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 same problem. Thus, in the nonadaptive setting, quantum algorithms can achieve at most a very limited speed-up over classical query algorithms.
We note that the question of putting lower bounds on nonadaptive quantum query algorithms has been studied previously. First, Zalka has obtained a tight lower bound on the nonadaptive quantum query complexity of the unordered search problem, which is a particular learning problem @cite_2 . Second, in @cite_13 , Nishimura and Yamakami give lower bounds on the nonadaptive quantum query complexity of a multiple-block variant of the ordered search problem. Finally, @cite_14 develop the weighted adversary argument of Ambainis @cite_12 to obtain lower bounds that are specific to the nonadaptive setting. Unlike the situation considered here, their bounds also apply to quantum algorithms for computing partial functions.
@cite_14 @cite_13 @cite_12 @cite_2
1888573181 2137637395 2770560274 2126338609
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, which is a natural generalization of the ordered search problem. 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 applied to the notions of computational complexity theory: quantum truth-table reducibility and quantum truth-table autoreducibility. We present two general methods for proving lower bounds on the query complexity of nonadaptive quantum algorithms. Both methods are based on the adversary method of Ambainis. We show that they yield optimal lower bounds for several natural problems, and we challenge the reader to determine the nonadaptive quantum query complexity of the ''1-to-1 versus 2-to-1'' problem and of Hidden Translation. In addition to the results presented at Wollic 2008 in the conference version of this paper, we show that the lower bound given by the second method is always at least as good (and sometimes better) as the lower bound given by the first method. We also compare these two quantum lower bounds to probabilistic lower bounds. 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 tight up to a sub-logarithmic factor. The extra factor comes from converting a continuous-time query algorithm into a discrete-query algorithm. We give a direct and simplified quantum algorithm based on the dual SDP, with a bounded-error query complexity that matches the general adversary bound. Therefore, the general adversary lower bound is tight; it is in fact an SDP for quantum query complexity. This implies that the quantum query complexity of the composition f(g,...,g) of two boolean functions f and g matches the product of the query complexities of f and g, without a logarithmic factor for error reduction. It further shows that span programs are equivalent to quantum query algorithms. The degree of a polynomial representing (or approximating) a function f is a lower bound for the quantum query complexity of 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 @W(M^1^.^3^2^1^...). This is the first superlinear separation between polynomial degree and quantum query complexity. The lower bound is shown by a generalized version of the quantum adversary method. Although a quantum state requires exponentially many classical bits to describe, the laws of quantum mechanics impose severe restrictions on how that state can be accessed. This paper shows in three settings that quantum messages have only limited advantages over classical ones. First, we show that BQP qpoly is contained in PP poly, where BQP qpoly is the class of problems solvable in quantum polynomial time, given a polynomial-size "quantum advice state" that depends only on the input length. This resolves a question of Buhrman, and means that we should not hope for an unrelativized separation between quantum and classical advice. Underlying our complexity result is a general new relation between deterministic and quantum one-way communication complexities, which applies to partial as well as total functions. Second, we construct an oracle relative to which NP is not contained in BQP qpoly. To do so, we use the polynomial method to give the first correct proof of a direct product theorem for quantum search. This theorem has other applications; for example, it can be used to fix a flawed result of Klauck about quantum time-space tradeoffs for sorting. Third, we introduce a new trace distance method for proving lower bounds on quantum one-way communication complexity. Using this method, we obtain optimal quantum lower bounds for two problems of Ambainis, for which no nontrivial lower bounds were previously known even for classical randomized protocols. 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 coincide with the general adversary lower bound. Second, we give a quantum algorithm for evaluating span programs with only a logarithmic query overhead on the witness size. The first result is motivated by a quantum algorithm for evaluating composed span programs. The algorithm is known to be optimal for evaluating a large class of formulas. The allowed gates include all constant-size functions for which there is an optimal span program. So far, good span programs have been found in an ad hoc manner, and the SDP automates this procedure. Surprisingly, the SDP's value equals the general adversary bound. A corollary is an optimal quantum algorithm for evaluating "balanced" formulas over any finite boolean gate set. The second result extends span programs' applicability beyond the formula-evaluation problem. We extend the analysis of the quantum algorithm for evaluating span programs. The previous analysis shows that a corresponding bipartite graph has a large spectral gap, but only works when applied to the composition of constant-size span programs. We show generally that properties of eigenvalue-zero eigenvectors in fact imply an "effective" spectral gap around zero. A strong universality result for span programs follows. A good quantum query algorithm for a problem implies a good span program, and vice versa. Although nearly tight, this equivalence is nontrivial. Span programs are a promising model for developing more quantum algorithms. A locally decodable code (LDC) encodes n-bit strings x in m-bit codewords C(x) in such a way that one can recover any bit xi from a corrupted codeword by querying only a few bits of that word. We use a quantum argument to prove that LDCs with 2 classical queries require exponential length: m = 2Ω(n). Previously, this was known only for linear codes (, in: Proceedings of 17th IEEE Conference on Computation Complexity, 2002, pp. 175-183). The proof proceeds by showing that a 2-query LDC can be decoded with a single quantum query, when defined in an appropriate sense. It goes on to establish an exponential lower bound on any 'l-query locally quantum-decodable code'. We extend our lower bounds to non-binary alphabets and also somewhat improve the polynomial lower bounds by Katz and Trevisan for LDCs with more than 2 queries. Furthermore, we show that q quantum queries allow more succinct LDCs than the best known LDCs with q classical queries. Finally, we give new classical lower bounds and quantum upper bounds for the setting of private information retrieval. In particular, we exhibit a quantum 2-server private information retrieval (PIR) scheme with O(n3 10) qubits of communication, beating the O(n1 3) bits of communication of the best known classical 2-server PIR. The quantum adversary method is one of the most successful techniques for proving lower bounds on quantum query complexity. It gives optimal lower bounds for many problems, has application to classical complexity in formula size lower bounds, and is versatile with equivalent formulations interms of weight schemes, eigen values, and Kolmogorov complexity. All these formulations rely on the principlethat if an algorithm successfully computes a function then, in particular, itis able to distinguish between inputs which map to different values. We present a stronger version of the adversary method which goes beyond this principle to make explicit use of the stronger condition that the algorithm actually computes the function. This new method, which we call ADV+-, has all the advantages ofthe old: it is a lower bound on bounded-error quantum query complexity, its square is a lower bound on formula size, and it behaves well with respect tofunction composition. Moreover ADV+- is always at least as large as the adversary method ADV, and we show an example of a monotone function forwhich ADV+-(f)=Omega(ADV(f)1.098). We also give examples showing that ADV+- does not face limitations of ADV like the certificate complexity barrier and the property testing barrier. We prove new lower bounds for bounded error quantum communication complexity. Our methods are based on the Fourier transform of the considered functions. First we generalize a method for proving classical communication complexity lower bounds developed by Raz to the quantum case. Applying this method we give an exponential separation between bounded error quantum communication complexity and nondeterministic quantum communication complexity. We develop several other lower bound methods based on the Fourier transform, notably showing that s (f) n , for the average sensitivity s (f) of a function f, yields a lower bound on the bounded error quantum communication complexity of f(x AND y XOR z), where x is a Boolean word held by Alice and y,z are Boolean words held by Bob. We then prove the first large lower bounds on the bounded error quantum communication complexity of functions, for which a polynomial quantum speedup is possible. For all the functions we investigate, the only previously applied general lower bound method based on discrepancy yields bounds that are O( n). The collision problem is to decide whether a function X: 1,..,n -> 1,..,n is one-to-one or two-to-one, given that one of these is the case. We show a lower bound of Theta(n^ 1 5 ) on the number of queries needed by a quantum computer to solve this problem with bounded error probability. The best known upper bound is O(n^ 1 3 ), but obtaining any lower bound better than Theta(1) was an open problem since 1997. Our proof uses the polynomial method augmented by some new ideas. We also give a lower bound of Theta(n^ 1 7 ) for the problem of deciding whether two sets are equal or disjoint on a constant fraction of elements. Finally we give implications of these results for quantum complexity theory. Given a function f as an oracle, the collision problem is to find two distinct indexes i and j such that f(i) = f(j), under the promise that such indexes exist. Since the security of many fundamental cryptographic primitives depends on the hardness of finding collisions, our lower bounds provide evidence for the existence of cryptographic primitives that are immune to quantum cryptanalysis. We prove that any quantum algorithm for finding a collision in an r-to-one function must evaluate the function Ω((n r)1 3) times, where n is the size of the domain and r|n. This matches an upper bound of Brassard, Hoyer, and Tapp. No lower bound better than constant was previously known. Our result also implies a quantum lower bound of Ω(n2 3) queries for the element distinctness problem, which is to determine whether n integers are all distinct. The best previous lower bound was Ω(sn) queries. In the k-junta testing problem, a tester has to efficiently decide whether a given function f: 0, 1 n → 0, 1 is a k-junta (i.e., depends on at most fc of its input bits) or is e-far from any k-junta. Our main result is a quantum algorithm for this problem with query complexity O([EQUATION]) and time complexity O(n[EQUATION]). This quadratically improves over the query complexity of the previous best quantum junta tester, due to Atici and Servedio. Our tester is based on a new quantum algorithm for a gapped version of the combinatorial group testing problem, with an up to quartic improvement over the query complexity of the best classical algorithm. For our upper bound on the time complexity we give a near-linear time implementation of a shallow variant of the quantum Fourier transform over the symmetric group, similar to the Schur-Weyl transform. We also prove a lower bound of Ω(k1 3) queries for junta-testing (for constant e). We consider the question of locality in distributed computing in the context of quantum information. Specifically, we focus on the round complexity of quantum distributed algorithms, with no bounds imposed on local computational power or on the bit size of messages. Linial's LOCAL model of a distributed system is augmented through two types of quantum extensions: (1) initialization of the system in a quantum entangled state, and or (2) application of quantum communication channels. For both types of extensions, we discuss proof-of-concept examples of distributed problems whose round complexity is in fact reduced through genuinely quantum effects. Nevertheless, we show that even such quantum variants of the LOCAL model have non-trivial limitations, captured by a very simple (purely probabilistic) notion which we call "physical locality" (φ-LOCAL). While this is strictly weaker than the "computational locality" of the classical LOCAL model, it nevertheless leads to a generic view-based analysis technique for constructing lower bounds on round complexity. It turns out that the best currently known lower time bounds for many distributed combinatorial optimization problems, such as Maximal Independent Set, bounds cannot be broken by applying quantum processing, in any conceivable way. The quantum adversary method is a very versatile method for proving lower bounds on quantum algorithms. It has many equivalent formulations, yields tight bounds for many computational problems, and has natural connections to classical lower bounds. One of its formulations is in terms of the spectral norm of some matrices. We define a weighted version of this spectral method and show that it possesses useful composition properties. The results generalize and unify previously known composition properties of adversary methods. We study a model of communication complexity that encompasses many well-studied problems, including classical and quantum communication complexity, the complexity of simulating distributions arising from bipartite measurements of shared quantum states, and XOR games. In this model, Alice gets an input x, Bob gets an input y, and their goal is to each produce an output a, b distributed according to some pre-specified joint distribution p(a, b|x, y). Our results apply to any non-signaling distribution, that is, those where Alice's marginal distribution does not depend on Bob's input, and vice versa. By taking a geometric view of the non-signaling distributions, we introduce a simple new technique based on affine combinations of lower-complexity distributions, and we give the first general technique to apply to all these settings, with elementary proofs and very intuitive interpretations. Specifically, we introduce two complexity measures, one which gives lower bounds on classical communication, and one for quantum communication. These measures can be expressed as convex optimization problems. We show that the dual formulations have a striking interpretation, since they coincide with maximum violations of Bell and Tsirelson inequalities. The dual expressions are closely related to the winning probability of XOR games. Despite their apparent simplicity, these lower bounds subsume many known communication complexity lower bound methods, most notably the recent lower bounds of Linial and Shraibman for the special case of Boolean functions. We show that as in the case of Boolean functions, the gap between the quantum and classical lower bounds is at most linear in the size of the support of the distribution, and does not depend on the size of the inputs. This translates into a bound on the gap between maximal Bell and Tsirelson inequality violations, which was previously known only for the case of distributions with Boolean outcomes and uniform marginals. It also allows us to show that for some distributions, information theoretic methods are necessary to prove strong lower bounds. Finally, we give an exponential upper bound on quantum and classical communication complexity in the simultaneous messages model, for any non-signaling distribution. One consequence of this is a simple proof that any quantum distribution can be approximated with a constant number of bits of communication. We introduce a new method to derive lower bounds on randomized and quantum communication complexity. Our method is based on factorization norms, a notion from Banach Space theory. This approach gives us access to several powerful tools from this area such as normed spaces duality and Grothendiek's inequality. This extends the arsenal of methods for deriving lower bounds in communication complexity. As we show, our method subsumes most of the previously known general approaches to lower bounds on communication complexity. Moreover, we extend all (but one) of these lower bounds to the realm of quantum communication complexity with entanglement. Our results also shed some light on the question how much communication can be saved by using entanglement. It is known that entanglement can save one of every two qubits, and examples for which this is tight are also known. It follows from our results that this bound on the saving in communication is tight almost always. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009 We prove new lower bounds for locally decodable codes and private information retrieval. We show that a 2-query LDC encoding n-bit strings over an l-bit alphabet, where the decoder only uses b bits of each queried position, needs code length @math Similarly, a 2-server PIR scheme with an n-bit database and t-bit queries, where the user only needs b bits from each of the two l-bit answers, unknown to the servers, satisfies @math This implies that several known PIR schemes are close to optimal. Our results generalize those of [8], who proved roughly the same bounds for linear LDCs and PIRs. Like earlier work by Kerenidis and de Wolf [12], our classical bounds are proved using quantum computational techniques. In particular, we give a tight analysis of how well a 2-input function can be computed from a quantum superposition of both inputs. We exploit a coding technique called position-based coding in the entanglement-assisted setting to establish lower bounds for error exponents, lower bounds on the second-order coding rate, and one-shot lower bounds, and we also demonstrate that position-based coding can be a powerful tool for analyzing other communication settings. In particular, we reduce the quantum simultaneous decoding conjecture for entanglement-assisted or unassisted communication over a quantum multiple access channel to open questions in multiple quantum hypothesis testing. We then determine an achievable rate region for entanglement-assisted or unassisted classical communication over a quantum multiple-access channel, when using a particular quantum simultaneous decoder. The achievable rate regions given in this latter case are generally suboptimal, involving differences of Renyi-two entropies and conditional quantum entropies. We develop a theory of the algorithmic information in bits contained in an individual pure quantum state. This extends classical Kolmogorov complexity to the quantum domain retaining classical descriptions. Quantum Kolmogorov complexity coincides with the classical Kolmogorov complexity on the classical domain. Quantum Kolmogorov complexity is upper-bounded and can be effectively approximated from above under certain conditions. With high probability, a quantum object is incompressible. Upper and lower bounds of the quantum complexity of multiple copies of individual pure quantum states are derived and may shed some light on the no-cloning properties of quantum states. In the quantum situation complexity is not subadditive. We discuss some relations with "no-cloning" and "approximate cloning" properties. Adaptivity is an important feature of data analysis---the choice of questions to ask about a dataset often depends on previous interactions with the same dataset. However, statistical validity is typically studied in a nonadaptive model, where all questions are specified before the dataset is drawn. Recent work by (STOC, 2015) and Hardt and Ullman (FOCS, 2014) initiated the formal study of this problem, and gave the first upper and lower bounds on the achievable generalization error for adaptive data analysis. Specifically, suppose there is an unknown distribution @math and a set of @math independent samples @math is drawn from @math . We seek an algorithm that, given @math as input, accurately answers a sequence of adaptively chosen queries about the unknown distribution @math . How many samples @math must we draw from the distribution, as a function of the type of queries, the number of queries, and the desired level of accuracy? In this work we make two new contributions: (i) We give upper bounds on the number of samples @math that are needed to answer statistical queries. The bounds improve and simplify the work of (STOC, 2015), and have been applied in subsequent work by those authors (Science, 2015, NIPS, 2015). (ii) We prove the first upper bounds on the number of samples required to answer more general families of queries. These include arbitrary low-sensitivity queries and an important class of optimization queries. As in , our algorithms are based on a connection with algorithmic stability in the form of differential privacy. We extend their work by giving a quantitatively optimal, more general, and simpler proof of their main theorem that stability implies low generalization error. We also study weaker stability guarantees such as bounded KL divergence and total variation distance.