On the energy landscape of symmetric quantum signal processing
Symmetric quantum signal processing provides a parameterized representation of a real polynomial, which can be translated into an efficient quantum circuit for performing a wide range of computational tasks on quantum computers. For a given polynomial $f$, the parameters (called phase factors) can b...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
2022-11-01
|
| Series: | Quantum |
| Online Access: | https://quantum-journal.org/papers/q-2022-11-03-850/pdf/ |
| _version_ | 1797990141793402880 |
|---|---|
| author | Jiasu Wang Yulong Dong Lin Lin |
| author_facet | Jiasu Wang Yulong Dong Lin Lin |
| author_sort | Jiasu Wang |
| collection | DOAJ |
| description | Symmetric quantum signal processing provides a parameterized representation of a real polynomial, which can be translated into an efficient quantum circuit for performing a wide range of computational tasks on quantum computers. For a given polynomial $f$, the parameters (called phase factors) can be obtained by solving an optimization problem. However, the cost function is non-convex, and has a very complex energy landscape with numerous global and local minima. It is therefore surprising that the solution can be robustly obtained in practice, starting from a fixed initial guess $\Phi^0$ that contains no information of the input polynomial. To investigate this phenomenon, we first explicitly characterize all the global minima of the cost function. We then prove that one particular global minimum (called the maximal solution) belongs to a neighborhood of $\Phi^0$, on which the cost function is strongly convex under the condition ${\left\lVert f\right\rVert}_{\infty}=\mathcal{O}(d^{-1})$ with $d=\mathrm{deg}(f)$. Our result provides a partial explanation of the aforementioned success of optimization algorithms. |
| first_indexed | 2024-04-11T08:31:48Z |
| format | Article |
| id | doaj.art-016311eccae14517949e7b5581666995 |
| institution | Directory Open Access Journal |
| issn | 2521-327X |
| language | English |
| last_indexed | 2024-04-11T08:31:48Z |
| publishDate | 2022-11-01 |
| publisher | Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften |
| record_format | Article |
| series | Quantum |
| spelling | doaj.art-016311eccae14517949e7b55816669952022-12-22T04:34:29ZengVerein zur Förderung des Open Access Publizierens in den QuantenwissenschaftenQuantum2521-327X2022-11-01685010.22331/q-2022-11-03-85010.22331/q-2022-11-03-850On the energy landscape of symmetric quantum signal processingJiasu WangYulong DongLin LinSymmetric quantum signal processing provides a parameterized representation of a real polynomial, which can be translated into an efficient quantum circuit for performing a wide range of computational tasks on quantum computers. For a given polynomial $f$, the parameters (called phase factors) can be obtained by solving an optimization problem. However, the cost function is non-convex, and has a very complex energy landscape with numerous global and local minima. It is therefore surprising that the solution can be robustly obtained in practice, starting from a fixed initial guess $\Phi^0$ that contains no information of the input polynomial. To investigate this phenomenon, we first explicitly characterize all the global minima of the cost function. We then prove that one particular global minimum (called the maximal solution) belongs to a neighborhood of $\Phi^0$, on which the cost function is strongly convex under the condition ${\left\lVert f\right\rVert}_{\infty}=\mathcal{O}(d^{-1})$ with $d=\mathrm{deg}(f)$. Our result provides a partial explanation of the aforementioned success of optimization algorithms.https://quantum-journal.org/papers/q-2022-11-03-850/pdf/ |
| spellingShingle | Jiasu Wang Yulong Dong Lin Lin On the energy landscape of symmetric quantum signal processing Quantum |
| title | On the energy landscape of symmetric quantum signal processing |
| title_full | On the energy landscape of symmetric quantum signal processing |
| title_fullStr | On the energy landscape of symmetric quantum signal processing |
| title_full_unstemmed | On the energy landscape of symmetric quantum signal processing |
| title_short | On the energy landscape of symmetric quantum signal processing |
| title_sort | on the energy landscape of symmetric quantum signal processing |
| url | https://quantum-journal.org/papers/q-2022-11-03-850/pdf/ |
| work_keys_str_mv | AT jiasuwang ontheenergylandscapeofsymmetricquantumsignalprocessing AT yulongdong ontheenergylandscapeofsymmetricquantumsignalprocessing AT linlin ontheenergylandscapeofsymmetricquantumsignalprocessing |