2,251 research outputs found
Oracles Are Subtle But Not Malicious
Theoretical computer scientists have been debating the role of oracles since
the 1970's. This paper illustrates both that oracles can give us nontrivial
insights about the barrier problems in circuit complexity, and that they need
not prevent us from trying to solve those problems.
First, we give an oracle relative to which PP has linear-sized circuits, by
proving a new lower bound for perceptrons and low- degree threshold
polynomials. This oracle settles a longstanding open question, and generalizes
earlier results due to Beigel and to Buhrman, Fortnow, and Thierauf. More
importantly, it implies the first nonrelativizing separation of "traditional"
complexity classes, as opposed to interactive proof classes such as MIP and
MA-EXP. For Vinodchandran showed, by a nonrelativizing argument, that PP does
not have circuits of size n^k for any fixed k. We present an alternative proof
of this fact, which shows that PP does not even have quantum circuits of size
n^k with quantum advice. To our knowledge, this is the first nontrivial lower
bound on quantum circuit size.
Second, we study a beautiful algorithm of Bshouty et al. for learning Boolean
circuits in ZPP^NP. We show that the NP queries in this algorithm cannot be
parallelized by any relativizing technique, by giving an oracle relative to
which ZPP^||NP and even BPP^||NP have linear-size circuits. On the other hand,
we also show that the NP queries could be parallelized if P=NP. Thus, classes
such as ZPP^||NP inhabit a "twilight zone," where we need to distinguish
between relativizing and black-box techniques. Our results on this subject have
implications for computational learning theory as well as for the circuit
minimization problem.Comment: 20 pages, 1 figur
A Full Characterization of Quantum Advice
We prove the following surprising result: given any quantum state rho on n
qubits, there exists a local Hamiltonian H on poly(n) qubits (e.g., a sum of
two-qubit interactions), such that any ground state of H can be used to
simulate rho on all quantum circuits of fixed polynomial size. In terms of
complexity classes, this implies that BQP/qpoly is contained in QMA/poly, which
supersedes the previous result of Aaronson that BQP/qpoly is contained in
PP/poly. Indeed, we can exactly characterize quantum advice, as equivalent in
power to untrusted quantum advice combined with trusted classical advice.
Proving our main result requires combining a large number of previous tools --
including a result of Alon et al. on learning of real-valued concept classes, a
result of Aaronson on the learnability of quantum states, and a result of
Aharonov and Regev on "QMA+ super-verifiers" -- and also creating some new
ones. The main new tool is a so-called majority-certificates lemma, which is
closely related to boosting in machine learning, and which seems likely to find
independent applications. In its simplest version, this lemma says the
following. Given any set S of Boolean functions on n variables, any function f
in S can be expressed as the pointwise majority of m=O(n) functions f1,...,fm
in S, such that each fi is the unique function in S compatible with O(log|S|)
input/output constraints.Comment: We fixed two significant issues: 1. The definition of YQP machines
needed to be changed to preserve our results. The revised definition is more
natural and has the same intuitive interpretation. 2. We needed properties of
Local Hamiltonian reductions going beyond those proved in previous works
(whose results we'd misstated). We now prove the needed properties. See p. 6
for more on both point
Quantum rejection sampling
Rejection sampling is a well-known method to sample from a target
distribution, given the ability to sample from a given distribution. The method
has been first formalized by von Neumann (1951) and has many applications in
classical computing. We define a quantum analogue of rejection sampling: given
a black box producing a coherent superposition of (possibly unknown) quantum
states with some amplitudes, the problem is to prepare a coherent superposition
of the same states, albeit with different target amplitudes. The main result of
this paper is a tight characterization of the query complexity of this quantum
state generation problem. We exhibit an algorithm, which we call quantum
rejection sampling, and analyze its cost using semidefinite programming. Our
proof of a matching lower bound is based on the automorphism principle which
allows to symmetrize any algorithm over the automorphism group of the problem.
Our main technical innovation is an extension of the automorphism principle to
continuous groups that arise for quantum state generation problems where the
oracle encodes unknown quantum states, instead of just classical data.
Furthermore, we illustrate how quantum rejection sampling may be used as a
primitive in designing quantum algorithms, by providing three different
applications. We first show that it was implicitly used in the quantum
algorithm for linear systems of equations by Harrow, Hassidim and Lloyd.
Secondly, we show that it can be used to speed up the main step in the quantum
Metropolis sampling algorithm by Temme et al.. Finally, we derive a new quantum
algorithm for the hidden shift problem of an arbitrary Boolean function and
relate its query complexity to "water-filling" of the Fourier spectrum.Comment: 19 pages, 5 figures, minor changes and a more compact style (to
appear in proceedings of ITCS 2012
AN ANALOGUE-DIGITAL CHURCH-TURING THESIS
info:eu-repo/semantics/publishedVersio
Oracles and query lower bounds in generalised probabilistic theories
We investigate the connection between interference and computational power
within the operationally defined framework of generalised probabilistic
theories. To compare the computational abilities of different theories within
this framework we show that any theory satisfying three natural physical
principles possess a well-defined oracle model. Indeed, we prove a subroutine
theorem for oracles in such theories which is a necessary condition for the
oracle to be well-defined. The three principles are: causality (roughly, no
signalling from the future), purification (each mixed state arises as the
marginal of a pure state of a larger system), and strong symmetry existence of
non-trivial reversible transformations). Sorkin has defined a hierarchy of
conceivable interference behaviours, where the order in the hierarchy
corresponds to the number of paths that have an irreducible interaction in a
multi-slit experiment. Given our oracle model, we show that if a classical
computer requires at least n queries to solve a learning problem, then the
corresponding lower bound in theories lying at the kth level of Sorkin's
hierarchy is n/k. Hence, lower bounds on the number of queries to a quantum
oracle needed to solve certain problems are not optimal in the space of all
generalised probabilistic theories, although it is not yet known whether the
optimal bounds are achievable in general. Hence searches for higher-order
interference are not only foundationally motivated, but constitute a search for
a computational resource beyond that offered by quantum computation.Comment: 17+7 pages. Comments Welcome. Published in special issue
"Foundational Aspects of Quantum Information" in Foundations of Physic
Classifying the computational power of stochastic physical oracles
Consider a computability and complexity theory in which the
classical set-theoretic oracle to a Turing machine is replaced by
a physical process, and oracle queries return measurements of
physical behaviour. The idea of such physical oracles is relevant
to many disparate situations, but research has focussed on physical
oracles that were classic deterministic experiments which
measure physical quantities. In this paper, we broaden the scope
of the theory of physical oracles by tackling non-deterministic
systems. We examine examples of three types of non-determinism,
namely systems that are: (1) physically nondeterministic,
as in quantum phenomena; (2) physically deterministic but
whose physical theory is non-deterministic, as in statistical mechanics;
and (3) physically deterministic but whose computational
theory is non-deterministic caused by error margins. Physical
oracles that have probabilistic theories we call stochastic
physical oracles. We propose a set SPO of axioms for a basic
form of stochastic oracles. We prove that Turing machines
equipped with a physical oracle satisfying the axioms SPO compute
precisely the non-uniform complexity class BPP//log* in
polynomial time. This result of BPP/log* is a computational
limit to a great range of classical and non-classical measurement,
and of analogue-digital computation in polynomial time under
general conditions.info:eu-repo/semantics/publishedVersio
Semantic Security and Indistinguishability in the Quantum World
At CRYPTO 2013, Boneh and Zhandry initiated the study of quantum-secure
encryption. They proposed first indistinguishability definitions for the
quantum world where the actual indistinguishability only holds for classical
messages, and they provide arguments why it might be hard to achieve a stronger
notion. In this work, we show that stronger notions are achievable, where the
indistinguishability holds for quantum superpositions of messages. We
investigate exhaustively the possibilities and subtle differences in defining
such a quantum indistinguishability notion for symmetric-key encryption
schemes. We justify our stronger definition by showing its equivalence to novel
quantum semantic-security notions that we introduce. Furthermore, we show that
our new security definitions cannot be achieved by a large class of ciphers --
those which are quasi-preserving the message length. On the other hand, we
provide a secure construction based on quantum-resistant pseudorandom
permutations; this construction can be used as a generic transformation for
turning a large class of encryption schemes into quantum indistinguishable and
hence quantum semantically secure ones. Moreover, our construction is the first
completely classical encryption scheme shown to be secure against an even
stronger notion of indistinguishability, which was previously known to be
achievable only by using quantum messages and arbitrary quantum encryption
circuits.Comment: 37 pages, 2 figure
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