Quantum Simulation: Solving Schrödinger Equation on a Quantum Computer

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1 Purdue Uiversity Purdue e-pubs Birc Poster Sessios Birc Naotechology Ceter Quatum Simulatio: Solvig Schrödiger Equatio o a Quatum Computer Hefeg Wag Purdue Uiversity, wag10@purdue.edu Sabre Kais Birc Naotechology Ceter ad Departmet of Chemistry, Purdue Uiversity, ais@purdue.edu Ala Aspuru-Guzi Harvard Uiversity Follow this ad additioal wors at: Wag, Hefeg; Kais, Sabre; ad Aspuru-Guzi, Ala, "Quatum Simulatio: Solvig Schrödiger Equatio o a Quatum Computer" (008). Birc Poster Sessios. Paper This documet has bee made available through Purdue e-pubs, a service of the Purdue Uiversity Libraries. Please cotact epubs@purdue.edu for additioal iformatio.

2 Quatum Simulatio: Solvig Schrödiger Equatio o a Quatum Computer Hefeg Wag ad Sabre Kais Departmet of Chemistry ad Birc Naotechology Ceter Ala Aspuru-Guzi Departmet of Chemistry ad Chemical Biology, Harvard

3 Why Quatum Computig? Come forth ito the light of thigs, let ature be your teacher --W. Wordsworth o Why ot? If ature is quatum mechaical o Rolf Ladauer: Iformatio is physical o Quatum computatio, quatum commuicatio, quatum teleportatio, superdese codig, quatum cryptography, etc.

5 Bit vs. Qubit Classical Bit State 0 or 1 Measuremets does t chage the state Determiistic result Ca mae a copy of bit Qubit State 0>, 1> or Superpositio Measuremets chage the system Probability results Ca ot cloe the qubit

6 Power of Quatum Computig Quatum parallelism: superpositio priciple 1 i 0 c i i A 1 i 0 c i A i Quatum etaglemet: olocal correlatio c 1, 1, 1 Bell state: Jozef Grusa, Quatum computig, 1999, McGraw-Hill Publishig Compay

7 Shor s Algorithm The difficulty of factorizatio uderpis the security of may commo methods of ecryptio

8 Quatum Simulatio Problems of simulatig quatum systems o a classical computer Expoetial complexity growth of quatum systems. The Schrödiger equatio, which i most cases is too complicated to be exactly solvable. As the Hilbert space icreases expoetially with the icrease of the size of the system, umerical methods lie Cofiguratio Iteractio (CI) are too expesive.

9 Cotiue 198, Feyma: The expoetial complexity of quatum systems might be put to good use to simulate dyamics of aother quatum system. Seth Lloyd: the coecture is correct, it is possible to build uiversal quatum simulators. For a fairly geeral class of quatum systems, especially discrete systems, it is possible to achieve a expoetial speedup. Richard P. Feyma, Simulatig physics with computers. Iter. J. Theo. Phys., 198, 1, Seth Lloyd, Uiversal Quatum Simulators. Sciece, 1996, 73,

10 Solvig Schrödiger Equatio The Schrödiger equatio The solutio i H ˆ t The eergy spectrum P P ( r, t) A u ( r) exp( ie t) () E A ( E ) () t ( r,0) ( r, t) A exp( ie t) E

11 Quatum Phase Estimatio [0,1) To estimate a umber give uitary operator U with eigevector, eigevalue Prepare two qubit registers: idex register ad target register U Perform the cotrolled- o the target register Iverse QFT o the first register, the mae a measuremet. e i

12 Quatum Algorithm to Obtai Eigeeergy through Phase Estimatio Costruct two registers: The idex register ad the target register, a approximate eigevector. 0 0 V a H 1 1 M M 1 0 V a CotrolU 1 M M 1 0 U V a U exp ( iht) V a c

13 Meas QFT M i M M e c M c M c U M 3., ) ( 1 ) ( 1 i e c p V a c t E e e U i iht / ;

14 Basis Set Method HF wave fuctio: oe determiat Mappig the Foc space of the FCI wave fuctio to the Hilbert space of the qubits. Direct mappig: each qubit represets the fermioic occupatio state of a atomic orbital. Occupied or ot. Compact mappig: oly a subspace of Foc space with fixed electros is mapped oto the qubits. Compact (spi) mappig Ala Aspuru-Guzi, et. al Sciece, 005, 309,

15 Problems of Usig HF Wave Fuctio as Iitial Guess Excited states have differet leadig cofiguratios For some complicated curves, several states have to be cosidered simultaeously i order to obtai a accurate descriptio. The success probability is small at regios far from the equilibrium geometry ad avoided crossig.

16 Geeral CI Scheme FCI is too expesive eve o a quatum computer for moderate size molecules. For most chemical problems, the iformatio of groud state ad a few low-lyig excited states is sufficiet. MCSCF provides good iitial guess wave fuctios, which ca be evolved efficietly to the CI wave fuctio. The success probability ca be icreased dramatically at critical regios by usig appropriate MCSCF wave fuctio as iitial guess.

17 MCSCF Wave Fuctio MCSCF wave fuctio MCSCF A K K K K A i ik i μ μ C μ i Averaged MCSCF wave fuctio E H E i w i E i

18 Implemet CI Based o MCSCF Wave Fuctio Mappig the Foc space of the averaged MCSCF wave fuctio to the Hilbert space of the qubits. More compact mappig, cosider the symmetry of the molecule, states belog to differet irreducible represetatio do ot iteract. Usig the MCSCF wave fuctio as the iitial guess, for either the groud or excited state. Probability of gettig the correspodig CI wave fuctio i i is MCSCF

19 Applicatio to Water Molecule Groud state: cc-pvdz basis set, siglet state, Cv symmetry. MCSCF method: freeze first two a1 orbitals, active space, 15 CSFs: MRCI usig the same active space, but the sigle ad double excitatios to the exteral space, 1387 CSFs.

20 Results Hefeg Wag, Sabre Kais, Ala Aspuru-Guzi ad Mar R. Hoffma, Quatum Algorithm for Solvig Schrödiger Equatio to obtai the eergy spectrum. submitted.

21 Cotiue

22 Cotiue

23 Summary Usig the MCSCF wave fuctio as the iitial guess ca improve the success probability dramatically, eve ust a few CSFs. The method ca be geeralized to geeral MRCI scheme ad the etire potetial eergy surface ca be explored for groud state ad excited states. Istead of startig from a sigle elemet as i the HF wave fuctio, MCSCF method starts from a small matrix. This maes the evolutio safer ad faster. The idea ca be geeralized to Fiite elemet method ad ay umerical method.

The time evolution of the state of a quantum system is described by the time-dependent Schrödinger equation (TDSE): ( ) ( ) 2m "2 + V ( r,t) (1.

The time evolution of the state of a quantum system is described by the time-dependent Schrödinger equation (TDSE): ( ) ( ) 2m 2 + V ( r,t) (1. Adrei Tokmakoff, MIT Departmet of Chemistry, 2/13/2007 1-1 574 TIME-DEPENDENT QUANTUM MECHANICS 1 INTRODUCTION 11 Time-evolutio for time-idepedet Hamiltoias The time evolutio of the state of a quatum system