Symmetry preserving generators in Lie Group












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In quantum computing, we can use finite types of generator to fill $U(N)$ space of all unitary operations, where $N$ is the number of qubits. e.g. $S={X, Z, CNOT}$, where $X$ and $Z$ are single qubit Pauli operators and $CNOT$ is a two qubit operator. With these three generators, we are able to represent a general $U(N)$ transformation as $G_{U(N)} = prod_k e^{-itheta_k s_k^{l_k}/2}, s_kin S.$ Here, the superscript $l_k$ denotes the qubit index.



Now, we put $U(1)$ symmetry restriction to operations, which means the total number of qubits are reserved, hence $X$ gate is not allowed. It turns out $S_{U(1)}={Z, SWAP, CZ}$ is enough as generators to represent all allowed $U(1)$ symmetric operations. [not sure]



Similarly, For $SU(2)$ symmetry, single generator $S_{SU(2)}={SWAP}$ is enough. [at least in $S^2=0$ block]



Constructing the symmetry preserving generator set is important to quantum chemistry. Is it possible to extend the above results to a general symmetry?










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  • Please be more consistent with your notation; for instance, $U(N)$ and U(N) are different sets of symbols.
    – Shaun
    Dec 23 at 13:59
















0














In quantum computing, we can use finite types of generator to fill $U(N)$ space of all unitary operations, where $N$ is the number of qubits. e.g. $S={X, Z, CNOT}$, where $X$ and $Z$ are single qubit Pauli operators and $CNOT$ is a two qubit operator. With these three generators, we are able to represent a general $U(N)$ transformation as $G_{U(N)} = prod_k e^{-itheta_k s_k^{l_k}/2}, s_kin S.$ Here, the superscript $l_k$ denotes the qubit index.



Now, we put $U(1)$ symmetry restriction to operations, which means the total number of qubits are reserved, hence $X$ gate is not allowed. It turns out $S_{U(1)}={Z, SWAP, CZ}$ is enough as generators to represent all allowed $U(1)$ symmetric operations. [not sure]



Similarly, For $SU(2)$ symmetry, single generator $S_{SU(2)}={SWAP}$ is enough. [at least in $S^2=0$ block]



Constructing the symmetry preserving generator set is important to quantum chemistry. Is it possible to extend the above results to a general symmetry?










share|cite|improve this question









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刘金国 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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  • Please be more consistent with your notation; for instance, $U(N)$ and U(N) are different sets of symbols.
    – Shaun
    Dec 23 at 13:59














0












0








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In quantum computing, we can use finite types of generator to fill $U(N)$ space of all unitary operations, where $N$ is the number of qubits. e.g. $S={X, Z, CNOT}$, where $X$ and $Z$ are single qubit Pauli operators and $CNOT$ is a two qubit operator. With these three generators, we are able to represent a general $U(N)$ transformation as $G_{U(N)} = prod_k e^{-itheta_k s_k^{l_k}/2}, s_kin S.$ Here, the superscript $l_k$ denotes the qubit index.



Now, we put $U(1)$ symmetry restriction to operations, which means the total number of qubits are reserved, hence $X$ gate is not allowed. It turns out $S_{U(1)}={Z, SWAP, CZ}$ is enough as generators to represent all allowed $U(1)$ symmetric operations. [not sure]



Similarly, For $SU(2)$ symmetry, single generator $S_{SU(2)}={SWAP}$ is enough. [at least in $S^2=0$ block]



Constructing the symmetry preserving generator set is important to quantum chemistry. Is it possible to extend the above results to a general symmetry?










share|cite|improve this question









New contributor




刘金国 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











In quantum computing, we can use finite types of generator to fill $U(N)$ space of all unitary operations, where $N$ is the number of qubits. e.g. $S={X, Z, CNOT}$, where $X$ and $Z$ are single qubit Pauli operators and $CNOT$ is a two qubit operator. With these three generators, we are able to represent a general $U(N)$ transformation as $G_{U(N)} = prod_k e^{-itheta_k s_k^{l_k}/2}, s_kin S.$ Here, the superscript $l_k$ denotes the qubit index.



Now, we put $U(1)$ symmetry restriction to operations, which means the total number of qubits are reserved, hence $X$ gate is not allowed. It turns out $S_{U(1)}={Z, SWAP, CZ}$ is enough as generators to represent all allowed $U(1)$ symmetric operations. [not sure]



Similarly, For $SU(2)$ symmetry, single generator $S_{SU(2)}={SWAP}$ is enough. [at least in $S^2=0$ block]



Constructing the symmetry preserving generator set is important to quantum chemistry. Is it possible to extend the above results to a general symmetry?







lie-groups symmetric-groups quantum-computation






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asked Dec 23 at 13:43









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  • Please be more consistent with your notation; for instance, $U(N)$ and U(N) are different sets of symbols.
    – Shaun
    Dec 23 at 13:59


















  • Please be more consistent with your notation; for instance, $U(N)$ and U(N) are different sets of symbols.
    – Shaun
    Dec 23 at 13:59
















Please be more consistent with your notation; for instance, $U(N)$ and U(N) are different sets of symbols.
– Shaun
Dec 23 at 13:59




Please be more consistent with your notation; for instance, $U(N)$ and U(N) are different sets of symbols.
– Shaun
Dec 23 at 13:59















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