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

Dogra, Shruti; Dorai, Kavita; Arvind, .

doi:10.1088/1361-6455/aaa69f

Cited by 14 publications

(16 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Using this basis set of QNSV, it is possible to define distinct units of quantum information that can be realized by a quantum system in terms of a superposition of mutually orthogonal quantum states 4,40 . These arbitrary vectors will be also called qunits, with n = 0, 1, 2, ..., instead of using the nomenclature qudits for a d-dimensional space (commonly employed in quantum information 41 ).…”

Section: The Integer Number Representationmentioning

confidence: 99%

“…This is known as the most general quantum pure state in the Majorana representation 45 (see, e.g., Ref. 4 ). Notice that in quantum information, Weyl operators 8 can be used as a type of Pauli basis for these qunits, which can be properly constructed from the Z components (cf.…”

Section: The Integer Number Representationmentioning

confidence: 99%

“…Algebraic structures in quantum mechanics are of two-fold interest, namely for explaining microscopic physical phenomena and for applications in quantum computing/information [1][2][3][4][5][6] . An interesting example of this successful mathematical structure refers to the problem of universal quantum gates [7][8][9] , which is recognized as a particular case of the Lie algebras applied to quantum computers 10 .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A quantum number theory

Daiha¹,

Rivelino²

2021

Preprint

View full text Add to dashboard Cite

We employ an algebraic procedure based on quantum mechanics to propose a 'quantum number theory' (QNT) as a possible extension of the 'classical number theory'. We built our QNT by defining pure quantum number operators (q-numbers) of a Hilbert space that generate classical numbers (c-numbers) belonging to discrete Euclidean spaces. To start with this formalism, we define a 2-component natural q-number N, such that N 2 ≡ N 2 1 +N 2 2 , satisfying a Heisenberg-Dirac algebra, which allows to generate a set of natural c-numbers n ∈ N. A probabilistic interpretation of QNT is then inferred from this representation.Furthermore, we define a 3-component integer q-number Z, such thatand obeys a Lie algebra structure. The eigenvalues of each Z component generate a set, albeit all components do not generate Z 3 simultaneously. We interpret the eigenvectors of the q-numbers as 'q-number state vectors' (QNSV), which form multidimensional orthonormal basis sets useful to describe state-vector superpositions defined here as qunits. To interconnect QNSV of different dimensions, associated to the same c-number, we propose a quantum mapping operation to relate distinct Hilbert subspaces, and its structure can generate a subset W ⊆ Q * , the field of non-zero rationals. In the present description, QNT is related to quantum computing theory and allows dealing with nontrivial computations in high dimensions.

show abstract

Section: The Integer Number Representationmentioning

confidence: 99%

Section: The Integer Number Representationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A quantum number theory

Daiha¹,

Rivelino²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Neither Solovay-Kitaev type approximations were investigated for three-level systems. Nevertheless, a recent work emphasized the significance of the subgroup SOð3Þ of SUð3Þ for qutrit-based quantum computation, showing that any state of a qutrit could be obtained from a oneparameter family of states through the action of SOð3Þ [21]. In this regard, one should not ignore the remarkable relationship between the groups SOð3Þ and SUð2Þ.…”

Section: Introductionmentioning

confidence: 99%

Solovay–Kitaev Approximations of Special Orthogonal Matrices

Mahasinghe

Bandaranayake

Silva

2020

Advances in Mathematical Physics

View full text Add to dashboard Cite

The circuit-gate framework of quantum computing relies on the fact that an arbitrary quantum gate in the form of a unitary matrix of unit determinant can be approximated to a desired accuracy by a fairly short sequence of basic gates, of which the exact bounds are provided by the Solovay–Kitaev theorem. In this work, we show that a version of this theorem is applicable to orthogonal matrices with unit determinant as well, indicating the possibility of using orthogonal matrices for efficient computation. We further develop a version of the Solovay–Kitaev algorithm and discuss the computational experience.

show abstract

“…In [31,32] the pure N-qubit states were expresses geometrically, using the mapping that associates them with a polynomial. In [33] the study of the Majorana geometrical representation of the qutrit is presented. The geometry of separable states is studied particularly active (cf.…”

Section: Introductionmentioning

confidence: 99%

Malevich’s Suprematist Composition Picture for Spin States

Man’ko

Markovich

2019

Entropy

View full text Add to dashboard Cite

This paper proposes an alternative geometric representation of single qudit states based on probability simplexes to describe the quantum properties of noncomposite systems. In contrast to the known high dimension pictures, we present the planar picture of quantum states, using the elementary geometry. The approach is based on, so called, Malevich square representation of the single qubit state. It is shown that the quantum statistics of the single qudit with some spin j and observables are formally equivalent to statistics of the classical system with N 2 - 1 random vector variables and N 2 - 1 classical probability distributions, obeying special constrains, found in this study. We present a universal inequality, that describes the single qudits state quantumness. The inequality provides a possibility to experimentally check up entanglement of the system in terms of the classical probabilities. The simulation study for the single qutrit and ququad systems, using the Metropolis Monte-Carlo method, is obtained. The geometrical representation of the single qudit states, presented in the paper, is useful in providing a visualization of quantum states and illustrating their difference from the classical ones.

show abstract

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

Cited by 14 publications

References 41 publications

A quantum number theory

A quantum number theory

Solovay–Kitaev Approximations of Special Orthogonal Matrices

Malevich’s Suprematist Composition Picture for Spin States

Contact Info

Product

Resources

About