Symmetry preserving generators in Lie Group
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
asked Dec 23 at 13:43
刘金国
1011
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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?
lie-groups symmetric-groups quantum-computation
asked Dec 23 at 13:43
刘金国
1011
New contributor
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
add a comment |
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
asked Dec 23 at 13:43
刘金国
1011
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
lie-groups symmetric-groups quantum-computation
asked Dec 23 at 13:43
刘金国
1011
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.
asked Dec 23 at 13:43
刘金国
1011
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.
edited 2 days ago
asked Dec 23 at 13:43
刘金国
1011
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.
asked Dec 23 at 13:43
刘金国
1011
asked Dec 23 at 13:43
刘金国
1011
1011
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.
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.
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.
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
add a comment |
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
add a comment |
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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