More Grover given In l y Grover's circuit applies a phase shift on t so that we are left with Iu Il S IUS are the input bits I or equivalently the Il S the phase is applied on the whole circuit because I7 is us entangled with the rest of the State We can ignore it in analysis we can think of I 7 as a local Variable where we use it the perform operations and discard 1 7 after the operation tamiltonian Simulations important for simulating quantum physics e.g simulating molecules physical laws are often differential equations describe the rate of change of the system solvin the eq determines behavior e g 45 axle where the solution is Nfl eat x o rate of change is a constant times current value Vectorized we have f AE with solution eat 6 It is some matrix eat will make sense when we explore the solution of Schrodinger's equation Schpj dinger's equation diaglacianas diag Cbc bubs diaglais azbuazbs describes quantum mechanical systems Ix CH iHWH LEEE IEEE qbjafga functions with matrix arguments let f X a ta X ta X't asX3t FEyt Iriesumeefptansion that converges to fix suppose X V diag W Wr Vt be the eigenvalue decomposition D diag w Nr and DE digg Cwp WE we see that X2s DVtJLVDvtj VD2vto.so XK VDKvt by induction XO V diag Wo w Vt VIVt I so f X dy V IV t ta VX'VttazVX2Vtt V LAI ta X taz ft Vt d ag a Wy down diag la Wii Iac Wr drag lazuli ya up Vt U diag cloth w ta w t dota Wutazwft Vt V diag flu f Wr Vt we can see that applying f to X applies f to the eigenvalues of X X must have eigenvalue decomposition
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
More Grovergiven In l y Grover's circuit applies a phase shift on t so that
we are left with Iu Il S
IUS are the input bits Ior equivalently the Il S the phase isapplied on
the wholecircuit
because I 7 is usentangled with therest of the State We can
ignore it in analysiswe can think of I 7 as a local Variable where
we use it
the perform operationsand discard 1 7 after the operation
tamiltonian Simulationsimportant forsimulatingquantum physics
e.g simulatingmolecules
physical laws are oftendifferential equations
describe the rate of changeof the system
solvin the eq determinesbehavior
e g45 axle where the solution
is Nfl eat x o
rate of changeis a constant times
current value
Vectorized we have f AE with solution eat 6It is some matrixeat will make sense when
we explorethesolution of Schrodinger's equation
Schpjdinger's equation diaglacianasdiagCbcbubs diaglais azbuazbsdescribesquantum
mechanical systems
Ix CH iHWH LEEE IEEE qbjafgafunctions with matrix argumentslet f X a ta X ta X't asX3t
FEyt Iriesumeefptansion that converges to fixsuppose X Vdiag W Wr Vtbe the eigenvalue decompositionD diag w Nr and DEdiggCwp WEwe see that X2s DVtJLVDvtj
VD2vto.soXK VDKvt by inductionXO Vdiag Wo w Vt VIVt I
so f X dyV IV t taVX'VttazVX2Vtt
V LAI ta X taz ftVt
dag a Wy down diaglaWii IacWr draglazuli ya up Vt
U diag cloth w ta w t dotaWutazwft Vt
V diag flu f Wr Vtwe can see that applying f to X applies f to theeigenvalues of XX must have eigenvalue
decomposition
1 4 eHtt1 6 is the solution to Schrodinger'sequation 1x'Lts j Hilts
thisworks because Taylor series expansion of eix converges
at H's eigenvalues tvtsooapply e it to each of H's eigenvalues to get
matrix H'It Vei
and so the solutionis 1 4 H'A x o
e i converges everywhereIt has real eigenvalues iff Ht H H is Hermitian
if w is an eigenvalue of H then EV is an eigenvalue of e IH
if WE1K then le int 1if all eigenvalues of it are real then
e IH is unitary
Ht Vdiag w w r f t Vdiag E Gr VtJi is the complex conjugate of Wi
diag Ic yTr diag w wr iff wi ERIHamiltonian simulationgivenHermitian matrix H initial state IX 7 and timet preparethe state lxltp e e.atxoSH is the quantumsystem changes with other systemsany reasonableHamiltonianwill beHermitianwe can take time f 1 and ignore t
if H is Hermitian then e IH is unitarythe reverse is true if U is unitary then everyeigenvalue
is
of the form z e in
Computing the log elog Ui log z flogCeow IL i w I 2 w
I togU is Hermitianeigenvalue are
real and theTaylorseriesof log converges
We can think in terms of unitaries or HamiltoniansTgatesif H is Hermitian then U e Hisunitary
Physics
if U is unitary then H itog U is Hermitian
we can think of any gate as an expontial of a Hermitian
every quantum computationis a Hamiltonian simulation
U e i lilog Uit
Related Documents
