Universal optimal cloning of qubits and quantum registers

We review our recent work on the universal (i.e. input state independent) optimal quantum copying (cloning) of qubits. We present unitary transformations which describe the optimal cloning of a qubit and we present the corresponding quantum logical network. We also present network for an optimal quantum copying ``machine'' (transformation) which produces N+1 identical copies from the original qubit. Here again the quality (fidelity) of the copies does not depend on the state of the original and is only a function of the number of copies, N. In addition, we present the machine which universaly and optimally clones states of quantum objects in arbitrary-dimensional Hilbert spaces. In particular, we discuss universal cloning of quantum registers.Comment: To be presented at the First NASA Conference on Quantum Computing and Quantum Communications, 17-20 February 1998, Palm Springs, US

A cloned qutrit and its utility in information processing tasks

We, in this paper, analyze the efficacy of an output as a resource from a universal quantum cloning machine in information processing tasks such as teleportation and dense coding. For this, we have considered the $3\otimes 3$ dimensional system (or qutrit system). The output states are found to be NPT states for certain ranges of machine parameters. Using the output state as an entangled resource, we have also studied the optimal fidelities of teleportation and capacities of dense coding protocols with respect to the machine parameters and have made a few interesting observations. Our work is mainly motivated from the fact that the cloning output can be used as a resource in quantum information processing and adds a valuable dimension to the applications of cloning machines.Comment: 18 pages, 4 figures, (The arxiv version of this article have been modified and some typos present in the earlier version have been corrected

An algorithm for DNA read alignment on quantum accelerators

With small-scale quantum processors transitioning from experimental physics labs to industrial products, these processors allow us to efficiently compute important algorithms in various fields. In this paper, we propose a quantum algorithm to address the challenging field of big data processing for genome sequence reconstruction. This research describes an architecture-aware implementation of a quantum algorithm for sub-sequence alignment. A new algorithm named QiBAM (quantum indexed bidirectional associative memory) is proposed, that uses approximate pattern-matching based on Hamming distances. QiBAM extends the Grover's search algorithm in two ways to allow for: (1) approximate matches needed for read errors in genomics, and (2) a distributed search for multiple solutions over the quantum encoding of DNA sequences. This approach gives a quadratic speedup over the classical algorithm. A full implementation of the algorithm is provided and verified using the OpenQL compiler and QX simulator framework. This represents a first exploration towards a full-stack quantum accelerated genome sequencing pipeline design. The open-source implementation can be found on https://github.com/prince-ph0en1x/QAGS.Comment: Keywords: quantum algorithms, quantum search, DNA read alignment, genomics, associative memory, accelerators, in-memory computin

An Introduction to Quantum Game Theory

This essay gives a self-contained introduction to quantum game theory, and is primarily oriented to economists with little or no acquaintance with quantum mechanics. It assumes little more than a basic knowledge of vector algebra. Quantum mechanical notation and results are introduced as needed. It is also shown that some fundamental problems of quantum mechanics can be formulated as games.Comment: 69 page

Broadcasting of entanglement at a distance using linear optics and telecloning of entanglement

We propose a scheme for broadcasting entanglement at a distance based on linear optics. We show that an initial polarization entangled state can be simultaneously split and transmitted to a pair of observers situated at different locations with the help of two conditional Bell-state analyzers based on two beam splitters characterized by the same reflectivity R. In particular for R=1/3 the final states coincide with the output states obtained by the broadcasting protocol proposed by Buzek et al. [Phys. Rev. A 55, 3327 (1997)]. Further we present a different protocol called telecloning of entanglement, which combines the many-to-many teleportation and nonlocal optimal asymmetric cloning of an arbitrary entangled state. This scheme allows the optimal transmission of the two nonlocal optimal clones of an entangled state to two pairs of spatially separated receivers.Comment: presented as poster at the Conference Quantum Optics VI, KRYNICA, Poland, June 200

Universality and optimality of programmable quantum processors

We analyze and compare the optimality of approximate and probabilistic universal programmable quantum processors. We define several characteristics how to quantify the optimality and we study in detail performance of three types of programmable quantum processors based on (1) the C-NOT gate, (2) the SWAP operation, and (3) the model of the quantum information distributor - the QID processor. We show under which conditions the measurement assisted QID processor is optimal. We also investigate optimality of the so-called U-processors and we also compare the optimal approximative implementation of U(1) qubit rotations with the known probabilistic implementation as introduced by Vidal, Masanes and Cirac [ {\em Phys. Rev. Lett.} {\bf 88}, 047905 (2002)].Comment: 9 page

An Introduction to Quantum Computing, Without the Physics

This paper is a gentle but rigorous introduction to quantum computing intended for discrete mathematicians. Starting from a small set of assumptions on the behavior of quantum computing devices, we analyze their main characteristics, stressing the differences with classical computers, and finally describe two well-known algorithms (Simon's algorithm and Grover's algorithm) using the formalism developed in previous sections. This paper does not touch on the physics of the devices, and therefore does not require any notion of quantum mechanics. Numerical examples on an implementation of Grover's algorithm using open-source software are provided.Comment: v5 simplifies a proo

Asymmetric quantum cloning machines in any dimension

A family of asymmetric cloning machines for $N$-dimensional quantum states is introduced. These machines produce two imperfect copies of a single state that emerge from two distinct Heisenberg channels. The tradeoff between the quality of these copies is shown to result from a complementarity akin to Heisenberg uncertainty principle. A no-cloning inequality is derived for isotropic cloners: if $\pi_a$ and $\pi_b$ are the depolarizing fractions associated with the two copies, the domain in $(\sqrt{\pi_a},\sqrt{\pi_b})$-space located inside a particular ellipse representing close-to-perfect cloning is forbidden. More generally, a no-cloning uncertainty relation is discussed, quantifying the impossibility of copying imposed by quantum mechanics. Finally, an asymmetric Pauli cloning machine is defined that makes two approximate copies of a quantum bit, while the input-to-output operation underlying each copy is a (distinct) Pauli channel. The class of symmetric Pauli cloning machines is shown to provide an upper bound on the quantum capacity of the Pauli channel of probabilities $p_x$, $p_y$ and $p_z$.Comment: 18 pages RevTeX, 3 Postscript figures; new discussion on no-cloning uncertainty relations, several corrections, added reference

Streaming universal distortion-free entanglement concentration

This paper presents a streaming (sequential) protocol for universal entanglement concentration at the Shannon bound. Alice and Bob begin with N identical (but unknown) two-qubit pure states, each containing E ebits of entanglement. They each run a reversible algorithm on their qubits, and end up with Y perfect EPR pairs, where Y = NE +- O(\sqrt N). Our protocol is streaming, so the N input systems are fed in one at a time, and perfect EPR pairs start popping out almost immediately. It matches the optimal block protocol exactly at each stage, so the average yield after n inputs is = nE - O(log n). So, somewhat surprisingly, there is no tradeoff between yield and lag -- our protocol optimizes both. In contrast, the optimal N-qubit block protocol achieves the same yield, but since no EPR pairs are produced until the entire input block is read, its lag is O(N). Finally, our algorithm runs in O(log N) space, so a lot of entanglement can be efficiently concentrated using a very small (e.g., current or near-future technology) quantum processor. Along the way, we find an optimal streaming protocol for extracting randomness from classical i.i.d. sources and a more space-efficient implementation of the Schur transform.Comment: 16 page

Optical Quantum Cloning - a Review

After a brief introduction to the quantum no-cloning theorem and its link with the linearity and causality of quantum mechanics, the concept of quantum cloning machines is sketched, following, whenever possible, the chronology of the main results. The important classes of quantum cloning machines are reviewed, in particular state-independent and state-dependent cloning machines. The 1-to-2 cloning problem is then studied from a formal point of view, using the isomorphism between completely positive maps and operators, which leads to the so-called double-Bell ansatz. This also yields an efficient numerical approach to quantum cloning, based on semidefinite programming methods. The derivation of the optimal N-to-M universal cloning machine in d dimensions is then detailed, as well as the notion of asymmetric cloning machines. In the second part of this review, the optical implementation of cloning machines is considered. It is shown that the universal cloning of photons can be achieved by parametric amplification of light or by symmetrization via the Hong-Ou-Mandel effect. The various experimental demonstrations of quantum cloning machines are reviewed. The cloning of orthogonally polarized photons is also considered, as well as the asymmetric and phase-covariant cloning of photons. Finally, the extension of quantum cloning to continuous variables is analyzed. The optimal cloning of coherent states of light by phase-insensitive amplification is explained, as well as the experimental realization of continuous-variable quantum cloning with linear optics, measurement, and feed-forward operations.Comment: 76 pages, 14 figures, invited contribution to Progress in Optics, Ed. E. Wolf, in pres