Definition of *\ast
H *=H¯ TH^\ast = \overline{H}^T
Definition of Hamilton
H *=HH^\ast = H
Definition of Unitary
U *U=IU^\ast U = I
Unitary does not change length
(UV) *(UV)=V *U *UV=V *IV=I(U V)^\ast (U V) = V^\ast U^\ast U V = V^\ast I V = I
Solution of dVdt=−iH⋅V\frac{d V}{d t} = - i H \cdot V is V=e −itHV 0V = e^{- i t H} V_0
V *VV^\ast V is a constant. Accordingly, if V 0V_0 is a unitary matrix or a vector then V tV_t is a unitary or a same-length vector:
dV *Vdt=dV *dtV+V *dVdt=V *(−iH) *V+V *(−iH)V=V *(H *i)V−iV *HV=iV *(H *−H)V=0\frac{d V^\ast V}{d t} = \frac{d V^\ast}{d t} V + V^\ast \frac{d V}{d t} = V^\ast(-i H)^\ast V + V^\ast (-i H) V = V^\ast (H^\ast i) V - i V^\ast H V = i V^\ast (H^\ast - H) V = 0
X
Y
Z
H
CCNOT
CNOT
Rx
Ry
Rz
CNOT
