4,341 research outputs found
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
Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy
We consider quantum computations comprising only commuting gates, known as
IQP computations, and provide compelling evidence that the task of sampling
their output probability distributions is unlikely to be achievable by any
efficient classical means. More specifically we introduce the class post-IQP of
languages decided with bounded error by uniform families of IQP circuits with
post-selection, and prove first that post-IQP equals the classical class PP.
Using this result we show that if the output distributions of uniform IQP
circuit families could be classically efficiently sampled, even up to 41%
multiplicative error in the probabilities, then the infinite tower of classical
complexity classes known as the polynomial hierarchy, would collapse to its
third level. We mention some further results on the classical simulation
properties of IQP circuit families, in particular showing that if the output
distribution results from measurements on only O(log n) lines then it may in
fact be classically efficiently sampled.Comment: 13 page
General framework for quantum search algorithms
Grover's quantum search algorithm drives a quantum computer from a prepared
initial state to a desired final state by using selective transformations of
these states. Here, we analyze a framework when one of the selective
trasformations is replaced by a more general unitary transformation. Our
framework encapsulates several previous generalizations of the Grover's
algorithm. We show that the general quantum search algorithm can be improved by
controlling the transformations through an ancilla qubit. As a special case of
this improvement, we get a faster quantum algorithm for the two-dimensional
spatial search.Comment: revised versio
Unbounded-error One-way Classical and Quantum Communication Complexity
This paper studies the gap between quantum one-way communication complexity
and its classical counterpart , under the {\em unbounded-error}
setting, i.e., it is enough that the success probability is strictly greater
than 1/2. It is proved that for {\em any} (total or partial) Boolean function
, , i.e., the former is always exactly one half
as large as the latter. The result has an application to obtaining (again an
exact) bound for the existence of -QRAC which is the -qubit random
access coding that can recover any one of original bits with success
probability . We can prove that -QRAC exists if and only if
. Previously, only the construction of QRAC using one qubit,
the existence of -RAC, and the non-existence of
-QRAC were known.Comment: 9 pages. To appear in Proc. ICALP 200
Dynamic Binding Communication Mechanism
Shastri & Ajjanagadde have proposed a biologically plausible connectionist rule-based reasoning system (hereafter referred to as a knowledge base, or KB), that represents a dynamic binding as the simultaneous, or in-phase, activity of the appropriate nodes [9]. This paper makes the first attempt at designing a biologically plausible connectionist interface mechanism between 2 distinct phase-based KB, as the next step toward providing a computational account of common-sense reasoning. The Dynamic Binding Communication Mechanism (DBCM) extracts a dynamic binding from a source KB and incorporates the binding into a destination KB so that it is consistent with the knowledge already represented in the latter. DBCM consists of several distinct, special-purpose modules. The Binding Memory (BM) is made up of several identical banks of nodes. Each time a temporally-encoded dynamic binding is extracted from the source KB, it is transferred into one of the banks, where the binding is converted to a spatially-encoded representation. The Phase Database (PD) monitors the target KB and produces a phased output that is out of phase with all other nodes in the target KB. The Phase Allocator (PA) synthesizes information from the Phase Database and from the target KB to determine the phase in which to introduce the new dynamic binding into the target KB. In turn, the PA extracts a single binding from one of the banks in the BM and introduces it into the target KB. The interface also utilizes 2 searchlight mechanisms: the first governs which bank in the BM receives bindings; the second mediates between the active banks (those which are currently representing bindings), and the Phase Allocator
Computation with narrow CTCs
We examine some variants of computation with closed timelike curves (CTCs),
where various restrictions are imposed on the memory of the computer, and the
information carrying capacity and range of the CTC. We give full
characterizations of the classes of languages recognized by polynomial time
probabilistic and quantum computers that can send a single classical bit to
their own past. Such narrow CTCs are demonstrated to add the power of limited
nondeterminism to deterministic computers, and lead to exponential speedup in
constant-space probabilistic and quantum computation. We show that, given a
time machine with constant negative delay, one can implement CTC-based
computations without the need to know about the runtime beforehand.Comment: 16 pages. A few typo was correcte
Can closed timelike curves or nonlinear quantum mechanics improve quantum state discrimination or help solve hard problems?
We study the power of closed timelike curves (CTCs) and other nonlinear
extensions of quantum mechanics for distinguishing nonorthogonal states and
speeding up hard computations. If a CTC-assisted computer is presented with a
labeled mixture of states to be distinguished--the most natural formulation--we
show that the CTC is of no use. The apparent contradiction with recent claims
that CTC-assisted computers can perfectly distinguish nonorthogonal states is
resolved by noting that CTC-assisted evolution is nonlinear, so the output of
such a computer on a mixture of inputs is not a convex combination of its
output on the mixture's pure components. Similarly, it is not clear that CTC
assistance or nonlinear evolution help solve hard problems if computation is
defined as we recommend, as correctly evaluating a function on a labeled
mixture of orthogonal inputs.Comment: 4 pages, 3 figures. Final version. Added several references, updated
discussion and introduction. Figure 1(b) very much enhance
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