From Sec. 2.1 to Sec. 2.3, we describe the dynamics of the qubits in the space which includes the degrees of freedom in the qubit-system only, and doesn't include those in the environment, that is, we treat the problems in a closed system. For an open quantum system, the dynamics of the qubits is described in the space which includes the qubit-system subspace (degrees of freedom in the qubit-qubit-system) and the environment subspace (degrees of freedom in the environment). Thus the total Hamiltonian of an open quantum system can be written as
H(t) = HS(t) + HCN(t) + HE(t) + HQN(t). (2.25)
Here HS(t) and HCN(t) are de ned in the qubit-system subspace, and HS(t) is the ideal qubit-system Hamiltonian and HCN(t)is the classical noise Hamiltonian, which correspond to HI(t) and HN(t), respectively, discussed in a closed system from Sec. 2.1 to Sec. 2.3.
HE(t) is the environment Hamiltonian and is de ned in the environment subspace. In an open quantum system, except the classical noise, the quantum noise also degrades the gate delity and is described by the quantum noise Hamiltonian HQN(t), coupling the qubit-system subspace and the environment subspace together. Detailed form of these Hamiltonians in Eq. (2.25) are shown below:
HS(t) = HS(t) ⊗ IE, (2.26) HCN(t) = [X
j
βj(t)SCNj(t)] ⊗ IE, (2.27)
HE(t) = IS⊗ HE(t), (2.28)
HQN(t) =X
j
SQNj(t) ⊗ Ej(t). (2.29)
Here IE and IS are the identity operators in the environment subspace and in the qubit-system subspace, respectively. HS(t)is the ideal qubit-system Hamiltonian operator in the qubit-system subspace, and HE(t)is the environment Hamiltonian operator in the environ-ment subspace. In Eq. (2.27), βj(t)is the strength of the j-th classical noise and SCNj(t) is the corresponding system coupling operator term. For the quantum noise Hamiltonian in Eq. (2.29), SQNj(t) and Ej(t) are the system-environment coupling operators in the qubit-system subspace and in the environment subspace, respectively. In fact, if we choose Ej(t) = βj(t)IE, quantum noise recovers to classical noise, but the treatment processes for classical noise and quantum noise are somewhat dierent, so we separate them.
Following treatment processes are similar to those in a closed system in Sec. 2.1. First, we transform the Hamiltonian to the interaction picture by US(t) ⊗ UE(t), where
US(t) = T+exp[−i ˆ t
0
HS(t0)dt0], (2.30)
UE(t) = T+exp[−i ˆ t
0
HE(t0)dt0] (2.31)
are the ideal qubit-system propagator and the environment propagator, respectively. Then the total Hamiltonian in the interaction picture becomes ˜HCN(t) + ˜HQN(t), where
H˜CN(t) = [X
j
βj(t)RCNj(t)] ⊗ IE, (2.32) H˜QN(t) =X
j
RQNj(t) ⊗ REj(t), (2.33)
and
RCNj(t) ≡ US†(t)SCNj(t)US(t), (2.34) RQNj(t) ≡ US†(t)SQNj(t)US(t), (2.35) REj(t) ≡ UE†(t)Ej(t)UE(t). (2.36)
The total propagator in the interaction picture at the gate operation time tf is
U (t˜ f) = T+exp[−i ˆ tf
0
( ˜HCN(t) + ˜HQN(t))dt0]. (2.37)
If the strength of both classical noise and quantum noise is not too large, we can expand U (t˜ f) by Dyson series [75] as the form ˜U (tf) = I + Ψ1+ Ψ2+ · · · , where the rst two terms of Ψj are
Ψ1 = −i ˆ tf
0
[ ˜HCN(t) + ˜HQN(t)]dt0, (2.38)
Ψ2 = − ˆ tf
0
dt1 ˆ t1
0
dt2[ ˜HCN(t1) + ˜HQN(t1)][ ˜HCN(t2) + ˜HQN(t2)]. (2.39)
The total propagator in the original frame becomes
U (tf) = [US(tf) ⊗ UE(tf)] · (I + Ψ1+ Ψ2+ · · · ). (2.40)
Next derivations are dierent from those in a closed system in Sec. 2.1. In general, the dynamics of an open quantum system is described by a density matrix
ρ(t) = U (t)ρ(0)U†(t). (2.41)
Here we assume the initial density matrix is separable, ρ(0) = ρS(0) ⊗ ρE(0), and ρS(0) is the initial density matrix in the qubit-system subspace, and ρE(0) is the initial density matrix in the environment subspace. To see the quantum noise contribution to the ensemble average gate in delity, we need to trace over the degrees of freedom in the environment to
obtain the reduced density matrix in the qubit-system subspace as
ρS(t) = TrEρ(t) = US(t) ρS(0) + ¯Ψ1+ ¯Ψ2+ · · · US†(t), (2.42)
where the de nitions of ¯Ψ1 and ¯Ψ2 are
Ψ¯1≡ TrE[Ψ1(ρS(0) ⊗ ρE(0)) + h.c.], (2.43) Ψ¯2≡ TrE[Ψ2(ρS(0) ⊗ ρE(0)) + h.c.] + TrE[Ψ1(ρS(0) ⊗ ρE(0))Ψ†1]. (2.44)
Here h.c. is the abbreviation of Hermitian conjugate. TrE denotes tracing over the degrees of freedom in the environment subspace only it acts on. To see the classical noise contribution to the ensemble average in delity, we can take the ensemble average of the reduced density matrices ρS(t) in Eq. (2.42) over dierent classical noise realizations or take the ensemble average of in delities later, and no matter the former method or the latter method, we can obtain the same results. And we use the latter method to derive the cost functions.
In order to obtain the propagator for the reduced density matrix ρS(t), we should vec-torize it as
vec[ρS(t)] = G(t)vec[ρS(0)]. (2.45)
Here vec [ρS(0)]is the initial vectorized reduced density matrix, vec [ρS(t)]is the vectorized reduced density matrix at time t, the symbol vec denotes vectorizing the matrix it acts on, for example, in one-qubit system vec [ρS(t)] = (ρS,11(t), ρS,21(t), ρS,12(t), ρS,22(t))T, and
G(t) ≡ [US?(t) ⊗ US(t)] · (I + ¯ψ1+ ¯ψ2+ · · · ). (2.46)
is the propagator of the vectorized reduced density matrix. The relation between ¯Ψj in Eq. (2.42) and ¯ψj in Eq. (2.46) is
vec[ ¯Ψj] ≡ ¯ψjvec[ρS(0)]. (2.47)
Now we can de ne the gate in delity in an open quantum system as
Iopen≡ 1 − 1
22nRe{Tr[G†TG(tf)]}, (2.48) where GT = UT?⊗ UT is the target gate for the vectorized reduced density matrix, and UT
is the target gate in the qubit-system subspace, n is the qubit number. Tr here denotes a trace over the matrix it acts on. The gate in delity de nition in an open quantum system Iopen in Eq. (2.48) can recover to the gate in delity de nition in a closed system I in Eq.
(2.2) if G(t) can be written as G(t) = V?(t) ⊗ V (t), where V (t) is a matrix with the same dimensions as UT. Substituting G(t) in Eq. (2.46) and GT into Iopen in Eq. (2.48), we can obtain the expanded Iopen as
Iopen= J1+ J2,open+ ξ + O( ˜HmCN, ˜HmQN, m ≥ 3), (2.49) J1≡ 1 − 1
22nRe{Tr[G†TGS(tf)]}, (2.50) J2,open≡ − 1
22nRe[Tr( ¯ψ1+ ¯ψ2)]. (2.51)
The forms of above equations are similar to those in a closed system from Eq. (2.12) to Eq. (2.14). Here J1 is the de nition of gate in delity for the ideal qubit-system, where GS(t) ≡ [US?(t) ⊗ US(t)] and US(t) is the ideal qubit-system propagator in Eq. (2.30).
J2,open is the lowest-order contribution of the classical noise and the quantum noise to the gate in delity; the function of ξ is equivalent to that of in a closed system as discussed in Appendix A, and O( ˜HmCN, ˜HmQN, m ≥ 3) represents other higher-order terms of noise excluding ξ. Substituting the de nition of ¯ψ1 and ¯ψ2 into J2,open in Eq. (2.51), we can easily obtain Tr( ¯ψ1) = 0without extra assumptions, and thus J2,open = −Re[Tr( ¯ψ2)]/22n . For quantum noise, Iopen has been the ensemble average in delity because the action of taking the ensemble average for quantum noise has been done when we trace the degrees of freedom in the environment to obtain the reduced density matrix in Eq. (2.42). Next, for classical noise, we take the ensemble average of Iopen over dierent classical noise realizations to obtain the complete ensemble average in delity for both classical noise and
quantum noise contribution of the classical noise to the ensemble average in delity hIopeni, and these two terms are exactly the same as hJ2i derived in a closed system in Eq. (2.16). The next two terms are the lowest-order contribution of the quantum noise to hIopeni, which have similar forms as those of the classical noise. The last three terms are the contribution from the combination of the classical noise and the quantum noise, and these three terms can be omitted if the classical noise has zero mean, hβj(t1)i = 0, or the quantum noise has zero mean, TrE REj(t1)ρE(0) = 0. For this case, only the classical noise and quantum noise correlation functions, CjkCN(t1, t2)and CjkQN(t1, t2), the coupling operators to the ideal qubit-system, SCNj(t) and SQNj(t), and the ideal qubit-system Hamiltonian, HS(t), are
required to evaluate J1 and hJ2,openifor optimization.
Once cost functions J1 and hJ2,openi are de ned clearly, then we can use our two-step optimization introduced in Sec. 2.2, with the rst step J1 optimization and the second step J1+ hJ2,openioptimization, to nd the optimal control pulses for suppressing both the classical noise and the quantum noise simultaneously.