A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.

Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer

All our former experience with application of quantum theory seems to say:
{\it what is predicted by quantum formalism must occur in laboratory} But the
essence of quantum formalism - entanglement, recognized by Einstein, Podolsky,
Rosen and Schr\"odinger - waited over 70 years to enter to laboratories as a
new resource as real as energy This holistic property of compound quantum systems, which involves
nonclassical correlations between subsystems, is a potential for many quantum
processes, including ``canonical'' ones: quantum cryptography, quantum
teleportation and dense coding However, it appeared that this new resource is
very complex and difficult to detect Being usually fragile to environment, it
is robust against conceptual and mathematical tools, the task of which is to
decipher its rich structure This article reviews basic aspects of entanglement including its
characterization, detection, distillation and quantifying In particular, the
authors discuss various manifestations of entanglement via Bell inequalities,
entropic inequalities, entanglement witnesses, quantum cryptography and point
out some interrelations They also discuss a basic role of entanglement in
quantum communication within distant labs paradigm and stress some
peculiarities such as irreversibility of entanglement manipulations including
its extremal form - bound entanglement phenomenon A basic role of entanglement
witnesses in detection of entanglement is emphasized

Quantum entanglement

In information processing, as in physics, our classical world view provides an incomplete approximation to an underlying quantum reality. Quantum effects like interference and entanglement play no direct role in conventional information processing, but they can--in principle now, but probably eventually in practice--be harnessed to break codes, create unbreakable codes, and speed up otherwise intractable computations.

Quantum information and computation

This beginning graduate textbook describes both recent achievements and classical results of computational complexity theory. Requiring essentially no background apart from mathematical maturity, the book can be used as a reference for self-study for anyone interested in complexity, including physicists, mathematicians, and other scientists, as well as a textbook for a variety of courses and seminars. More than 300 exercises are included with a selected hint set.

/pdf/computational-complexity-a-modern-approach-5gtk8urev9.pdf

Computational Complexity: A Modern Approach

A digital computer is generally believed to be an efficient universal computing device; that is, it is believed to be able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems that are generally thought to be hard on classical computers and that have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, for example, the number of digits of the integer to be factored.

/pdf/polynomial-time-algorithms-for-prime-factorization-and-rtdp7rr21z.pdf

Entanglement appears under two different forms in quantum theory, namely, as a property of states of joint systems and as a property of measurement eigenstates in joint measurements. By combining these two aspects of entanglement, it is possible to generate nonlocality between particles that never interacted, using the protocol of entanglement swapping. We show that even in the more constraining bilocal scenario where distant sources of particles are assumed to be independent, i.e., to share no prior randomness, entanglement swapping can be simulated classically with bounded communication, using only 9 bits in total. Our result thus provides an upper bound on the nonlocality of the entanglement swapping process.

/pdf/classical-simulation-of-entanglement-swapping-with-bounded-4qlql2wkvf.pdf

Classical Simulation of Entanglement Swapping with Bounded Communication

We investigate the structure of EXP-complete and hard sets under various kinds of reductions. In particular, we are interested in the way in which information that makes the set complete is stored in the set. We study for various types of reductions the question of whether the set difference A-S for a hard set A and a sparse set S is still hard. We also address the question of which complete sets A can be split into sets A1 and A2 such that $A\equiv^P_r A_1\equiv^P_r A_2$ for reduction type r, i.e., which complete sets are mitotic. We obtain both positive and negative answers to these questions depending on the reduction type and the structure of the sparse set.

/pdf/splittings-robustness-and-structure-of-complete-sets-rxomszmksj.pdf

Splittings, Robustness, and Structure of Complete Sets

We construct an oracle A such that PA = PA and NPA = EXPA: This relativized world has several amazing properties: The oracle A gives the first relativized world where one can solve satisfiability on formulae with at most one assignment yet P 6= NP. The oracle A is the first where PA = UPA 6= NPA = coNPA: The construction gives a much simpler proof than that of Fenner, Fortnow and Kurtz of a relativized world where all the NP-complete sets are polynomial-time isomorphic. It is the first such computable oracle. Relative to A we have a collapse of EXPA ZPPA PA/poly. We also create a different relativized world where there exists a set L in NP that is NP-complete under reductions that make one query to L but not complete under traditional many-one reductions. This contrasts with the result of Buhrman, Spaan and Torenvliet showing that these two completeness notions for NEXP coincide. Research done at Yale and the University of Maryland. Address: Elect Eng Comp Science, 19 Memorial Dr W Ste 2, Bethlehem PA 180153084, USA. Supported in part by the National Science Foundation under grants CCR-8958528, CCR-9415410 and CCR-9700417 and by NASA under grant NAG 52895. Email: beigel@eecs.lehigh.edu. http://www.eecs.lehigh.edu/ beigel/ xAddress: CWI, Kruislaan 413, 1098SJ Amsterdam, The Netherlands. Partially supported by the Dutch foundation for scientific research (NWO) by SION project 612-34-002, and by the European Union through NeuroCOLT ESPRIT Working Group Nr. 8556, and HC&M grant nr. ERB4050PL93-0516. Email: buhrman@cwi.nl. http://www.cwi.nl/ buhrman/ {Research done while on leave at CWI. Address: University of Chicago, Department of Computer Science, 1100 E. 58th. St., Chicago, IL 60637, USA. Supported in part by NSF grant CCR 92-53582, the Dutch Foundation for Scientific Research (NWO) and a Fulbright Scholar award. Email: fortnow@cs.uchicago.edu. http://www.cs.uchicago.edu/ fortnow

Might not be as easy as detecting unique solutions.

An Excursion to the Kolmogorov Random Strings

The quantum version of communication complexity allows the two communicating parties to exchange qubits and/or to make use of prior entanglement (shared EPR-pairs). Some lower bound techniques are available for qubit communication complexity, but except for the inner product function, no bounds are known for the model with unlimited prior entanglement. We show that the log-rank lower bound extends to the strongest model (qubit communication + unlimited prior entanglement). By relating the rank of the communication matrix to properties of polynomials, we are able to derive some strong bounds for exact protocols. In particular, we prove both the "log-rank conjecture" and the polynomial equivalence of quantum and classical communication complexity for various classes of functions. We also derive some weaker bounds for bounded-error quantum protocols.

Harry Buhrman

Papers

Classical Simulation of Entanglement Swapping with Bounded Communication

Splittings, Robustness, and Structure of Complete Sets

Might not be as easy as detecting unique solutions.

An Excursion to the Kolmogorov Random Strings

Communication Complexity Lower Bounds by Polynomials