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Wolfram`QuantumFramework`

QuantumChannel

QuantumChannel[{ B 1 , B 2 ,…},order,basis]
	represents a quantum channel with Kraus operators B i , in basis , to be applied onto the qubits indexed in order .

QuantumChannel[name[params],...]
	represents a the named quantum channel with name , and potential parameters specified by params .

Details and Options

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QuantumChannel[…]

describes a transformation of a density matrix ρ into

∑



. In our framework, it is define by specifying the operators

with the function QuantumChannel. The transformation due to a channel can be trace preserving (i.e.

∑



) or not.

▪

When

basis

isn't specified, the default is taken to be the computational basis.

▪

When the order order isn't specified, the default is taken to be the range of the arity of the operators.

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The following named operators are supported:

	"BitFlip"[p]	the bit flip channel with probability p
	"PhaseFlip"[p]	the phase flip channel with probability p
	"BitPhaseFlip"[p]	the bit phase flip channel with probability p
	"Depolarizing"[p]	the depolarizing channel with probability p
	"AmplitudeDamping"[γ]	the amplitude damping channel with decay γ
	"GeneralizedAmplitudeDamping"[γ,p]	the generalized amplitude damping channel, with decay γ and probability p
	"PhaseDamping"[λ]	the Phase damping channel with decay λ
	"ResetError"[ p 0 , p 1 ]	the reset error channel, with probability p 0 for \|0〉 state and p 1 for \|1〉

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The following properties specific to channels are supported:

	"TracePreservingQ"	tests if the channel preserves the trace of any state ∑ k † B k B k I
	"UnitalQ"	tests if the channel fixes the identity (I)=I
	"Unitality"	gives the operator ∑ k B k † B k
	"Operator"	returns the operator with extra qudits according to Stinespring dilation

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Equivalent description of the channels in terms of the Kraus operators that builds them:

"BitFlip"[p]	 p "X", 1-p "I"	The bit-flip channel represents bit-flip operation with probability p and remains unchanged with probability 1 - p.
"PhaseFlip"[p]	 p "Z", 1-p "I"	The phase-flip channel is represents a qubit undergoes a phase-flip operation with probability p and remains unchanged with probability 1 - p
"BitPhaseFlip"[p]	 p "Y", 1-p "I"	The bit-phase-flip channel represents a qubit undergoes a combined bit and phase-flip operation with probability p and remains unchanged with probability 1 - p.
"Depolarizing"[p]	 p/4 "X", p/4 "Y", p/4 "Z", 1-3p/4 "I"	The depolarizing channel represents a qubit undergoes a random Pauli operation (X, Y, or Z) with equal probability p/4 or remains unchanged with probability 1 - 3p/4.
"AmplitudeDamping"[γ]	{1,0},0, 1-γ ,0, γ ,{0,0}	The amplitude damping channel models the energy relaxation of a qubit, where γ is the damping rate.
"GeneralizedAmplitudeDamping"[γ,p]	 p {1,0},0, 1-γ , p 0, γ ,{0,0}, 1-p  1-γ ,0,{0,1}, 1-p {0,0}, γ ,0	The generalized amplitude damping models the energy relaxation of a qubit in the presence of a thermal environment, where p is the probability of the qubit being excited and γ is the damping rate.
"PhaseDamping"[λ]	{1,0},0, 1-λ ,{0,0},0, λ 	The phase damping channel models the dephasing of a qubit, where λ is the dephasing rate.
"ResetError"[ p 0 , p 1 ]	 1- p 0 - p 1 "I", p 0 {{1,0},{0,0}}, p 0 {{0,1},{0,0}}, p 1 {{0,0},{1,0}}, p 1 {{0,0},{0,1}}	The reset error channel describes with p 0 the probability for reset error of \|0〉 and p 1 of \|1〉