16,421 research outputs found
Black-box Hamiltonian simulation and unitary implementation
We present general methods for simulating black-box Hamiltonians using
quantum walks. These techniques have two main applications: simulating sparse
Hamiltonians and implementing black-box unitary operations. In particular, we
give the best known simulation of sparse Hamiltonians with constant precision.
Our method has complexity linear in both the sparseness D (the maximum number
of nonzero elements in a column) and the evolution time t, whereas previous
methods had complexity scaling as D^4 and were superlinear in t. We also
consider the task of implementing an arbitrary unitary operation given a
black-box description of its matrix elements. Whereas standard methods for
performing an explicitly specified N x N unitary operation use O(N^2)
elementary gates, we show that a black-box unitary can be performed with
bounded error using O(N^{2/3} (log log N)^{4/3}) queries to its matrix
elements. In fact, except for pathological cases, it appears that most
unitaries can be performed with only O(sqrt{N}) queries, which is optimal.Comment: 19 pages, 3 figures, minor correction
Maximum Inner-Product Search using Tree Data-structures
The problem of {\em efficiently} finding the best match for a query in a
given set with respect to the Euclidean distance or the cosine similarity has
been extensively studied in literature. However, a closely related problem of
efficiently finding the best match with respect to the inner product has never
been explored in the general setting to the best of our knowledge. In this
paper we consider this general problem and contrast it with the existing
best-match algorithms. First, we propose a general branch-and-bound algorithm
using a tree data structure. Subsequently, we present a dual-tree algorithm for
the case where there are multiple queries. Finally we present a new data
structure for increasing the efficiency of the dual-tree algorithm. These
branch-and-bound algorithms involve novel bounds suited for the purpose of
best-matching with inner products. We evaluate our proposed algorithms on a
variety of data sets from various applications, and exhibit up to five orders
of magnitude improvement in query time over the naive search technique.Comment: Under submission in KDD 201
Learning and Testing Variable Partitions
Let be a multivariate function from a product set to an
Abelian group . A -partition of with cost is a partition of
the set of variables into non-empty subsets such that is -close to
for some with
respect to a given error metric. We study algorithms for agnostically learning
partitions and testing -partitionability over various groups and error
metrics given query access to . In particular we show that
Given a function that has a -partition of cost , a partition
of cost can be learned in time
for any .
In contrast, for and learning a partition of cost is NP-hard.
When is real-valued and the error metric is the 2-norm, a
2-partition of cost can be learned in time
.
When is -valued and the error metric is Hamming
weight, -partitionability is testable with one-sided error and
non-adaptive queries. We also show that even
two-sided testers require queries when .
This work was motivated by reinforcement learning control tasks in which the
set of control variables can be partitioned. The partitioning reduces the task
into multiple lower-dimensional ones that are relatively easier to learn. Our
second algorithm empirically increases the scores attained over previous
heuristic partitioning methods applied in this context.Comment: Innovations in Theoretical Computer Science (ITCS) 202
Max-stable sketches: estimation of Lp-norms, dominance norms and point queries for non-negative signals
Max-stable random sketches can be computed efficiently on fast streaming
positive data sets by using only sequential access to the data. They can be
used to answer point and Lp-norm queries for the signal. There is an intriguing
connection between the so-called p-stable (or sum-stable) and the max-stable
sketches. Rigorous performance guarantees through error-probability estimates
are derived and the algorithmic implementation is discussed
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