LX-mixers for QAOA: Optimal mixers restricted to subspaces and the stabilizer formalism
We present a novel formalism to both understand and construct mixers that preserve a given subspace. The method connects and utilizes the stabilizer formalism that is used in error correcting codes. This can be useful in the setting when the quantum approximate optimization algorithm (QAOA), a popul...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
2024-11-01
|
| Series: | Quantum |
| Online Access: | https://quantum-journal.org/papers/q-2024-11-25-1535/pdf/ |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850158851523870720 |
|---|---|
| author | Franz G. Fuchs Ruben Pariente Bassa |
| author_facet | Franz G. Fuchs Ruben Pariente Bassa |
| author_sort | Franz G. Fuchs |
| collection | DOAJ |
| description | We present a novel formalism to both understand and construct mixers that preserve a given subspace. The method connects and utilizes the stabilizer formalism that is used in error correcting codes. This can be useful in the setting when the quantum approximate optimization algorithm (QAOA), a popular meta-heuristic for solving combinatorial optimization problems, is applied in the setting where the constraints of the problem lead to a feasible subspace that is large but easy to specify. The proposed method gives a systematic way to construct mixers that are resource efficient in the number of controlled not gates and can be understood as a generalization of the well-known X and XY mixers and a relaxation of the Grover mixer: Given a basis of any subspace, a resource efficient mixer can be constructed that preserves the subspace. The numerical examples provided show a dramatic reduction of CX gates when compared to previous results. We call our approach logical X-Mixer or logical X QAOA ($\textbf{LX-QAOA}$), since it can be understood as dividing the subspace into code spaces of stabilizers S and consecutively applying logical rotational X gates associated with these code spaces. Overall, we hope that this new perspective can lead to further insight into the development of quantum algorithms. |
| format | Article |
| id | doaj-art-62cf04db95484b6ebcc8459266340d53 |
| institution | OA Journals |
| issn | 2521-327X |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften |
| record_format | Article |
| series | Quantum |
| spelling | doaj-art-62cf04db95484b6ebcc8459266340d532025-08-20T02:23:45ZengVerein zur Förderung des Open Access Publizierens in den QuantenwissenschaftenQuantum2521-327X2024-11-018153510.22331/q-2024-11-25-153510.22331/q-2024-11-25-1535LX-mixers for QAOA: Optimal mixers restricted to subspaces and the stabilizer formalismFranz G. FuchsRuben Pariente BassaWe present a novel formalism to both understand and construct mixers that preserve a given subspace. The method connects and utilizes the stabilizer formalism that is used in error correcting codes. This can be useful in the setting when the quantum approximate optimization algorithm (QAOA), a popular meta-heuristic for solving combinatorial optimization problems, is applied in the setting where the constraints of the problem lead to a feasible subspace that is large but easy to specify. The proposed method gives a systematic way to construct mixers that are resource efficient in the number of controlled not gates and can be understood as a generalization of the well-known X and XY mixers and a relaxation of the Grover mixer: Given a basis of any subspace, a resource efficient mixer can be constructed that preserves the subspace. The numerical examples provided show a dramatic reduction of CX gates when compared to previous results. We call our approach logical X-Mixer or logical X QAOA ($\textbf{LX-QAOA}$), since it can be understood as dividing the subspace into code spaces of stabilizers S and consecutively applying logical rotational X gates associated with these code spaces. Overall, we hope that this new perspective can lead to further insight into the development of quantum algorithms.https://quantum-journal.org/papers/q-2024-11-25-1535/pdf/ |
| spellingShingle | Franz G. Fuchs Ruben Pariente Bassa LX-mixers for QAOA: Optimal mixers restricted to subspaces and the stabilizer formalism Quantum |
| title | LX-mixers for QAOA: Optimal mixers restricted to subspaces and the stabilizer formalism |
| title_full | LX-mixers for QAOA: Optimal mixers restricted to subspaces and the stabilizer formalism |
| title_fullStr | LX-mixers for QAOA: Optimal mixers restricted to subspaces and the stabilizer formalism |
| title_full_unstemmed | LX-mixers for QAOA: Optimal mixers restricted to subspaces and the stabilizer formalism |
| title_short | LX-mixers for QAOA: Optimal mixers restricted to subspaces and the stabilizer formalism |
| title_sort | lx mixers for qaoa optimal mixers restricted to subspaces and the stabilizer formalism |
| url | https://quantum-journal.org/papers/q-2024-11-25-1535/pdf/ |
| work_keys_str_mv | AT franzgfuchs lxmixersforqaoaoptimalmixersrestrictedtosubspacesandthestabilizerformalism AT rubenparientebassa lxmixersforqaoaoptimalmixersrestrictedtosubspacesandthestabilizerformalism |