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arXiv:0711.1733

Clifford quantum computer and the Mathieu groups

Michel R. P. Planat

Published Nov 12, 2007• 2 citations

Abstract

One learned from Gottesman-Knill theorem that the Clifford model of quantum computing \cite{Clark07} may be generated from a few quantum gates, the Hadamard, Phase and Controlled-Z gates, and efficiently simulated on a classical computer. We employ the group theoretical package GAP\cite{GAP} for simulating the two qubit Clifford group $\mathcal{C}_2$. We already found that the symmetric group S(6), aka the automorphism group of the generalized quadrangle W(2), controls the geometry of the two-qubit Pauli graph \cite{Pauligraphs}. Now we find that the {\it inner} group ${Inn}(\mathcal{C}_2)=\mathcal{C}_2/{Center}(\mathcal{C}_2)$ exactly contains two normal subgroups, one isomorphic to $\mathcal{Z}_2^{\times 4}$ (of order 16), and the second isomorphic to the parent $A'(6)$ (of order 5760) of the alternating group A(6). The group $A'(6)$ stabilizes an {\it hexad} in the Steiner system $S(3,6,22)$ attached to the Mathieu group M(22). Both groups A(6) and $A'(6)$ have an {\it outer} automorphism group $\mathcal{Z}_2\times \mathcal{Z}_2$, a feature we associate to two-qubit quantum entanglement.

DOI: 10.48550/arxiv.0711.1733

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One learned from Gottesman-Knill theorem that the Clifford model of quantum computing \cite{Clark07} may be generated from a few quantum gates, the Hadamard, Phase and Controlled-Z gates, and efficiently simulated on a classical computer. We employ the group theoretical package GAP\cite{GAP} for simulating the two qubit Clifford group

\mathcal{C}_2

. We already found that the symmetric group S(6), aka the automorphism group of the generalized quadrangle W(2), controls the geometry of the two-qubit Pauli graph \cite{Pauligraphs}. Now we find that the {\it inner} group

{Inn}(\mathcal{C}_2)=\mathcal{C}_2/{Center}(\mathcal{C}_2)

exactly contains two normal subgroups, one isomorphic to

\mathcal{Z}_2^{\times 4}

(of order 16), and the second isomorphic to the parent

A'(6)

(of order 5760) of the alternating group A(6). The group

A'(6)

stabilizes an {\it hexad} in the Steiner system

S(3,6,22)

attached to the Mathieu group M(22). Both groups A(6) and

A'(6)

have an {\it outer} automorphism group

\mathcal{Z}_2\times \mathcal{Z}_2

, a feature we associate to two-qubit quantum entanglement.

Quantum computations starting with computational basis states and involving only Clifford operations, are classically simulable despite the fact that they generate highly entangled states; this is the content of the Gottesman-Knill theorem. Here we isolate the ingredients of the theorem and provide generalisations of some of them with the aim of identifying new classes of simulable quantum computations. In the usual construction, Clifford operations arise as projective normalisers of the first and second tensor powers of the Pauli group. We consider replacing the Pauli group by an arbitrary finite subgroup

G

of

U(d)

. In particular we seek

G

such that

G\otimes G

has an entangling normaliser. Via a generalisation of the Gottesman-Knill theorem the resulting normalisers lead to classes of quantum circuits that can be classically efficiently simulated. For the qubit case

d = 2

we exhaustively treat all finite irreducible subgroups of

U(2)

and find that the only ones (up to unitary equivalence and trivial phase extensions) with entangling normalisers are the groups generated by X and the

n^{\rm th}

root of

Z

for

n \in \N

.

Quantum computing has developed with the goal of classically simulating any unitary time evolution. The Clifford group model deals about stabilizing the time evolution of any Pauli operator, the latter accounting for environmental errors and measurements as well. The size of Clifford group increases drastically with the number of qubits, being related to the allowed symmetries of permutation groups. The zoo of subgroups includes: - the well known gate generators (Hadamard, Phase and Controlled NOT), - entangling (dihedral) teleportation groups, and – in the two-qubit case, a maximal subgroup of the large Mathieu groups. The six-fold peculiar symmetry of the two-qubit system corresponds to an outer property which one may associate to quantum entanglement (see M Planat, Clifford quantum computer and the Mathieu groups, 0711.1733 [quant-ph]).

Quantum computations that involve only Clifford operations are classically simulable despite the fact that they generate highly entangled states; this is the content of the Gottesman-Knill theorem. Here we isolate the ingredients of the theorem and provide generalisations of some of them with the aim of identifying new classes of simulable quantum computations. In the usual construction, Clifford operations arise as projective normalisers of the first and second tensor powers of the Pauli group. We consider replacing the Pauli group by an arbitrary finite subgroup G of U(d). In particular we seek G such that G tensor G has an entangling normaliser. Via a generalisation of the Gottesman-Knill theorem the resulting normalisers lead to classes of quantum circuits that can be classically efficiently simulated. For the qubit case d=2 we exhaustively treat all finite subgroups of U(2) and find that the only ones (up to unitary equivalence and trivial phase extensions) with entangling normalisers are the groups G_n generated by X and the n^th root of Z.

Abstract Associated to every finite abelian group G , there is a generalization of qubit Clifford group associated to G . We demonstrate that every G -Clifford gate can be decomposed into some distinguished elementary types: Fourier transforms, quadratic phase gates, and automorphism gates. Combined with earlier results on classical simulation of normalizer circuits, this implies a generalized Gottesman-Knill-type result. We additionally provide circuits for a universal quantum computing scheme based on local two-qudit G -Clifford gates and magic states.

Finite group extensions offer a natural language to quantum computing. In a nutshell, one roughly describes the action of a quantum computer as consisting of two finite groups of gates: error gates from the general Pauli group and stabilizing gates within an extension group . In this communication we explore the nice adequacy between group theoretical concepts such as commutators, normal subgroups, groups of automorphisms, short exact sequences, wreath products etc. and the coherent quantum computational primitives. The structure of the single-qubit and two-qubit Clifford groups is analyzed in detail. As a byproduct, we discover that M 20 , the smallest perfect group for which the commutator subgroup departs from the set of commutators, underlies quantum coherence of the two-qubit system. We recover similar results by looking at the automorphisms of a complete set of mutually unbiased bases.