An improved formalism for quantum computation based on geometric algebra—case study: Grover’s search algorithm
The Grover search algorithm is one of the two key algorithms in the field of quantum computing, and hence it is desirable to represent it in the simplest and most intuitive formalism possible. We show firstly, that Clifford’s geometric algebra, provides a significantly simpler representation than th...
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| Veröffentlicht in: | Quantum information processing Jg. 12; H. 4; S. 1719 - 1735 |
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| Abstract | The Grover search algorithm is one of the two key algorithms in the field of quantum computing, and hence it is desirable to represent it in the simplest and most intuitive formalism possible. We show firstly, that Clifford’s geometric algebra, provides a significantly simpler representation than the conventional bra-ket notation, and secondly, that the basis defined by the states of maximum and minimum weight in the Grover search space, allows a simple visualization of the Grover search analogous to the precession of a spin-
particle. Using this formalism we efficiently solve the exact search problem, as well as easily representing more general search situations. We do not claim the development of an improved algorithm, but show in a tutorial paper that geometric algebra provides extremely compact and elegant expressions with improved clarity for the Grover search algorithm. Being a key algorithm in quantum computing and one of the most studied, it forms an ideal basis for a tutorial on how to elucidate quantum operations in terms of geometric algebra—this is then of interest in extending the applicability of geometric algebra to more complicated problems in fields of quantum computing, quantum decision theory, and quantum information. |
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| AbstractList | The Grover search algorithm is one of the two key algorithms in the field of quantum computing, and hence it is desirable to represent it in the simplest and most intuitive formalism possible. We show firstly, that Clifford’s geometric algebra, provides a significantly simpler representation than the conventional bra-ket notation, and secondly, that the basis defined by the states of maximum and minimum weight in the Grover search space, allows a simple visualization of the Grover search analogous to the precession of a spin-
particle. Using this formalism we efficiently solve the exact search problem, as well as easily representing more general search situations. We do not claim the development of an improved algorithm, but show in a tutorial paper that geometric algebra provides extremely compact and elegant expressions with improved clarity for the Grover search algorithm. Being a key algorithm in quantum computing and one of the most studied, it forms an ideal basis for a tutorial on how to elucidate quantum operations in terms of geometric algebra—this is then of interest in extending the applicability of geometric algebra to more complicated problems in fields of quantum computing, quantum decision theory, and quantum information. |
| Author | Iqbal, Azhar Lohe, M. A. von Smekal, Lorenz Chappell, James M. Abbott, Derek |
| Author_xml | – sequence: 1 givenname: James M. surname: Chappell fullname: Chappell, James M. email: james.m.chappell@adelaide.edu.au organization: School of Electrical and Electronic Engineering, University of Adelaide – sequence: 2 givenname: Azhar surname: Iqbal fullname: Iqbal, Azhar organization: School of Electrical and Electronic Engineering, University of Adelaide – sequence: 3 givenname: M. A. surname: Lohe fullname: Lohe, M. A. organization: School of Chemistry and Physics, University of Adelaide – sequence: 4 givenname: Lorenz surname: von Smekal fullname: von Smekal, Lorenz organization: Institut für Kernphysik, Technische Universität Darmstadt – sequence: 5 givenname: Derek surname: Abbott fullname: Abbott, Derek organization: School of Electrical and Electronic Engineering, University of Adelaide |
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| Cites_doi | 10.1103/PhysRevA.60.2746 10.1103/PhysRevA.63.054302 10.1103/PhysRevA.60.2742 10.1103/PhysRevLett.79.325 10.1371/journal.pone.0036404 10.1143/PTP.116.783 10.1088/0305-4470/34/4/312 10.1007/s10773-008-9826-7 10.1088/1751-8113/40/13/F01 10.1103/PhysRevA.66.062301 10.1017/CBO9780511807497 10.1016/S0375-9601(98)00010-3 10.1016/S0026-2692(01)00116-1 10.1002/(SICI)1521-3978(199806)46:4/5<493::AID-PROP493>3.0.CO;2-P 10.1103/PhysRevA.71.042320 10.1088/1751-8113/42/13/135307 10.1103/PhysRevA.62.052304 10.1007/978-94-009-6292-7 10.1109/ISTCS.1997.595153 10.1007/s00006-010-0206-z 10.1103/PhysRevLett.95.150501 10.1103/PhysRevA.77.012316 10.1145/276698.276712 10.1103/PhysRevA.63.012310 10.1119/1.1359518 10.1103/PhysRevLett.80.4329 10.1103/PhysRevA.65.052322 10.1143/JPSJ.78.054801 10.1103/PhysRevA.65.034305 10.1007/978-94-009-4728-3_27 10.1016/S0375-9601(99)00631-3 |
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| Issue | 4 |
| Keywords | Geometric algebra Grover search algorithm Quantum algorithms Quantum computing Courseware Visualization Quantum algorithm Decision making Ideal Quantum computer Grover algorithm Search algorithm Clifford algebra Quantum information Algebraic geometry |
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| References | LongG.LiY.ZhangW.NiuL.Phase matching in quantum searchingPhys. Lett. A19992621273417320771999PhLA..262...27L1059.8151810.1016/S0375-9601(99)00631-3 Brassard, G., Hoyer, P.: An exact quantum polynomial-time algorithm for simon’s problem. In: Proceedings of the 5th Israeli Symposium on Theory of Computing and Systems ISTCS, pp. 12–23 (1997) HestenesD.SobczykG.Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, vol. 51984BerlinSpringer10.1007/978-94-009-6292-7 AertsD.CzachorM.OrlowskiL.Teleportation of geometric structures in 3DJ. Phys. A Math. Theor.200942131353072009JPhA...42m5307A10.1088/1751-8113/42/13/135307 BihamE.KenigsbergD.Grover’s quantum search algorithm for an arbitrary initial mixed statePhys. Rev. A2002660623012002PhRvA..66f2301B10.1103/PhysRevA.66.062301 GregoričM.Mankoč BorštnikN.Quantum gates and quantum algorithms with clifford algebra techniquesInt. J. Theor. Phys.20094825075151162.8134210.1007/s10773-008-9826-7 ShapiraD.ShimoniY.BihamO.Algebraic analysis of quantum search with pure and mixed statesPhys. Rev. A20057104232021437692005PhRvA..71d2320S10.1103/PhysRevA.71.042320 KorepinV.E.ValliloB.C.Group theoretical formulation of a quantum partial search algorithmProg. Theor. Phys.200611678379322984112006PThPh.116..783K1115.8132610.1143/PTP.116.783 Grover, L.: A framework for fast quantum mechanical algorithms. In: Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, ACM, pp. 53–62 (1998) ParkerR.DoranC.Analysis of One and Two Particle Quantum Systems Using Geometric Algebra2002Boston, MABirkhäuser213226 HøyerP.Arbitrary phases in quantum amplitude amplificationPhys. Rev. A20006250523040523092000PhRvA..62e2302Y10.1103/PhysRevA.62.052304 AlvesR.LavorC.Clifford algebra applied to Grover’s algorithmAdv. Appl. Clifford Algebras20102047748827376011210.8101710.1007/s00006-010-0206-z ChappellJ.IqbalA.LoheM.Von SmekalL.An analysis of the quantum penny flip game using geometric algebraJ. Phys. Soc. Jpn.200978554801548042009JPSJ...78E4801C10.1143/JPSJ.78.054801 SomarooS.CoryD.HavelT.Expressing the operations of quantum computing in multiparticle geometric algebraPhys. Lett. A19982401–21716196161998PhLA..240....1S1044.8153110.1016/S0375-9601(98)00010-3 AertsD.CzachorM.Tensor-product versus geometric-product codingPhys. Rev. A20087701231624871772008PhRvA..77a2316A10.1103/PhysRevA.77.012316 ChappellJ.M.IqbalA.AbbottD.N-player quantum games in an EPR settingPLoS ONE201275e364042012PLoSO...736404C10.1371/journal.pone.0036404 De SabbataV.DattaB.Geometric Algebra and Applications to Physics2007LondonTaylor & Francis Group1128.81002 GroverL.Quantum computers can search rapidly by using almost any transformationPhys. Rev. Lett.19988019432943321998PhRvL..80.4329G10.1103/PhysRevLett.80.4329 NgJ.AbbottD.Introduction to solid-state quantum computation for engineersMicroelectron. J.2002331–217117710.1016/S0026-2692(01)00116-1 DoranC.LasenbyA.Geometric Algebra for Physicists2003CambridgeCambridge University Press1078.5300110.1017/CBO9780511807497 Hestenes, D.: Clifford Algebras and Their Applications in Mathematical Physics (Reidel, Dordrecht/Boston, 1986), chap. Clifford Algebra and the interpretation of quantum mechanics (1986) LiC.HwangC.HsiehJ.WangK.General phase-matching condition for a quantum searching algorithmPhys. Rev. A20026530343052002PhRvA..65c4305L10.1103/PhysRevA.65.034305 BoyerM.BrassardG.HøyerP.TappaA.Tight bounds on quantum searchingFortsch. Phys.199846/494935061998ForPh..46..493B10.1002/(SICI)1521-3978(199806)46:4/5<493::AID-PROP493>3.0.CO;2-P NielsenM.ChuangI.Quantum Computation and Quantum Information, 1st edn2002CambridgeAddison-Wesley GroverL.Quantum mechanics helps in searching for a needle in a haystackPhys. Rev. Lett.19977923253281997PhRvL..79..325G10.1103/PhysRevLett.79.325 LongG.TuC.LiY.ZhangW.YanH.An SO(3) picture for quantum searchingJ. Phys. A Math. Gen.200134486186618260302001JPhA...34..861L1034.8100710.1088/0305-4470/34/4/312 Vlasov, A.Y.: eprint arXiv:quant-ph/9907079 (1999) BihamE.BihamO.BironD.GrasslM.LidarD.Grover’s quantum search algorithm for an arbitrary initial amplitude distributionPhys. Rev. A199960427421999PhRvA..60.2742B10.1103/PhysRevA.60.2742 BihamE.BihamO.BironD.GrasslM.LidarD.A.ShapiraD.Analysis of generalized Grover quantum search algorithms using recursion equationsPhys. Rev. A2000630123102001PhRvA..63a2310B10.1103/PhysRevA.63.012310 VlasovA.Y.Clifford algebras and universal sets of quantum gatesPhys. Rev. A20016305430218424252001PhRvA..63e4302V10.1103/PhysRevA.63.054302 GroverL.From schrödingers equation to the quantum search algorithmAm. J. Phys.20016977697772001AmJPh..69..769G10.1119/1.1359518 GroverL.K.Fixed-point quantum searchPhys. Rev. Lett.2005951505011505042005PhRvL..95o0501G10.1103/PhysRevLett.95.150501 HsiehJ.LiC.General su(2) formulation for quantum searching with certaintyPhys. Rev. A2002650523222002PhRvA..65e2322H10.1103/PhysRevA.65.052322 ZalkaC.Grover’s quantum searching algorithm is optimalPhys. Rev. A199960274627511999PhRvA..60.2746Z10.1103/PhysRevA.60.2746 AertsD.CzachorM.Cartoon computation: quantum-like computing without quantum mechanicsJ. Phys. A Math. Theor.20074013F25923233512007JPhA...40..259A1114.8101910.1088/1751-8113/40/13/F01 D. Shapira (483_CR28) 2005; 71 V.E. Korepin (483_CR34) 2006; 116 D. Aerts (483_CR16) 2009; 42 J. Hsieh (483_CR20) 2002; 65 G. Long (483_CR8) 2001; 34 E. Biham (483_CR32) 1999; 60 J.M. Chappell (483_CR19) 2012; 7 M. Gregorič (483_CR11) 2009; 48 L. Grover (483_CR3) 2001; 69 S. Somaroo (483_CR10) 1998; 240 P. Høyer (483_CR30) 2000; 62 J. Chappell (483_CR7) 2009; 78 E. Biham (483_CR33) 2002; 66 483_CR13 483_CR1 M. Nielsen (483_CR5) 2002 D. Hestenes (483_CR12) 1984 D. Aerts (483_CR14) 2007; 40 L. Grover (483_CR2) 1998; 80 483_CR17 A.Y. Vlasov (483_CR18) 2001; 63 C. Zalka (483_CR26) 1999; 60 R. Parker (483_CR23) 2002 C. Li (483_CR24) 2002; 65 C. Doran (483_CR21) 2003 M. Boyer (483_CR27) 1998; 46/49 E. Biham (483_CR29) 2000; 63 483_CR31 J. Ng (483_CR6) 2002; 33 L.K. Grover (483_CR35) 2005; 95 D. Aerts (483_CR15) 2008; 77 R. Alves (483_CR9) 2010; 20 G. Long (483_CR25) 1999; 262 L. Grover (483_CR4) 1997; 79 V. De Sabbata (483_CR22) 2007 |
| References_xml | – reference: GroverL.Quantum computers can search rapidly by using almost any transformationPhys. Rev. Lett.19988019432943321998PhRvL..80.4329G10.1103/PhysRevLett.80.4329 – reference: ParkerR.DoranC.Analysis of One and Two Particle Quantum Systems Using Geometric Algebra2002Boston, MABirkhäuser213226 – reference: VlasovA.Y.Clifford algebras and universal sets of quantum gatesPhys. Rev. A20016305430218424252001PhRvA..63e4302V10.1103/PhysRevA.63.054302 – reference: HøyerP.Arbitrary phases in quantum amplitude amplificationPhys. Rev. A20006250523040523092000PhRvA..62e2302Y10.1103/PhysRevA.62.052304 – reference: AertsD.CzachorM.Cartoon computation: quantum-like computing without quantum mechanicsJ. Phys. A Math. Theor.20074013F25923233512007JPhA...40..259A1114.8101910.1088/1751-8113/40/13/F01 – reference: GroverL.Quantum mechanics helps in searching for a needle in a haystackPhys. Rev. Lett.19977923253281997PhRvL..79..325G10.1103/PhysRevLett.79.325 – reference: HestenesD.SobczykG.Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, vol. 51984BerlinSpringer10.1007/978-94-009-6292-7 – reference: Vlasov, A.Y.: eprint arXiv:quant-ph/9907079 (1999) – reference: ShapiraD.ShimoniY.BihamO.Algebraic analysis of quantum search with pure and mixed statesPhys. Rev. A20057104232021437692005PhRvA..71d2320S10.1103/PhysRevA.71.042320 – reference: HsiehJ.LiC.General su(2) formulation for quantum searching with certaintyPhys. Rev. A2002650523222002PhRvA..65e2322H10.1103/PhysRevA.65.052322 – reference: Brassard, G., Hoyer, P.: An exact quantum polynomial-time algorithm for simon’s problem. In: Proceedings of the 5th Israeli Symposium on Theory of Computing and Systems ISTCS, pp. 12–23 (1997) – reference: LiC.HwangC.HsiehJ.WangK.General phase-matching condition for a quantum searching algorithmPhys. Rev. A20026530343052002PhRvA..65c4305L10.1103/PhysRevA.65.034305 – reference: ZalkaC.Grover’s quantum searching algorithm is optimalPhys. Rev. A199960274627511999PhRvA..60.2746Z10.1103/PhysRevA.60.2746 – reference: GroverL.From schrödingers equation to the quantum search algorithmAm. J. Phys.20016977697772001AmJPh..69..769G10.1119/1.1359518 – reference: KorepinV.E.ValliloB.C.Group theoretical formulation of a quantum partial search algorithmProg. Theor. Phys.200611678379322984112006PThPh.116..783K1115.8132610.1143/PTP.116.783 – reference: AertsD.CzachorM.Tensor-product versus geometric-product codingPhys. Rev. A20087701231624871772008PhRvA..77a2316A10.1103/PhysRevA.77.012316 – reference: BihamE.BihamO.BironD.GrasslM.LidarD.Grover’s quantum search algorithm for an arbitrary initial amplitude distributionPhys. Rev. A199960427421999PhRvA..60.2742B10.1103/PhysRevA.60.2742 – reference: NielsenM.ChuangI.Quantum Computation and Quantum Information, 1st edn2002CambridgeAddison-Wesley – reference: ChappellJ.IqbalA.LoheM.Von SmekalL.An analysis of the quantum penny flip game using geometric algebraJ. Phys. Soc. Jpn.200978554801548042009JPSJ...78E4801C10.1143/JPSJ.78.054801 – reference: ChappellJ.M.IqbalA.AbbottD.N-player quantum games in an EPR settingPLoS ONE201275e364042012PLoSO...736404C10.1371/journal.pone.0036404 – reference: GregoričM.Mankoč BorštnikN.Quantum gates and quantum algorithms with clifford algebra techniquesInt. J. Theor. Phys.20094825075151162.8134210.1007/s10773-008-9826-7 – reference: GroverL.K.Fixed-point quantum searchPhys. Rev. Lett.2005951505011505042005PhRvL..95o0501G10.1103/PhysRevLett.95.150501 – reference: AertsD.CzachorM.OrlowskiL.Teleportation of geometric structures in 3DJ. Phys. A Math. Theor.200942131353072009JPhA...42m5307A10.1088/1751-8113/42/13/135307 – reference: De SabbataV.DattaB.Geometric Algebra and Applications to Physics2007LondonTaylor & Francis Group1128.81002 – reference: NgJ.AbbottD.Introduction to solid-state quantum computation for engineersMicroelectron. J.2002331–217117710.1016/S0026-2692(01)00116-1 – reference: BoyerM.BrassardG.HøyerP.TappaA.Tight bounds on quantum searchingFortsch. Phys.199846/494935061998ForPh..46..493B10.1002/(SICI)1521-3978(199806)46:4/5<493::AID-PROP493>3.0.CO;2-P – reference: BihamE.KenigsbergD.Grover’s quantum search algorithm for an arbitrary initial mixed statePhys. Rev. A2002660623012002PhRvA..66f2301B10.1103/PhysRevA.66.062301 – reference: Hestenes, D.: Clifford Algebras and Their Applications in Mathematical Physics (Reidel, Dordrecht/Boston, 1986), chap. Clifford Algebra and the interpretation of quantum mechanics (1986) – reference: LongG.TuC.LiY.ZhangW.YanH.An SO(3) picture for quantum searchingJ. Phys. A Math. 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| SubjectTerms | Algebra Algorithmics. Computability. Computer arithmetics Applied sciences Associative rings and algebras Classical and quantum physics: mechanics and fields Computer science; control theory; systems Data Structures and Information Theory Exact sciences and technology Mathematical Physics Mathematics Physics Physics and Astronomy Quantum computation Quantum Computing Quantum information Quantum Information Technology Quantum Physics Sciences and techniques of general use Spintronics Theoretical computing |
| Title | An improved formalism for quantum computation based on geometric algebra—case study: Grover’s search algorithm |
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