Fixed-Point Grover Adaptive Search for Quadratic Binary Optimization Problems
In this article, we study a Grover-type method for quadratic unconstrained binary optimization (QUBO) problems. For an <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula>-dimensional QUBO problem with <inline-formula><tex-math notatio...
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2024-01-01
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| Series: | IEEE Transactions on Quantum Engineering |
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| Online Access: | https://ieeexplore.ieee.org/document/10726869/ |
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| author | Akos Nagy Jaime Park Cindy Zhang Atithi Acharya Alex Khan |
| author_facet | Akos Nagy Jaime Park Cindy Zhang Atithi Acharya Alex Khan |
| author_sort | Akos Nagy |
| collection | DOAJ |
| description | In this article, we study a Grover-type method for quadratic unconstrained binary optimization (QUBO) problems. For an <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula>-dimensional QUBO problem with <inline-formula><tex-math notation="LaTeX">$m$</tex-math></inline-formula> nonzero terms, we construct a marker oracle for such problems with a tunable parameter, <inline-formula><tex-math notation="LaTeX">$\Lambda \in [ 1, m ] \cap \mathbb {Z}$</tex-math></inline-formula>. At <inline-formula><tex-math notation="LaTeX">$d \in \mathbb {Z}_+$</tex-math></inline-formula> precision, the oracle uses <inline-formula><tex-math notation="LaTeX">$O (n + \Lambda d)$</tex-math></inline-formula> qubits and has total depth of <inline-formula><tex-math notation="LaTeX">$O (\frac{m}{\Lambda } \log _{2} (n) + \log _{2} (d))$</tex-math></inline-formula> and a non-Clifford depth of <inline-formula><tex-math notation="LaTeX">$O (\frac{m}{\Lambda })$</tex-math></inline-formula>. Moreover, each qubit is required to be connected to at most <inline-formula><tex-math notation="LaTeX">$O (\log _{2} (\Lambda + d))$</tex-math></inline-formula> other qubits. In the case of a maximum graph cuts, as <inline-formula><tex-math notation="LaTeX">$d = 2 \left\lceil \log _{2} (n) \right\rceil$</tex-math></inline-formula> always suffices, the depth of the marker oracle can be made as shallow as <inline-formula><tex-math notation="LaTeX">$O (\log _{2} (n))$</tex-math></inline-formula>. For all values of <inline-formula><tex-math notation="LaTeX">$\Lambda$</tex-math></inline-formula>, the non-Clifford gate count of these oracles is strictly lower (at least by a factor of <inline-formula><tex-math notation="LaTeX">$\sim 2$</tex-math></inline-formula>) than previous constructions. Furthermore, we introduce a novel fixed-point Grover adaptive search for QUBO problems, using our oracle design and a hybrid fixed-point Grover search, motivated by the works of Boyer et al. (1988) and Li et al. (2019). This method has better performance guarantees than previous Grover adaptive search methods. Some of our results are novel and useful for any method based on the fixed-point Grover search. Finally, we give a heuristic argument that, with high probability and in <inline-formula><tex-math notation="LaTeX">$O (\frac{\log _{2} (n)}{\sqrt{\epsilon }})$</tex-math></inline-formula> time, this adaptive method finds a configuration that is among the best <inline-formula><tex-math notation="LaTeX">$\epsilon 2^{n}$</tex-math></inline-formula> ones. |
| format | Article |
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| institution | Kabale University |
| issn | 2689-1808 |
| language | English |
| publishDate | 2024-01-01 |
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| series | IEEE Transactions on Quantum Engineering |
| spelling | doaj-art-4d92e74dd4834638b0156df17a41694f2025-01-25T00:03:42ZengIEEEIEEE Transactions on Quantum Engineering2689-18082024-01-01511210.1109/TQE.2024.348465010726869Fixed-Point Grover Adaptive Search for Quadratic Binary Optimization ProblemsAkos Nagy0https://orcid.org/0000-0002-1799-7631Jaime Park1https://orcid.org/0009-0009-7806-5211Cindy Zhang2Atithi Acharya3Alex Khan4https://orcid.org/0000-0003-0324-1566BEIT Canada, Toronto, ON, CanadaVanderbilt University, Nashville, TN, USACindy Zhang resides in, Piscataway, NJ, USARutgers, The State University of New Jersey, New Brunswick, NJ, USANational Quantum Laboratory, University of Maryland, College Park, MD, USAIn this article, we study a Grover-type method for quadratic unconstrained binary optimization (QUBO) problems. For an <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula>-dimensional QUBO problem with <inline-formula><tex-math notation="LaTeX">$m$</tex-math></inline-formula> nonzero terms, we construct a marker oracle for such problems with a tunable parameter, <inline-formula><tex-math notation="LaTeX">$\Lambda \in [ 1, m ] \cap \mathbb {Z}$</tex-math></inline-formula>. At <inline-formula><tex-math notation="LaTeX">$d \in \mathbb {Z}_+$</tex-math></inline-formula> precision, the oracle uses <inline-formula><tex-math notation="LaTeX">$O (n + \Lambda d)$</tex-math></inline-formula> qubits and has total depth of <inline-formula><tex-math notation="LaTeX">$O (\frac{m}{\Lambda } \log _{2} (n) + \log _{2} (d))$</tex-math></inline-formula> and a non-Clifford depth of <inline-formula><tex-math notation="LaTeX">$O (\frac{m}{\Lambda })$</tex-math></inline-formula>. Moreover, each qubit is required to be connected to at most <inline-formula><tex-math notation="LaTeX">$O (\log _{2} (\Lambda + d))$</tex-math></inline-formula> other qubits. In the case of a maximum graph cuts, as <inline-formula><tex-math notation="LaTeX">$d = 2 \left\lceil \log _{2} (n) \right\rceil$</tex-math></inline-formula> always suffices, the depth of the marker oracle can be made as shallow as <inline-formula><tex-math notation="LaTeX">$O (\log _{2} (n))$</tex-math></inline-formula>. For all values of <inline-formula><tex-math notation="LaTeX">$\Lambda$</tex-math></inline-formula>, the non-Clifford gate count of these oracles is strictly lower (at least by a factor of <inline-formula><tex-math notation="LaTeX">$\sim 2$</tex-math></inline-formula>) than previous constructions. Furthermore, we introduce a novel fixed-point Grover adaptive search for QUBO problems, using our oracle design and a hybrid fixed-point Grover search, motivated by the works of Boyer et al. (1988) and Li et al. (2019). This method has better performance guarantees than previous Grover adaptive search methods. Some of our results are novel and useful for any method based on the fixed-point Grover search. Finally, we give a heuristic argument that, with high probability and in <inline-formula><tex-math notation="LaTeX">$O (\frac{\log _{2} (n)}{\sqrt{\epsilon }})$</tex-math></inline-formula> time, this adaptive method finds a configuration that is among the best <inline-formula><tex-math notation="LaTeX">$\epsilon 2^{n}$</tex-math></inline-formula> ones.https://ieeexplore.ieee.org/document/10726869/Quantum computingquantum algorithmoptimization |
| spellingShingle | Akos Nagy Jaime Park Cindy Zhang Atithi Acharya Alex Khan Fixed-Point Grover Adaptive Search for Quadratic Binary Optimization Problems IEEE Transactions on Quantum Engineering Quantum computing quantum algorithm optimization |
| title | Fixed-Point Grover Adaptive Search for Quadratic Binary Optimization Problems |
| title_full | Fixed-Point Grover Adaptive Search for Quadratic Binary Optimization Problems |
| title_fullStr | Fixed-Point Grover Adaptive Search for Quadratic Binary Optimization Problems |
| title_full_unstemmed | Fixed-Point Grover Adaptive Search for Quadratic Binary Optimization Problems |
| title_short | Fixed-Point Grover Adaptive Search for Quadratic Binary Optimization Problems |
| title_sort | fixed point grover adaptive search for quadratic binary optimization problems |
| topic | Quantum computing quantum algorithm optimization |
| url | https://ieeexplore.ieee.org/document/10726869/ |
| work_keys_str_mv | AT akosnagy fixedpointgroveradaptivesearchforquadraticbinaryoptimizationproblems AT jaimepark fixedpointgroveradaptivesearchforquadraticbinaryoptimizationproblems AT cindyzhang fixedpointgroveradaptivesearchforquadraticbinaryoptimizationproblems AT atithiacharya fixedpointgroveradaptivesearchforquadraticbinaryoptimizationproblems AT alexkhan fixedpointgroveradaptivesearchforquadraticbinaryoptimizationproblems |