q. Any bounded-error classical algorithm for this task requires Omega(n^d) queries to the polynomial. We give an exact quantum algorithm that uses O(n^(d-1)) queries for constant d, which is optimal. In the case q=2, this gives a quantum algorithm that uses O(n^(d-1)) queries to identify a codeword picked from the binary Reed-Muller code of order d.
We study classical query algorithms with post-selection, and find that they are closely connected to rational functions with nonnegative coefficients. We show that the post-selected classical query complexity of a Boolean function is equal to the minimal degree of a rational function with nonnegative coefficients that approximates it (up to a factor of two). For post-selected quantum query algorithms, a similar relationship was shown by Mahadev and de Wolf, where the rational approximations are allowed to have negative coefficients. Using our characterisation, we find an exponentially large se…
We study the composition question for bounded-error randomized query complexity: Is R(f o g) = Omega(R(f) R(g)) for all Boolean functions f and g? We show that inserting a simple Boolean function h, whose query complexity is only Theta(log R(g)), in between f and g allows us to prove R(f o h o g) = Omega(R(f) R(h) R(g)). We prove this using a new lower bound measure for randomized query complexity we call randomized sabotage complexity, RS(f). Randomized sabotage complexity has several desirable properties, such as a perfect composition theorem, RS(f o g) >= RS(f) RS(g), and a composition theo…
The quantum query complexity of Boolean matrix multiplication is typically studied as a function of the matrix dimension, n, as well as the number of 1s in the output, \ell. We prove an upper bound of O (n\sqrt{\ell}) for all values of \ell. This is an improvement over previous algorithms for all values of \ell. On the other hand, we show that for any \eps < 1 and any \ell <= \eps n^2, there is an Ω(n\sqrt{\ell}) lower bound for this problem, showing that our algorithm is essentially tight. We first reduce Boolean matrix multiplication to several instances of graph collision. We then provide a…
Inspired by the Elitzur-Vaidman bomb testing problem [arXiv:hep-th/9305002], we introduce a new query complexity model, which we call bomb query complexity $B(f)$. We investigate its relationship with the usual quantum query complexity $Q(f)$, and show that $B(f)=Θ(Q(f)^2)$. This result gives a new method to upper bound the quantum query complexity: we give a method of finding bomb query algorithms from classical algorithms, which then provide nonconstructive upper bounds on $Q(f)=Θ(\sqrt{B(f)})$. We subsequently were able to give explicit quantum algorithms matching our upper bound method. We…
Simon in his FOCS'94 paper was the first to show an exponential gap between classical and quantum computation. The problem he dealt with is now part of a well-studied class of problems, the hidden subgroup problems. We study Simon's problem from the point of view of quantum query complexity and give here a first nontrivial lower bound on the query complexity of a hidden subgroup problem, namely Simon's problem. Our bound is optimal up to a constant factor.
Exponential separation in quantum query complexity of the quantum switch with respect to simulations with standard quantum circuits
本文证明了量子开关作用于两个n量子比特信道时，任何因果有序电路若对一信道调用M≤max(2,2^n-1)次、对另一信道调用一次，都无法确定性精确模拟，从而揭示了不定因果序在量子查询复杂度上相对标准电路的指数级优势。
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 first give an $Ø(2^{n/3})$ quantum algorithm for the 0-1 Knapsack problem with $n$ variables. More generally, for 0-1 Integer Linear Programs with $n$ variables and $d$ inequalities we give an $Ø(2^{n/3}n^d)$ quantum algorithm. For $d =o(n/\log n)$ this running time is bounded by $Ø(2^{n(1/3+ε)})$ for every $ε>0$ and in particular it is better than the $Ø(2^{n/2})$ upper bound for general quantum search. To investigate whether better algorithms for these NP-hard problems are possible, we formulate a \emph{symmetric} claw problem corresponding to 0-1 Knapsack and study its quantum query comp…
We compare classical and quantum query complexities of total Boolean functions. It is known that for worst-case complexity, the gap between quantum and classical can be at most polynomial. We show that for average-case complexity under the uniform distribution, quantum algorithms can be exponentially faster than classical algorithms. Under non-uniform distributions the gap can even be super-exponential. We also prove some general bounds for average-case complexity and show that the average-case quantum complexity of MAJORITY under the uniform distribution is nearly quadratically better than th…
A strong direct product theorem (SDPT) states that solving n instances of a problem requires Omega(n) times the resources for a single instance, even to achieve success probability exp(-Omega(n)). We prove that quantum communication complexity obeys an SDPT whenever the communication lower bound for a single instance is proved by the generalized discrepancy method, the strongest technique in that model. We prove that quantum query complexity obeys an SDPT whenever the query lower bound for a single instance is proved by the polynomial method, one of the two main techniques in that model. In bo…
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.
Krylov Polynomials and Quantum Query Complexity
本文提出通过Krylov多项式方法确定量子查询复杂性的最优逼近度，揭示了状态依赖谱结构对查询效率的提升作用。
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。
Query complexity is a model of computation in which we have to compute a function $f(x_1, \ldots, x_N)$ of variables $x_i$ which can be accessed via queries. The complexity of an algorithm is measured by the number of queries that it makes. Query complexity is widely used for studying quantum algorithms, for two reasons. First, it includes many of the known quantum algorithms (including Grover's quantum search and a key subroutine of Shor's factoring algorithm). Second, one can prove lower bounds on the query complexity, bounding the possible quantum advantage. In the last few years, there hav…
Separations in query complexity for total search problems
本文研究总搜索问题的查询复杂度，通过总搜索问题与部分函数的相互转换，在量子查询复杂度、近似度、正量子对手、锥 junta 度及通信复杂度等多个测度之间建立了新的指数级分离，并重新证明了部分函数的相关分离结果。
We establish a lower bound of $Ω{(\sqrt{n})}$ on the bounded-error quantum query complexity of read-once Boolean functions, providing evidence for the conjecture that $Ω(\sqrt{D(f)})$ is a lower bound for all Boolean functions. Our technique extends a result of Ambainis, based on the idea that successful computation of a function requires ``decoherence'' of initially coherently superposed inputs in the query register, having different values of the function. The number of queries is bounded by comparing the required total amount of decoherence of a judiciously selected set of input-output pair…
We show a power 2.5 separation between bounded-error randomized and quantum query complexity for a total Boolean function, refuting the widely believed conjecture that the best such separation could only be quadratic (from Grover's algorithm). We also present a total function with a power 4 separation between quantum query complexity and approximate polynomial degree, showing severe limitations on the power of the polynomial method. Finally, we exhibit a total function with a quadratic gap between quantum query complexity and certificate complexity, which is optimal (up to log factors). These …
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。
We present a trichotomy theorem for the quantum query complexity of regular languages. Every regular language has quantum query complexity Theta(1), ~Theta(sqrt n), or Theta(n). The extreme uniformity of regular languages prevents them from taking any other asymptotic complexity. This is in contrast to even the context-free languages, which we show can have query complexity Theta(n^c) for all computable c in [1/2,1]. Our result implies an equivalent trichotomy for the approximate degree of regular languages, and a dichotomy---either Theta(1) or Theta(n)---for sensitivity, block sensitivity, ce…
We show that any quantum algorithm deciding whether an input function $f$ from $[n]$ to $[n]$ is 2-to-1 or almost 2-to-1 requires $Θ(n)$ queries to $f$. The same lower bound holds for determining whether or not a function $f$ from $[2n-2]$ to $[n]$ is surjective. These results yield a nearly linear $Ω(n/\log n)$ lower bound on the quantum query complexity of $\cl{AC}^0$. The best previous lower bound known for any $\cl{AC^0}$ function was the $Ω((n/\log n)^{2/3})$ bound given by Aaronson and Shi's $Ω(n^{2/3})$ lower bound for the element distinctness problem.
Improved Quantum Query Complexity on Easier Inputs
本文提出改进的量子跨度程序算法，使结构化输入带来的查询复杂度改进无需事先承诺即可保持，并将其推广到状态转换，从而在多个搜索问题上证明指数级和超多项式级平均查询复杂度量子优势。
We study the query complexity of determining if a graph is connected with global queries. The first model we look at is matrix-vector multiplication queries to the adjacency matrix. Here, for an $n$-vertex graph with adjacency matrix $A$, one can query a vector $x \in \{0,1\}^n$ and receive the answer $Ax$. We give a randomized algorithm that can output a spanning forest of a weighted graph with constant probability after $O(\log^4(n))$ matrix-vector multiplication queries to the adjacency matrix. This complements a result of Sun et al.\ (ICALP 2019) that gives a randomized algorithm that can …
We study nondeterministic quantum algorithms for Boolean functions f. Such algorithms have positive acceptance probability on input x iff f(x)=1. In the setting of query complexity, we show that the nondeterministic quantum complexity of a Boolean function is equal to its ``nondeterministic polynomial'' degree. We also prove a quantum-vs-classical gap of 1 vs n for nondeterministic query complexity for a total function. In the setting of communication complexity, we show that the nondeterministic quantum complexity of a two-party function is equal to the logarithm of the rank of a nondetermini…
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。