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