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.
Simon's 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, Simon's 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 Simon's problem have also been proposed, which deterministically solve the problem with O(n) queries. Also the quantum lower bound Ω(n) for Simon's problem is known. Although these algorithms are either complicated or specialized, their resu…
Simon's 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 Simon's problem has been proved by Cai $\&$ Qiu. Generalized Simon's 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 $x \oplus y\in S$, for any $x, y\in {\{0, 1\}}^n$, where $|S|=2^k$ for some $0\leq k\leq n$. The goal is to find $S$. For the…
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…
The quantum security of lightweight block ciphers is receiving more and more attention. However, the existing quantum attacks on lightweight block ciphers mainly focused on the quantum exhaustive search, while the quantum dedicated attacks combined with classical cryptanalysis methods haven't been well studied. In this paper, we study quantum key recovery attack on SIMON block cipher using Quantum Amplitude Amplification algorithm in Q1 model. At first, we reanalyze the quantum circuit complexity of quantum master key exhaustive search on SIMON block cipher. The Clifford gates count is estimat…
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 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.
Random Garbage Separates XOR from Forward-Only Queries
Khalifa University of Science and Technology（哈利法科学技术大学）；Center for Quantum and Topological Systems, NYUAD Research Institute（NYUAD研究院量子与拓扑系统中心）；Division of Science, New York University Abu Dhabi（纽约大学阿布扎比分校科学系）
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…
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…
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 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.
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…
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 …
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.
The first quantum algorithm to offer an exponential speedup (in the query complexity setting) over classical algorithms was Simon's algorithm for identifying a hidden exclusive-or mask. Here we observe how part of Simon's algorithm can be interpreted as a Clebsch-Gordan transform. Inspired by this we show how Clebsch-Gordan transforms can be used to efficiently find a hidden involution on the group G^n where G is the dihedral group of order eight (the group of symmetries of a square.) This problem previously admitted an efficient quantum algorithm but a connection to Clebsch-Gordan transforms …
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…
We define a new query measure we call quantum distinguishing complexity, denoted QD(f) for a Boolean function f. Unlike a quantum query algorithm, which must output a state close to |0> on a 0-input and a state close to |1> on a 1-input, a "quantum distinguishing algorithm" can output any state, as long as the output states for any 0-input and 1-input are distinguishable. Using this measure, we establish a new relationship in query complexity: For all total functions f, Q_0(f)=O~(Q(f)^5), where Q_0(f) and Q(f) denote the zero-error and bounded-error quantum query complexity of f respectively, …
We study the power of nonadaptive quantum query algorithms, which are algorithms whose queries to the input do not depend on the result of previous queries. First, we show that any bounded-error nonadaptive quantum query algorithm that computes some total boolean function depending on n variables must make Omega(n) queries to the input in total. Second, we show that, if there exists a quantum algorithm that uses k nonadaptive oracle queries to learn which one of a set of m boolean functions it has been given, there exists a nonadaptive classical algorithm using O(k log m) queries to solve the …
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 …
In the exact quantum query model a successful algorithm must always output the correct function value. We investigate the function that is true if exactly $k$ or $l$ of the $n$ input bits given by an oracle are 1. We find an optimal algorithm (for some cases), and a nontrivial general lower and upper bound on the minimum number of queries to the black box.
We combine the classical notions and techniques for bounded query classes with those developed in quantum computing. We give strong evidence that quantum queries to an oracle in the class NP does indeed reduce the query complexity of decision problems. Under traditional complexity assumptions, we obtain an exponential speedup between the quantum and the classical query complexity of function classes. For decision problems and function classes we obtain the following results: o P_||^NP[2k] is included in EQP_||^NP[k] o P_||^NP[2^(k+1)-2] is included in EQP^NP[k] o FP_||^NP[2^(k+1)-2] is include…
Span programs are an important model of quantum computation due to their tight correspondence with quantum query complexity. For any decision problem $f$, the minimum complexity of a span program for $f$ is equal, up to a constant factor, to the quantum query complexity of $f$. Moreover, this correspondence is constructive. A span program for $f$ with complexity $C$ can be compiled into a bounded error quantum algorithm for $f$ with query complexity $O(C)$, and vice versa. In this work, we prove an analogous connection for quantum time complexity. In particular, we show how to convert a quantu…
In this note we investigate the relationship between worst-case quantum query complexity and average-case classical query complexity. Specifically, we show that if a quantum computer can evaluate a total Boolean function f with bounded error using T queries in the worst case, then a deterministic classical computer can evaluate f using O(T^5) queries in the average case, under a uniform distribution of inputs. If f is monotone, we show furthermore that only O(T^3) queries are needed. Previously, Beals et al. (1998) showed that if a quantum computer can evaluate f with bounded error using T que…
We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order $\ell\geq1$ sum to at most $c^{\ell}\sqrt{\binom{d}{\ell}(1+\log n)^{\ell-1}},$ where $n$ is the number of variables, $d$ is the tree depth, and $c>0$ is an absolute constant. This bound is essentially tight and settles a conjecture due to Tal (arxiv 2019; FOCS 2020). The bounds prior to our work degraded rapidly with $\ell,$ becoming trivial already at $\ell=\sqrt{d}.$ As an application, we obtain, for every integer $k\geq1,$ a partial Boolean function on $n$ bits that has bounded-error qua…
We consider the problem of determining if a sequence of parentheses is well parenthesized, with a depth of at most h. We denote this language as $Dyck_h$. We study the quantum query complexity of this problem for different h as function of the length n of the word. It has been known from a recent paper by Aaronson et al. that, for any constant h, since $Dyck_h$ is star-free, it has quantum query complexity $\tildeΘ(\sqrt{n})$, where the hidden logarithm factors in $\tildeΘ$ depend on h. Their proof does not give rise to an algorithm. When h is not a constant, $Dyck_h$ is not even context-free.…
We show that quantum query complexity satisfies a strong direct product theorem. This means that computing $k$ copies of a function with less than $k$ times the quantum queries needed to compute one copy of the function implies that the overall success probability will be exponentially small in $k$. For a boolean function $f$ we also show an XOR lemma---computing the parity of $k$ copies of $f$ with less than $k$ times the queries needed for one copy implies that the advantage over random guessing will be exponentially small. We do this by showing that the multiplicative adversary method, whic…
We study the query complexity of quantum learning problems in which the oracles form a group $G$ of unitary matrices. In the simplest case, one wishes to identify the oracle, and we find a description of the optimal success probability of a $t$-query quantum algorithm in terms of group characters. As an application, we show that $Ω(n)$ queries are required to identify a random permutation in $S_n$. More generally, suppose $H$ is a fixed subgroup of the group $G$ of oracles, and given access to an oracle sampled uniformly from $G$, we want to learn which coset of $H$ the oracle belongs to. We c…
Simon's problem asks the following: determine if a function $f: \{0,1\}^n \rightarrow \{0,1\}^n$ is one-to-one or if there exists a unique $s \in \{0,1\}^n$ such that $f(x) = f(x \oplus s)$ for all $x \in \{0,1\}^n$, given the promise that exactly one of the two holds. A classical algorithm that can solve this problem for every $f$ requires $2^{Ω(n)}$ queries to $f$. Simon showed that there is a quantum algorithm that can solve this promise problem for every $f$ using only $\mathcal O(n)$ quantum queries to $f$. A matching lower bound on the number of quantum queries was given by Koiran et al.…
Let $H$ be a fixed graph on $n$ vertices. Let $f_H(G) = 1$ iff the input graph $G$ on $n$ vertices contains $H$ as a (not necessarily induced) subgraph. Let $α_H$ denote the cardinality of a maximum independent set of $H$. In this paper we show: \[Q(f_H) = Ω\left(\sqrt{α_H \cdot n}\right),\] where $Q(f_H)$ denotes the quantum query complexity of $f_H$. As a consequence we obtain a lower bounds for $Q(f_H)$ in terms of several other parameters of $H$ such as the average degree, minimum vertex cover, chromatic number, and the critical probability. We also use the above bound to show that $Q(f_H)…
The negative weight adversary method, $\mathrm{ADV}^\pm(g)$, is known to characterize the bounded-error quantum query complexity of any Boolean function $g$, and also obeys a perfect composition theorem $\mathrm{ADV}^\pm(f \circ g^n) = \mathrm{ADV}^\pm(f) \mathrm{ADV}^\pm(g)$. Belovs gave a modified version of the negative weight adversary method, $\mathrm{ADV}_{rel}^\pm(f)$, that characterizes the bounded-error quantum query complexity of a relation $f \subseteq \{0,1\}^n \times [K]$, provided the relation is efficiently verifiable. A relation is efficiently verifiable if $\mathrm{ADV}^\pm(f_…
We show that for a relation $f\subseteq \{0,1\}^n\times \mathcal{O}$ and a function $g:\{0,1\}^{m}\times \{0,1\}^{m} \rightarrow \{0,1\}$ (with $m= O(\log n)$), $$\mathrm{R}_{1/3}(f\circ g^n) = Ω\left(\mathrm{R}_{1/3}(f) \cdot \left(\log\frac{1}{\mathrm{disc}(M_g)} - O(\log n)\right)\right),$$ where $f\circ g^n$ represents the composition of $f$ and $g^n$, $M_g$ is the sign matrix for $g$, $\mathrm{disc}(M_g)$ is the discrepancy of $M_g$ under the uniform distribution and $\mathrm{R}_{1/3}(f)$ ($\mathrm{R}_{1/3}(f\circ g^n)$) denotes the randomized query complexity of $f$ (randomized communica…
A quantum algorithm is exact if it always produces the correct answer, on any input. Coming up with exact quantum algorithms that substantially outperform the best classical algorithm has been a quite challenging task. In this paper, we present two new exact quantum algorithms for natural problems: 1) for the problem EXACT_k^n in which we have to determine whether the sequence of input bits x_1, ..., x_n contains exactly k values x_i=1; 2) for the problem THRESHOLD_k^n in which we have to determine if at least k of n input bits are equal to 1.
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 …
Quantum algorithm is a key tool for cryptanalysis. At present, people are committed to building powerful quantum algorithms and tapping the potential of quantum algorithms, so as to further analyze the security of cryptographic algorithms under quantum computing. Recombining classical quantum algorithms is one of the current ideas to construct quantum algorithms. However, they cannot be easily combined, the feasibility of quantum algorithms needs further analysis in quantum environment. This paper reanalyzes the existing combined algorithm Grover-meets-Simon algorithm in terms of the principle…
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 $O(n^{1+{1\over 2}})$.
We give and prove an optimal exact quantum query algorithm with complexity $k+1$ for computing the promise problem (i.e., symmetric and partial Boolean function) $DJ_n^k$ defined as: $DJ_n^k(x)=1$ for $|x|=n/2$, $DJ_n^k(x)=0$ for $|x|$ in the set $\{0, 1,\ldots, k, n-k, n-k+1,\ldots,n\}$, and it is undefined for the rest cases, where $n$ is even, $|x|$ is the Hamming weight of $x$. The case of $k=0$ is the well-known Deutsch-Jozsa problem. We outline all symmetric (and partial) Boolean functions with degrees 1 and 2, and prove their exact quantum query complexity. Then we prove that any symmet…
The quantum query models is one of the most important models in quantum computing. Several well-known quantum algorithms are captured by this model, including the Deutsch-Jozsa algorithm, the Simon algorithm, the Grover algorithm and others. In this paper, we characterize the computational power of exact one-query quantum algorithms. It is proved that a total Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$ can be exactly computed by a one-query quantum algorithm if and only if $f(x)=x_{i_1}$ or ${x_{i_1} \oplus x_{i_2} }$ (up to isomorphism). Note that unlike most work in the literature bas…
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.
A long-standing aim of quantum information research is to understand what gives quantum computers their advantage. This requires separating problems that need genuinely quantum resources from those for which classical resources are enough. Two examples of quantum speed-up are the Deutsch-Jozsa and Simon's problem, both efficiently solvable on a quantum Turing machine, and both believed to lack efficient classical solutions. Here we present a framework that can simulate both quantum algorithms efficiently, solving the Deutsch-Jozsa problem with probability 1 using only one oracle query, and Sim…
