29,855 research outputs found
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 -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
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