1,690 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
Computational Complexity for Physicists
These lecture notes are an informal introduction to the theory of
computational complexity and its links to quantum computing and statistical
mechanics.Comment: references updated, reprint available from
http://itp.nat.uni-magdeburg.de/~mertens/papers/complexity.shtm
On the Hardness of Almost All Subset Sum Problems by Ordinary Branch-and-Bound
Given positive integers , and a positive integer right
hand side , we consider the feasibility version of the subset sum
problem which is the problem of determining whether a subset of
adds up to . We show that if the right hand side
is chosen as for a constant and if the 's are independentand identically distributed from a
discrete uniform distribution taking values , then the probability that the instance of the subset sum problem
generated requires the creation of an exponential number of branch-and-bound
nodes when one branches on the individual variables in any order goes to as
goes to infinity.Comment: 5 page
Certification with an NP Oracle
In the certification problem, the algorithm is given a function with
certificate complexity and an input , and the goal is to find a
certificate of size for 's value at . This
problem is in , and assuming , is not in . Prior works, dating back to Valiant in
1984, have therefore sought to design efficient algorithms by imposing
assumptions on such as monotonicity.
Our first result is a algorithm for the general
problem. The key ingredient is a new notion of the balanced influence of
variables, a natural variant of influence that corrects for the bias of the
function. Balanced influences can be accurately estimated via uniform
generation, and classic algorithms are known for
the latter task.
We then consider certification with stricter instance-wise guarantees: for
each , find a certificate whose size scales with that of the smallest
certificate for . In sharp contrast with our first result, we show
that this problem is -hard even to approximate. We
obtain an optimal inapproximability ratio, adding to a small handful of
problems in the higher levels of the polynomial hierarchy for which optimal
inapproximability is known. Our proof involves the novel use of bit-fixing
dispersers for gap amplification.Comment: 25 pages, 2 figures, ITCS 202
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