12,244 research outputs found
Hardness of approximation for quantum problems
The polynomial hierarchy plays a central role in classical complexity theory.
Here, we define a quantum generalization of the polynomial hierarchy, and
initiate its study. We show that not only are there natural complete problems
for the second level of this quantum hierarchy, but that these problems are in
fact hard to approximate. Using these techniques, we also obtain hardness of
approximation for the class QCMA. Our approach is based on the use of
dispersers, and is inspired by the classical results of Umans regarding
hardness of approximation for the second level of the classical polynomial
hierarchy [Umans, FOCS 1999]. The problems for which we prove hardness of
approximation for include, among others, a quantum version of the Succinct Set
Cover problem, and a variant of the local Hamiltonian problem with hybrid
classical-quantum ground states.Comment: 21 pages, 1 figure, extended abstract appeared in Proceedings of the
39th International Colloquium on Automata, Languages and Programming (ICALP),
pages 387-398, Springer, 201
BQP-completeness of Scattering in Scalar Quantum Field Theory
Recent work has shown that quantum computers can compute scattering
probabilities in massive quantum field theories, with a run time that is
polynomial in the number of particles, their energy, and the desired precision.
Here we study a closely related quantum field-theoretical problem: estimating
the vacuum-to-vacuum transition amplitude, in the presence of
spacetime-dependent classical sources, for a massive scalar field theory in
(1+1) dimensions. We show that this problem is BQP-hard; in other words, its
solution enables one to solve any problem that is solvable in polynomial time
by a quantum computer. Hence, the vacuum-to-vacuum amplitude cannot be
accurately estimated by any efficient classical algorithm, even if the field
theory is very weakly coupled, unless BQP=BPP. Furthermore, the corresponding
decision problem can be solved by a quantum computer in a time scaling
polynomially with the number of bits needed to specify the classical source
fields, and this problem is therefore BQP-complete. Our construction can be
regarded as an idealized architecture for a universal quantum computer in a
laboratory system described by massive phi^4 theory coupled to classical
spacetime-dependent sources.Comment: 41 pages, 7 figures. Corrected typo in foote
Limitations of semidefinite programs for separable states and entangled games
Semidefinite programs (SDPs) are a framework for exact or approximate
optimization that have widespread application in quantum information theory. We
introduce a new method for using reductions to construct integrality gaps for
SDPs. These are based on new limitations on the sum-of-squares (SoS) hierarchy
in approximating two particularly important sets in quantum information theory,
where previously no -round integrality gaps were known: the set of
separable (i.e. unentangled) states, or equivalently, the
norm of a matrix, and the set of quantum correlations; i.e. conditional
probability distributions achievable with local measurements on a shared
entangled state. In both cases no-go theorems were previously known based on
computational assumptions such as the Exponential Time Hypothesis (ETH) which
asserts that 3-SAT requires exponential time to solve. Our unconditional
results achieve the same parameters as all of these previous results (for
separable states) or as some of the previous results (for quantum
correlations). In some cases we can make use of the framework of
Lee-Raghavendra-Steurer (LRS) to establish integrality gaps for any SDP, not
only the SoS hierarchy. Our hardness result on separable states also yields a
dimension lower bound of approximate disentanglers, answering a question of
Watrous and Aaronson et al. These results can be viewed as limitations on the
monogamy principle, the PPT test, the ability of Tsirelson-type bounds to
restrict quantum correlations, as well as the SDP hierarchies of
Doherty-Parrilo-Spedalieri, Navascues-Pironio-Acin and Berta-Fawzi-Scholz.Comment: 47 pages. v2. small changes, fixes and clarifications. published
versio
Quantum Commuting Circuits and Complexity of Ising Partition Functions
Instantaneous quantum polynomial-time (IQP) computation is a class of quantum
computation consisting only of commuting two-qubit gates and is not universal
in the sense of standard quantum computation. Nevertheless, it has been shown
that if there is a classical algorithm that can simulate IQP efficiently, the
polynomial hierarchy (PH) collapses at the third level, which is highly
implausible. However, the origin of the classical intractability is still less
understood. Here we establish a relationship between IQP and computational
complexity of the partition functions of Ising models. We apply the established
relationship in two opposite directions. One direction is to find subclasses of
IQP that are classically efficiently simulatable in the strong sense, by using
exact solvability of certain types of Ising models. Another direction is
applying quantum computational complexity of IQP to investigate (im)possibility
of efficient classical approximations of Ising models with imaginary coupling
constants. Specifically, we show that there is no fully polynomial randomized
approximation scheme (FPRAS) for Ising models with almost all imaginary
coupling constants even on a planar graph of a bounded degree, unless the PH
collapses at the third level. Furthermore, we also show a multiplicative
approximation of such a class of Ising partition functions is at least as hard
as a multiplicative approximation for the output distribution of an arbitrary
quantum circuit.Comment: 36 pages, 5 figure
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