braket.cpp running
notation  (x,y) is complex number x+iy
          vectors are printed   vec[j] = (x,y)
          matrices are printed  mat[i][j] = (x,y)
          kets are printed (x,y)|0> + (p,q)|1>
          bras are printed (x,y)<0| + (p,q)<1|

to be a valid physical qubit, ket notation  a|0> + b|1> 
abs(a^2 + b^2) must equal 1
to be a valid physical qubit, bra notation  a<0| + b<1| 
abs(a^2 + b^2) must equal 1
 a and b are shown as (x,y) x real part, y imaginary part
 a is first element of vector, b is second element of vector

From math we know e^i theta is a complex number with magnitude 1
From Euler's formula e^i theta = cos(theta) + i sin(theta)
Thus a qubit can be written as two angles, theta and phi:
cos(theta) |0> + e^-i phi sin(theta) |1>) (not used in this simulation)

In computer code and printout:
vector = matrix * vector  ketout = matvec(mat, ketin)
state vector |0>|1> = |01> = ket01 = outp(ket0,ket1)
outer product, outp, result is four complex numbers
|a>|b>|c> = |abc> is a state vector having 8 complex numbers

a ket represents a qubit, stored as a vector of two complex numbers
logical zero = |0> = ket0 = a column vector
vec[0] = (1,0)
vec[1] = (0,0)

logical one = |1> = ket1 =
vec[0] = (0,0)
vec[1] = (1,0)


logical zero bra = <0| = bra0 =  row vector
bra0 <0|  =(1,0)(0,0)

logical one bra = <1| = bra1 =  row vector
bra1 <0|  =(0,0)(1,0)


<a||b> a braket is the inner=dot product producing a complex number
<0||0> = (1,0)
<0||1> = (0,0)
<1||0> = (0,0)
<1||1> = (1,0)


state of qubits are kets combined with tensor product=outer product
A two qubit system will be on the surface of Bloch unit sphere
 |x> = a|0> + b|1> , |y> = c|0> + d|1>, a,b,c,d complex
 then |x>|y> = |xy> = four entry vector =
                      using complex conjugate of c and d
 ac|00> + ad|01> + bc|10> + bd|11> written as a vector =
  | ac  ad  bc  bd | just four complex numbers, a 'state'
   mathematically, complex tensor product of  |a b| with |c d|

many examples are shown both in bra ket notation and as vectors

|0>|0> tensor product resulting in a vector, outp()
|0>|0> = |00> =
vec[0] = (1,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (0,0)

|0>|1> = |01> =
vec[0] = (0,0)
vec[1] = (1,0)
vec[2] = (0,0)
vec[3] = (0,0)

|1>|0> = |10> =
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (1,0)
vec[3] = (0,0)

|1>|1> = |11> =
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (1,0)


Note: reading state |011> binary as 3, subscript [3] is (1,0)
outer product, outp function is general
|0>|0>|0> outp(outp(ket0, ket0), ket0)=|000>
vec[0] = (1,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (0,0)
vec[4] = (0,0)
vec[5] = (0,0)
vec[6] = (0,0)
vec[7] = (0,0)

|0>|0>|1> outp(outp(ket0, ket0), ket1)=|001>
vec[0] = (0,0)
vec[1] = (1,0)
vec[2] = (0,0)
vec[3] = (0,0)
vec[4] = (0,0)
vec[5] = (0,0)
vec[6] = (0,0)
vec[7] = (0,0)

|0>|1>|0> outp(outp(ket0, ket1), ket0)=|010>
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (1,0)
vec[3] = (0,0)
vec[4] = (0,0)
vec[5] = (0,0)
vec[6] = (0,0)
vec[7] = (0,0)

|0>|1>|1> outp(outp(ket0, ket1), ket1)=|011>
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (1,0)
vec[4] = (0,0)
vec[5] = (0,0)
vec[6] = (0,0)
vec[7] = (0,0)

|1>|0>|0> outp(outp(ket1, ket0), ket0)=|100>
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (0,0)
vec[4] = (1,0)
vec[5] = (0,0)
vec[6] = (0,0)
vec[7] = (0,0)

|1>|0>|1> outp(outp(ket1, ket0), ket1)=|101>
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (0,0)
vec[4] = (0,0)
vec[5] = (1,0)
vec[6] = (0,0)
vec[7] = (0,0)

|1>|1>|0> outp(outp(ket1, ket1), ket0)=|110>
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (0,0)
vec[4] = (0,0)
vec[5] = (0,0)
vec[6] = (1,0)
vec[7] = (0,0)

|1>|1>|1> outp(outp(ket1, ket1), ket1)=|111>
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (0,0)
vec[4] = (0,0)
vec[5] = (0,0)
vec[6] = (0,0)
vec[7] = (1,0)


tensor product to make matrix
operator |0> tensor |0> 
mat[0][0] = (1,0)
mat[0][1] = (0,0)
mat[1][0] = (0,0)
mat[1][1] = (0,0)

operator |0> tensor |1> 
mat[0][0] = (0,0)
mat[0][1] = (1,0)
mat[1][0] = (0,0)
mat[1][1] = (0,0)

operator |1> tensor |0> 
mat[0][0] = (0,0)
mat[0][1] = (0,0)
mat[1][0] = (1,0)
mat[1][1] = (0,0)

operator |1> tensor |1> 
mat[0][0] = (0,0)
mat[0][1] = (0,0)
mat[1][0] = (0,0)
mat[1][1] = (1,0)


a=ket0+ket1 =
vec[0] = (1,0)
vec[1] = (1,0)

should be a|0> + b|1>  where a^2+b^2=1

a=0.7071, b=0.7071  or scale sum
vec[0] = (0.7071,0)
vec[1] = (0.7071,0)


included ket00 is |00>, etc
included ket000 is |000>, etc
haramard operator ket out = |H| ket in
mat[0][0] = (0.707107,0)
mat[0][1] = (0.707107,0)
mat[1][0] = (0.707107,0)
mat[1][1] = (-0.707107,0)

keth =  |H||0> 
vec[0] = (0.707107,0)
vec[1] = (0.707107,0)

keth =  |H||1> 
vec[0] = (0.707107,0)
vec[1] = (-0.707107,0)


Pauli-X is a single qubit  logical not  operator
Pauli-X operator ket out = |PX| ket in
mat[0][0] = (0,0)
mat[0][1] = (1,0)
mat[1][0] = (1,0)
mat[1][1] = (0,0)

keth =  |PX||0> 
vec[0] = (0,0)
vec[1] = (1,0)

keth =  |PX||1> 
vec[0] = (1,0)
vec[1] = (0,0)


Pauli-Y operator ket out = |PY| ket in
mat[0][0] = (0,0)
mat[0][1] = (0,-1)
mat[1][0] = (0,1)
mat[1][1] = (0,0)

keth =  |PY||0> 
vec[0] = (0,0)
vec[1] = (0,1)

keth =  |PY||1> 
vec[0] = (0,-1)
vec[1] = (0,0)


Pauli-Z operator ket out = |PZ| ket in
mat[0][0] = (1,0)
mat[0][1] = (0,0)
mat[1][0] = (0,0)
mat[1][1] = (0,-1)

keth =  |PZ||0> 
vec[0] = (1,0)
vec[1] = (0,0)

keth =  |PZ||1> 
vec[0] = (0,0)
vec[1] = (0,-1)


cnot operator two input, two output
              the first entry in ket is the control
mat[0][0] = (1,0)
mat[0][1] = (0,0)
mat[0][2] = (0,0)
mat[0][3] = (0,0)
mat[1][0] = (0,0)
mat[1][1] = (1,0)
mat[1][2] = (0,0)
mat[1][3] = (0,0)
mat[2][0] = (0,0)
mat[2][1] = (0,0)
mat[2][2] = (0,0)
mat[2][3] = (1,0)
mat[3][0] = (0,0)
mat[3][1] = (0,0)
mat[3][2] = (1,0)
mat[3][3] = (0,0)

cnot |00> ket00 =
vec[0] = (1,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (0,0)
               check  |00>

cnot |01> ket01 =
vec[0] = (0,0)
vec[1] = (1,0)
vec[2] = (0,0)
vec[3] = (0,0)
               check |01>

cnot |10> ket10 =
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (1,0)
               check |11>

cnot |11> ket11 =
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (1,0)
vec[3] = (0,0)
               check |10>


Now Repeat using cnot2 function
cnot |0>|0> =
vec[0] = (1,-0)
vec[1] = (0,-0)
               check |0>

cnot |0>|1> =
vec[0] = (0,-0)
vec[1] = (1,-0)
               check |1>

cnot |1>|0> =
vec[0] = (0,-0)
vec[1] = (1,-0)
               check |1>

cnot |1>|1> =
vec[0] = (1,-0)
vec[1] = (0,-0)
               check |0>


Now Repeat using register, pair of kets |0>,|1>
rather than |0>|1> = |01>
reg00 =
vec[0] = (1,0)
vec[1] = (0,0)
vec[2] = (1,0)
vec[3] = (0,0)
reg01 =
vec[0] = (1,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (1,0)
reg10 =
vec[0] = (0,0)
vec[1] = (1,0)
vec[2] = (1,0)
vec[3] = (0,0)
reg11 =
vec[0] = (0,0)
vec[1] = (1,0)
vec[2] = (0,0)
vec[3] = (1,0)


cnot |0>,|0> reg00 =
vec[0] = (1,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (1,0)
reg[0]= |0>
reg[1]= |1>

cnot |0>,|1> reg01 =
vec[0] = (1,0)
vec[1] = (0,0)
vec[2] = (1,0)
vec[3] = (0,0)
reg[0]= |0>
reg[1]= |0>

cnot |1>,|0> reg10 =
vec[0] = (0,0)
vec[1] = (1,0)
vec[2] = (0,0)
vec[3] = (1,0)
reg[0]= |1>
reg[1]= |1>

cnot |1>,|1> reg11 =
vec[0] = (0,0)
vec[1] = (1,0)
vec[2] = (1,0)
vec[3] = (0,0)
reg[0]= |1>
reg[1]= |0>


ccnot is also known as Toffoli
ccnot operator three input, three output
              the first two ket are the control
mat[0][0] = (1,0)
mat[0][1] = (0,0)
mat[0][2] = (0,0)
mat[0][3] = (0,0)
mat[0][4] = (0,0)
mat[0][5] = (0,0)
mat[0][6] = (0,0)
mat[0][7] = (0,0)
mat[1][0] = (0,0)
mat[1][1] = (1,0)
mat[1][2] = (0,0)
mat[1][3] = (0,0)
mat[1][4] = (0,0)
mat[1][5] = (0,0)
mat[1][6] = (0,0)
mat[1][7] = (0,0)
mat[2][0] = (0,0)
mat[2][1] = (0,0)
mat[2][2] = (1,0)
mat[2][3] = (0,0)
mat[2][4] = (0,0)
mat[2][5] = (0,0)
mat[2][6] = (0,0)
mat[2][7] = (0,0)
mat[3][0] = (0,0)
mat[3][1] = (0,0)
mat[3][2] = (0,0)
mat[3][3] = (1,0)
mat[3][4] = (0,0)
mat[3][5] = (0,0)
mat[3][6] = (0,0)
mat[3][7] = (0,0)
mat[4][0] = (0,0)
mat[4][1] = (0,0)
mat[4][2] = (0,0)
mat[4][3] = (0,0)
mat[4][4] = (1,0)
mat[4][5] = (0,0)
mat[4][6] = (0,0)
mat[4][7] = (0,0)
mat[5][0] = (0,0)
mat[5][1] = (0,0)
mat[5][2] = (0,0)
mat[5][3] = (0,0)
mat[5][4] = (0,0)
mat[5][5] = (1,0)
mat[5][6] = (0,0)
mat[5][7] = (0,0)
mat[6][0] = (0,0)
mat[6][1] = (0,0)
mat[6][2] = (0,0)
mat[6][3] = (0,0)
mat[6][4] = (0,0)
mat[6][5] = (0,0)
mat[6][6] = (0,0)
mat[6][7] = (1,0)
mat[7][0] = (0,0)
mat[7][1] = (0,0)
mat[7][2] = (0,0)
mat[7][3] = (0,0)
mat[7][4] = (0,0)
mat[7][5] = (0,0)
mat[7][6] = (1,0)
mat[7][7] = (0,0)

ccnot |0>|0>|0> ket000 =
vec[0] = (1,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (0,0)
vec[4] = (0,0)
vec[5] = (0,0)
vec[6] = (0,0)
vec[7] = (0,0)
            check |000>

ccnot |0>|0>|1> ket001 =
vec[0] = (0,0)
vec[1] = (1,0)
vec[2] = (0,0)
vec[3] = (0,0)
vec[4] = (0,0)
vec[5] = (0,0)
vec[6] = (0,0)
vec[7] = (0,0)
            check |001>

ccnot |0>|1>|0> ket010 =
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (1,0)
vec[3] = (0,0)
vec[4] = (0,0)
vec[5] = (0,0)
vec[6] = (0,0)
vec[7] = (0,0)
            check |010>

ccnot |0>|1>|1> ket011 =
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (1,0)
vec[4] = (0,0)
vec[5] = (0,0)
vec[6] = (0,0)
vec[7] = (0,0)
            check |011>

ccnot |1>|0>|0> ket100 =
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (0,0)
vec[4] = (1,0)
vec[5] = (0,0)
vec[6] = (0,0)
vec[7] = (0,0)
            check |100>

ccnot |1>|0>|1> ket101 =
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (0,0)
vec[4] = (0,0)
vec[5] = (1,0)
vec[6] = (0,0)
vec[7] = (0,0)
            check |101>

ccnot |1>|1>|0> ket110 =
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (0,0)
vec[4] = (0,0)
vec[5] = (0,0)
vec[6] = (0,0)
vec[7] = (1,0)
            check |111>

ccnot |1>|1>|1> ket111 =
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (0,0)
vec[4] = (0,0)
vec[5] = (0,0)
vec[6] = (1,0)
vec[7] = (0,0)
            check |110>


swap2 operator two input, two output
mat[0][0] = (1,0)
mat[0][1] = (0,0)
mat[0][2] = (0,0)
mat[0][3] = (0,0)
mat[1][0] = (0,0)
mat[1][1] = (0,0)
mat[1][2] = (1,0)
mat[1][3] = (0,0)
mat[2][0] = (0,0)
mat[2][1] = (1,0)
mat[2][2] = (0,0)
mat[2][3] = (0,0)
mat[3][0] = (0,0)
mat[3][1] = (0,0)
mat[3][2] = (0,0)
mat[3][3] = (1,0)
swap2 |0>|0> ket00 =
vec[0] = (1,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (0,0)
            check |00>

swap2 |0>|1> ket01 =
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (1,0)
vec[3] = (0,0)
            check |10>

swap2 |1>|0> ket10 =
vec[0] = (0,0)
vec[1] = (1,0)
vec[2] = (0,0)
vec[3] = (0,0)
            check |01>

swap2 |1>|1> ket11 =
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (1,0)
            check |11>


cswap operator three input, three output
              the first ket is the control
mat[0][0] = (1,0)
mat[0][1] = (0,0)
mat[0][2] = (0,0)
mat[0][3] = (0,0)
mat[0][4] = (0,0)
mat[0][5] = (0,0)
mat[0][6] = (0,0)
mat[0][7] = (0,0)
mat[1][0] = (0,0)
mat[1][1] = (1,0)
mat[1][2] = (0,0)
mat[1][3] = (0,0)
mat[1][4] = (0,0)
mat[1][5] = (0,0)
mat[1][6] = (0,0)
mat[1][7] = (0,0)
mat[2][0] = (0,0)
mat[2][1] = (0,0)
mat[2][2] = (1,0)
mat[2][3] = (0,0)
mat[2][4] = (0,0)
mat[2][5] = (0,0)
mat[2][6] = (0,0)
mat[2][7] = (0,0)
mat[3][0] = (0,0)
mat[3][1] = (0,0)
mat[3][2] = (0,0)
mat[3][3] = (1,0)
mat[3][4] = (0,0)
mat[3][5] = (0,0)
mat[3][6] = (0,0)
mat[3][7] = (0,0)
mat[4][0] = (0,0)
mat[4][1] = (0,0)
mat[4][2] = (0,0)
mat[4][3] = (0,0)
mat[4][4] = (1,0)
mat[4][5] = (0,0)
mat[4][6] = (0,0)
mat[4][7] = (0,0)
mat[5][0] = (0,0)
mat[5][1] = (0,0)
mat[5][2] = (0,0)
mat[5][3] = (0,0)
mat[5][4] = (0,0)
mat[5][5] = (0,0)
mat[5][6] = (1,0)
mat[5][7] = (0,0)
mat[6][0] = (0,0)
mat[6][1] = (0,0)
mat[6][2] = (0,0)
mat[6][3] = (0,0)
mat[6][4] = (0,0)
mat[6][5] = (1,0)
mat[6][6] = (0,0)
mat[6][7] = (0,0)
mat[7][0] = (0,0)
mat[7][1] = (0,0)
mat[7][2] = (0,0)
mat[7][3] = (0,0)
mat[7][4] = (0,0)
mat[7][5] = (0,0)
mat[7][6] = (0,0)
mat[7][7] = (1,0)

cswap |0>|0>|0> =
vec[0] = (1,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (0,0)
vec[4] = (0,0)
vec[5] = (0,0)
vec[6] = (0,0)
vec[7] = (0,0)
            check |000>

cswap |0>|0>|1> =
vec[0] = (0,0)
vec[1] = (1,0)
vec[2] = (0,0)
vec[3] = (0,0)
vec[4] = (0,0)
vec[5] = (0,0)
vec[6] = (0,0)
vec[7] = (0,0)
            check |001>

cswap |0>|1>|0> =
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (1,0)
vec[3] = (0,0)
vec[4] = (0,0)
vec[5] = (0,0)
vec[6] = (0,0)
vec[7] = (0,0)
            check |010>

cswap |0>|1>|1> =
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (1,0)
vec[4] = (0,0)
vec[5] = (0,0)
vec[6] = (0,0)
vec[7] = (0,0)
            check |011>

cswap |1>|0>|0> =
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (0,0)
vec[4] = (1,0)
vec[5] = (0,0)
vec[6] = (0,0)
vec[7] = (0,0)
            check |100>

cswap |1>|0>|1> =
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (0,0)
vec[4] = (0,0)
vec[5] = (0,0)
vec[6] = (1,0)
vec[7] = (0,0)
            check |110>

cswap |1>|1>|0> =
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (0,0)
vec[4] = (0,0)
vec[5] = (1,0)
vec[6] = (0,0)
vec[7] = (0,0)
            check |101>

cswap |1>|1>|1> =
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (0,0)
vec[4] = (0,0)
vec[5] = (0,0)
vec[6] = (0,0)
vec[7] = (1,0)
            check |111>

rotation Rk, c_mat = Rk_make(k) 
p=4, x=(0,1.5708), exp(x)=(6.12323e-17,1)
Rk2 =
mat[0][0] = (1,0)
mat[0][1] = (0,0)
mat[1][0] = (0,0)
mat[1][1] = (6.12323e-17,1)

keta =  |Rk2||0> 
vec[0] = (1,0)
vec[1] = (0,0)

keta =  |Rk2||1> 
vec[0] = (0,0)
vec[1] = (6.12323e-17,1)

p=8, x=(0,0.785398), exp(x)=(0.707107,0.707107)
Rk3 =
mat[0][0] = (1,0)
mat[0][1] = (0,0)
mat[1][0] = (0,0)
mat[1][1] = (0.707107,0.707107)

keta =  |Rk3||0> 
vec[0] = (1,0)
vec[1] = (0,0)

keta =  |Rk3||1> 
vec[0] = (0,0)
vec[1] = (0.707107,0.707107)

p=16, x=(0,0.392699), exp(x)=(0.92388,0.382683)
Rk4 =
mat[0][0] = (1,0)
mat[0][1] = (0,0)
mat[1][0] = (0,0)
mat[1][1] = (0.92388,0.382683)

keta =  |Rk4||0> 
vec[0] = (1,0)
vec[1] = (0,0)

keta =  |Rk4||1> 
vec[0] = (0,0)
vec[1] = (0.92388,0.382683)

controlled Rk, cRk(k) 
cRk4 =
mat[0][0] = (1,0)
mat[0][1] = (0,0)
mat[0][2] = (0,0)
mat[0][3] = (0,0)
mat[1][0] = (0,0)
mat[1][1] = (1,0)
mat[1][2] = (0,0)
mat[1][3] = (0,0)
mat[2][0] = (0,0)
mat[2][1] = (0,0)
mat[2][2] = (1,0)
mat[2][3] = (0,0)
mat[3][0] = (0,0)
mat[3][1] = (0,0)
mat[3][2] = (0,0)
mat[3][3] = (0.92388,0.382683)

reg =  |cRk4||0>|0> 
vec[0] = (1,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (0,0)

reg =  |cRk4||0>|1> 
vec[0] = (0,0)
vec[1] = (1,0)
vec[2] = (0,0)
vec[3] = (0,0)

reg =  |cRk4||1>|0> 
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (1,0)
vec[3] = (0,0)

reg =  |cRk4||1>|1> 
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (0.92388,0.382683)

get_ket(0, ket000) =
vec[0] = (1,0)
vec[1] = (0,0)

get_ket(1, ket000) =
vec[0] = (0,0)
vec[1] = (0,0)

get_ket(2, ket000) =
vec[0] = (0,0)
vec[1] = (0,0)


get_ket(0, ket010) =
vec[0] = (0,0)
vec[1] = (0,0)

get_ket(1, ket010) =
vec[0] = (1,0)
vec[1] = (0,0)

get_ket(2, ket010) =
vec[0] = (0,0)
vec[1] = (0,0)


get_ket(0, ket111) =
vec[0] = (0,0)
vec[1] = (0,0)

get_ket(1, ket111) =
vec[0] = (0,0)
vec[1] = (0,0)

get_ket(2, ket111) =
vec[0] = (0,0)
vec[1] = (0,0)


root sigma x operator two input, two output 
              the first entry in ket is the control
mat[0][0] = (1,0)
mat[0][1] = (0,0)
mat[0][2] = (0,0)
mat[0][3] = (0,0)
mat[1][0] = (0,0)
mat[1][1] = (1,0)
mat[1][2] = (0,0)
mat[1][3] = (0,0)
mat[2][0] = (0,0)
mat[2][1] = (0,0)
mat[2][2] = (0.5,0.5)
mat[2][3] = (0.5,-0.5)
mat[3][0] = (0,0)
mat[3][1] = (0,0)
mat[3][2] = (0.5,-0.5)
mat[3][3] = (0.5,0.5)

rsx |00> ket00 =
vec[0] = (1,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (0,0)

rsx |01> ket01 =
vec[0] = (0,0)
vec[1] = (1,0)
vec[2] = (0,0)
vec[3] = (0,0)

rsx |10> ket10 =
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (0.5,0.5)
vec[3] = (0.5,-0.5)

rsx |11> ket11 =
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (0.5,-0.5)
vec[3] = (0.5,0.5)


root sigma x transpose operator two input, two output 
              the first entry in ket is the control
mat[0][0] = (1,0)
mat[0][1] = (0,0)
mat[0][2] = (0,0)
mat[0][3] = (0,0)
mat[1][0] = (0,0)
mat[1][1] = (1,0)
mat[1][2] = (0,0)
mat[1][3] = (0,0)
mat[2][0] = (0,0)
mat[2][1] = (0,0)
mat[2][2] = (0.5,-0.5)
mat[2][3] = (0.5,0.5)
mat[3][0] = (0,0)
mat[3][1] = (0,0)
mat[3][2] = (0.5,0.5)
mat[3][3] = (0.5,-0.5)

rsxt |00> ket00 =
vec[0] = (1,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (0,0)

rsxt |01> ket01 =
vec[0] = (0,0)
vec[1] = (1,0)
vec[2] = (0,0)
vec[3] = (0,0)

rsxt |10> ket10 =
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (0.5,-0.5)
vec[3] = (0.5,0.5)

rsxt |11> ket11 =
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (0.5,0.5)
vec[3] = (0.5,-0.5)


Now Repeat using register, pair of kets |0>,|1>
rather than |0>|1> = |01>
reg00 =
vec[0] = (1,0)
vec[1] = (0,0)
vec[2] = (1,0)
vec[3] = (0,0)
reg01 =
vec[0] = (1,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (1,0)
reg10 =
vec[0] = (0,0)
vec[1] = (1,0)
vec[2] = (1,0)
vec[3] = (0,0)
reg11 =
vec[0] = (0,0)
vec[1] = (1,0)
vec[2] = (0,0)
vec[3] = (1,0)


cnot |0>,|0> reg00 =
vec[0] = (1,0)
vec[1] = (0,0)
vec[2] = (0.5,-0.5)
vec[3] = (0.5,0.5)

cnot |0>,|1> reg01 =
vec[0] = (1,0)
vec[1] = (0,0)
vec[2] = (0.5,0.5)
vec[3] = (0.5,-0.5)

cnot |1>,|0> reg10 =
vec[0] = (0,0)
vec[1] = (1,0)
vec[2] = (0.5,-0.5)
vec[3] = (0.5,0.5)

cnot |1>,|1> reg11 =
vec[0] = (0,0)
vec[1] = (1,0)
vec[2] = (0.5,0.5)
vec[3] = (0.5,-0.5)
keta = 
vec[0] = (0,0)
vec[1] = (1,0)
ketb = 
vec[0] = (0.5,0.5)
vec[1] = (0.5,-0.5)


check that ket00, ket01, ket10, ket11 are still OK
vec[0] = (1,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (0,0)
vec[0] = (0,0)
vec[1] = (1,0)
vec[2] = (0,0)
vec[3] = (0,0)
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (1,0)
vec[3] = (0,0)
vec[0] = (0,0)
vec[1] = (0,0)
vec[2] = (0,0)
vec[3] = (1,0)


braket.cpp finished