Considering the open system effect by introducing the auxiliary density matrix make the dimension of the matrix of the propagator big. For example, fitting the bath correlation function with 1 exponential terms for the lowest cutoff frequency ωc = 2π × 1GHz has 1280 × 1280 superoperators, which is very difficult to deal with using matlab. The result is in Figure 4.14. And Figure 4.13 shows the optimal control pulses obtained in the closed system with leakage states. We evaluate the error by evolving the open system with the closed-system optimal control pulses. One iteration takes about 45 minutes. The error affect one iteration with the parameters used is small:
J2 = 7.5266 × 10−5. That is, the closed system optimal control are good enough to achieve a high-fidelity CNOT gates.
The gate errors for different temperature obtained by evolving the non-Markovian master equation with the optimal control pulses in closed system case are presented in Fig. 4.14. Since the optimal control calculation for two qubits with leakage states, interacting with a non-Markovian bath is quite computationally expensive. We choose to take the closed-system control pulses for the open system. The gate error obtained in one iteration will be taken as the error for the open system. The gate error obtained this way also same as a measure about how good the closed-system optimal control pulses are against the environment induced decoherence. We find that the error is only around 10−3 ∼ 10−4even at high cutoff frequency for the parameters used in our calculations. This indicates that the closed-system optimal control pulses are pretty robust against decoherence for the model we investigate.
In Figure 4.15, we insert the different error J2 of closed system into T = 10mK
Figure 4.13: Optimized pulses obtain in a closed system with leakage states for tf = 55ps, which are used to evolve the open system with leakage states.
open system. It shows that the error can be further improved if we have the extreme low error result in closed system. Since the dimension is so big and the iteration of closed system is slow enough, we only demonstrate the possibility of the improvement in open system.
Compare to the gate error for the optimal control case of the open system without leakage states shown in Fig. 4.8, the presence of the leakage states seems to prevent the gate error at low cutoff frequencies from decreasing. But keep in mind that we do not actually perform optimal control algorithm for the gate error shown in Fig. 4.14.
0 20 40 60 80 100 10−5
10−4 10−3
ω
2π
(GHz)
error
10mK 100mK 500mK 1000mK
Figure 4.14: The error J2 as a function of cutoff frequency ωc for different temper-atures at tf = 55ps for the open quantum system with leakage states. The gate error is obtained by inserting the optimal control pulses of the closed system into the non-Markovian master equation of the open system.
0 20 40 60 80 100 10−5
10−4 10−3
ω
2π
(GHz)
error
insert closed sys. error = 2e−5 insert closed sys. error = 6e−6
Figure 4.15: Demonstrate the effect of different insert errors into open system. In 10mK environment , the error J2 is possible to get lower level in open system if we insert lower error pulse shapes of closed system. The blue line insert the pulse shapes of error J2 = 2 × 10−5 of closed system into open system, and the green line insert the pulse shapes of error J2 = 6 × 10−6 of closed system into open system.
Chapter 5 Conclusion
We give a short conclusion in this Chapter. In order to perform optimal control of gate operations in closed and open quantum system, we choose the Krotov optimal control method. Krotov optimal control method is effective when we are dealing with closed system with no leakage level. Using J2 = 2N1 T r[(Q − G(T ))†(Q − G(T ))] is a better way to define the error than J1 = 1 − N1T r(Q†G(T )) since the closed system result will make J2 = J1 and in the open system result G(T ) will not be unitary and the error function J2 take the distance of every matrix element in the Q − G(T ) into account. However, the definition of J2 will cause the derivation of Krotov optimal control not able to be monotonically convergence anymore. But since the Krotov method only guarantee the convergence of the cost function and we only need the convergence of the error. So the try and error of penalty become a practical issue during the simulation. Using optimal control to optimize J2 in the closed system with no leakage states can give extreme low error, e.g. 10−10. When we insert this control parameters (closed system control pulses) into non-Markovian open system and do the optimal control again, we find that the iteration just stops at very low numbers, e.g. 10 with error J2 about 10−3 ∼ 10−5, depending on the bath parameters used in the calculations. This shows that the optimal control is efficient and robust in the closed system. The optimal control of error J2 in the closed system with leakage states is hard to reach low error (but possible if the iteration increases). With 5000 to 10000 iterations we could only reach error J2 ∼ 10−5. With the same iteration
number, we could reach error of 10−10 for the closed system without leakage states.
For the case of open system with leakage states, we simply use the optimal control pulses obtained in the closed system to evolve the open system to obtain the gate error as the optimal control calculation in this case is rather computationally expensive.
The result of the gate error is around 10−3 ∼ 10−4 for the bath parameters extracted from the experimental noise spectrum.
The reason why the improvement for the open system optimal control is not great may be the following. In our charge qubit model, it is assumed that we have control only on the σz terms or on the operators with diagonal matrix elements in the charge state basis in our Hamiltonian, Thus we may be able to correct the environment induced relaxation efficiently. As a result, the optimal control can not improve much the gate error for our open system model. If control over both σz and σx terms or over both diagonal and off-diagonal matrix element operators in the Hamiltonian were allowed, the optimal control theory might be able to do a much better job on gate error improvement.
Appendix A
RK4 and expm
In this appendix, we compare the Runge-Kutta method (RK4) with matrix exponen-tial (expm) method of solving the differenexponen-tial equation of the propagator:
dG(t)
dt = Λ(t)G(t). (A.1)
This problem of solving the differential equation is hard when you want to directly solve it. However, we can use the assumption that the superoperator Hamiltonian can be treated as time independent in every small time step dt.
That is,
G(t + dt) = eΛ(t)dtG(t), (A.2)
which is exact in every small time step.
This can lead us to the evolution of the propagator G
G(tf) = eΛ(tf−dt)dt· · · eΛ(2dt)dteΛ(dt)dteΛ(0)dtG(0). (A.3)
The definition of matrix exponential is
But the above definition cannot be directly turned into codes when the norm of A is bigger than 1. Since the Taylor series do not converge. One may use the Pade approximation to calculate the matrix exponential.
Another method to solve the differential equation is Runge-Kutta method (RK4).
For a differential equation dydt = f (t, y) with the initial condition y(0) = y0, yn+1
at the time tn+1 = tn+ h can be written as
yn+1= yn+ h
6(k1+ 2k2+ 2k3+ k4), (A.5)
where h is a small time step
k1 = f (tn, yn)
Since we already make our control be constant in each of the small time intervals. By comparing the expm of matlab and our own rk4. We figure out that every time step of the differential equation must solve separately. That is, choose ng1(tn+ h2) and ng1(tn+ h) both as ng1(tn) and choose ng2(tn+ h2) and ng2(tn+ h) both as ng2(tn).
Which means that the control does not change at all in this time interval.
Remember that RK4 gives only approximated results. When the norm of Λ(t)dt is too big, using RK4 to approximate the matrix exponential result would fail. Why is this issue important? Consider the closed system where is no environment noise.
Theoretically, the optimal control of a two-qubit CNOT gate should be perfect and only bounded by the computer precision limit. Somehow the error of RK4 may cause problem such as error re-bounce in early iteration or make the final error lower than it used to be when using expm.
The lowest final time we choose with good fidelity is55ps. We divide it into 551 time steps in the closed system problem. Other paper [6] choose only 55 time steps. In the first time step, the propagator elements only have error around 10−6 as compared to the answer of expm. However, the initial error could cause a slightly difference after all 551 points and make the propagator non-unitary. The propagator should be unitary under the condition of being in a closed system. And this causes the values of two definitions of the error functions different.
For example:
and B = C =
Compute the differential equation dG(t)
dt = aΛ(t)G(t) (A.12)
and compare each element of the matrix at the final time for different values of ”a”.
When the norm of the aΛ(t) is too big, e.g. a = 100, 1000, the RK4 method fails.
Appendix B
Derivatives of Matrix, Traces
The following matrix properties are extracted from [17]. These properties and tech-niques are necessary for our calculations.
B.1 Matrix Multiplication
A matrix multiplication is simple.
AB = C,
and the dimension of these real matrices are
A (m × n), B (n × p), C (m × p).
The elements of matrix C can be represented by those of the original two matrices:
Cij = (AB)ij =
n
X
r=1
AirBrj.