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Evolution: the quantum computer paradigm

Michel R. P. Planat

Published Jan 21, 2008• 0 citations
OpenAlex

Abstract

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]).

Related papers

Clifford quantum computer and the Mathieu groups

Michel R. P. Planat

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.

arXiv:0711.1733•Nov 12, 2007•2 citations

On Clifford groups in quantum computing

J. Tolar

The term Clifford group was introduced in 1998 by D. Gottesmann in his investigation of quantum error-correcting codes. The simplest Clifford group in multiqubit quantum computation is generated by a restricted set of unitary Clifford gates - the Hadamard, π /4-phase and controlled-X gates. Because of this restriction the Clifford model of quantum computation can be efficiently simulated on a classical computer (the Gottesmann-Knill theorem). However, this fact does not diminish the importance of the Clifford model, since it may serve as a suitable starting point for a full-fledged quantum computation. In the general case of a single or composite quantum system with finite-dimensional Hilbert space the finite Weyl-Heisenberg group of unitary operators defines the quantum kinematics and the states of the quantum register. Then the corresponding Clifford group is defined as the group of unitary operators leaving the Weyl-Heisenberg group invariant. The aim of this contribution is to show that our comprehensive results on symmetries of the Pauli gradings of quantum operator algebras - covering any single as well as composite finite quantum systems - directly correspond to Clifford groups defined as quotients with respect to U(1).

arXiv:1810.10259•Journal of Physics Conference Series•Aug 1, 2018•22 citations

Quantum circuits

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Abstract In this chapter, we describe the circuit model of quantum computation, focusing on the initialization of a set of qubits, their unitary evolution, and final projective measurement. We outline Di Vincenzo’s criteria for constructing a functional quantum computer. We describe single-qubit gates and measurements, and we present two-qubit gates. We prove that single-qubit gates and a particular two-qubit gate, the CNOT gate, are universal for quantum computation. In the last part of the chapter, we present the Clifford group of unitary transformations that stabilize the Pauli group, and the Gottesman-Knill theorem.

•Aug 25, 2023•0 citations

Generalized Clifford groups and simulation of associated quantum circuits

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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 Information and Computation•Jan 1, 2008•22 citations

Generalised Clifford groups and simulation of associated quantum circuits

Sean Clark, Richard Jozsa, Noah Linden

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.

arXiv:quant-ph/0701103•Jan 15, 2007•4 citations

Clifford quantum computing

Michel R. P. Planat

One reports about the embedding of the Pauli group into the Clifford group, and one explicits the concept of Clifford quantum computation, following Gottesman-Knill approach. Normal subgroups of the two-qubit Clifford group are investigated and a link to the generalized quadrangle of other two and hexads in the Steiner system S(3,6,22) is found.

•Nov 21, 2007•0 citations
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]).
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 C2\mathcal{C}_2C2​. 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(C2)=C2/Center(C2){Inn}(\mathcal{C}_2)=\mathcal{C}_2/{Center}(\mathcal{C}_2)Inn(C2​)=C2​/Center(C2​) exactly contains two normal subgroups, one isomorphic to Z2×4\mathcal{Z}_2^{\times 4}Z2×4​ (of order 16), and the second isomorphic to the parent A′(6)A'(6)A′(6) (of order 5760) of the alternating group A(6). The group A′(6)A'(6)A′(6) stabilizes an {\it hexad} in the Steiner system S(3,6,22)S(3,6,22)S(3,6,22) attached to the Mathieu group M(22). Both groups A(6) and A′(6)A'(6)A′(6) have an {\it outer} automorphism group Z2×Z2\mathcal{Z}_2\times \mathcal{Z}_2Z2​×Z2​, a feature we associate to two-qubit quantum entanglement.
The term Clifford group was introduced in 1998 by D. Gottesmann in his investigation of quantum error-correcting codes. The simplest Clifford group in multiqubit quantum computation is generated by a restricted set of unitary Clifford gates - the Hadamard, π /4-phase and controlled-X gates. Because of this restriction the Clifford model of quantum computation can be efficiently simulated on a classical computer (the Gottesmann-Knill theorem). However, this fact does not diminish the importance of the Clifford model, since it may serve as a suitable starting point for a full-fledged quantum computation. In the general case of a single or composite quantum system with finite-dimensional Hilbert space the finite Weyl-Heisenberg group of unitary operators defines the quantum kinematics and the states of the quantum register. Then the corresponding Clifford group is defined as the group of unitary operators leaving the Weyl-Heisenberg group invariant. The aim of this contribution is to show that our comprehensive results on symmetries of the Pauli gradings of quantum operator algebras - covering any single as well as composite finite quantum systems - directly correspond to Clifford groups defined as quotients with respect to U(1).
Abstract In this chapter, we describe the circuit model of quantum computation, focusing on the initialization of a set of qubits, their unitary evolution, and final projective measurement. We outline Di Vincenzo’s criteria for constructing a functional quantum computer. We describe single-qubit gates and measurements, and we present two-qubit gates. We prove that single-qubit gates and a particular two-qubit gate, the CNOT gate, are universal for quantum computation. In the last part of the chapter, we present the Clifford group of unitary transformations that stabilize the Pauli group, and the Gottesman-Knill theorem.
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 GGG of U(d)U(d)U(d). In particular we seek GGG such that G⊗GG\otimes GG⊗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=2d = 2d=2 we exhaustively treat all finite irreducible subgroups of U(2)U(2)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 nthn^{\rm th}nth root of ZZZ for n∈Nn \in \Nn∈N.
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.