Minimal Representations of Natural Numbers Under a Set of Operators

This paper studies the minimal length representation of the natural numbers. Let O be a fixed set of integer-valued functions (primarily hyperoperations). For each n, what is the shortest way of expressing n as a combinations of functions in O to the constant 1? For example, if O contains the two functions Sx (successor of x) and *xy (x times y) then the shortest representation of 12 is *SSS1SS1, with 8 symbols. This is taken to mean that 8 is complexity of 12 under O. We make a study of such minimal representations and complexities in this paper, proving and/or rightly predicting bounds on complexities, discussing some relevant patterns in the complexities and minimal representations of the natural numbers and listing the results gleaned from computational analysis. Computationally, the first 4.5 million natural numbers were probed to verify our mathematically obtained results. Due to the finiteness of the problem, we used the method of exhaustion of possibilities to state some other results as well.Comment: 13 pages, 2 figures, 2 table

Experimental construction of generic three-qubit states and their reconstruction from two-party reduced states on an NMR quantum information processor

We experimentally explore the state space of three qubits on an NMR quantum information processor. We construct a scheme to experimentally realize a canonical form for general three-qubit states up to single-qubit unitaries. This form involves a non-trivial combination of GHZ and W-type maximally entangled states of three qubits. The general circuit that we have constructed for the generic state reduces to those for GHZ and W states as special cases. The experimental construction of a generic state is carried out for a nontrivial set of parameters and the good fidelity of preparation is confirmed by complete state tomography. The GHZ and W-states are constructed as special cases of the general experimental scheme. Further, we experimentally demonstrate a curious fact about three-qubit states, where for almost all pure states, the two-qubit reduced states can be used to reconstruct the full three-qubit state. For the case of a generic state and for the W-state, we demonstrate this method of reconstruction by comparing it with the directly tomographed three-qubit state.Comment: Revised version to appear in PRA new results adde

Determining the parity of a permutation using an experimental NMR qutrit

We present the NMR implementation of a recently proposed quantum algorithm to find the parity of a permutation. In the usual qubit model of quantum computation, speedup requires the presence of entanglement and thus cannot be achieved by a single qubit. On the other hand, a qutrit is qualitatively more quantum than a qubit because of the existence of quantum contextuality and a single qutrit can be used for computing. We use the deuterium nucleus oriented in a liquid crystal as the experimental qutrit. This is the first experimental exploitation of a single qutrit to carry out a computational task.Comment: 6 pages 4 figures revte

Experimental demonstration of quantum contextuality on an NMR qutrit

We experimentally test quantum contextuality of a single qutrit using NMR. The contextuality inequalities based on nine observables developed by Kurzynski et. al. are first reformulated in terms of traceless observables which can be measured in an NMR experiment. These inequalities reveal the contextuality of almost all single-qutrit states. We demonstrate the violation of the inequality on four different initial states of a spin-1 deuterium nucleus oriented in a liquid crystal matrix, and follow the violation as the states evolve in time. We also describe and experimentally perform a single-shot test of contextuality for a subclass of qutrit states whose density matrix is diagonal in the energy basis.Comment: 7 pages revtex 4 figure

Majorana representation, qutrit Hilbert space and NMR implementation of qutrit gates

We report a study of the Majorana geometrical representation of a qutrit, where a pair of points on a unit sphere represents its quantum states. A canonical form for qutrit states is presented, where every state can be obtained from a one-parameter family of states via $SO(3)$ action. The notion of spin-1 magnetization which is invariant under $SO(3)$ is geometrically interpreted on the Majorana sphere. Furthermore, we describe the action of several quantum gates in the Majorana picture and experimentally implement these gates on a spin-1 system (an NMR qutrit) oriented in a liquid crystalline environment. We study the dynamics of the pair of points representing a qutrit state under various useful quantum operations and connect them to different NMR operations. Finally, using the Gell Mann matrix picture we experimentally implement a scheme for complete qutrit state tomography.Comment: replaced with final version 3 figures adde

Quadratic Chabauty and rational points II: Generalised height functions on Selmer varieties

We give new instances where Chabauty--Kim sets can be proved to be finite, by developing a notion of "generalised height functions" on Selmer varieties. We also explain how to compute these generalised heights in terms of iterated integrals and give the first explicit nonabelian Chabauty result for a curve $X/\mathbb{Q}$ whose Jacobian has Mordell-Weil rank larger than its genus

A comparative study of system size dependence of the effect of non-unitary channels on different classes of quantum states

We investigate the effect of different types of non-unitary quantum channels on multi-qubit quantum systems. For an $n$-qubit system and a particular channel, in order to draw unbiased conclusions about the system as a whole as opposed to specific states, we evolve a large number of randomly generated states under the given channel. We increase the number of qubits and study the effect of system size on the decoherence processes. The entire scheme is repeated for various types of environments which include dephasing channel, depolarising channel, collective dephasing channel and zero temperature bath. Non-unitary channels representing the environments are modeled via their Karus operator decomposition or master equation approach. Further, for a given $n$ we restrict ourselves to the study of particular subclasses of entangled states, namely the GHZ-type and W-type states. We generate random states within these classes and study the class behaviors under different quantum channels for various values of $n$.Comment: 10 pages revtex 10 pdf figure

Implementation of the quantum Fourier transform on a hybrid qubit-qutrit NMR quantum emulator

The quantum Fourier transform (QFT) is a key ingredient of several quantum algorithms and a qudit-specific implementation of the QFT is hence an important step toward the realization of qudit-based quantum computers. This work develops a circuit decomposition of the QFT for hybrid qudits based on generalized Hadamard and generalized controlled-phase gates, which can be implemented using selective rotations in NMR. We experimentally implement the hybrid qudit QFT on an NMR quantum emulator, which uses four qubits to emulate a single qutrit coupled to two qubits.Comment: 7 pages, 5 figure

Energy and System Size Dependence of Photon Production at Forward Rapidities at RHIC

The energy and system size dependence of pseudorapidity ($\eta$) and multiplicity distributions of photons are measured in the region -2.3 $\leq$ $\eta$ $\leq$ -3.7 for Cu + Cu collisions at $\sqrt{s_{NN}}$ = 200 and 62.4 GeV. Photon multiplicity measurements at forward rapidity have been carried out using a Photon Multiplicity Detector (PMD) in the STAR experiment. Photons are found to follow longitudinal scaling for Cu + Cu collisions for 0-10% centrality. A Comparison of pseudorapidity distributions with HIIJING model is also presented.Comment: 6 PAGES, 6 FIGURES, Poster presented in QM 200

Quantum chaos, thermalization and tunneling in an exactly solvable few body system

Exactly solvable models that exhibit quantum signatures of classical chaos are both rare as well as important - more so in view of the fact that the mechanisms for ergodic behavior and thermalization in isolated quantum systems and its connections to non-integrability are under active investigation. In this work, we study quantum systems of few qubits collectively modeled as a kicked top, a textbook example of quantum chaos. In particular, we show that the 3 and 4 qubit cases are exactly solvable and yet, interestingly, can display signatures of ergodicity and thermalization. Deriving analytical expressions for entanglement entropy and concurrence, we see agreement in certain parameter regimes between long-time average values and ensemble averages of random states with permutation symmetry. Comparing with results using the data of a recent transmons based experiment realizing the 3-qubit case, we find agreement for short times, including a peculiar step-like behaviour in correlations of some states. In the case of 4-qubits we point to a precursor of dynamical tunneling between what in the classical limit would be two stable islands. Numerical results for larger number of qubits show the emergence of the classical limit including signatures of a bifurcation.Comment: 6+5 pages, 7 figure