Lower bounds in the quantum cell probe model

We introduce a new model for studying quantum data structure problems --- the "quantum cell probe model". We prove a lower bound for the static predecessor problem in the 'address-only' version of this model where, essentially, we allow quantum parallelism only over the 'address lines' of the queries. This model subsumes the classical cell probe model, and many quantum query algorithms like Grover's algorithm fall into this framework. We prove our lower bound by obtaining a round elimination lemma for quantum communication complexity. A similar lemma was proved by Miltersen, Nisan, Safra and Wigderson for classical communication complexity, but their proof does not generalise to the quantum setting. We also study the static membership problem in the quantum cell probe model. Generalising a result of Yao, we show that if the storage scheme is 'implicit', that is it can only store members of the subset and 'pointers', then any quantum query scheme must make \Omega(\log n) probes. We also consider the one-round quantum communication complexity of set membership and show tight bounds

Lower Bounds on Time-Space Trade-Offs for Approximate Near Neighbors

We show tight lower bounds for the entire trade-off between space and query time for the Approximate Near Neighbor search problem. Our lower bounds hold in a restricted model of computation, which captures all hashing-based approaches. In articular, our lower bound matches the upper bound recently shown in [Laarhoven 2015] for the random instance on a Euclidean sphere (which we show in fact extends to the entire space $\mathbb{R}^d$ using the techniques from [Andoni, Razenshteyn 2015]). We also show tight, unconditional cell-probe lower bounds for one and two probes, improving upon the best known bounds from [Panigrahy, Talwar, Wieder 2010]. In particular, this is the first space lower bound (for any static data structure) for two probes which is not polynomially smaller than for one probe. To show the result for two probes, we establish and exploit a connection to locally-decodable codes.Comment: 47 pages, 2 figures; v2: substantially revised introduction, lots of small corrections; subsumed by arXiv:1608.03580 [cs.DS] (along with arXiv:1511.07527 [cs.DS]

Optimal Hashing-based Time-Space Trade-offs for Approximate Near Neighbors

[See the paper for the full abstract.] We show tight upper and lower bounds for time-space trade-offs for the $c$-Approximate Near Neighbor Search problem. For the $d$-dimensional Euclidean space and $n$-point datasets, we develop a data structure with space $n^{1 + \rho_u + o(1)} + O(dn)$ and query time $n^{\rho_q + o(1)} + d n^{o(1)}$ for every $\rho_u, \rho_q \geq 0$ such that: \begin{equation} c^2 \sqrt{\rho_q} + (c^2 - 1) \sqrt{\rho_u} = \sqrt{2c^2 - 1}. \end{equation} This is the first data structure that achieves sublinear query time and near-linear space for every approximation factor $c > 1$, improving upon [Kapralov, PODS 2015]. The data structure is a culmination of a long line of work on the problem for all space regimes; it builds on Spherical Locality-Sensitive Filtering [Becker, Ducas, Gama, Laarhoven, SODA 2016] and data-dependent hashing [Andoni, Indyk, Nguyen, Razenshteyn, SODA 2014] [Andoni, Razenshteyn, STOC 2015]. Our matching lower bounds are of two types: conditional and unconditional. First, we prove tightness of the whole above trade-off in a restricted model of computation, which captures all known hashing-based approaches. We then show unconditional cell-probe lower bounds for one and two probes that match the above trade-off for $\rho_q = 0$, improving upon the best known lower bounds from [Panigrahy, Talwar, Wieder, FOCS 2010]. In particular, this is the first space lower bound (for any static data structure) for two probes which is not polynomially smaller than the one-probe bound. To show the result for two probes, we establish and exploit a connection to locally-decodable codes.Comment: 62 pages, 5 figures; a merger of arXiv:1511.07527 [cs.DS] and arXiv:1605.02701 [cs.DS], which subsumes both of the preprints. New version contains more elaborated proofs and fixed some typo

Data Structures in Classical and Quantum Computing

This survey summarizes several results about quantum computing related to (mostly static) data structures. First, we describe classical data structures for the set membership and the predecessor search problems: Perfect Hash tables for set membership by Fredman, Koml\'{o}s and Szemer\'{e}di and a data structure by Beame and Fich for predecessor search. We also prove results about their space complexity (how many bits are required) and time complexity (how many bits have to be read to answer a query). After that, we turn our attention to classical data structures with quantum access. In the quantum access model, data is stored in classical bits, but they can be accessed in a quantum way: We may read several bits in superposition for unit cost. We give proofs for lower bounds in this setting that show that the classical data structures from the first section are, in some sense, asymptotically optimal - even in the quantum model. In fact, these proofs are simpler and give stronger results than previous proofs for the classical model of computation. The lower bound for set membership was proved by Radhakrishnan, Sen and Venkatesh and the result for the predecessor problem by Sen and Venkatesh. Finally, we examine fully quantum data structures. Instead of encoding the data in classical bits, we now encode it in qubits. We allow any unitary operation or measurement in order to answer queries. We describe one data structure by de Wolf for the set membership problem and also a general framework using fully quantum data structures in quantum walks by Jeffery, Kothari and Magniez

The Quantum Complexity of Set Membership

We study the quantum complexity of the static set membership problem: given a subset S (|S| \leq n) of a universe of size m (m \gg n), store it as a table of bits so that queries of the form `Is x \in S?' can be answered. The goal is to use a small table and yet answer queries using few bitprobes. This problem was considered recently by Buhrman, Miltersen, Radhakrishnan and Venkatesh, where lower and upper bounds were shown for this problem in the classical deterministic and randomized models. In this paper, we formulate this problem in the "quantum bitprobe model" and show tradeoff results between space and time.In this model, the storage scheme is classical but the query scheme is quantum.We show, roughly speaking, that similar lower bounds hold in the quantum model as in the classical model, which imply that the classical upper bounds are more or less tight even in the quantum case. Our lower bounds are proved using linear algebraic techniques.Comment: 19 pages, a preliminary version appeared in FOCS 2000. This is the journal version, which will appear in Algorithmica (Special issue on Quantum Computation and Quantum Cryptography). This version corrects some bugs in the parameters of some theorem

Error-Correcting Data Structures

We study data structures in the presence of adversarial noise. We want to encode a given object in a succinct data structure that enables us to efficiently answer specific queries about the object, even if the data structure has been corrupted by a constant fraction of errors. This new model is the common generalization of (static) data structures and locally decodable error-correcting codes. The main issue is the tradeoff between the space used by the data structure and the time (number of probes) needed to answer a query about the encoded object. We prove a number of upper and lower bounds on various natural error-correcting data structure problems. In particular, we show that the optimal length of error-correcting data structures for the Membership problem (where we want to store subsets of size s from a universe of size n) is closely related to the optimal length of locally decodable codes for s-bit strings.Comment: 15 pages LaTeX; an abridged version will appear in the Proceedings of the STACS 2009 conferenc

Minimum length uncertainty relations in the presence of dark energy

We introduce a dark energy-modified minimum length uncertainty relation (DE-MLUR) or dark energy uncertainty principle (DE-UP) for short. The new relation is structurally similar to the MLUR introduced by K{\' a}rolyh{\' a}zy (1968), and reproduced by Ng and van Dam (1994) using alternative arguments, but with a number of important differences. These include a dependence on the de Sitter horizon, which may be expressed in terms of the cosmological constant as $l_{\rm dS} \sim 1/\sqrt{\Lambda}$. Applying the DE-UP to both charged and neutral particles, we obtain estimates of two limiting mass scales, expressed in terms of the fundamental constants $\left\{G,c,\hbar,\Lambda, e\right\}$. Evaluated numerically, the charged particle limit corresponds to the order of magnitude value of the electron mass ($m_e$), while the neutral particle limit is consistent with current experimental bounds on the mass of the electron neutrino ($m_{\nu_e}$). Possible cosmological consequences of the DE-UP are considered and we note that these lead naturally to a holographic relation between the bulk and the boundary of the Universe. Low and high energy regimes in which dark energy effects may dominate canonical quantum behaviour are identified and the possibility of testing the model using near-future experiments is briefly discussed.Comment: 27 pages, 3 figures, 1 table, 1 appendix. Major revisions, invited contribution to the Galaxies special issue "The dark side of the Universe", T. Harko and F. Lobo eds. (v3). Published version, https://doi.org/10.3390/galaxies701001

A Lower Bound for Sampling Disjoint Sets

Suppose Alice and Bob each start with private randomness and no other input, and they wish to engage in a protocol in which Alice ends up with a set x subseteq[n] and Bob ends up with a set y subseteq[n], such that (x,y) is uniformly distributed over all pairs of disjoint sets. We prove that for some constant beta0 of the uniform distribution over all pairs of disjoint sets of size sqrt{n}