586 research outputs found
Improved Soundness for QMA with Multiple Provers
We present three contributions to the understanding of QMA with multiple
provers:
1) We give a tight soundness analysis of the protocol of [Blier and Tapp,
ICQNM '09], yielding a soundness gap Omega(1/N^2). Our improvement is achieved
without the use of an instance with a constant soundness gap (i.e., without
using a PCP).
2) We give a tight soundness analysis of the protocol of [Chen and Drucker,
ArXiV '10], thereby improving their result from a monolithic protocol where
Theta(sqrt(N)) provers are needed in order to have any soundness gap, to a
protocol with a smooth trade-off between the number of provers k and a
soundness gap Omega(k^2/N), as long as k>=Omega(log N). (And, when
k=Theta(sqrt(N)), we recover the original parameters of Chen and Drucker.)
3) We make progress towards an open question of [Aaronson et al., ToC '09]
about what kinds of NP-complete problems are amenable to sublinear
multiple-prover QMA protocols, by observing that a large class of such examples
can easily be derived from results already in the PCP literature - namely, at
least the languages recognized by a non-deterministic RAMs in quasilinear time.Comment: 24 pages; comments welcom
Largest separable balls around the maximally mixed bipartite quantum state
For finite-dimensional bipartite quantum systems, we find the exact size of
the largest balls, in spectral norms for , of
separable (unentangled) matrices around the identity matrix. This implies a
simple and intutively meaningful geometrical sufficient condition for
separability of bipartite density matrices: that their purity \tr \rho^2 not
be too large. Theoretical and experimental applications of these results
include algorithmic problems such as computing whether or not a state is
entangled, and practical ones such as obtaining information about the existence
or nature of entanglement in states reached by NMR quantum computation
implementations or other experimental situations.Comment: 7 pages, LaTeX. Motivation and verbal description of results and
their implications expanded and improved; one more proof included. This
version differs from the PRA version by the omission of some erroneous
sentences outside the theorems and proofs, which will be noted in an erratum
notice in PRA (and by minor notational differences
Quantum XOR Games
We introduce quantum XOR games, a model of two-player one-round games that
extends the model of XOR games by allowing the referee's questions to the
players to be quantum states. We give examples showing that quantum XOR games
exhibit a wide range of behaviors that are known not to exist for standard XOR
games, such as cases in which the use of entanglement leads to an arbitrarily
large advantage over the use of no entanglement. By invoking two deep
extensions of Grothendieck's inequality, we present an efficient algorithm that
gives a constant-factor approximation to the best performance players can
obtain in a given game, both in case they have no shared entanglement and in
case they share unlimited entanglement. As a byproduct of the algorithm we
prove some additional interesting properties of quantum XOR games, such as the
fact that sharing a maximally entangled state of arbitrary dimension gives only
a small advantage over having no entanglement at all.Comment: 43 page
Quantum de Finetti Theorems under Local Measurements with Applications
Quantum de Finetti theorems are a useful tool in the study of correlations in
quantum multipartite states. In this paper we prove two new quantum de Finetti
theorems, both showing that under tests formed by local measurements one can
get a much improved error dependence on the dimension of the subsystems. We
also obtain similar results for non-signaling probability distributions. We
give the following applications of the results:
We prove the optimality of the Chen-Drucker protocol for 3-SAT, under the
exponential time hypothesis.
We show that the maximum winning probability of free games can be estimated
in polynomial time by linear programming. We also show that 3-SAT with m
variables can be reduced to obtaining a constant error approximation of the
maximum winning probability under entangled strategies of O(m^{1/2})-player
one-round non-local games, in which the players communicate O(m^{1/2}) bits all
together.
We show that the optimization of certain polynomials over the hypersphere can
be performed in quasipolynomial time in the number of variables n by
considering O(log(n)) rounds of the Sum-of-Squares (Parrilo/Lasserre) hierarchy
of semidefinite programs. As an application to entanglement theory, we find a
quasipolynomial-time algorithm for deciding multipartite separability.
We consider a result due to Aaronson -- showing that given an unknown n qubit
state one can perform tomography that works well for most observables by
measuring only O(n) independent and identically distributed (i.i.d.) copies of
the state -- and relax the assumption of having i.i.d copies of the state to
merely the ability to select subsystems at random from a quantum multipartite
state.
The proofs of the new quantum de Finetti theorems are based on information
theory, in particular on the chain rule of mutual information.Comment: 39 pages, no figure. v2: changes to references and other minor
improvements. v3: added some explanations, mostly about Theorem 1 and
Conjecture 5. STOC version. v4, v5. small improvements and fixe
Testing product states, quantum Merlin-Arthur games and tensor optimisation
We give a test that can distinguish efficiently between product states of n
quantum systems and states which are far from product. If applied to a state
psi whose maximum overlap with a product state is 1-epsilon, the test passes
with probability 1-Theta(epsilon), regardless of n or the local dimensions of
the individual systems. The test uses two copies of psi. We prove correctness
of this test as a special case of a more general result regarding stability of
maximum output purity of the depolarising channel. A key application of the
test is to quantum Merlin-Arthur games with multiple Merlins, where we obtain
several structural results that had been previously conjectured, including the
fact that efficient soundness amplification is possible and that two Merlins
can simulate many Merlins: QMA(k)=QMA(2) for k>=2. Building on a previous
result of Aaronson et al, this implies that there is an efficient quantum
algorithm to verify 3-SAT with constant soundness, given two unentangled proofs
of O(sqrt(n) polylog(n)) qubits. We also show how QMA(2) with log-sized proofs
is equivalent to a large number of problems, some related to quantum
information (such as testing separability of mixed states) as well as problems
without any apparent connection to quantum mechanics (such as computing
injective tensor norms of 3-index tensors). As a consequence, we obtain many
hardness-of-approximation results, as well as potential algorithmic
applications of methods for approximating QMA(2) acceptance probabilities.
Finally, our test can also be used to construct an efficient test for
determining whether a unitary operator is a tensor product, which is a
generalisation of classical linearity testing.Comment: 44 pages, 1 figure, 7 appendices; v6: added references, rearranged
sections, added discussion of connections to classical CS. Final version to
appear in J of the AC
On the role of entanglement in quantum computational speed-up
For any quantum algorithm operating on pure states we prove that the presence
of multi-partite entanglement, with a number of parties that increases
unboundedly with input size, is necessary if the quantum algorithm is to offer
an exponential speed-up over classical computation. Furthermore we prove that
the algorithm can be classically efficiently simulated to within a prescribed
tolerance \eta even if a suitably small amount of global entanglement
(depending on \eta) is present. We explicitly identify the occurrence of
increasing multi-partite entanglement in Shor's algorithm. Our results do not
apply to quantum algorithms operating on mixed states in general and we discuss
the suggestion that an exponential computational speed-up might be possible
with mixed states in the total absence of entanglement. Finally, despite the
essential role of entanglement for pure state algorithms, we argue that it is
nevertheless misleading to view entanglement as a key resource for quantum
computational power.Comment: Main proofs simplified. A few further explanatory remarks added. 22
pages, plain late
Quantum Proofs
Quantum information and computation provide a fascinating twist on the notion
of proofs in computational complexity theory. For instance, one may consider a
quantum computational analogue of the complexity class \class{NP}, known as
QMA, in which a quantum state plays the role of a proof (also called a
certificate or witness), and is checked by a polynomial-time quantum
computation. For some problems, the fact that a quantum proof state could be a
superposition over exponentially many classical states appears to offer
computational advantages over classical proof strings. In the interactive proof
system setting, one may consider a verifier and one or more provers that
exchange and process quantum information rather than classical information
during an interaction for a given input string, giving rise to quantum
complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum
analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit
some properties from their classical counterparts, they also possess distinct
and uniquely quantum features that lead to an interesting landscape of
complexity classes based on variants of this model.
In this survey we provide an overview of many of the known results concerning
quantum proofs, computational models based on this concept, and properties of
the complexity classes they define. In particular, we discuss non-interactive
proofs and the complexity class QMA, single-prover quantum interactive proof
systems and the complexity class QIP, statistical zero-knowledge quantum
interactive proof systems and the complexity class \class{QSZK}, and
multiprover interactive proof systems and the complexity classes QMIP, QMIP*,
and MIP*.Comment: Survey published by NOW publisher
Dimension Independent Disentanglers from Unentanglement and Applications
Quantum entanglement is a key enabling ingredient in diverse applications.
However, the presence of unwanted adversarial entanglement also poses
challenges in many applications.
In this paper, we explore methods to "break" quantum entanglement.
Specifically, we construct a dimension-independent k-partite disentangler
(like) channel from bipartite unentangled input. We show: For every , there is an efficient channel such that for every bipartite separable
state , the output is
close to a k-partite separable state. Concretely, for some distribution
on states from , Moreover, . Without the
bipartite unentanglement assumption, the above bound is conjectured to be
impossible.
Leveraging our disentanglers, we show that unentangled quantum proofs of
almost general real amplitudes capture NEXP, greatly relaxing the nonnegative
amplitudes assumption in the recent work of QMA^+(2)=NEXP. Specifically, our
findings show that to capture NEXP, it suffices to have unentangled proofs of
the form where has non-negative amplitudes, only has negative amplitudes and with . Additionally, we present a protocol achieving an almost largest
possible gap before obtaining QMA^R(k)=NEXP$, namely, a 1/poly(n) additive
improvement to the gap results in this equality.Comment: 28 page
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