This project is developing algorithms and communication primitives for quantum information processing, with a focus on secure communication, quantum error correction, and image processing applications. Developing quantum optical implementations for selected applications is another goal of this project.This project is a joint effort of the Departments of Computer Science, Mathematics, and Physics.

- Course on Quantum Algorithms Spring 2003
- Quantum Computing Seminar Fall 2002-Spring 2003

- A. Klappenecker and M. Rötteler
**On the Monomiality of Nice Error Bases**

[all formats]**Status:**Submitted, Jan 16, 2003.Unitary error bases generalize the Pauli matrices to higher dimensional systems. 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 (see Problem 6) relates these two constructions and asks whether each nice error basis is equivalent to a shift-and-multiply basis. We solve this problem and show that the answer is negative. However, we also show that it is always possible to find a fairly sparse representation of a nice error basis.

@Unpublished{preprint030116, author = {Klappenecker, A. and R{\"o}tteler, M.}, title = {On the monomiality of nice error bases}, note = {Preprint}, month = {January}, year = 2003 }

- M. S. Zubairy, M. Kim, and M.O. Scully
**A cavity QED based quantum phase gate****Status:**submitted.We describe a quantum phase gate in which the two qubits are represented by the photons in the two modes of the cavity field. The gate is implemented by passing a three-level atom in a cascade configuration through the cavity. The upper levels of the atom are resonant with one of the cavity modes whereas the lower levels are appropriately detuned from the other mode of the cavity. A $\pi $ phase shift is introduced when there is one photon each in the two modes and the atom is initially in the ground state. We also discuss the one-bit unitary gate in such a system and discuss potential applications.

@Unpublished{zubairy03, author = {Zubairy, M.S. and Kim, M. and Scully, M.O.}, title = {A cavity QED based quantum phase gate}, note = {Submitted to Phys. Rev. Lett.}, year = 2003 }

- A. Klappenecker and M. Rötteler
**Quantum Software Reusability****Status:**Submitted.The design of efficient quantum circuits is an important issue in quantum computing. It is in general a formidable task to find a highly optimized quantum circuit for a given unitary matrix. We propose a quantum circuit design method that has the following unique feature: It allows to construct efficient quantum circuits in a systematic way by reusing and combining a set of highly optimized quantum circuits. Specifically, the method realizes a quantum circuit for a given unitary matrix by implementing a linear combination of representing matrices of a group, which have known fast quantum circuits. We motivate and illustrate this method by deriving extremely efficient quantum circuits for the discrete Hartley transform and for the fractional Fourier transforms. The sound mathematical basis of this design method allows to give meaningful and natural interpretations of the resulting circuits. We demonstrate this aspect by giving a natural interpretation of known teleportation circuits.

@Unpublished{klappenecker, author = {Klappenecker, A. and R{\"o}tteler, M.}, title = {Quantum Software Reusability}, note = {Submitted to Intl. J. Found. Comp. Science}, year = 2002 }

- G. Chen, Z. Ding, C.-R. Hu, W.-M. Ni and J. Zhou
**A note on the elliptic Sine-Gordon equation**

In: "Variational Methods for Nonlinear Elliptic Partial Differential Equations: Open Problems, Recent Progress and Numerical Algorithms", J.M. Neuberger (ed.), Contemporary Mathematics, AMS, 2003. Accepted, to appear.The elliptic sine-Gordon equation originates from the static case of the hyperbolic sine-Gordon equation modeling the Josephson junction in superconductivity. However, the elliptic sine-Gordon boundary value problem as studied in the mathematical literature actually has an opposite sign in front of the sine nonlinearity; it models not the ``usual'' Josephson junction but rather the Josephson $\pi$-junction, which is of contemporary interest to physicists. We first furnish this physical backdrop that has motivated our study here. Then we aim to establish the existence of nonconstant solutions of the semilinear elliptic sine-Gordon equation subject to homogeneous Neumann and Dirichlet boundary conditions by using critical point theory. Positive numerical solutions of the Dirichlet case, which are global minima of the variational problem, are computed on a dumbbell-shaped 2D domain for visualization.

@Unpublished{chen03a, author = {Chen, G. and Ding, Z. and Hu, C.-R. and Ni, W.-M. and Zhou, J.}, title = {A note on the elliptic Sine-Gordon equation}, note = {Submitted}, year = 2003 }

- G. Chen, D. A. Church, B.-G. Englert, and M.S. Zubairy
**Mathematical Models of Contemporary Elementary Quantum Computing Devices****Status:**Accepted, to appear in CRM Proceedings and Lecture NotesComputations with a future quantum computer will be implemented through the operations by elementary quantum gates. It is now well known that the collection of 1-bit and 2-bit quantum gates are universal for quantum computation, i.e., any n-bit unitary operation can be carried out by concatenations of 1-bit and 2-bit elementary quantum gates. Three contemporary quantum devices - cavity QED, ion traps and quantum dots - have been widely regarded as perhaps the most promising candidates for the construction of elementary quantum gates. In this paper, we describe the physical properties of these devices, and show the mathematical derivations based on the interaction of the laser field as control with atoms, ions or electron spins, leading to the following: (i) the 1-bit unitary rotation gates; and (ii) the 2-bit quantum phase gates and the controlled-not gate. This paper is aimed at providing a sufficiently self-contained survey account of analytical nature for mathematicians, physicists and computer scientists to aid interdisciplinary understanding in the research of quantum computation.

@Unpublished{chen03b, author = {Chen, G. and Church, D.A. and Englert, B.-G. and M.S. Zubairy}, title = {Mathematical Models of Contemporary Elementary Quantum Computing Devices}, note = {To appear in CRM Proceedings and Lecture Notes}, year = 2003 }

- G. Chen, B.-G. Englert, and J. Zhou
**Convergence analysis of an optimal scaling algorithm for semilinear elliptic boundary value problems**

In: "Variational Methods for Nonlinear Elliptic Partial Differential Equations: Open Problems, Recent Progress and Numerical Algorithms", J.M. Neuberger (ed.), Contemporary Mathematics, AMS, 2003. Accepted, to appear.Proof of convergence for iterative schemes for finding unstable solutions of semilinear elliptic boundary value problems is an arduous task. In perspective is a special iterative algorithm using the idea of scaling. In the form called

*Scaling Iterative Algorithm (SIA)*based on normalizing each iterate's function value to be 1 at a given interior point of the domain, it is found that SIA is computationally quite advantageous. Yet no convergence analysis is available. In this paper, we present a different idea of scaling which is an optimal scaling in the sense that the first integral is optimized. For this*Optimal Scaling Iterative Algorithm (OSIA)*, we prove the convergence under certain assumptions on the nonlinearity and stipulated stepsize rule.@Unpublished{chen03c, author = {Chen, G. and Englert, B.-G. and Zhou, J.}, title = {Convergence analysis of an optimal scaling algorithm for semilinear elliptic boundary value problems}, note = {To appear}, year = 2003 }

- A. Klappenecker and M. Rötteler
**Nice Error Bases: Constructions, Equivalence, and Applications**

Applied Algebra, Algebraic Algorithms, and Error Correcting Codes, Toulouse, France, 2003. Accepted, to appear.

[summary][ps]Unitary error bases are fundamental primitives in quantum computing, which are instrumental for quantum error-correcting codes and the design of teleportation and super-dense coding schemes. There are two prominent constructions of such bases: an algebraic construction using projective representations of finite groups and a combinatorial construction using Latin squares and Hadamard matrices. An open problem posed by Schlingemann and Werner relates these two constructions, and asks whether each algebraic construction is equivalent to a combinatorial construction. We answer this question by giving an explicit counterexample in dimension 165 which has been constructed with the help of a computer algebra system.

@Unpublished{preprint021217, author = {Klappenecker, A. and R{\"o}tteler, M.}, title = {Nice error bases: {C}onstructions, equivalence, and applications}, note = {Applied Algebra, Algebraic Algorithms, and Error Correcting Codes, Toulouse, France, 2003.}, year = 2003 }

- A.B. Matsko, I. Novikova, M.S. Zubairy, and G.R. Welch
**Nonlinear magneto-optical rotation of elliptically polarized light**

Phys. Rev A 67, 2003, to appear.We predict theoretically and demonstrate experimentally an ellipticity-dependent nonlinear magneto-optic rotation of elliptically-polarized light propagating in a coherent atomic medium. We show that this effect results from a hexadecapole and higher order momenta of atomic coherence, and is associated with an enhancement of Kerr and higher orders nonlinearities accompanied by suppression of the other linear and nonlinear susceptibility terms of the medium. These nonlinearities might be useful for quantum signal processing. In particular, we report an observation of an enhancement the polarization rotation of elliptically polarized light resonant with the 5S

_{1/2}F=2 -> 5P_{1/2}F=1 transition of Rb87.@Unpublished{matsko03, author = {Matsko, A.B. and Novikova, I. and Zubairy, M.S. and Welch, G.R.}, title = {Nonlinear magneto-optical rotation of elliptically polarized light}, note = {To appear in Phys. Rev A}, year = 2003 }

- M. Ahmad, S. Qamar, and M.S. Zubairy
**Reconstruction of a multimode entangled state using two-photon phase-sensitiuve linear amplifier**

Phys. Rev. A 67, **** (2003), to appear.We propose a model for the measurement of an arbitrary multimode entangled state of the cavity field using two-photon correlated emission laser (CEL). We consider two cases: (a) The modes have different frequencies and detected separately and (b) the modes consist of two orthogonal polarization states and are detected using a single balanced homodyne detector. The basic idea is to amplify the initial multimode state such that there is no-noise in the quadrature of interest and all the noise is fed into the conjugate quadrature component. The amplified noise-free quadrature is prepared in different phases and then corresponding quadrature distribution is measured. The Wigner function of the initial multimode entangled state is then reconstructed by using inverse Radon transformation. This scheme is insensitive to the noise associated with the non-unit efficiency of the detector in the homodyne detection measurement scheme.

@Unpublished{ahmad03a, author = {Ahmad, M. and Qamar, S. and Zubairy, M.S.}, title = {Reconstruction of a multimode entangled state using a two-photon phase-sensitive linear amplifier}, note = {To appear in Phys. Rev. A}, year = 2003 }

- S. Qamar, S.-Y. Zhu, and M. S. Zubairy
**Teleportation of an atomic momentum state**

Phys. Rev. A 67, **** (2003), to appear.In this paper, we propose a scheme for teleportating a superposition of atomic center of mass momentum states to a superposition of the cavity field using quantum controlled NOT gate via atomic scattering in the Bragg's regime and cavity quantum electrodynamics.

@Unpublished{qamar03, author = {Qamar, S. and Zhu, S.-Y. and Zubairy, M.S.}, title = {Teleportation of an atomic momentum state}, note = {To appear in Phys. Rev. A}, year = 2003 }

- O. Schulz, R. Steinhübl, M. Weber, B.-G. Englert,
C. Kurtsiefer, and H. Weinfurter
**Ascertaining the values of σ**_{x}, σ_{y}, and σ_{z}of a polarization qubit

Phys. Rev. Letters, 2003, to appear.In the 1987 spin retrodiction puzzle of Vaidman, Aharonov, and Albert one is challenged to ascertain the values of σ

_{x}, σ_{y}, and &sigma_{z}of a spin-1/2 particle by utilizing entanglement. We report the experimental realization of a quantum-optical version in which the outcome of an intermediate polarization projection is inferred by exploiting single-photon two-qubit quantum gates. The experimental success probability is consistently above the 90.2% threshold of the optimal one-qubit strategy, with an average success probability of 95.6%.@Unpublished{schulz03, author = {Schulz, O. and Steinh{\"u}bl, R. and Weber, M. and Englert, B.-G. and Kurtsiefer, C. and Weinfurter, H.}, title = {Ascertaining the values of $sigma_x$, $sigma_y$, and $sigma_z$ of a polarization qubit}, note = {Accepted, to appear in Physical Review Letters}, year = 2003 }

- V. Rostovstev, A.B. Matsko, N. Nayak, M.S. Zubairy, and M.O. Scully
**The quantum afterburner: Improving engine efficiency by extracting laser energy from wasted gas**

Phys. Rev. A 67, **** (2003), to appear.We show that it is possible to improve the efficiency of a classical Otto-cycle heat engine by adding a high-Q microwave cavity and a laser system which can extract coherent laser energy from thermally excited "exhaust" atoms. This improvement does not violate the second law of thermodynamics, i.e. we show that a combined high-Q microwave cavity/laser system does not improve the efficiency of a classical Carnot-cycle heat engine.

@Article{rostovstev03, author = {Rostovstev, V. and Matsko, A.B. and Nayak, N. and Zubairy, M.S. and Scully, M.O.}, title = {The quantum afterburner: Improving engine efficiency by extracting laser energy from wasted gas}, journal = {Physical Review A}, year = 2003, volume = 67, note = {To appear} }

- M.O. Scully, M.S. Zubairy, G.S. Agarwal, and H. Walther
**Extracting work from a single heat bath via vanishing quantum coherence**

Science 299, p. 862-864, 2003.We present here a quantum Carnot engine in which the atoms in the heat bath are given a small bit of quantum coherence. The induced quantum coherence becomes vanishingly small in the high-temperature limit at which we operate and the heat bath is essentially thermal. However, the phase associated with the atomic coherence, provides a new control parameter that can be varied to increase the temperature of the radiation field and to extract work from a single heat bath. The deep physics behind the second law of thermodynamics is not violated; nevertheless, the quantum Carnot engine has certain features that are not possible in a classical engine.

@Article{scully03, author = {Scully, M.O. and Zubairy, M.S. and Agarwal, G.S. and Walther, H. }, title = {Extracting work from a single heat bath via vanishing quantum coherence}, journal = {Science}, year = 2003, volume = 299, pages = {862-864} }

- S. Qamar, S.-Y. Zhu, and M.S. Zubairy
**Time-dependent Autler-Townes spectroscopy**

J. Opt. B: Quantum Semiclass. Opt. 5(2), pp. 175-193, 2003.Autler-Townes spontaneous emission spectroscopy is revisited for a time-dependent case. We report the results of spontaneous emission spectra for nonstationary scattered light signals using the definition of the time-dependent physical spectrum. This is a rare example of problems where time-dependent spectra can be calculated exactly.

@Article{qamar03, author = {Qamar, S. and Zhu, S.-Y. and Zubairy, M.S.}, title = {Time-dependent Autler-Townes spectroscopy}, journal = {J. Opt. B: Quantum Semiclass. Opt.}, year = 2003, volume = 5, number = 2, pages = {175-193} }

- K.T. Kapale, M.O. Scully, S.-Y. Zhu, and M.S. Zubairy
**Quenching of spontaneous emission through interference of incoherent pump processes**

Physical Review A 67, 023804, 2003We investigate the steady-state spontaneous emission of a V-type three-level atom, with the coherence between the two upper levels modified and controlled via incoherent pumping to a fourth auxiliary level. The external pumping gives us an easily controllable handle in manipulating the spontaneous emission to such an extent that, under certain conditions, complete quenching of spontaneous emission is possible. We also show that even the interference between the decay channels, which is considered a key requirement in spontaneous emission quenching through quantum interference, is not essential to achieve near 100% trapping and almost complete suppression of spontaneous emission. Thus we provide a scheme for spontaneous emission quenching which can be easily realized experimentally.

@Article{kapale03, author = {Kapale, K.T. and Scully, M.O. and Zhu, S.-Y. and Zubairy, M.S.}, title = {Quenching of spontaneous emission through interference of incoherent pump processes}, journal = {Physical Review A}, year = 2003, volume = 67, pages = 023804 }

- K.T. Kapale, S. Qamar, M.S. Zubairy
**Spectroscopic measurement of an atomic wave function**

Physical Review A 67, 023805, 2003We present a simple spectroscopic method based on Autler-Townes spectroscopy to determine the centerof- mass atomic wave function. The detection of spontaneously emitted photons from a three-level atom, in which two upper levels are driven by a classical standing light, yields information about the position and momentum distribution of the atom A.M. Herkommer, W.P. Schleich, and M.S. Zubairy, J. Mod. Opt. 44, 2507, 1997. In this paper, we show that both the amplitude and phase information of the center-of-mass atomic wave function can be obtained from these distributions after a series of conditional measurements on the atom and the emitted photon.

@Article{kapale03b, author = {Kapale, K.T. and Qamar, S. and Zubairy, M.S.}, title = {Spectroscopic measurement of an atomic wave function}, journal = {Physical Review A}, year = 2003, volume = 67, pages = 023805 }

- A. B. Matsko, I. Novikova, G.R. Welch, M.S. Zubairy
**Enhancement of Kerr nonlinearity by multiphoton coherence**

Optics Letter 28(2), pp. 96-98, 2003We propose a new method of resonant enhancement of optical Kerr nonlinearity that uses multilevel atomic coherence. The enhancement is accompanied by suppression of the other linear and nonlinear susceptibility terms of the medium. We show that the effect results in a modification of the nonlinear Faraday rotation of light propagating in an 87 Rb vapor cell by changing the ellipticity of the light.

@Article{matsko03, author = {Matsko, A.B. and Novikova, I. and Welch, G.R. and Zubairy, M.S.}, title = {Enhancement of Kerr nonlinearity by multiphoton coherence}, journal = {Optics Letters}, year = 2003, volume = 28, number = 2, pages = {96-98} }

- A. Klappenecker and M. Rötteler
**Engineering Functional Quantum Algorithms**

Physical Review A, 67, 010302, 2003.

[postscript] [pdf][prola]Suppose that a quantum circuit with

*K*elementary gates is known for a unitary matrix*U*, and assume that*U*^{m}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+m*elementary gates. The functions of^{2}ln *m*)*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.@Article{klappenecker037, author = {Klappenecker, A. and R{\"o}tteler, M.}, title = {Engineering functional quantum algorithms}, journal = {Physical Review A}, year = 2003, volume = 67, pages = 010302 }

- G. Song and A. Klappenecker
**Optimal Realizations of Controlled Unitary Gates**

Journal of Quantum Information and Computation, 3(2), pages 139-155, 2003.

[postscript][pdf]The controlled-not gate and the single qubit gates are considered elementary gates in quantum computing. It is natural to ask how many such elementary gates are needed to implement more elaborate gates or circuits. Recall that a controlled-

*U*gate can be realized with two controlled-not gates and four single qubit gates. We prove that this implementation is optimal if and only if the matrix*U*satisfies the conditions tr*U*≠ 0, tr(*UX*) ≠ 0, and det*U*≠ 1. We also derive optimal implementations in the remaining non-generic cases.@Article{klappenecker036, author = {Song, G. and Klappenecker, A.}, title = {Optimal Realizations of Controlled Unitary Gates}, journal = {Journal of Quantum Information and Computation}, year = 2003, volume = 3, number = 2, pages = {139-155}, }

- S.A. Fulling, B.-G. Englert, M.D. Pilloff
**Interacting Bosons at Finite Temperature: How Bogolugov Visited a Black Hole and Came Home Again**

Found. Phys. 33, pp. 87-110, 2003The structure of the thermal equilibrium state of a weakly interacting Bose gas is of current interest. We calculate the density matrix of that state in two ways. The most effective method, in terms of yielding a simple, explicit answer, is to construct a generating function within the traditional framework of quantum statistical mechanics. The alternative method, arguably more interesting, is to construct the thermal state as a vector state in an artificial system with twice as many degrees of freedom. It is well known that this construction has an actual physical realization in the quantum thermodynamics of black holes, where the added degrees of freedom correspond to the second sheet of the Kruskal manifold and the thermal vector state is a state of the Unruh or the Hartle-Hawking type. What is unusual about the present work is that the Bogolubov transformation used to construct the thermal state combines in a rather symmetrical way with Bogolubov's original transformation of the same form, used to implement the interaction of the nonideal gas in linear approximation. In addition to providing a density matrix, the method makes it possible to calculate efficiently certain expectation values directly in terms of the thermal vector state of the doubled system.

@Article{fulling03, author = {Fulling, S.A. and Englert, B.-G. and Pilloff, M.D.}, title = {Interacting Bosons at Finite Temperature: How Bogolugov Visited a Black Hole and Came Home Again}, journal = {Found. Phys.}, year = 2003, volume = 33, pages = {87-110} }

- B.-G. Englert, S.A. Fulling, M.D. Pilloff
**Statistics of dressed modes in a thermal state**

Opt. Comm., 208, pp. 139-144, 2002By a Wigner-function calculation, we evaluate the trace of a certain Gaussian operator arising in the theory of a boson system subject to both finite temperature and (weak) interaction. Thereby we rederive (and generalize) a recent result by Kocharovsky, Kocharovsky, and Scully [Phys. Rev. A, vol. 61, art. 053606 (2000)] in a way that is technically much simpler. One step uses a special case of the response of Wigner functions to linear transformations, and we demonstrate the general case by simple means. As an application we extract the counting statistics for each mode of the Bose gas.

@Article{englert02, author = {Englert, B.-G. and Fulling, S.A. and Pilloff, M.D.}, title = {Statistics of dressed modes in a thermal state}, journal = {Opt. Comm.}, year = 2002, volume = 208, pages = {139-144} }

- B.-G. Englert and G. Morigi
**Five Lectures on Dissipative Master Equations**

In "Coherent Evolution in Noisy Environments", Lecture Notes in Physics 611, pp. 55-106After giving a heuristic derivation of the master equation that is commonly employed for modeling the dynamics of a damped harmonic oscillator, we focus on the algebraic properties of this master equation. In particular, we report the eigenvalues of its Liouville operator and the corresponding right and left eigenvectors. These tools are then used for a study of micromaser dynamics, where the harmonic oscillator is a privileged cavity mode of the radiation field. In addition to being damped, the mode is also driven by atoms that traverse the cavity one by one and interact strongly with the quantized radiation. The most important statistical properties of the exiting atoms are derived. For the sake of pedagogy, the treatment advances from the simple to the complicated, and the reader may benefit from numerous homework assignments.

@InProceedings{englert02, author = {Englert, B.-G. and Morigi, G.}, title = {Five Lectures on Dissipative Master Equations}, booktitle = {Coherent Evolution in Noisy Environments}, pages = {55-106}, year = 2002, editor = {Buchleitner, A. and Hornberger, K.}, volume = 611, series = {LNP}, address = {Berlin}, publisher = {Springer-Verlag} }

- Z. Diao, M.S. Zubairy, and G. Chen
**Quantum Circuit Design for Grover's algorithm**

Z. Naturforschung, 57a, pp. 701-708, 2002We present a circuit design realizing Grover's algorithm based on 1-bit unitary gates and 2-bit quantum phase gates. In the first step, we express the circuit block which performs a key unitary transformation that flips only the sign of the state |11...11>$ using 1-bit and 2-bit gates. The Grover's iteration operator can then be constructed using this key unitary transformation twice, plus other operations involving only 1-bit unitary gates on each qubit. Mathematical proofs are given to justify that the circuiting satisfies the desired operator properties.

@Article{diao02, author = {Diao, Z. and Zubairy, M.S. and Chen, G.}, title = {Quantum Circuit Design for Grover's algorithm}, journal = {Z. Naturforschung}, year = 2002, volume = {57a}, pages = {701-708} }

- C. Bowden, G. Chen, Z. Diao, A. Klappenecker
**The Universality of the Quantum Fourier Transform in Forming the Basis of Quantum Computing Algorithms**

Journal of Mathematical Analysis and Applications, 274, pages 69-80, 2002.

[pdf] (on IDEALFirst)The quantum Fourier transform (QFT) is a powerful tool in quantum computing. The main ingredients of QFT are formed by the Walsh-Hadamard transform H and phase shifts P(w), w in

**R**, which are 2×2 unitary matrices. We show that H and P(w), w in**R**, suffice to generate the unitary group U(2) and, consequently, through controlled-U operations and their concatenations, the entire unitary group U(2^{n}) on n qubits can be generated.@Article{klappenecker035, author = {Bowden, C. and Chen, G. and Diao, Z. and Klappenecker, A.}, title = {The Universality of the Quantum Fourier Transform in Forming the Basis of Quantum Computing Algorithms}, journal = {Journal of Mathematical Analysis and Applications}, year = 2002, volume = 274, pages = {69-80} }