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.

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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

How much do we have to change a string to increase its Kolmogorov complexity? We show that we can increase the complexity of any non-random string of length n by flipping $O(\sqrt{n})$ bits and some strings require $\Omega(\sqrt{n})$ bit flips. For a given m, we also give bounds for increasing the complexity of a string by flipping m bits.

Increasing Kolmogorov Complexity

A survey of classical simulation methods for quantum computation

The goal of position-based cryptography is to use the geographical position of a party as its only “credential”. For example, one would like to send a message to a party at a geographical position pos with the guarantee that the party can decrypt the message only if he or she is physically present at pos. The general concept of position-based cryptography was introduced by Chandran, Goyal, Moriarty and Ostrovsky [1]. A central task in position-based cryptography is the problem of position-verification. We have a prover P at position pos, wishing to convince a set of verifiers V0, . . . , Vk (at different points in geographical space) that P is indeed at that position pos. The prover can run an interactive protocol with the verifiers in order to convince them. The main technique for such a protocol is known as distance bounding [2]. In this technique, a verifier sends a random nonce to P and measures the time taken for P to reply back with this value. Assuming that the speed of communication is bounded by the speed of light, this technique gives an upper bound on the distance of P from the verifier. The problem of secure position-verification has been studied before in the field of wireless security, and there have been several proposals for this task ([2–9]). However, [1] shows that there exists no protocol for secure position-verification that offers security in the presence of multiple colluding adversaries. In other words, the set of verifiers cannot distinguish between the case when they are interacting with an honest prover at pos and the case when they are interacting with multiple colluding dishonest provers, none of which is at position pos. The impossibility result of [1] relies heavily on the fact that an adversary can locally store all information he receives and at the same time share this information with other colluding adversaries, located elsewhere. Due to the no-cloning theorem, such a strategy will not work in the quantum setting, which opens the door to secure protocols that use quantum information. The quantum model was first studied by Kent et al. under the name of “quantum tagging” [10, 11]. Several schemes were developed [11–15] and proven later to be insecure. Finally in [16] it was shown that in general no unconditionally secure quantum position-verification scheme is possible. Any scheme can be broken using a double exponential amount of EPR pairs in the size of the messages of the protocol. Later, Beigi and König improved in [17] the double exponential dependence to single exponential making use of port-based teleportation [18, 19]. Due to the exponential overhead in EPR pairs, the general no-go theorem does not rule out the existence of quantum schemes that are secure for all practical purposes. Such schemes should have the property that the protocol, when followed honestly, is feasible, but cheating the protocol requires unrealistic amounts of resources, for example EPR pairs or time.

The Garden-Hose Game and Application to Position-Based Quantum Cryptography

Many computational problems are subject to a quantum speed-up: one might find that a problem having an O(n^3)-time or O(n^2)-time classic algorithm can be solved by a known O(n^1.5)-time or O(n)-time quantum algorithm. The question naturally arises: how much quantum speed-up is possible? 
The area of fine-grained complexity allows us to prove optimal lower-bounds on the complexity of various computational problems, based on the conjectured hardness of certain natural, well-studied problems. This theory has recently been extended to the quantum setting, in two independent papers by Buhrman, Patro, and Speelman (arXiv:1911.05686), and by Aaronson, Chia, Lin, Wang, and Zhang (arXiv:1911.01973). 
In this paper, we further extend the theory of fine-grained complexity to the quantum setting. A fundamental conjecture in the classical setting states that the 3SUM problem cannot be solved by (classical) algorithms in time O(n^{2-a}), for any a>0. We formulate an analogous conjecture, the Quantum-3SUM-Conjecture, which states that there exist no sublinear O(n^{1-b})-time quantum algorithms for the 3SUM problem. 
Based on the Quantum-3SUM-Conjecture, we show new lower-bounds on the time complexity of quantum algorithms for several computational problems. Most of our lower-bounds are optimal, in that they match known upper-bounds, and hence they imply tight limits on the quantum speedup that is possible for these problems.

Limits of quantum speed-ups for computational geometry and other problems: Fine-grained complexity via quantum walks

In this paper we consider a quantum version of Pascal's triangle. Pascal's triangle is a well-known triangular array of numbers and when these numbers are plotted modulo 2, a fractal known as the Sierpinski triangle appears. We first prove the appearance of more general fractals when Pascal's triangle is considered modulo prime powers. The numbers in Pascal's triangle can be obtained by scaling the probabilities of the simple symmetric random walk on the line. In this paper we consider a quantum version of Pascal's triangle by replacing the random walk by the quantum walk known as the Hadamard walk. We show that when the amplitudes of the Hadamard walk are scaled to become integers and plotted mo

Harry Buhrman

Papers

Increasing Kolmogorov Complexity

A survey of classical simulation methods for quantum computation

The Garden-Hose Game and Application to Position-Based Quantum Cryptography

Limits of quantum speed-ups for computational geometry and other problems: Fine-grained complexity via quantum walks

Quantum Pascal's triangle and Sierpinski's carpet