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

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

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

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

An optimal quantum algorithm for the oracle identification problem

In the oracle identification problem, we are given oracle access to an unknown N-bit string x promised to belong to a known set C of size M and our task is to identify x. We present a quantum algorithm for the problem that is optimal in its dependence on N and M. Our algorithm considerably simplifies and improves the previous best algorithm due to Ambainis et al. Our algorithm also has applications in quantum learning theory, where it improves the complexity of exact learning with membership queries, resolving a conjecture of Hunziker et al. The algorithm is based on ideas from classical learning theory and a new composition theorem for solutions of the filtered $\gamma_2$-norm semidefinite program, which characterizes quantum query complexity. Our composition theorem is quite general and allows us to compose quantum algorithms with input-dependent query complexities without incurring a logarithmic overhead for error reduction. As an application of the composition theorem, we remove all log factors from the best known quantum algorithm for Boolean matrix multiplication.Comment: 16 pages; v2: minor change

Statistical Zero Knowledge and quantum one-way functions

One-way functions are a very important notion in the field of classical cryptography. Most examples of such functions, including factoring, discrete log or the RSA function, can be, however, inverted with the help of a quantum computer. In this paper, we study one-way functions that are hard to invert even by a quantum adversary and describe a set of problems which are good such candidates. These problems include Graph Non-Isomorphism, approximate Closest Lattice Vector and Group Non-Membership. More generally, we show that any hard instance of Circuit Quantum Sampling gives rise to a quantum one-way function. By the work of Aharonov and Ta-Shma, this implies that any language in Statistical Zero Knowledge which is hard-on-average for quantum computers, leads to a quantum one-way function. Moreover, extending the result of Impagliazzo and Luby to the quantum setting, we prove that quantum distributionally one-way functions are equivalent to quantum one-way functions. Last, we explore the connections between quantum one-way functions and the complexity class QMA and show that, similarly to the classical case, if any of the above candidate problems is QMA-complete then the existence of quantum one-way functions leads to the separation of QMA and AvgBQP.Comment: 20 pages; Computational Complexity, Cryptography and Quantum Physics; Published version, main results unchanged, presentation improve

Faithful Squashed Entanglement

Squashed entanglement is a measure for the entanglement of bipartite quantum states. In this paper we present a lower bound for squashed entanglement in terms of a distance to the set of separable states. This implies that squashed entanglement is faithful, that is, strictly positive if and only if the state is entangled. We derive the bound on squashed entanglement from a bound on quantum conditional mutual information, which is used to define squashed entanglement and corresponds to the amount by which strong subadditivity of von Neumann entropy fails to be saturated. Our result therefore sheds light on the structure of states that almost satisfy strong subadditivity with equality. The proof is based on two recent results from quantum information theory: the operational interpretation of the quantum mutual information as the optimal rate for state redistribution and the interpretation of the regularised relative entropy of entanglement as an error exponent in hypothesis testing. The distance to the set of separable states is measured by the one-way LOCC norm, an operationally-motivated norm giving the optimal probability of distinguishing two bipartite quantum states, each shared by two parties, using any protocol formed by local quantum operations and one-directional classical communication between the parties. A similar result for the Frobenius or Euclidean norm follows immediately. The result has two applications in complexity theory. The first is a quasipolynomial-time algorithm solving the weak membership problem for the set of separable states in one-way LOCC or Euclidean norm. The second concerns quantum Merlin-Arthur games. Here we show that multiple provers are not more powerful than a single prover when the verifier is restricted to one-way LOCC operations thereby providing a new characterisation of the complexity class QMA.Comment: 24 pages, 1 figure, 1 table. Due to an error in the published version, claims have been weakened from the LOCC norm to the one-way LOCC nor

The power of symmetric extensions for entanglement detection

In this paper, we present new progress on the study of the symmetric extension criterion for separability. First, we show that a perturbation of order O(1/N) is sufficient and, in general, necessary to destroy the entanglement of any state admitting an N Bose symmetric extension. On the other hand, the minimum amount of local noise necessary to induce separability on states arising from N Bose symmetric extensions with Positive Partial Transpose (PPT) decreases at least as fast as O(1/N^2). From these results, we derive upper bounds on the time and space complexity of the weak membership problem of separability when attacked via algorithms that search for PPT symmetric extensions. Finally, we show how to estimate the error we incur when we approximate the set of separable states by the set of (PPT) N -extendable quantum states in order to compute the maximum average fidelity in pure state estimation problems, the maximal output purity of quantum channels, and the geometric measure of entanglement.Comment: see Video Abstract at http://www.quantiki.org/video_abstracts/0906273