Phase sensitivity of entanglement in the Quantum Phase Estimation algorithm
Abstract We study entanglement in the steps before Quantum Fourier Transform (pre-QFT) part of the Quantum Phase Estimation and the Quantum Counting algorithms (QPEA—QCA) with the use of three entanglement detection tools. In particular we focus on the sensitivity of entanglement to the input value...
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Published in | Physica scripta Vol. 99; no. 9; pp. 95122 - 95135 |
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Format | Journal Article |
Language | English |
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Abstract | Abstract We study entanglement in the steps before Quantum Fourier Transform (pre-QFT) part of the Quantum Phase Estimation and the Quantum Counting algorithms (QPEA—QCA) with the use of three entanglement detection tools. In particular we focus on the sensitivity of entanglement to the input value (the phase ϕ and the ratio of marked elements M N ) in some basic cases. One starts from numerical observations and deduce some general results in particular regarding the classes of entanglement. More precisely, when the second register of both algorithms (i.e. the register on which a specific unitary operator act, see section 2) is initialized in the non-entangled superposition of two (separable) eigenvectors, one proves that the QPEA and QCA curves of entanglement evolution are the same up to a scalar multiplication of the parameter ϕ . One demonstrates that a local minimum is obtained and corresponds to an EPR (Einstein-Podolsky-Rosen) state and finally one proves that, up to Stochastic Local Operation and Classical Communication (SLOCC), all states, except for a few values of ϕ , are equivalent to the product of a separable state and a generalized GHZ (Greenberger-Horen-Zeilinger) state , i.e. GHZ n + 1 = 1 2 ( 00 ... 0 + 11 ... 1 ) . |
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AbstractList | Abstract We study entanglement in the steps before Quantum Fourier Transform (pre-QFT) part of the Quantum Phase Estimation and the Quantum Counting algorithms (QPEA—QCA) with the use of three entanglement detection tools. In particular we focus on the sensitivity of entanglement to the input value (the phase ϕ and the ratio of marked elements M N ) in some basic cases. One starts from numerical observations and deduce some general results in particular regarding the classes of entanglement. More precisely, when the second register of both algorithms (i.e. the register on which a specific unitary operator act, see section 2) is initialized in the non-entangled superposition of two (separable) eigenvectors, one proves that the QPEA and QCA curves of entanglement evolution are the same up to a scalar multiplication of the parameter ϕ . One demonstrates that a local minimum is obtained and corresponds to an EPR (Einstein-Podolsky-Rosen) state and finally one proves that, up to Stochastic Local Operation and Classical Communication (SLOCC), all states, except for a few values of ϕ , are equivalent to the product of a separable state and a generalized GHZ (Greenberger-Horen-Zeilinger) state , i.e. GHZ n + 1 = 1 2 ( 00 ... 0 + 11 ... 1 ) . |
Author | Atchonouglo, Kossi Amouzou, Grâce Holweck, Frédéric |
Author_xml | – sequence: 1 givenname: Grâce orcidid: 0009-0008-2002-7768 surname: Amouzou fullname: Amouzou, Grâce organization: Université de Technologie de Belfort-Montbéliard Laboratoire Interdisciplinaire Carnot de Bourgogne, ICB/UTBM, UMR 6303 CNRS, 90010 Belfort Cedex, France – sequence: 2 givenname: Kossi orcidid: 0000-0002-6483-8780 surname: Atchonouglo fullname: Atchonouglo, Kossi organization: Université de Lomé Laboratoire de Modélisations Mathématiques et Applications, Togo – sequence: 3 givenname: Frédéric orcidid: 0000-0001-6287-1782 surname: Holweck fullname: Holweck, Frédéric organization: Université de Technologie de Belfort-Montbéliard Laboratoire Interdisciplinaire Carnot de Bourgogne, ICB/UTBM, UMR 6303 CNRS, 90010 Belfort Cedex, France |
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References | Einstein (psad6cafbib9) 1935; 47 Bell (psad6cafbib26) 1964; 1 Grover (psad6cafbib3) 1997; 79 Holweck (psad6cafbib13) 2016; 15 Nielsen (psad6cafbib20) 2002; 2 Rossi (psad6cafbib12) 2013; 87 Mermin (psad6cafbib27) 1990; 65 Gnatenko (psad6cafbib29) 2022; 136 Amouzou (psad6cafbib28) 2022; 20 Horodecki (psad6cafbib7) 2009; 81 Jozsa (psad6cafbib6) 1997 Martin (psad6cafbib21) 2021; 3 Most (psad6cafbib24) 2010; 81 Shimoni (psad6cafbib17) 2005; 72 Rebentrost (psad6cafbib22) 2018; 98 Tan (psad6cafbib19) 2020; 59 Zhao (psad6cafbib25) 2015; 14 Susulovska (psad6cafbib30) 2021 Deutsch (psad6cafbib2) 1992; 439 Jaffali (psad6cafbib15) 2019; 18 Aspect (psad6cafbib10) 1982; 49 Kendon (psad6cafbib16) 2004 Bruß (psad6cafbib11) 2011; 83 Bernstein (psad6cafbib1) 1997; 26 Dabrowski (psad6cafbib8) 2017; 4 Bönsel (psad6cafbib18) 2024 Stamatopoulos (psad6cafbib23) 2020; 4 Brassard (psad6cafbib5) 1998; 95 Shor (psad6cafbib4) 1999; 41 de Boutray (psad6cafbib14) 2021; 20 |
References_xml | – volume: 4 start-page: 272 year: 2017 ident: psad6cafbib8 article-title: Einstein-Podolsky-Rosen paradox in a hybrid bipartite system publication-title: Optical Society of America doi: 10.1364/OPTICA.4.000272 contributor: fullname: Dabrowski – volume: 4 start-page: 291 year: 2020 ident: psad6cafbib23 article-title: Option pricing using quantum computers publication-title: Quantum doi: 10.22331/q-2020-07-06-291 contributor: fullname: Stamatopoulos – volume: 15 start-page: 4391 year: 2016 ident: psad6cafbib13 article-title: Grover’s algorithm and the secant varieties publication-title: Quantum Inf. Process. doi: 10.1007/s11128-016-1445-2 contributor: fullname: Holweck – volume: 47 start-page: 777 year: 1935 ident: psad6cafbib9 article-title: Can quantum-mechanical description of physical reality be considered complete? publication-title: Phys. Rev. doi: 10.1103/PhysRev.47.777 contributor: fullname: Einstein – volume: 72 year: 2005 ident: psad6cafbib17 article-title: Entangled quantum states generated by shor’s factoring algorithm publication-title: Phys. Rev. A doi: 10.1103/PhysRevA.72.062308 contributor: fullname: Shimoni – volume: 26 start-page: 1411 year: 1997 ident: psad6cafbib1 article-title: Quantum complexity theory publication-title: SIAM J. Comput. doi: 10.1137/S0097539796300921 contributor: fullname: Bernstein – volume: 2 start-page: 16 year: 2002 ident: psad6cafbib20 article-title: Quantum computation and quantum information publication-title: American Association of Physics Teachers doi: 10.1119/1.1463744 contributor: fullname: Nielsen – volume: 81 year: 2010 ident: psad6cafbib24 article-title: Entanglement of periodic states, the quantum fourier transform, and shor’s factoring algorithm publication-title: Phys. Rev. A doi: 10.1103/PhysRevA.81.052306 contributor: fullname: Most – volume: 41 start-page: 303 year: 1999 ident: psad6cafbib4 article-title: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer publication-title: SIAM Rev. doi: 10.1137/S0036144598347011 contributor: fullname: Shor – volume: 87 year: 2013 ident: psad6cafbib12 article-title: Scale invariance of entanglement dynamics in grover’s quantum search algorithm publication-title: Phys. Rev. A doi: 10.1103/PhysRevA.87.022331 contributor: fullname: Rossi – volume: 79 start-page: 325 year: 1997 ident: psad6cafbib3 article-title: Quantum mechanics helps in searching for a needle in a haystack publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.79.325 contributor: fullname: Grover – year: 2024 ident: psad6cafbib18 article-title: Generating multipartite nonlocality to benchmark quantum computers contributor: fullname: Bönsel – volume: 65 start-page: 1838 year: 1990 ident: psad6cafbib27 article-title: Extreme quantum entanglement in a superposition of macroscopically distinct states publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.65.1838 contributor: fullname: Mermin – volume: 3 year: 2021 ident: psad6cafbib21 article-title: Toward pricing financial derivatives with an ibm quantum computer publication-title: Physical Review Research doi: 10.1103/PhysRevResearch.3.013167 contributor: fullname: Martin – start-page: 465 year: 2021 ident: psad6cafbib30 article-title: Quantifying geometric measure of entanglement of multi-qubit graph states on the ibm’s quantum computer doi: 10.1109/QCE52317.2021.00080 contributor: fullname: Susulovska – volume: 14 start-page: 2861 year: 2015 ident: psad6cafbib25 article-title: A multipartite entanglement measure based on coefficient matrices publication-title: Quantum Inf. Process. doi: 10.1007/s11128-015-1023-z contributor: fullname: Zhao – volume: 49 start-page: 1804 year: 1982 ident: psad6cafbib10 article-title: Experimental test of bell’s inequalities using time-varying analyzers publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.49.1804 contributor: fullname: Aspect – volume: 1 start-page: 195 year: 1964 ident: psad6cafbib26 article-title: On the einstein podolsky rosen paradox publication-title: Physics Physique Fizika doi: 10.1103/PhysicsPhysiqueFizika.1.195 contributor: fullname: Bell – volume: 18 start-page: 1 year: 2019 ident: psad6cafbib15 article-title: Quantum entanglement involved in grover’s and shor’s algorithms: the four-qubit case publication-title: Quantum Inf. Process. doi: 10.1007/s11128-019-2249-y contributor: fullname: Jaffali – year: 1997 ident: psad6cafbib6 contributor: fullname: Jozsa – volume: 98 year: 2018 ident: psad6cafbib22 article-title: Quantum computational finance: Monte carlo pricing of financial derivatives publication-title: Phys. Rev. A doi: 10.1103/PhysRevA.98.022321 contributor: fullname: Rebentrost – volume: 81 start-page: 865 year: 2009 ident: psad6cafbib7 article-title: Quantum entanglement publication-title: Rev. Mod. Phys. doi: 10.1103/RevModPhys.81.865 contributor: fullname: Horodecki – volume: 59 start-page: 1372 year: 2020 ident: psad6cafbib19 article-title: Entanglement in phase estimation algorithm and quantum counting algorithm publication-title: Int. J. Theor. Phys. doi: 10.1007/s10773-019-04341-y contributor: fullname: Tan – volume: 439 start-page: 553 year: 1992 ident: psad6cafbib2 article-title: Rapid solution of problems by quantum computation publication-title: The Royal Society London doi: 10.1098/rspa.1992.0167 contributor: fullname: Deutsch – volume: 20 start-page: 1 year: 2021 ident: psad6cafbib14 article-title: Mermin polynomials for non-locality and entanglement detection in grover’s algorithm and quantum fourier transform publication-title: Quantum Inf. Process. doi: 10.1007/s11128-020-02976-z contributor: fullname: de Boutray – volume: 136 start-page: 40003 year: 2022 ident: psad6cafbib29 article-title: Geometric measure of entanglement of multi-qubit graph states and its detection on a quantum computer publication-title: Europhys. Lett. doi: 10.1209/0295-5075/ac419b contributor: fullname: Gnatenko – volume: 83 year: 2011 ident: psad6cafbib11 article-title: Multipartite entanglement in quantum algorithms publication-title: Phys. Rev. A doi: 10.1103/PhysRevA.83.052313 contributor: fullname: Bruß – volume: 95 start-page: 11032 year: 1998 ident: psad6cafbib5 article-title: Quantum computing publication-title: Proc. Natl Acad. Sci. doi: 10.1073/pnas.95.19.11032 contributor: fullname: Brassard – year: 2004 ident: psad6cafbib16 article-title: Entanglement and its role in shor’s algorithm contributor: fullname: Kendon – volume: 20 year: 2022 ident: psad6cafbib28 article-title: Entanglement and nonlocality of four-qubit connected hypergraph states publication-title: International Journal of Quantum Information doi: 10.1142/S0219749922500010 contributor: fullname: Amouzou |
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Snippet | Abstract We study entanglement in the steps before Quantum Fourier Transform (pre-QFT) part of the Quantum Phase Estimation and the Quantum Counting algorithms... |
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SubjectTerms | coefficient matrices entanglement geometric measure of entanglement mermin polynomials quantum counting algorithm quantum phase estimation algorithm |
Title | Phase sensitivity of entanglement in the Quantum Phase Estimation algorithm |
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