http://www-anw.cs.umass.edu/~barto/courses/cs687/Sutton-Precup-Singh-AIJ99.pdf

Between MDPs and semi-MDPs: a framework for temporal abstraction in reinforcement learning

Usually, a proof of a theorem contains more knowledge than the mere fact that the theorem is true. For instance, to prove that a graph is Hamiltonian it suffices to exhibit a Hamiltonian tour in it; however, this seems to contain more knowledge than the single bit Hamiltonian/non-Hamiltonian.In this paper a computational complexity theory of the “knowledge” contained in a proof is developed. Zero-knowledge proofs are defined as those proofs that convey no additional knowledge other than the correctness of the proposition in question. Examples of zero-knowledge proof systems are given for the languages of quadratic residuosity and 'quadratic nonresiduosity. These are the first examples of zero-knowledge proofs for languages not known to be efficiently recognizable.

/pdf/the-knowledge-complexity-of-interactive-proof-systems-31apre0ecf.pdf

The knowledge complexity of interactive proof-systems

We initiate the cryptographic study of order-preserving symmetric encryption (OPE), a primitive suggested in the database community by Agrawal et al. (SIGMOD '04) for allowing efficient range queries on encrypted data. Interestingly, we first show that a straightforward relaxation of standard security notions for encryption such as indistinguishability against chosen-plaintext attack (IND-CPA) is unachievable by a practical OPE scheme. Instead, we propose a security notion in the spirit of pseudorandom functions (PRFs) and related primitives asking that an OPE scheme look "as-random-as-possible" subject to the order-preserving constraint. We then design an efficient OPE scheme and prove its security under our notion based on pseudorandomness of an underlying blockcipher. Our construction is based on a natural relation we uncover between a random order-preserving function and the hypergeometric probability distribution. In particular, it makes black-box use of an efficient sampling algorithm for the latter.

/pdf/order-preserving-symmetric-encryption-2xrb4ww6qk.pdf

Order-Preserving Symmetric Encryption

Journal of the ACM

The field of quantum algorithms aims to find ways to speed up the solution of computational problems by using a quantum computer. A key milestone in this field will be when a universal quantum computer performs a computational task that is beyond the capability of any classical computer, an event known as quantum supremacy. This would be easier to achieve experimentally than full-scale quantum computing, but involves new theoretical challenges. Here we present the leading proposals to achieve quantum supremacy, and discuss how we can reliably compare the power of a classical computer to the power of a quantum computer. Proposals for demonstrating quantum supremacy, when a quantum computer supersedes any possible classical computer at a specific task, are reviewed.

/pdf/quantum-computational-supremacy-44xs7qgoko.pdf

Quantum computational supremacy

We prove the following facts about the language recognition power of quantum Turing machines (QTMs) in the unbounded error setting: QTMs are strictly more powerful than probabilistic Turing machines for any common space bound s satisfying s(n)=o(loglogn). For ''one-way'' Turing machines, where the input tape head is not allowed to move left, the above result holds for s(n)=o(logn). We also give a characterization for the class of languages recognized with unbounded error by real-time quantum finite automata (QFAs) with restricted measurements. It turns out that these automata are equal in power to their probabilistic counterparts, and this fact does not change when the QFA model is augmented to allow general measurements and mixed states. Unlike the case with classical finite automata, when the QFA tape head is allowed to remain stationary in some steps, more languages become recognizable. We define and use a QTM model that generalizes the other variants introduced earlier in the study of quantum space complexity.

Unbounded-error quantum computation with small space bounds

We introduce a new model of two-way finite automaton, which is endowed with the capability of resetting the position of the tape head to the left end of the tape in a single move during the computation. Several variants of this model are examined, with the following results: The weakest known model of computation where quantum computers recognize more languages with bounded error than their classical counterparts is identified. We prove that two-way probabilistic and quantum finite automata (2PFAs and 2QFAs) can be considerably more concise than both their one-way versions (1PFAs and 1QFAs), and two-way nondeterministic finite automata (2NFAs). For this purpose, we demonstrate several infinite families of regular languages which can be recognized with some fixed probability greater than 1 2 by just tuning the transition amplitudes of a 2QFA (and, in one case, a 2PFA) with a constant number of states, whereas the sizes of the corresponding 1PFAs, 1QFAs and 2NFAs grow without bound. We also show that 2QFAs with mixed states can support highly efficient probability amplification.

https://hal.inria.fr/hal-00990436/document

Succinctness of two-way probabilistic and quantum finite automata

We present five examples where quantum finite automata (QFAs) outperform their classical counterparts. This may be useful as a relatively simple technique to introduce quantum computation concepts to computer scientists. We also describe a modern QFA model involving superoperators that is able to simulate all known QFA and classical finite automaton variants.

Quantum Finite Automata: A Modern Introduction

The nondeterministic quantum finite automaton (NQFA) is the only known case wherea one-way quantum finite automaton (QFA) model has been shown to be strictly superiorin terms of language recognition power to its probabilistic counterpart. We give acharacterization of the class of languages recognized by NQFAs, demonstrating that itis equal to the class of exclusive stochastic languages. We also characterize the class oflanguages that are recognized necessarily by two-sided error by QFAs. It is shown thatthese classes remain the same when the QFAs used in their definitions are replaced byseveral different model variants that have appeared in the literature. We prove severalclosure properties of the related classes. The ramifications of these results about classicaland quantum sublogarithmic space complexity classes are examined.

A. C. Cem Say

Papers

Unbounded-error quantum computation with small space bounds

Succinctness of two-way probabilistic and quantum finite automata

Quantum Finite Automata: A Modern Introduction

Languages recognized by nondeterministic quantum finite automata

Qualitative system identification: deriving structure from behavior