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# 量子模拟

See here.

### qPCA

Compute

[ operatorname{Tr}_{1} left(e^{-iSDelta t} rho otimes sigma e^{iSDelta t} right) ]

where (rho) and (sigma) are two density matrix, (S) is the swap operator.

Note that (e^{-iSDelta t} = cos (Delta t) I_1 otimes I_2 - isin(Delta t) S).

First considering that (rho) and (sigma) are density matrix of pure state, i.e, (rho = ket{psi_1} bra{psi_1}, sigma = ket{psi_2} bra{psi_2}),then

[begin{aligned} e^{-iSDelta t} ket{psi_1} ket{psi_2} end{aligned} = cos (Delta t)ket{psi_1} ket{psi_2} - i sin(Delta t) ket{psi_2} ket{psi_1}. ]

So we have

[begin{aligned} & e^{-iSDelta t} rho otimes sigma e^{iSDelta t} \ =& e^{-iSDelta t} ket{psi_1} ket{psi_2} cdot bra{psi_1} bra{psi_2} e^{iSDelta t} \ = & left[cos (Delta t)ket{psi_1} ket{psi_2} - i sin(Delta t) ket{psi_2} ket{psi_1}right] \ cdot & left[ cos (Delta t)bra{psi_1} bra{psi_2} + i sin(Delta t) bra{psi_2} bra{psi_1} right] \ =& cos^2 (Delta t)ket{psi_1} ket{psi_2}bra{psi_1} bra{psi_2} + sin^2(Delta t)ket{psi_2} ket{psi_1} bra{psi_2} bra{psi_1} \ &- i sin(Delta t)cos (Delta t)left[ ket{psi_2} ket{psi_1}bra{psi_1} bra{psi_2} - ket{psi_1} ket{psi_2} bra{psi_2} bra{psi_1} right] end{aligned} ]

Note that (ket{psi_1} ket{psi_2}bra{psi_1} bra{psi_2} = rho otimes sigma, ket{psi_2} ket{psi_1} bra{psi_2} bra{psi_1} = sigma otimes rho), so

[ operatorname{Tr}_{1}ket{psi_1} ket{psi_2}bra{psi_1} bra{psi_2} = sigma, operatorname{Tr}_{1} ket{psi_2} ket{psi_1} bra{psi_2} bra{psi_1} = rho. ]

Now we consider (operatorname{Tr}_{2} ket{psi_2} ket{psi_1}bra{psi_1} bra{psi_2}). We have

[begin{aligned} operatorname{Tr}_{1} ket{psi_2} ket{psi_1}bra{psi_1} bra{psi_2} &= sum_{j} left( bra{j}otimes I right)ket{psi_2} ket{psi_1}bra{psi_1} bra{psi_2}left( ket{j}otimes I right) \ &=sum_{j} ket{psi_1}bra{psi_2} otimes (bra{j} ket{psi_2} bra{psi_1} ket{j} ) \ &= braket{psi_1 mid psi_2} ket{psi_1}bra{psi_2} \ &= rhosigma end{aligned} ]

Similarly, \$ operatorname{Tr}_{1} ket{psi_1} ket{psi_2}bra{psi_2} bra{psi_1} = rho sigma\$. So we have

[begin{aligned} operatorname{Tr}_{1} left(e^{-iSDelta t} rho otimes sigma e^{iSDelta t} right) &= cos^2 (Delta t)sigma + sin^2(Delta t)rho - i sin(Delta t)cos (Delta t)left[ rho, sigma right] \ &=sigma-i Delta t[rho, sigma]+Oleft(Delta t^2right) end{aligned} ]