Code Conversion using Stabilizer Formalism

[Fe-r-oz]

Motivation

By applying these operations on a Stabilizer of[[5, 1 3]]code (assuming that it's the Perfect5() code) -- By using this conversion scheme, the paper "Fault-tolerant quantum error correction code conversion" converts it into [[7, 1, 3]] Steane code. We can test whether at the end, we get equivalent Steane Code.

Caption: Fig1. Circuit diagram of conversion between error correction codes. Solid circles with a line connecting them represent control-sign (CZ) gates. Single-qubit rotations are labelled by the axis of rotation and a subscripted angle of rotation. Hadamard gates are labelled by ‘H’.

We need a two qubit CZ gate, X rotation of pi/2 and Z rotation of pi/2. Do we have access to these gates at the moment? I am not sure they are present in SymbolicCliffords.jl, there is some note on boolean formulas of phase here:

Is this for internal use or should we have it in the documentation as well?

QuantumClifford.jl/src/symbolic_cliffords.jl

Lines 293 to 303 in c64492b

	To get the boolean formulas for the phase, it is easiest to first write down the truth table for the phase:
	for i in 0:15
	s=S"__"
	d=tuple(Bool.(digits(i,base=2,pad=4))...)
	s[1,1]=d[1:2]
	s[1,2]=d[3:4]
	println("$(Int.(d)) $(phases(explicit_tableaux*s)[1]÷2)")
	end
	Then we can use a truth-table to boolean formula converter, e.g. https://www.dcode.fr/boolean-expressions-calculator
	(by just writing the initial unsimplified formula as the OR of all true rows of the table)
	=#

Code Conversion using Stabilizer Formalism

Can we apply the entire circuit (shown above, generally as well) to the Stabilizer state? Is that a fair question to ask? Or we apply each gate step by step to the Stabilizer state which we can do using apply! ? I also checked out the naive_encoding_circuit in circuits.jl and it seems there the gates are applied one by one, in a sequential manner.

It can be pretty useful when converting one Stabilizer code to another (via a sequence of applying gates, represented as a circuit) as conversion schemes may be described in the form of a circuit. I tried to apply this example given in doc.

julia> s = S"+ XZZX_
       + _XZZX
       + X_XZZ
       + ZX_XZ"
+ XZZX_
+ _XZZX
+ X_XZZ
+ ZX_XZ

julia> circuit = [sCNOT(2,4),sHadamard(2),sCPHASE(1,3),sSWAP(2,4)]
4-element Vector{AbstractSymbolicOperator}:
 sCNOT(2,4)
 sHadamard(2)
 sCPHASE(1,3)
 sSWAP(2,4)

julia> apply!(s, circuit)  #  not be the correct way but I hope it conveys the idea
ERROR: MethodError: no method matching apply!(::Stabilizer{QuantumClifford.Tableau{Vector{UInt8}, Matrix{UInt64}}}, ::Vector{AbstractSymbolicOperator})

Closest candidates are:
  apply!(::QuantumClifford.AbstractStabilizer, ::sMZ)
   @ QuantumClifford ~/GitHub/QuantumClifford/QuantumClifford.jl/src/symbolic_cliffords.jl:427
  apply!(::QuantumClifford.AbstractStabilizer, ::sMX)
   @ QuantumClifford ~/GitHub/QuantumClifford/QuantumClifford.jl/src/symbolic_cliffords.jl:419
  apply!(::QuantumClifford.AbstractStabilizer, ::sMY)
   @ QuantumClifford ~/GitHub/QuantumClifford/QuantumClifford.jl/src/symbolic_cliffords.jl:423
  ...

Stacktrace:
 [1] top-level scope
   @ REPL[21]:1

Thanks for your help, @Krastanov!

[Fe-r-oz]

Pardon me as the question was not clear. ZCZ/CZ is present! I think single qubit rotations, Rₓ, R_y, R_z are not yet implemented. The above circuit (FIG 1 of paper) utilizes these gates towards the end.

circuit = [
              sHadamard(6), sCNOT(1, 6), sHadamard(6),
              sHadamard(7), sCNOT(2, 7), sHadamard(7),
              sHadamard(8), sCNOT(2, 8), sHadamard(8),
              sHadamard(9), sCNOT(3, 9), sHadamard(9),
              sHadamard(10), sCNOT(4, 10), sHadamard(10),
              sHadamard(4), sCNOT(2, 4), sHadamard(4),
              sHadamard(8), sCNOT(4, 8), sHadamard(8),
              sHadamard(8), sCNOT(7, 8), sHadamard(8),
              sHadamard(7), sCNOT(5, 7), sHadamard(7),
              sHadamard(5), sCNOT(1, 5), sHadamard(5),
              sHadamard(6), sCNOT(1, 6), sHadamard(6),
              sHadamard(10), sCNOT(4, 10), sHadamard(10),
              sHadamard(9), sCNOT(3, 9), sHadamard(9),
             ....missing single qubit rotation gates...
          ]

Before applying the rotation gates:

From Appendix, step 14, pg. 7: https://arxiv.org/pdf/1112.2417v1

After applying the rotation gates, stated in point 15:

From Appendix, pg. 7: https://arxiv.org/pdf/1112.2417v1