Progress towards Analytically Optimal Angles in Quantum Approximate Optimisation
The quantum approximate optimisation algorithm is a <i>p</i> layer, time variable split operator method executed on a quantum processor and driven to convergence by classical outer-loop optimisation. The classical co-processor varies individual application times of a problem/driver propa...
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MDPI AG
2022-07-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/15/2601 |
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| author | Daniil Rabinovich Richik Sengupta Ernesto Campos Vishwanathan Akshay Jacob Biamonte |
| author_facet | Daniil Rabinovich Richik Sengupta Ernesto Campos Vishwanathan Akshay Jacob Biamonte |
| author_sort | Daniil Rabinovich |
| collection | DOAJ |
| description | The quantum approximate optimisation algorithm is a <i>p</i> layer, time variable split operator method executed on a quantum processor and driven to convergence by classical outer-loop optimisation. The classical co-processor varies individual application times of a problem/driver propagator sequence to prepare a state which approximately minimises the problem’s generator. Analytical solutions to choose optimal application times (called parameters or angles) have proven difficult to find, whereas outer-loop optimisation is resource intensive. Here we prove that the optimal quantum approximate optimisation algorithm parameters for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> layer reduce to one free variable and in the thermodynamic limit, we recover optimal angles. We moreover demonstrate that conditions for vanishing gradients of the overlap function share a similar form which leads to a linear relation between circuit parameters, independent of the number of qubits. Finally, we present a list of numerical effects, observed for particular system size and circuit depth, which are yet to be explained analytically. |
| first_indexed | 2024-03-09T10:07:13Z |
| format | Article |
| id | doaj.art-3739cccc74754d8abc37818f2842b65f |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T10:07:13Z |
| publishDate | 2022-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-3739cccc74754d8abc37818f2842b65f2023-12-01T23:01:53ZengMDPI AGMathematics2227-73902022-07-011015260110.3390/math10152601Progress towards Analytically Optimal Angles in Quantum Approximate OptimisationDaniil Rabinovich0Richik Sengupta1Ernesto Campos2Vishwanathan Akshay3Jacob Biamonte4Laboratory of Quantum Algorithms for Machine Learning and Optimisation, Skolkovo Institute of Science and Technology, 3 Nobel Street, 121205 Moscow, RussiaLaboratory of Quantum Algorithms for Machine Learning and Optimisation, Skolkovo Institute of Science and Technology, 3 Nobel Street, 121205 Moscow, RussiaLaboratory of Quantum Algorithms for Machine Learning and Optimisation, Skolkovo Institute of Science and Technology, 3 Nobel Street, 121205 Moscow, RussiaLaboratory of Quantum Algorithms for Machine Learning and Optimisation, Skolkovo Institute of Science and Technology, 3 Nobel Street, 121205 Moscow, RussiaLaboratory of Quantum Algorithms for Machine Learning and Optimisation, Skolkovo Institute of Science and Technology, 3 Nobel Street, 121205 Moscow, RussiaThe quantum approximate optimisation algorithm is a <i>p</i> layer, time variable split operator method executed on a quantum processor and driven to convergence by classical outer-loop optimisation. The classical co-processor varies individual application times of a problem/driver propagator sequence to prepare a state which approximately minimises the problem’s generator. Analytical solutions to choose optimal application times (called parameters or angles) have proven difficult to find, whereas outer-loop optimisation is resource intensive. Here we prove that the optimal quantum approximate optimisation algorithm parameters for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> layer reduce to one free variable and in the thermodynamic limit, we recover optimal angles. We moreover demonstrate that conditions for vanishing gradients of the overlap function share a similar form which leads to a linear relation between circuit parameters, independent of the number of qubits. Finally, we present a list of numerical effects, observed for particular system size and circuit depth, which are yet to be explained analytically.https://www.mdpi.com/2227-7390/10/15/2601variatonal algorithmsQAOAquantum circuit optimization |
| spellingShingle | Daniil Rabinovich Richik Sengupta Ernesto Campos Vishwanathan Akshay Jacob Biamonte Progress towards Analytically Optimal Angles in Quantum Approximate Optimisation Mathematics variatonal algorithms QAOA quantum circuit optimization |
| title | Progress towards Analytically Optimal Angles in Quantum Approximate Optimisation |
| title_full | Progress towards Analytically Optimal Angles in Quantum Approximate Optimisation |
| title_fullStr | Progress towards Analytically Optimal Angles in Quantum Approximate Optimisation |
| title_full_unstemmed | Progress towards Analytically Optimal Angles in Quantum Approximate Optimisation |
| title_short | Progress towards Analytically Optimal Angles in Quantum Approximate Optimisation |
| title_sort | progress towards analytically optimal angles in quantum approximate optimisation |
| topic | variatonal algorithms QAOA quantum circuit optimization |
| url | https://www.mdpi.com/2227-7390/10/15/2601 |
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