Fall 2004
We discuss a scheme for teleporting an arbitrary superposition of entangled Dicke states of any number of atoms (qubits) between two distant cavities. Our method relies on adiabatic passage using multi-atom dark states in each cavity, and a conditional detection of photons leaking out of both cavities.
Much recent progress has been made in the development of quantum computers. For solid-state designs, optically-addressed qubits are currently in the lead. In this talk I will discuss the status of the most promising of these optically-addressed solid-state designs, including a potential room-temperature solid-state quantum computer in diamond.
We begin with an abstract review of how quantum entanglement is applied to teleportation. Then we present the naive description of a teleportation paradox in the context of special relativity. Naturally, every paradox is based on a fallacy, but this one does not depend on the classical transmission of information. We examine the naive description more critically and discuss the mathematical modeling problems for quantum-mechanical measurement in the relativistic setting.
A simple encryption scheme is shown which may have the potential to provide an absolutely secure encryption with classical information. The Sender and Receiver use a one-time-used key which they randomly generate. They do not share this key with the other side or with anybody else and delete it after the use. The message is travelling between them three times and it is always encrypted differently. A simple operator formalism is shown to achieve the secure encryption. However, there is an open question. What are the proper operators? The answer is not known at the moment, thus the audience is invited to help to answer it.If the operators exist, quantum encryption may not be necessary. The good news is that there is a mechanical analogy of the method which is known for a long time.
The Discrete Fourier Transform can be defined on finite abelian groups and we are going to define it, in a similar manner, on finite-dimensional complex vector spaces. Our approach to Fourier analysis depends on a mixed-radix representation of integers and we use that representation to define special vectors, called characters. The characters form an orthonormal basis of the vector space, we call it Fourier basis, and Fourier transform is defined as a change of basis. Thus the quantum Fourier transform and Hadamard transform appear as special cases depending on the type of our Fourier basis.
We analyze entanglement of formation and negative domains of Wigner function in their dependence on the number of dimensions. Here we focus on s-waves, that is, wave functions that depend on a hyperradius. The special example of a Gaussian and a shell wave function allows us to perform the calculations analytically.
In Scully and Druhl (Phys. Rev. A 25(1982)), the authors showed an interesting concept of a quantum eraser. For example, the famous Young double-slit experiment demonstrates that a ``particle'' manifests wavelike behavior by displaying interference after passing both slits; but the interference disappears and a particle-like behavior is observed if the which-path information becomes available. What would happen if we erase the which-path information after the particle has passed through the slits? Would a quantum eraser process restore the fringes? Scully and Druhl answered the question affirmatively.Many experiments have confirmed the above idea. Garisto and Hardy (Phys. Rev. A 60(1999)) used a tag bit to describe this argument elegantly. Zubairy, Agarwal and Scully (Phys. Rev. A 70(2004)) used a cavity-QED design to illustrate this quantum disentanglement eraser; cf. a talk given by M.S. Zubairy in Spring Semester 2004 in the Quantum Computing Seminar.
In this talk, we want to clarify a few issues for the quantum disentanglement eraser. We will use quantum circuits to illustrate the design in lieu of any specific devices. This provides a somewhat more mathematical viewpoint of the quantum eraser.
The speaker is mostly aided by Dr. Zijian Diao in the preparation of this talk.
Cyclic codes are an important class of error correcting codes that are extensively studied in classical coding theory. In this talk we will present a general result that allows us to construct linear cyclic quantum codes. This enables us to construct nonbinary quantum BCH codes, thus allowing us to design codes with a desired error correcting capability. Time permitting we will illustrate this method by constructing a family of single error correcting codes which are perfect in the sense of meeting the sphere packing bound.The talk is based on joint work with Avanti Ketkar, Santosh Kumar, and Andreas Klappenecker.
In quantum computing, each 1-bit unitary operation is a matrix (or an operator) in SU(2). One may talk about shining laser pulses with certain phase angles theta in order to acheive such unitary operations. But, what is the actual geometrical meaning of angles theta in such theta-pulses? Physicists have a clear understanding of their interpretations whereas mathematicians seem to feel somewhat confused. In this talk, we will clarify such an issue by using the homomorphism from SU(2) to SO(3).This talk is prepared with the assistance of Mr. Zhigang Zhang.
A basic problem in quantum communication theory is to establish the existence or nonexistence of quantum error correcting codes with certain parameters. An [[n,k,d]]_q code is a quantum code that can encode the state of k quantum systems with q-levels into the state of n such systems such that all errors affecting less than d systems can be detected and all errors affecting floor(d-1/2) systems can be corrected.We derive a general method that allows us to find upper bounds on k for given length n and minimum distance d. We show that several known upper bounds can be derived as a consequence of our method. And we give a lower bound method that establishes the existence of stabilizer codes. Combining the two methods, we prove the existence of optimal quantum MDS codes for short lengths.
This is joint work with Pradeep Kiran Sarvepalli.
Spring 2004
Programs on a quantum computer are often expressed as a sequence of quantum gates. Feynman introduced a pictorial notation that is adopted in many papers of quantum computing. Drawing quantum circuits by hand is cumbersome, and modifying old figures is often nearly as time consuming as creating new figures. I wrote a software package qcg that allows a convenient textual specification of quantum circuits. It is possible to use the package in combination with typesetting software such as LaTeX or plain TeX.I will explain how the package is used, provide examples, and explain my design decisions. I will show how one can embed the text in a LaTeX document, so that it suffices to maintain a single document. The package is written in Metapost, a declarative language that deserves to be better known. The talk will be accessible for a general audience.
In this talk we shall describe the concept of a correlated emission laser in which the phase noise associated with natural linewidth is quenched. We shall then discuss how such systems can be used as the generator of macroscopic entanglement.
Evariste Galois related a lattice of field extensions with a lattice of automorphism groups. The connection bearing his name was so fruitful that this elegant theory is still the pinnacle of most courses in abstract algebra. Birkhoff and others generalized the notion of Galois connections to other data structures. These concepts occur today in a wide range of applications, including data mining and denotational semantics of programming languages.We establish a Galois connection between the lattice of quantum error correcting codes and a lattice of matrix groups. The interesting objects in Galois theory are the so-called closed objects. For example, in the traditional theory of fields and groups, the closed objects are the normal field extensions and the normal subgroups. We show that the closed objects in our Galois theory of quantum error correcting codes are the so-called stabilizer codes. Incidentally, stabilizer codes are the most widely studied class of quantum error correcting codes in the current literature. We develop the theory of nonbinary stabilizer codes within this elegant framework and prove numerous new results.
Several years ago a quantum computer was proposed based Very Large Scale Integration (VLSI) of single phosphorous atoms in silicon. Efforts to fabricate this computer have met with mixed results. I will describe how substituting nitrogen-vacancy (NV) color centers in diamond for P in Si can overcome the remaining technical barriers to the implementation of a VLSI quantum computer.
Algorithms for quantum computers are expressible in various equivalent forms, including unitary matrices and so-called quantum circuits. However, the matrix form is often unwieldy (e.g., a quantum computation on just 20 quantum bits requires matrices with over a trillion complex elements each). The same computation can be expressed much more compactly as a quantum circuit composed of elementary quantum gates such as the Hadamard gate; Pauli X, Y, and Z matrices; S; T; control-NOT; etc.QLS is a Windows-based application which allows the user to graphically create, edit, and simulate a quantum circuit. In addition to the elementary gates, quantum measurement, swap, arbitrary true/false controls, arbitrary function gates, and discrete Fourier transform gates are supported. The circuit may be executed, i.e., simulated, as a whole or in single-step mode. The resulting superimposed quantum states may be displayed as a list, a bar graph, or a "topographic map" which includes a graphical representation of amplitude and phase.
Steven Sensarn is a senior at Texas A&M University majoring in Electrical Engineering. This work is part of his Honors thesis research under the direction of Dr. Walter C. Daugherity, Department of Computer Science, Texas A&M University.
We give an introduction to nonbinary stabilizer codes. We discuss various different characterizations of such codes, explain numerous code constructions, and derive linear programming bounds on the achievable minimum distance.This is joint work with Andreas Klappenecker.
Quantum physics and information science are merged into quantum information science and making another revolution in science and technology. Quantum computers can do computations exponentially faster and larger than digital computers can handle. Quantum teleportation moves things around without continuously passing through space. Quantum cryptography provides absolute security to communications. Some basic ideas of quantum information science will be disscussed in this talk.Dr. Jaewan Kim is Professor at the Korea Institute for Advanced Study in Seoul.
Vandersypen et al. have factorized the number 15 successfully with a liquid NMR quantum computer. In this talk, we will begin with a quick review of Shor's algorithm and give a simple introduction to the circuit design. Then we will show in detail how the NMR computer accomplishes this task. We will discuss the realization of qubits, pulses, and the output of the experiment. The experiments differ from Shor's original algorithm, but some specific characteristics of NMR actually allowed to simplify the implementation. Two cases were tried in the experimental work: the so-called difficult case (a = 7), and the so-called easy case (a = 11). Both have succeeded.
Fall 2003
We review the development of continuum and kinetic models of vehicular traffic flow, and compare and contrast these development to analogous developments in fluid flow. The development of microscopic models of vehicular traffic flow is similarly reviewed, with some emphasis on the recent introduction of cellular automata (CA) models. Ongoing work relating to the application of a simple CA model to achieving some insight into the alleged "two-regime" phenomenon in traffic flow is briefly described.
C.S. Lent, W. Porod, et al., proposed a Quantum Dots Cellular Automata (QDCA) architecture for quantum computing. Binary information is encoded in the electric charge configuration of the quantum dot cells, where each dot cell consists of five quantum dots. Complex array of cells can then be formed to constitute basic gates as the building blocks for the computer. In this talk, we discuss part of the theory of QDCA related to the modeling of quantum dot cells and the formation of elementary gates.
The most popular class of quantum error correcting codes are stabilizer codes. Knill introduced in 1996 an apparent generalization of stabilizer codes. It was shown by Klappenecker and Rötteler that, surprisingly, many error models force Clifford codes to be stabilizer codes. As a result, only a single Clifford code was known to date that is not a stabilizer code. We derive a result which suggests that for most error models there exist an abundance of Clifford codes that are not equal to stabilizer codes.
Fundamental lower limits of power dissipation in general-purpose classical and quantum computation are studied and compared. Some conclusions and many questions arise.
A model of parallel optical computer for performing Fourier transform is presented. The computer operates in both entanlement and non-entanglement modes. The performance of these different working modes regarding information content, physical complexity, minimal power consumption, operating temperature and wavelength range is compared.
Conventional computational fluid dynamics (CFD) has been done using the Navier-Stokes equations which are a set of quasi-linear second-order PDE's that require a lot of compute time to solve many practical problems. A novel and innovative approach builds on Boltzmann's equation from which the Navier-Stokes can be derived. In this talk, I will discuss the origins of the Lattice-Boltzmann Method (LBM). The reasons for choosing this method are: it has a simple kinetic nature, (ii) it can easily incorporate complex geometries, (iii) it has several intrinsic physical advantages for performing large scale simulations and (iv) it is highly amenable to parallelization since local communication to the nearest neighbor is required as opposed to solving the Navier-Stokes equations where communication across the entire problem domain may be required. Other advantages of LBM include its simplicity, linearity and fewer computations per grid point. Our long-term objective is to compute the flows being investigated on the Earth simulator with LBM at much reduced computational expense. Even more challenging problems are possible with the coming advances in computing with LBM than with many currently popular CFD methods.
Quantum dots are one of the most promising candidates as quantum logic gates for the future quantum computer. In this talk we first introduce what quantum dots are and how they work. We formulate and derive the mathematical equations for the modeling of quantum dots based on the work of Burkard, Loss, and DiVincenzo.
Two observables are called complementary if precise knowledge of one of them implies that all possible outcomes are equally probable when measuring the other. This concept was introduced by Bohr in 1928. It finds practical applications in modern protocols of quantum cryptography.At least d+1 observables are required to determine the density matrix of an ensemble of d-dimensional quantum systems. It is possible to attain this lower bound with the help of pairwise complementary observables when d is a power of a prime. We use Weil sums to derive particularly lucid constructions of such sets of complementary observables. Numerous conjectures and open problems will be discussed, including two open problems for which I offer Challenges in Quantum Computing awards.
This is joint work with Martin Roetteler (University of Waterloo).
Prospects for using rare earth doped crystals to perform short-term demonstration experiments for quantum computing and storage are discussed. These materials have many similarities with liquid NMR except that they can be initialized optically with near 100% efficiency and so do not suffer from the limitations imposed by pseudovectors. The ability to manipulate spins optically has a number of implications for quantum storage and communication as well as computing.
So far, the most successful demonstration of quantum computation is done by NMR (nuclear magnetic resonance) experiments. In this talk, we will introduce the basic ideas of NMR, including the derivation of the Hamiltonian and the ensuing quantum gates. We then use a lattice-gas system as a model for the numerical computation of the diffusion equation as an example.This talk is part of a survey paper being prepared by Z. Zhang, G. Chen and P. Hemmer.
We shall discuss a possible experimental scheme of the Garisto-Hardy disentanglement eraser based on cavity QED system. This scheme can be used for a delayed choice quantum eraser. It also allows us to acquire single qubit control over teleportation of an arbitrary binary state.
Spring 2003
Several quantum and classical communications systems (e.g. quantum communications driven by memoryless sources, packet transmission over lossy networks, and multi-user communications by CDMA) deal with surpassingly similar problems of the mathematical frame theory for different reasons. We look into how some of the questions have been solved in different areas as well as the questions that are still unanswered. In particular, we discuss problems associated with quantum memoryless sources, and quantum measurements.Dr. Soljanin received her Ph.D. in 1994 from Texas A&M University. She is currently Member of Technical Staff at Bell-Labs, Lucent Technologies.
Slides of the talk.
Shor's algorithm for factorization requires the determination of the periodicity of a function. We shall discuss the possibility of doing so in cavity QED systems.
Schwinger introduced in 1960 the concept of mutually unbiased observables. A maximal set of such observables are ideal when determining the density matrix of a quantum system. The eigenbases corresponding to these observables are called mutually unbiased bases. The mutually unbiased bases are important primitives in quantum information processing. For instance, they are used in quantum key distribution protocols, and in the solution of the mean king's problem.In d dimensions, there can be at most d+1 mutually unbiased bases. It was shown in 1989 by Wootters and Fields that if d is a prime power, then this upper bound is attained. In all other dimensions, the construction of a maximal set of mutually unbiased bases remained elusive. The existence question of maximal sets of unbiased bases is one of the most profound open problems in quantum information processing.
I will show in this talk that maximal sets of mutually unbiased bases correspond to certain affine 2-designs in complex projective space CPd-1, and vice versa. The relation between these two hitherto unrelated areas is enormously fruitful, and some combinatorial conjectures suggest that maximal sets of mutually unbiased bases do not exist unless the dimension d is a prime power. We will also speculate about a relation between mutually unbiased bases and mutually orthogonal latin squares.
Apparently, it is difficult to design efficient quantum circuits. The design principles for classical algorithms are often not adequat or cannot be applied to quantum algorithms. We introduce a principle of software engineering to quantum computing, which allows to reuse existing efficient quantum circuits. The method is based on a solid theoretical foundation, and the resulting quantum circuits are easily interpretable.The talk is based on joint work with Martin Rötteler (Institute for Quantum Computing, Waterloo, Canada).
We show how spectroscopic methods can be employed in the sub-wavelength localization of atoms. We also discuss a possible spectroscopic scheme for the measurement of center-of-mass wavefunction of atoms.
In this second part of the talk, further design details and issues associated with developing scalable quantum computers in solids will be addressed. For concreteness, nitrogen-vacancy (NV) color centers in diamond will be used as the main example because of their good short-term potential. In NV diamond, nuclear and/or electron spins act as qubits, but initialization and readout is optical. Therefore, there are similarities to quantum computers based on liquid NMR, ion traps, and Kane's design.
Design details and issues associated with developing scalable quantum computers in solids will be addressed. For concreteness, nitrogen-vacancy (NV) color centers in diamond will be used as the main example because of their good short-term potential. In NV diamond, nuclear and/or electron spins act as qubits, but initialization and readout is optical. Therefore, there are similarities to quantum computers based on liquid NMR, ion traps, and Kane's design.
Quantum dots are regarded as one of the most promising quantum devices for the future quantum computer. In this talk, we will look at the physical design of quantum dots, give a heuristic derivation of the corresponding Hamiltonian and then derive elementary 1-bit and 2-bit quantum gates for universal quantum computation.This talk is based on the paper by G. Chen, D.A. Church, B.G. Englert and M.S. Zubairy, ``Mathematical Models of Contemporary Elementary Quantum Computing Devices''. In: Quantum Control, CRM Lecture Notes, AMS, 2003, to appear.
The quantum factoring algorithm proposed by P. Shor is usually applied in the context of binary logic, based on qubits. We extend the logic gates and FFT in the algorithm to multivalued systems (qudits), and propose an implementation based on wavepackets in multilevel Rydberg atoms. Possible realizations in a linear ion trap will be mentioned.
Quantum dots are regarded as one of the most promising quantum devices for the future quantum computer. In this talk, we will look at the physical design of quantum dots, give a heuristic derivation of the corresponding Hamiltonian and then derive elementary 1-bit and 2-bit quantum gates for universal quantum computation.This talk is based on the paper by G. Chen, D.A. Church, B.G. Englert and M.S. Zubairy entitled ``Mathematical Models of Contemporary Elementary Quantum Computing Devices'', to appear in the volume of ``Quantum Control'', CRM (Centre de Recherche Mathematiques, Universite de Montreal) Lecture Notes, published by the American Mathematical Society.
Mathematica is a well-known software package for performing mathematical computation (both symbolic and numerical), doing graphical animation, etc. Many add-on packages have been written to support particular fields, and quantum computing is no exception. Colin Williams and Scott Clearwater have written a number of useful Mathematica routines to support simulating quantum computation, and made them freely available as a supplement to their book. I will demonstrate typical basic operations of quantum circuits, simulations of Feynman's cursor computer, Shor's factorization algorithm, and quantum teleportation.
The presentation will discuss some basic problems in quantum computing and quantum control: how one may prepare any V in the Lie group SU(4) for a weakly J-coupled pair of spin half particles. While the method uses selective excitation it treats the J-coupling term exactly. Further, the input field produced is guaranteed to be bounded thereby respecting the regime for selective excitation approximation. This is in sharp contrast to other works. The methodology is based on (i) a certain Cartan factorization; (ii) the usage of Givens factorizationin conjunction with this Cartan decomposition; and (iii) the explicit factorization of single spin exponentials in a manner which makes it possible to treat J-coupling precisely.The point in (ii) enables the determination of the SU(4) ``Euler angles". This yields a fully constructive method to prepare V. The presentation will discuss one illustration of this methodology which proves that the method frequently requires times much smaller than even the spin-spin relaxation times.
The speaker will give a quick introduction of Shor's quantum computing algorithm, including number theoretic methods such as Euler's phi function, continued fractions, etc., in combination with the quantum Fourier transform and estimates, for the factoring of large integers.This lecture is mostly based on the speaker's short course material delivered at the Centre de Recherche Mathematiques, Université de Montreal, August 2002.
Unitary error bases generalize the Pauli matrices to higher dimensional systems. These bases are important primitives in quantum information processing, since they form the foundation of teleportation methods, dense coding schemes, and quantum error correcting codes. Two basic constructions of unitary error bases are known: An algebraic construction by Knill, which yields nice error bases, and a combinatorial construction by Werner, which yields shift-and-multiply bases. An open problem posed by Schlingemann and Werner relates these two constructions and asks whether each nice error basis is equivalent to a shift-and-multiply basis. We solve this open problem and show that the answer is negative. We also show that it is always possible to find a fairly sparse representation of a nice error basis, a result that could not have been anticipated from the definitions.This talk is based on joint work with Martin Rötteler.
We consider a generalization of Grover's search problem, viz., to find any one element in a set of acceptable choices which constitute a (known) fraction f of the total number of choices in an unsorted data base. An infinite family of sure-success quantum algorithms are introduced to solve this problem, each member for a different range of f. The nth member An of this family involves n queries of the data base, and so the lowest few members of this family should be very convenient algorithms within their ranges of validity. They are also easier to realize since they do not require long coherence time in the quantum system. The even member A2n of the family covers ever larger range of f for larger n, which in the limit of n -> ∞ becomes the full range between 0 and 1. The odd-member subfamily has similar properties, except that the f range of each A2n+1 extends to unity. Sure-success algorithms are particularly useful when the cost of failure of a search is very high. There are also non-search-related applications of these algorithms, such as the creation of specified coherent states (i.e., coherence engineering), including Schrodinger-cat states, etc. These algorithms can also contribute toward multi-stage searches, but another ingredient is needed.
We describe a quantum phase gate in which the two qubits are represented by the photons in the two modes of the cavity field. Application of this phase gate to quantum search algorithms and quantum teleportation will be discussed.
I will describe a program to develop scalable quantum and coherence-based computers. For quantum bits we will use the spin transitions of dopant atoms in various solid-state hosts. Unlike similar proposals, the goal here will be to control these spins using optical Raman excitation. Recently, similar techniques have been used to demonstrate ultraslow and stopped light in a crystal. One of the short-term goals is to demonstrate a working quantum computer, composed of an array of quantum nodes, having two qubits per node, but classical communication between the nodes. Here, an ensemble-based system will be used, in analogy with liquid NMR. In parallel, experiments are being conducted using nitrogen-vacancy (NV) color centers in diamond to demonstrate scalable quantum logic. In addition to quantum computers, coherence-based computers are also being developed to demonstrate quadratic speedup in searches, especially related to target recognition.
Suppose that a quantum circuit with K elementary gates is known for a unitary matrix U, and assume that Um is a scalar matrix for some positive integer m. We show that a function of U can be realized on a quantum computer with at most O(mK+m2log m) elementary gates. The functions of U are realized by a generic quantum circuit, which has a particularly simple structure. Among other results, we obtain efficient circuits for the fractional Fourier transform. Moreover, we will discuss two related challenge problems (each with US $100 cash award).This talk is based on joint work with Martin Rötteler, see quant-ph/0208130.
For multi-object quantum search problems, a basic question is that the number of search targets is not known a priori. The determination or estimation of the number of search targets, say k, is called the quantum counting problem. In this talk, the speaker will use the quantum Fourier transform and eigenvalue kickback technique to estimate k, which is hidden in the phase of two orthogonal eigenstates of the unitary operator corresponding to the search algorithm.This talk is based on a recent paper by G. Chen and Z. Diao published in Zeitschrift für Naturforschung 56a (2001), 879-888.
We study the quantum circuit design using 1-bit and 2-bit unitary gates for the iterations of the multi-object quantum search algorithm. The oracle block is designed in such a way that it can accommodate any sign-flipping operations. A chief ingredient in the design is the permutation operator which is realized by a sequence of transpositions where in each transposition, only one bit is flipped.The talk is based on joint work with Zijian Diao from the University of Illinois, Urbana.
Spring 2002
Fall 2001