Group theory for quantum gates and quantum coherence

Planat, Michel; Jorrand, Philippe

doi:10.1088/1751-8113/41/18/182001

Cited by 17 publications

(26 citation statements)

References 37 publications

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“…The said restriction is that, for large N , only a, b values from a finite vicinity of 0 matter, which is to say that we break the cyclic nature of the labels a, b, 37) and take for granted that all relevant values of a, a ′ and b, b ′ are such that we stay inside the range −(π/ǫ − ǫ/2) · · · (π/ǫ − ǫ/2). Put differently, we give up the periodicity that would force us to identify a = +∞ with a = −∞ in the ǫ → 0 limit.…”

Section: The Continuous Limit Of N → ∞mentioning

confidence: 99%

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On Mutually Unbiased Bases

Durt

Englert

Bengtsson

et al. 2010

Int. J. Quantum Inform.

681

796

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Mutually unbiased bases for quantum degrees of freedom are central to all theoretical investigations and practical exploitations of complementary properties. Much is known about mutually unbiased bases, but there are also a fair number of important questions that have not been answered in full as yet. In particular, one can find maximal sets of N + 1 mutually unbiased bases in Hilbert spaces of prime-power dimension N = p m , with p prime and m a positive integer, and there is a continuum of mutually unbiased bases for a continuous degree of freedom, such as motion along a line. But not a single example of a maximal set is known if the dimension is another composite number (N = 6, 10, 12, . . . ).In this review, we present a unified approach in which the basis states are labeled by numbers 0, 1, 2, . . . , N − 1 that are both elements of a Galois field and ordinary integers. This dual nature permits a compact systematic construction of maximal sets of mutually unbiased bases when they are known to exist but throws no light on the open existence problem in other cases. We show how to use the thus constructed mutually unbiased bases in quantum-informatics applications, including dense coding, teleportation, entanglement swapping, covariant cloning, and state tomography, all of which rely on an explicit set of maximally entangled states (generalizations of the familiar two-q-bit Bell states) that are related to the mutually unbiased bases.There is a link to the mathematics of finite affine planes. We also exploit the one-toone correspondence between unbiased bases and the complex Hadamard matrices that turn the bases into each other. The Hadamard-matrix approach is instrumental in the very recent advance, surveyed here, of our understanding of the N = 6 situation. All evidence indicates that a maximal set of seven mutually unbiased bases does not exist -one can find no more than three pairwise unbiased bases -although there is currently no clear-cut demonstration of the case.

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Section: The Continuous Limit Of N → ∞mentioning

confidence: 99%

“…The terminology "Clifford operators" refers to the Clifford group, 37 which consists of all unitary operators that map the Heisenberg-Weyl group onto itself under conjugation, that is:…”

Section: The Remaining N − 1 Basesmentioning

confidence: 99%

On Mutually Unbiased Bases

Durt

Englert

Bengtsson

et al. 2010

Int. J. Quantum Inform.

681

796

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“…The maximal subgroup of the largest cardinality in W ′ (E 6 ) is isomorphic to the perfect group M 20 = Z 4 2 ⋊ A 5 of order 960, and is described in [23,24].…”

Section: Three-qubit Entanglement and The Crystallogaphic Group W (E 8 )mentioning

confidence: 99%

Balanced tripartite entanglement, the alternating group A4 and the lie algebra sl(3,ℂ)⊕u(1)

Planat¹,

Lévay²,

Санига³

2011

Reports on Mathematical Physics

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Abstract. We discuss three important classes of three-qubit entangled states and their encoding into quantum gates, finite groups and Lie algebras. States of the GHZ and W-type correspond to pure tripartite and bipartite entanglement, respectively. We introduce another generic class B of three-qubit states, that have balanced entanglement over two and three parties. We show how to realize the largest cristallographic group W (E 8 ) in terms of three-qubit gates (with real entries) encoding states of type GHZ or W [M. Planat, Clifford group dipoles and the enactment of Weyl/Coxeter group W (E 8 ) by entangling gates, Preprint 0904.3691 (quant-ph)]. Then, we describe a peculiar "condensation" of W (E 8 ) into the four-letter alternating group A 4 , obtained from a chain of maximal subgroups. Group A 4 is realized from two B-type generators and found to correspond to the Lie algebra sl(3, C) ⊕ u(1). Possible applications of our findings to particle physics and the structure of genetic code are also mentioned.

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“…within the context of two-qubit systems (see [3] and Secs 3.5 and 4.4). ¶ A Steiner system S(a, b, c) with parameters a, b, c, is a c-element set together with a set of b-element subsets of S (called blocks) with the property that each a-element subset of S is contained in exactly one block.…”

Section: The Single Qubit Casementioning

confidence: 99%

“…This generalizes our result concerning the symplectic generalized quadrangle GQ 2 associated with the two-qubit Pauli group P 2 . The isomorphism of W ′ (E 6 ) to the groups SU(4, 2) and PSp (4,3) provides an example of a group with two different BN pair structures (see [28] and [37] for the meaning of this group structure).…”

Section: Topological Entanglement the Yang-baxter Equation The Bellmentioning

confidence: 99%

Unitary Reflection Groups for Quantum Fault Tolerance

Planat¹,

Kibler²

2010

Jnl of Comp & Theo Nano

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Abstract.This paper explores the representation of quantum computing in terms of unitary reflections (unitary transformations that leave invariant a hyperplane of a vector space). The symmetries of qubit systems are found to be supported by Euclidean real reflections (i.e., Coxeter groups) or by specific imprimitive reflection groups, introduced (but not named) in a recent paper [Planat M and Jorrand Ph 2008, J Phys A: Math Theor 41, 182001]. The automorphisms of multiple qubit systems are found to relate to some Clifford operations once the corresponding group of reflections is identified. For a short list, one may point out the Coxeter systems of type B 3 and G 2 (for single qubits), D 5 and A 4 (for two qubits), E 7 and E 6 (for three qubits), the complex reflection groups G(2 l , 2, 5) and groups No 9 and 31 in the Shephard-Todd list. The relevant fault tolerant subsets of the Clifford groups (the Bell groups) are generated by the Hadamard gate, the π/4 phase gate and an entangling (braid) gate [Kauffman L H and Lomonaco S J 2004 New J. of Phys. 6, 134]. Links to the topological view of quantum computing, the lattice approach and the geometry of smooth cubic surfaces are discussed.

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Group theory for quantum gates and quantum coherence

Cited by 17 publications

References 37 publications

On Mutually Unbiased Bases

On Mutually Unbiased Bases

Balanced tripartite entanglement, the alternating group A4 and the lie algebra sl(3,ℂ)⊕u(1)

Unitary Reflection Groups for Quantum Fault Tolerance

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