A conjectures on Stabilizer states
Notation: We will denote the 2 x 2 matrix with first row (a,b) and second row (c,d) as ((a,b),(c,d))
The Clifford group is the group generated by the phase gate ((1,0),(0,I)), the Hadamard gate H and the non local gate controlled-Z gates. If we disregard the phase, this is, if we consider the gates c A=B for any unit complex number c, then there are 24 local gates acting on 1-qubits. Here they are:
((1, 0), (0, 1)), ((1, 0), (0, I)), ((1, 0), (0, -I)), ((1, 0), (0, -1)), ((1/2 + I/2, 1/2 + I/2), (1/2 + I/2, -(1/2) - I/2)), ((1/2 + I/2, 1/2 + I/2), (1/2 - I/2, -(1/2) + I/2)), ((1/2 + I/2, 1/2 + I/2), (-(1/2) + I/2, 1/2 - I/2)), ((1/2 + I/2, 1/2 + I/2), (-(1/2) - I/2, 1/2 + I/2)), ((1/2 + I/2, 1/2 - I/2), (1/2 + I/2, -(1/2) + I/2)), ((1/2 + I/2, 1/2 - I/2), (1/2 - I/2, 1/2 + I/2)), ((1/2 + I/2, 1/2 - I/2), (-(1/2) + I/2, -(1/2) - I/2)), ((1/2 + I/2, 1/2 - I/2), (-(1/2) - I/2, 1/2 - I/2)), ((1/2 + I/2, -(1/2) + I/2), (1/2 + I/2, 1/2 - I/2)), ((1/2 + I/2, -(1/2) + I/2), (1/2 - I/2, -(1/2) - I/ 2)), ((1/2 + I/2, -(1/2) + I/2), (-(1/2) + I/2, 1/2 + I/2)), ((1/2 + I/2, -(1/2) + I/2), (-(1/2) - I/2, -(1/2) + I/2)), ((1/2 + I/2, -(1/2) - I/2), (1/2 + I/2, 1/2 + I/2)), ((1/2 + I/2, -(1/2) - I/2), (1/2 - I/2, 1/2 - I/2)), ((1/2 + I/2, -(1/2) - I/2), (-(1/2) + I/2, -(1/2) + I/2)), ((1/2 + I/2, -(1/2) - I/2), (-(1/2) - I/2, -(1/2) - I/ 2)), ((0, 1), (1, 0)), ((0, 1), (I, 0)), ((0, 1), (-I, 0)), ((0, 1), (-1, 0))
For stabilizer states there are 8 global phases: -1, -I, I, 1, -((1 + I)/Sqrt[2]), -(( 1 - I)/Sqrt[2]), (1 - I)/Sqrt[2], (1 + I)/Sqrt[2])
Latour-Perdomo Conjecture:
If v1,... ,vk are vectors that represent all the stabilizer states for n particles (here we need to consider two states different if they differ by a global phase) and we write vi=ui+Iwi where ui, wi have real entries then the vector (ui, wi) are stabilizer states for n+1 particles. Moreover, every stabilizer state for n+1 particles with real entries can be written as a (ui,wi)
Let us see how this conjecture work for n equal 1
The stabilizer states for one particle (if we consider two states that differ by a the global phase the same) are represented by the following 6 vectors
(-1, 0), (-(1/2) - I/2, -(1/2) - I/2), (-(1/2) - I/2, -(1/2) + I/ 2), (-(1/2) - I/2, 1/2 - I/2), (-(1/2) - I/2, 1/2 + I/2), (0, -1)
If we consider two stabilizer states to be different when they differ by a global phase, then we get the following 48 states: (each one of the 6 vectors above are multiplied by the 8 global phases)
(-1, 0), (-(1/2) - I/2, -(1/2) - I/2), (-(1/2) - I/2, -(1/2) + I/ 2), (-(1/2) - I/2, 1/2 - I/2), (-(1/2) - I/2, 1/2 + I/2), (-(1/2) + I/2, -(1/2) - I/2), (-(1/2) + I/2, -(1/2) + I/ 2), (-(1/2) + I/2, 1/2 - I/2), (-(1/2) + I/2, 1/2 + I/2), (0, -1), (0, -I), (0, I), (0, 1), (0, -((1 + I)/Sqrt[2])), (0, -((1 - I)/Sqrt[2])), (0, (1 - I)/ Sqrt[2]), (0, (1 + I)/Sqrt[2]), (-I, 0), (I, 0), (1/2 - I/2, -(1/2) - I/2), (1/2 - I/2, -(1/2) + I/2), (1/2 - I/ 2, 1/2 - I/2), (1/2 - I/2, 1/2 + I/2), (1/2 + I/2, -(1/2) - I/2), (1/2 + I/2, -(1/2) + I/ 2), (1/2 + I/2, 1/2 - I/2), (1/2 + I/2, 1/2 + I/2), (1, 0), (-(1/Sqrt[2]), -(1/Sqrt[2])), (-(1/Sqrt[2]), -(I/Sqrt[2])), (-( 1/Sqrt[2]), I/Sqrt[2]), (-(1/Sqrt[2]), 1/Sqrt[ 2]), (-((1 + I)/Sqrt[2]), 0), (-((1 - I)/Sqrt[2]), 0), (-(I/Sqrt[2]), -(1/Sqrt[2])), (-(I/Sqrt[2]), -(I/Sqrt[2])), (-( I/Sqrt[2]), I/Sqrt[2]), (-(I/Sqrt[2]), 1/Sqrt[2]), (I/Sqrt[ 2], -(1/Sqrt[2])), (I/Sqrt[2], -(I/Sqrt[2])), (I/Sqrt[2], I/Sqrt[ 2]), (I/Sqrt[2], 1/Sqrt[2]), (1/Sqrt[2], -(1/Sqrt[2])), (1/Sqrt[ 2], -(I/Sqrt[2])), (1/Sqrt[2], I/Sqrt[2]), (1/Sqrt[2], 1/Sqrt[ 2]), ((1 - I)/Sqrt[2], 0), ((1 + I)/Sqrt[2], 0)
The first vector in the list above is (-1, 0), its real part is (-1,0) and its imaginary part is (0,0). Using these two vectors we can built the vector (-1,0,0,0). Let us see this process again, the second vector in the list above is (-(1/2) - I/2, -(1/2) - I/2), its real part is (-(1/2), -(1/2)) and its imaginary part is (- 1/2, - 1/2). Using these 2 vectors we build the vector (- 1/2, - 1/2,-1/2,-1/2). If we do the same with all the 48 vector above we will get 48 vectors, each one with 4 real entries.
If we now think about the stabilizer states for 2 particles, it is known that there are 60 stabilizer states if we disregard the global phase and, 480 if we do not.
Since the conjecture is true for n equal 1, we have that out of the 480 stabilizer states (which are represented by vectors with 4 complex entries), there are exactly 48 that have all the entries given by real numbers. They are exactly the 48 vectors with 4 real entries explained above. Those obtained using the real and imaginary part of the stabilizer states for 1 particle.
We have a list of the 24 local Clifford gates for 1 particle. Let us label these matrices as lc1, lc2, ... , lc24.
We have a list of the 48 stabilizer states for 2 particles. Let us label these vectors with real entries as s1,...s48.
We can use these 48 stabilizer states for 2 particles with real entries to generate all the stabilizer states. We just need to multiply each one of these 48 states with the local Clifford gates for 2 particles. This is, all the stabilizer stater for 2 particles are of the form the 4 by 4 matrix --tensor product lci with lcj-- times the vector sk. Here i and j move from 1 to 24 and k move from 1 to 48.
In this conjecture it is important that we consider two states different if they differ by a global phase. The list of all 2 qubit stabilizer states obtained above using the matrices lci, lcj and the real states sk, even though it has all the stabilizer states for 2 particle, it may not have a list of all the 480 states that are obtained when we consider the global phase. It is easy to obtain all 480 states using the 8 global phases listed above
Once we have all the 480 stabilizer states, we can use them to find all the 8640 stabilizer states for 3 particles (There are 1080 stabilizer states for 3 particles if we disregard the global phase and 8640 if we do not disregard the global phase)
We have shown that the conjecture is true for n equal 1, 2, 3, 4 ,5. If the conjecture is true for all n then we have a new procedure to compute all the stabilizer states for any n.