Lou Pagnucco
Aug19-04, 12:40 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Can we use Landau-Zener jumps for quantum computing? -i.e,\n\nIs there a family of time-varying Hamiltonians H(n,t)\nimplementable with polynomial resources such that:\n\n(1) \'n\' is an integer representing a problem instance,\n\'t\' represents time [0 <= t <= 1]\n\n(2) for every instance \'n\', the first three energy eigenvalues\nof H(n,t) are e0(n,t) <= e1(n,t) <= e2(n,t) with\nassociated eigenstates |e0(n,t)>, |e1(n,t)>, |e2(n,t)>\n\n(3) the ground state, |e0(n,0)>, is easy to prepare, and\n|e0(n,1)> encodes the solution of problem instance \'n\'\n\n(4) at times t1 and t2 (0 < t1 < t2 < 1), e0(n,t) and e1(n,t)\nundergo narrowly avoided collisions,\nand for t1<t<t2, e2(n,t) is polynomially bounded above e1(n,t)\n\nIf the system is prepared in |e0(n,0)>, then (with high probability}\nit will hop up to |e1(n,t1)> and then back down to |e0(n,t2)>,\nand on to the \'solution\' |e0(n,1)>.\n\nViewing this as a variant of adiabatic q-computing, the second\nLandau-Zener jump at t2 undoes the damage done by the first jump.\n\n(Note: For brevity, I did not scale time, which should be done.)\n\nDo such families of Hamiltonians exist for real problems,\nor is this just a fancy description of the null set?\n\nThanks,\nLou Pagnucco\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Can we use Landau-Zener jumps for quantum computing? -i.e,
Is there a family of time-varying Hamiltonians H(n,t)
implementable with polynomial resources such that:
(1) 'n' is an integer representing a problem instance,
't' represents time [0 <= t <= 1]
(2) for every instance 'n', the first three energy eigenvalues
of H(n,t) are e0(n,t) <= e1(n,t) <= e2(n,t) with
associated eigenstates |e0(n,t)>, |e1(n,t)>, |e2(n,t)>
(3) the ground state, |e0(n,0)>, is easy to prepare, and
|e0(n,1)> encodes the solution of problem instance 'n'
(4) at times t1 and t2 (0 < t1 < t2 < 1), e0(n,t) and e1(n,t)
undergo narrowly avoided collisions,
and for t1<t<t2, e2(n,t) is polynomially bounded above e1(n,t)
If the system is prepared in |e0(n,0)>, then (with high probability}
it will hop up to |e1(n,t1)> and then back down to |e0(n,t2)>,
and on to the 'solution' |e0(n,1)>.
Viewing this as a variant of adiabatic q-computing, the second
Landau-Zener jump at t2 undoes the damage done by the first jump.
(Note: For brevity, I did not scale time, which should be done.)
Do such families of Hamiltonians exist for real problems,
or is this just a fancy description of the null set?
Thanks,
Lou Pagnucco
Is there a family of time-varying Hamiltonians H(n,t)
implementable with polynomial resources such that:
(1) 'n' is an integer representing a problem instance,
't' represents time [0 <= t <= 1]
(2) for every instance 'n', the first three energy eigenvalues
of H(n,t) are e0(n,t) <= e1(n,t) <= e2(n,t) with
associated eigenstates |e0(n,t)>, |e1(n,t)>, |e2(n,t)>
(3) the ground state, |e0(n,0)>, is easy to prepare, and
|e0(n,1)> encodes the solution of problem instance 'n'
(4) at times t1 and t2 (0 < t1 < t2 < 1), e0(n,t) and e1(n,t)
undergo narrowly avoided collisions,
and for t1<t<t2, e2(n,t) is polynomially bounded above e1(n,t)
If the system is prepared in |e0(n,0)>, then (with high probability}
it will hop up to |e1(n,t1)> and then back down to |e0(n,t2)>,
and on to the 'solution' |e0(n,1)>.
Viewing this as a variant of adiabatic q-computing, the second
Landau-Zener jump at t2 undoes the damage done by the first jump.
(Note: For brevity, I did not scale time, which should be done.)
Do such families of Hamiltonians exist for real problems,
or is this just a fancy description of the null set?
Thanks,
Lou Pagnucco