11 research outputs found
Quantum walk sampling by growing seed sets
This work describes a new algorithm for creating a superposition over the edge set of a graph, encoding a quantum sample of the random walk stationary distribution. The algorithm requires a number of quantum walk steps scaling as Ă(m1/3ÎŽâ1/3), with m the number of edges and ÎŽ the random walk spectral gap. This improves on existing strategies by initially growing a classical seed set in the graph, from which a quantum walk is then run. The algorithm leads to a number of improvements: (i) it provides a new bound on the setup cost of quantum walk search algorithms, (ii) it yields a new algorithm for st-connectivity, and (iii) it allows to create a superposition over the isomorphisms of an n-node graph in time Ă(2n/3), surpassing the Ω(2n/2) barrier set by index erasure
(Quantum) complexity of testing signed graph clusterability
This study examines clusterability testing for a signed graph in the
bounded-degree model. Our contributions are two-fold. First, we provide a
quantum algorithm with query complexity for testing
clusterability, which yields a polynomial speedup over the best classical
clusterability tester known [arXiv:2102.07587]. Second, we prove an
classical query lower bound for testing
clusterability, which nearly matches the upper bound from [arXiv:2102.07587].
This settles the classical query complexity of clusterability testing, and it
shows that our quantum algorithm has an advantage over any classical algorithm
On the Power of Nonstandard Quantum Oracles
We study how the choices made when designing an oracle affect the complexity of quantum property testing problems defined relative to this oracle. We encode a regular graph of even degree as an invertible function f, and present f in different oracle models. We first give a one-query QMA protocol to test if a graph encoded in f has a small disconnected subset. We then use representation theory to show that no classical witness can help a quantum verifier efficiently decide this problem relative to an in-place oracle. Perhaps surprisingly, a simple modification to the standard oracle prevents a quantum verifier from efficiently deciding this problem, even with access to an unbounded witness
A classical oracle separation between QMA and QCMA
It is a long-standing open question in quantum complexity theory whether the
definition of quantum computation requires quantum
witnesses or if classical witnesses suffice .
We make progress on this question by constructing a randomized classical oracle
separating the respective computational complexity classes. Previous
separations [Aaronson-Kuperberg (CCC'07), Fefferman-Kimmel (MFCS'18)] required
a quantum unitary oracle. The separating problem is deciding whether a
distribution supported on regular un-directed graphs either consists of
multiple connected components (yes instances) or consists of one expanding
connected component (no instances) where the graph is given in an
adjacency-list format by the oracle. Therefore, the oracle is a distribution
over -bit boolean functions.Comment: 43 page
A Survey of Quantum Property Testing
The area of property testing tries to design algorithms that can efficiently handle very large amounts of data: given a large object that either has a certain property or is somehow âfarâ from having that property, a tester should efficiently distinguish between these two cases. In this survey we describe recent results obtained for quantum property testing. This area naturally falls into three parts. First, we may consider quantum testers for properties of classical objects. We survey the main examples known where quantum testers can be much (sometimes exponentially) more efficient than classical testers. Second, we may consider classical testers of quantum objects. This is the situation that arises for instance when one is trying to determine if quantum states or operations do what they are supposed to do, based only on classical input-output behavior. Finally, we may also consider quantum testers for properties of quantum objects, such as states or operations. We survey known bounds on testing various natural properties, such as whether two states are equal, whether a state is separable, whether two operations commute, etc. We also highlight connections to other areas of quantum information theory and mention a number of open questions. Contents
A survey of quantum property testing
The area of property testing tries to design algorithms that can efficiently
handle very large amounts of data: given a large object that either has a
certain property or is somehow "far" from having that property, a tester should
efficiently distinguish between these two cases. In this survey we describe
recent results obtained for quantum property testing. This area naturally falls
into three parts. First, we may consider quantum testers for properties of
classical objects. We survey the main examples known where quantum testers can
be much (sometimes exponentially) more efficient than classical testers.
Second, we may consider classical testers of quantum objects. This is the
situation that arises for instance when one is trying to determine if quantum
states or operations do what they are supposed to do, based only on classical
input-output behavior. Finally, we may also consider quantum testers for
properties of quantum objects, such as states or operations. We survey known
bounds on te