arXiv subjects

Amnon Ta-Shma

Publications and source records attributed to Amnon Ta-Shma.

15 recordsLinked to original sources

arXiv2018

On the Hardness of Satisfiability with Bounded Occurrences in the Polynomial-Time Hierarchy

$ \newcommand{\eps}ε \newcommand{\NP}{\mathsf{NP}} \newcommand{\YES}{\mathsf{YES}} \newcommand{\NO}{\mathsf{NO}} \newcommand{\myminus}{\text{-}}\newcommand{\Bsat}{\mathsf{B}} \newcommand{\threesat}{\rm{3}\myminus\mathsf{SAT}} \newcommand{\gapthreesat}{\mathsf{\forall\exists}\myminus{\rm{3}}\myminus\mathsf{SAT}} $In 1991, Papadimitriou and Yannakakis gave a reduction implying the $\NP$-hardness of approximating the problem $\threesat$ with bounded occurrences. Their reduction is based on expander graphs. We present an analogue of this result for the second level of the polynomial-time hierarchy based on superconcentrator graphs. This resolves an open question of Ko and Lin (1995) and should be useful in deriving inapproximability results in the polynomial-time hierarchy. More precisely, we show that given an instance of $\gapthreesat$ in which every variable occurs at most $\Bsat$ times (for some absolute constant $\Bsat$), it is $Π_2$-hard to distinguish between the following two cases: $\YES$ instances, in which for any assignment to the universal variables there exists an assignment to the existential variables that satisfies all the clauses, and $\NO$ instances in which there exists an assignment to the universal variables such that any assignment to the existential variables satisfies at most a $1-\eps$ fraction of the clauses. We also generalize this result to any level of the polynomial-time hierarchy.

cs.CC↗

arXiv2012

Limits of privacy amplification against non-signalling memory attacks

The task of privacy amplification, in which Alice holds some partially secret information with respect to an adversary Eve and wishes to distill it until it is completely secret, is known to be solvable almost optimally both in the classical and quantum world. Unfortunately, when considering an adversary who is only limited by non-signalling constraints such a statement cannot be made in general. We here prove that under the natural assumptions of time-ordered non-signalling system, which allow past subsystems to signal future subsystems (using the device's memory for example), super-polynomial privacy amplification by any hashing is impossible. This is in great relevance when considering practical device independent key distribution protocols which assume a super-quantum adversary.

quant-ph↗

arXiv2012

Towards the Impossibility of Non-Signalling Privacy Amplification from Time-Like Ordering Constraints

In the past few years there was a growing interest in proving the security of cryptographic protocols, such as key distribution protocols, from the sole assumption that the systems of Alice and Bob cannot signal to each other. This can be achieved by making sure that Alice and Bob perform their measurements in a space-like separated way (and therefore signalling is impossible according to the non-signalling postulate of relativity theory) or even by shielding their apparatus. Unfortunately, it was proven in [E. Haenggi, R. Renner, and S. Wolf. The impossibility of non-signaling privacy amplification] that, no matter what hash function we use, privacy amplification is impossible if we only impose non-signalling conditions between Alice and Bob and not within their systems. In this letter we reduce the gap between the assumptions of Haenggi et al. and the physical relevant assumptions, from an experimental point of view, which say that the systems can only signal forward in time within the systems of Alice and Bob. We consider a set of assumptions which is very close to the conditions above and prove that the impossibility result of Haenggi et al. still holds.

quant-ph↗

arXiv2011

Better short-seed quantum-proof extractors

We construct a strong extractor against quantum storage that works for every min-entropy $k$, has logarithmic seed length, and outputs $Ω(k)$ bits, provided that the quantum adversary has at most $βk$ qubits of memory, for any $β< \half$. The construction works by first condensing the source (with minimal entropy-loss) and then applying an extractor that works well against quantum adversaries when the source is close to uniform. We also obtain an improved construction of a strong quantum-proof extractor in the high min-entropy regime. Specifically, we construct an extractor that uses a logarithmic seed length and extracts $Ω(n)$ bits from any source over $\B^n$, provided that the min-entropy of the source conditioned on the quantum adversary's state is at least $(1-β) n$, for any $β< \half$.

quant-ph↗

arXiv2009

On the complexity of approximating the diamond norm

The diamond norm is a norm defined over the space of quantum transformations. This norm has a natural operational interpretation: it measures how well one can distinguish between two transformations by applying them to a state of arbitrarily large dimension. This interpretation makes this norm useful in the study of quantum interactive proof systems. In this note we exhibit an efficient algorithm for computing this norm using convex programming. Independently of us, Watrous recently showed a different algorithm to compute this norm. An immediate corollary of this algorithm is a slight simplification of the argument of Kitaev and Watrous [STOC 2000] that QIP is contained in EXP.

quant-ph↗

arXiv2009

Approximate quantum error correction for correlated noise

Most of the research done on quantum error correction studies an error model in which each qubit is affected by noise, independently of the other qubits. In this paper we study a different noise model -- one in which the noise may be correlated with the qubits it acts upon. We show both positive and negative results. On the one hand, we show controlled-X errors cannot be perfectly corrected, yet can be approximately corrected with sub-constant approximation error. On the other hand, we show that no non-trivial quantum error correcting code can approximately correct controlled phase error with sub-constant approximation error.

quant-ph↗

arXiv2008

Short seed extractors against quantum storage

Some, but not all, extractors resist adversaries with limited quantum storage. In this paper we show that Trevisan's extractor has this property, thereby showing an extractor against quantum storage with logarithmic seed length.

quant-ph↗

arXiv2007

An Explicit Construction of Quantum Expanders

Quantum expanders are a natural generalization of classical expanders. These objects were introduced and studied by Ben-Aroya and Ta-Shma and by Hastings. In this note we show how to construct explicit, constant-degree quantum expanders. The construction is essentially the classical Zig-Zag expander construction, applied to quantum expanders.

quant-ph↗

arXiv2007

On the power of quantum, one round, two prover interactive proof systems

We analyze quantum two prover one round interactive proof systems, in which noninteracting provers can share unlimited entanglement. The maximum acceptance probability is characterized as a superoperator norm. We get some partial results about the superoperator norm, and in particular we analyze the "rank one" case.

quant-ph↗

arXiv2007

Quantum expanders and the quantum entropy difference problem

We define quantum expanders in a natural way. We show that under certain conditions classical expander constructions generalize to the quantum setting, and in particular so does the Lubotzky, Philips and Sarnak construction of Ramanujan expanders from Cayley graphs of the group PGL. We show that this definition is exactly what is needed for characterizing the complexity of estimating quantum entropies.

quant-ph↗

arXiv2006

Interaction in Quantum Communication

In some scenarios there are ways of conveying information with many fewer, even exponentially fewer, qubits than possible classically. Moreover, some of these methods have a very simple structure--they involve only few message exchanges between the communicating parties. It is therefore natural to ask whether every classical protocol may be transformed to a ``simpler'' quantum protocol--one that has similar efficiency, but uses fewer message exchanges. We show that for any constant k, there is a problem such that its k+1 message classical communication complexity is exponentially smaller than its k message quantum communication complexity. This, in particular, proves a round hierarchy theorem for quantum communication complexity, and implies, via a simple reduction, an Omega(N^{1/k}) lower bound for k message quantum protocols for Set Disjointness for constant k. Enroute, we prove information-theoretic lemmas, and define a related measure of correlation, the informational distance, that we believe may be of significance in other contexts as well.

quant-ph↗

arXiv2003

Adiabatic Quantum State Generation and Statistical Zero Knowledge

The design of new quantum algorithms has proven to be an extremely difficult task. This paper considers a different approach to the problem, by studying the problem of 'quantum state generation'. This approach provides intriguing links between many different areas: quantum computation, adiabatic evolution, analysis of spectral gaps and groundstates of Hamiltonians, rapidly mixing Markov chains, the complexity class statistical zero knowledge, quantum random walks, and more. We first show that many natural candidates for quantum algorithms can be cast as a state generation problem. We define a paradigm for state generation, called 'adiabatic state generation' and develop tools for adiabatic state generation which include methods for implementing very general Hamiltonians and ways to guarantee non negligible spectral gaps. We use our tools to prove that adiabatic state generation is equivalent to state generation in the standard quantum computing model, and finally we show how to apply our techniques to generate interesting superpositions related to Markov chains.

quant-ph↗

arXiv2000

Interaction in Quantum Communication Complexity

One of the most intriguing facts about communication using quantum states is that these states cannot be used to transmit more classical bits than the number of qubits used, yet there are ways of conveying information with exponentially fewer qubits than possible classically. Moreover, these methods have a very simple structure---they involve little interaction between the communicating parties. We look more closely at the ways in which information encoded in quantum states may be manipulated, and consider the question as to whether every classical protocol may be transformed to a ``simpler'' quantum protocol of similar efficiency. By a simpler protocol, we mean a protocol that uses fewer message exchanges. We show that for any constant k, there is a problem such that its k+1 message classical communication complexity is exponentially smaller than its k message quantum communication complexity, thus answering the above question in the negative. Our result builds on two primitives, local transitions in bi-partite states (based on previous work) and average encoding which may be of significance in other applications as well.

quant-ph↗

arXiv2000

Quantum Bit Escrow

Unconditionally secure bit commitment and coin flipping are known to be impossible in the classical world. Bit commitment is known to be impossible also in the quantum world. We introduce a related new primitive - {\em quantum bit escrow}. In this primitive Alice commits to a bit $b$ to Bob. The commitment is {\em binding} in the sense that if Alice is asked to reveal the bit, Alice can not bias her commitment without having a good probability of being detected cheating. The commitment is {\em sealing} in the sense that if Bob learns information about the encoded bit, then if later on he is asked to prove he was playing honestly, he is detected cheating with a good probability. Rigorously proving the correctness of quantum cryptographic protocols has proved to be a difficult task. We develop techniques to prove quantitative statements about the binding and sealing properties of the quantum bit escrow protocol. A related primitive we construct is a quantum biased coin flipping protocol where no player can control the game, i.e., even an all-powerful cheating player must lose with some constant probability, which stands in sharp contrast to the classical world where such protocols are impossible.

quant-ph↗

arXiv1998

Dense Quantum Coding and a Lower Bound for 1-way Quantum Automata

We consider the possibility of encoding m classical bits into much fewer n quantum bits so that an arbitrary bit from the original m bits can be recovered with a good probability, and we show that non-trivial quantum encodings exist that have no classical counterparts. On the other hand, we show that quantum encodings cannot be much more succint as compared to classical encodings, and we provide a lower bound on such quantum encodings. Finally, using this lower bound, we prove an exponential lower bound on the size of 1-way quantum finite automata for a family of languages accepted by linear sized deterministic finite automata.

quant-ph↗