arXiv subjects

Sevag Gharibian

Publications and source records attributed to Sevag Gharibian.

45 records · Page 3Linked to original sources

arXiv2013

Approximation, Proof Systems, and Correlations in a Quantum World

This thesis studies three topics in quantum computation and information: The approximability of quantum problems, quantum proof systems, and non-classical correlations in quantum systems. In the first area, we demonstrate a polynomial-time (classical) approximation algorithm for dense instances of the canonical QMA-complete quantum constraint satisfaction problem, the local Hamiltonian problem. In the opposite direction, we next introduce a quantum generalization of the polynomial-time hierarchy, and define problems which we prove are not only complete for the second level of this hierarchy, but are in fact hard to approximate. In the second area, we study variants of the interesting and stubbornly open question of whether a quantum proof system with multiple unentangled quantum provers is equal in expressive power to a proof system with a single quantum prover. Our results concern classes such as BellQMA(poly), and include a novel proof of perfect parallel repetition for SepQMA(m) based on cone programming duality. In the third area, we study non-classical quantum correlations beyond entanglement, often dubbed "non-classicality". Among our results are two novel schemes for quantifying non-classicality: The first proposes the new paradigm of exploiting local unitary operations to study non-classical correlations, and the second introduces a protocol through which non-classical correlations in a starting system can be "activated" into distillable entanglement with an ancilla system. An introduction to all required linear algebra and quantum mechanics is included.

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arXiv2012

Hardness of approximation for quantum problems

The polynomial hierarchy plays a central role in classical complexity theory. Here, we define a quantum generalization of the polynomial hierarchy, and initiate its study. We show that not only are there natural complete problems for the second level of this quantum hierarchy, but that these problems are in fact hard to approximate. Using these techniques, we also obtain hardness of approximation for the class QCMA. Our approach is based on the use of dispersers, and is inspired by the classical results of Umans regarding hardness of approximation for the second level of the classical polynomial hierarchy [Umans, FOCS 1999]. The problems for which we prove hardness of approximation for include, among others, a quantum version of the Succinct Set Cover problem, and a variant of the local Hamiltonian problem with hybrid classical-quantum ground states.

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arXiv2012

QMA variants with polynomially many provers

We study three variants of multi-prover quantum Merlin-Arthur proof systems. We first show that the class of problems that can be efficiently verified using polynomially many quantum proofs, each of logarithmic-size, is exactly MQA (also known as QCMA), the class of problems which can be efficiently verified via a classical proof and a quantum verifier. We then study the class BellQMA(poly), characterized by a verifier who first applies unentangled, nonadaptive measurements to each of the polynomially many proofs, followed by an arbitrary but efficient quantum verification circuit on the resulting measurement outcomes. We show that if the number of outcomes per nonadaptive measurement is a polynomially-bounded function, then the expressive power of the proof system is exactly QMA. Finally, we study a class equivalent to QMA(m), denoted SepQMA(m), where the verifier's measurement operator corresponding to outcome "accept" is a fully separable operator across the m quantum proofs. Using cone programming duality, we give an alternate proof of a result of Harrow and Montanaro [FOCS, pp. 633--642 (2010)] that shows a perfect parallel repetition theorem for SepQMA(m) for any m.

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arXiv2011

All non-classical correlations can be activated into distillable entanglement

We devise a protocol in which general non-classical multipartite correlations produce a physically relevant effect, leading to the creation of bipartite entanglement. In particular, we show that the relative entropy of quantumness, which measures all non-classical correlations among subsystems of a quantum system, is equivalent to and can be operationally interpreted as the minimum distillable entanglement generated between the system and local ancillae in our protocol. We emphasize the key role of state mixedness in maximizing non-classicality: Mixed entangled states can be arbitrarily more non-classical than separable and pure entangled states.

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arXiv2011

Characterizing quantumness via entanglement creation

In [M. Piani et al., arXiv:1103.4032 (2011)] an activation protocol was introduced which maps the general non-classical (multipartite) correlations between given systems into bipartite entanglement between the systems and local ancillae by means of a potentially highly entangling interaction. Here, we study how this activation protocol can be used to entangle the starting systems themselves via entanglement swapping through a measurement on the ancillae. Furthermore, we bound the relative entropy of quantumness (a naturally arising measure of non-classicality in the scheme of Piani et al. above) for a special class of separable states, the so-called classical-quantum states. In particular, we fully characterize the classical-quantum two-qubit states that are maximally non-classical.

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arXiv2011

Approximation algorithms for QMA-complete problems

Approximation algorithms for classical constraint satisfaction problems are one of the main research areas in theoretical computer science. Here we define a natural approximation version of the QMA-complete local Hamiltonian problem and initiate its study. We present two main results. The first shows that a non-trivial approximation ratio can be obtained in the class NP using product states. The second result (which builds on the first one), gives a polynomial time (classical) algorithm providing a similar approximation ratio for dense instances of the problem. The latter result is based on an adaptation of the "exhaustive sampling method" by Arora et al. [J. Comp. Sys. Sci. 58, p.193 (1999)] to the quantum setting, and might be of independent interest.

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arXiv2009

Strong NP-Hardness of the Quantum Separability Problem

Given the density matrix rho of a bipartite quantum state, the quantum separability problem asks whether rho is entangled or separable. In 2003, Gurvits showed that this problem is NP-hard if rho is located within an inverse exponential (with respect to dimension) distance from the border of the set of separable quantum states. In this paper, we extend this NP-hardness to an inverse polynomial distance from the separable set. The result follows from a simple combination of works by Gurvits, Ioannou, and Liu. We apply our result to show (1) an immediate lower bound on the maximum distance between a bound entangled state and the separable set (assuming P != NP), and (2) NP-hardness for the problem of determining whether a completely positive trace-preserving linear map is entanglement-breaking.

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arXiv2009

Signatures of non-classicality in mixed-state quantum computation

We investigate signatures of non-classicality in quantum states, in particular, those involved in the DQC1 model of mixed-state quantum computation [Phys. Rev. Lett. 81, 5672 (1998)]. To do so, we consider two known non-classicality criteria. The first quantifies disturbance of a quantum state under locally noneffective unitary operations (LNU), which are local unitaries acting invariantly on a subsystem. The second quantifies measurement induced disturbance (MID) in the eigenbasis of the reduced density matrices. We study the role of both figures of non-classicality in the exponential speedup of the DQC1 model and compare them vis-a-vis the interpretation provided in terms of quantum discord. In particular, we prove that a non-zero quantum discord implies a non-zero shift under LNUs. We also use the MID measure to study the locking of classical correlations [Phys. Rev. Lett. 92, 067902 (2004)] using two mutually unbiased bases (MUB). We find the MID measure to exactly correspond to the number of locked bits of correlation. For three or more MUBs, it predicts the possibility of superior locking effects.

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arXiv2008

On Global Effects Caused by Locally Noneffective Unitary Operations

Given a bipartite quantum state rho with subsystems A and B of arbitrary dimensions, we study the entanglement detecting capabilities of locally noneffective, or cyclic, unitary operations [L. B. Fu, Europhys. Lett., vol. 75, pp. 1-7, 2006]. Local cyclic unitaries have the special property that they leave their target subsystem invariant. We investigate the distance between rho and the global state after local application of such unitaries as a possible indicator of entanglement. To this end, we derive and discuss closed formulae for the maximal such distance achievable for three cases of interest: (pseudo)pure quantum states, Werner states, and two-qubit states. What makes this criterion interesting, as we show here, is that it surprisingly displays behavior similar to recent anomalies observed for non-locality measures in higher dimensions, as well as demonstrates an equivalence to the CHSH inequality for certain classes of two-qubit states. Yet, despite these similarities, the criterion is not itself a non-locality measure. We also consider entanglement detection in bound entangled states.

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