Qutrit-Based Orthogonal Approximations with Inverse-Free Quantum Gate Sets
The efficient compiling of arbitrary single-qubit gates into a sequence of gates from a finite gate set is of fundamental importance in quantum computation. The exact bounds of this compilation are given by the Solovay–Kitaev theorem, which serves as a powerful tool in compiling quantum algorithms t...
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MDPI AG
2023-06-01
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| Series: | Computer Sciences & Mathematics Forum |
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| Online Access: | https://www.mdpi.com/2813-0324/7/1/12 |
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| author | Anuradha Mahasinghe Dulmi Fernando Kaushika De Silva |
| author_facet | Anuradha Mahasinghe Dulmi Fernando Kaushika De Silva |
| author_sort | Anuradha Mahasinghe |
| collection | DOAJ |
| description | The efficient compiling of arbitrary single-qubit gates into a sequence of gates from a finite gate set is of fundamental importance in quantum computation. The exact bounds of this compilation are given by the Solovay–Kitaev theorem, which serves as a powerful tool in compiling quantum algorithms that require many qubits. However, the inverse closure condition it imposes on the gate set adds a certain complexity to the experimental compilation, making the process less efficient. This was recently resolved by a version of the Solovay–Kitaev theorem for inverse-free gate sets, yielding a significant gain. Considering the recent progress in the field of three-level quantum systems, in which qubits are replaced by qutrits, it is possible to achieve the quantum speedup guaranteed by the Solovay–Kitaev theorem simply from orthogonal gates. Nevertheless, it has not been investigated previously whether the condition of inverse closure can be relaxed for these qutrit-based orthogonal compilations as well. In this work, we answer this positively, by obtaining improved Solovay–Kitaev approximations to an arbitrary orthogonal qutrit gate, with an accuracy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula> from a sequence of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><mi>l</mi><mi>o</mi><msup><mi>g</mi><mrow><mn>8.62</mn></mrow></msup><mrow><mo>(</mo><mn>1</mn><mo>/</mo><mi>ε</mi><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula> orthogonal gates taken from an inverse-free set. |
| first_indexed | 2024-03-08T20:52:41Z |
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| language | English |
| last_indexed | 2024-03-08T20:52:41Z |
| publishDate | 2023-06-01 |
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| series | Computer Sciences & Mathematics Forum |
| spelling | doaj.art-6160bac195a342c295e001db14917b022023-12-22T14:02:04ZengMDPI AGComputer Sciences & Mathematics Forum2813-03242023-06-01711210.3390/IOCMA2023-14416Qutrit-Based Orthogonal Approximations with Inverse-Free Quantum Gate SetsAnuradha Mahasinghe0Dulmi Fernando1Kaushika De Silva2Department of Mathematics, University of Colombo, Colombo 00300, Sri LankaDepartment of Mathematics, University of Colombo, Colombo 00300, Sri LankaDepartment of Mathematics, University of Sri Jayewardenepura, Nugegoda 10250, Sri LankaThe efficient compiling of arbitrary single-qubit gates into a sequence of gates from a finite gate set is of fundamental importance in quantum computation. The exact bounds of this compilation are given by the Solovay–Kitaev theorem, which serves as a powerful tool in compiling quantum algorithms that require many qubits. However, the inverse closure condition it imposes on the gate set adds a certain complexity to the experimental compilation, making the process less efficient. This was recently resolved by a version of the Solovay–Kitaev theorem for inverse-free gate sets, yielding a significant gain. Considering the recent progress in the field of three-level quantum systems, in which qubits are replaced by qutrits, it is possible to achieve the quantum speedup guaranteed by the Solovay–Kitaev theorem simply from orthogonal gates. Nevertheless, it has not been investigated previously whether the condition of inverse closure can be relaxed for these qutrit-based orthogonal compilations as well. In this work, we answer this positively, by obtaining improved Solovay–Kitaev approximations to an arbitrary orthogonal qutrit gate, with an accuracy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula> from a sequence of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><mi>l</mi><mi>o</mi><msup><mi>g</mi><mrow><mn>8.62</mn></mrow></msup><mrow><mo>(</mo><mn>1</mn><mo>/</mo><mi>ε</mi><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula> orthogonal gates taken from an inverse-free set.https://www.mdpi.com/2813-0324/7/1/12quantum computationquantum gatesapplications of group representations to physics |
| spellingShingle | Anuradha Mahasinghe Dulmi Fernando Kaushika De Silva Qutrit-Based Orthogonal Approximations with Inverse-Free Quantum Gate Sets Computer Sciences & Mathematics Forum quantum computation quantum gates applications of group representations to physics |
| title | Qutrit-Based Orthogonal Approximations with Inverse-Free Quantum Gate Sets |
| title_full | Qutrit-Based Orthogonal Approximations with Inverse-Free Quantum Gate Sets |
| title_fullStr | Qutrit-Based Orthogonal Approximations with Inverse-Free Quantum Gate Sets |
| title_full_unstemmed | Qutrit-Based Orthogonal Approximations with Inverse-Free Quantum Gate Sets |
| title_short | Qutrit-Based Orthogonal Approximations with Inverse-Free Quantum Gate Sets |
| title_sort | qutrit based orthogonal approximations with inverse free quantum gate sets |
| topic | quantum computation quantum gates applications of group representations to physics |
| url | https://www.mdpi.com/2813-0324/7/1/12 |
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