SCIENTIFIC REPpRTS

Received: 13 May 2016 Accepted: 27 June 2016 Published: 22 July 2016

Reverse engineering of a Hamiltonian by designing the evolution operators

Yi-Hao Kang1, Ye-Hong Chen1, Qi-Cheng Wu1, Bi-Hua Huang1, Yan Xia1 & Jie Song2

We propose an effective and flexible scheme for reverse engineering of a Hamiltonian by designing the evolution operators to eliminate the terms of Hamiltonian which are hard to be realized in practice. Different from transitionless quantum driving (TQD), the present scheme is focus on only one or parts of moving states in a D-dimension (D > 3) system. The numerical simulation shows that the present scheme not only contains the results of TQD, but also has more free parameters, which make this scheme more flexible. An example is given by using this scheme to realize the population transfer for a Rydberg atom. The influences of various decoherence processes are discussed by numerical simulation and the result shows that the scheme is fast and robust against the decoherence and operational imperfection. Therefore, this scheme may be used to construct a Hamiltonian which can be realized in experiments.

Executing computation and communication tasks1-4 with time-dependent interactions in quantum information processing (QIP)5-12 have attracted more and more interests in recent years. It has been shown that, the adiabatic passage, resonant pulses, and some other methods can be used to realize the evolution process. Among of them, the adiabatic passage techniques are known for their robustness against variations of experimental parameters. Therefore, many schemes have been proposed with adiabatic passage techniques in quantum information processing field. For example, rapid adiabatic passage, stimulated Raman adiabatic passage, and their variants13-22 have been widely used to perform population transfers in two- or three-level systems. The system keeps in the instantaneous ground state of its time-dependent Hamiltonian during the entire evolution process under an adiabatic control of a quantum system. To ensure that the adiabatic condition is always satisfied, the control parameters in the Hamiltonian should be well designed, which usually issue in relatively long execution time. Although little heating or friction will be created when the system remains in the instantaneous ground state, the long time required may make the operation useless or even impossible to implement because decoherence would spoil the intended dynamics. On the other hand, using resonant pulses, the scheme may has a relatively high speed, but it requires exact pulse areas and resonances. Therefore, accelerating the adiabatic passage towards the perfect final outcome is a good idea and perhaps the most reasonable way to actually fight against the decoherence that is accumulated during a long operation time. Consequently, some alternative approaches have been put forward by combining the virtues of adiabatic techniques and resonant pulses together for achieving controlled quantum state evolutions with both high speed and fidelity, such as optimal control theory23-25 and composite pulses26,27. Recently, by designing nonadiabatic shortcuts to speed up quantum adiabatic process, a new technique named "shortcuts to adiabaticity" (STA)28-39 opens a new chapter in the fast and robust quantum state control. As two famous methods of STA, "Transitionless quantum driving" (TQD)31-34 and inverse engineering34-38 based on Lewis-Riesenfeld invariants40 have been intensively focused, They have been applied in different kinds of fields including "fast quantum information processing", "fast cold-atom", "fast ion transport", "fast wave-packet splitting", "fast expansion", etc.41-60. For example, with invariant-based inverse engineering, a fast population transfer in a three-level system has been achieved by Chen and Muga56. Chen et at.57 have proposed a scheme for fast generation of three-atom singlet states by TQD. These schemes have shown the powerful application for invariant-based inverse engineering and TQD in QIP.

It has been pointed out in ref. 34 that, invariant-based inverse engineering and TQD are strongly related and potentially equivalent to each other. Invariant-based method is convenience and effective with a Hamiltonian

■department of Physics, Fuzhou University, Fuzhou 350002, China. 2Department of Physics, Harbin Institute of Technology, Harbin 150001, China. Correspondence and requests for materials should be addressed to Y.X. (email: xia-208@163.com)

which admits known structures for the invariants. But for most systems, the invariants are unknown or hard to be solved. As for TQD, it will not meet this difficult point. However, some terms of Hamiltonian constructed by TQD, which are difficult to be realized in experiments, may appear when we accelerate adiabatic schemes. Therefore, how to avoid these problematic terms is a notable problem. Till now, some schemes61-70 have been proposed to solve the problem of the TQD method recently. For example, Ibanez et al.6i have produced a sequence of STA by examining the limitations and capabilities of superadiabatic iterations. Ibanez et al.65 have also studied the STA for a two-level system with multiple Schrodinger pictures, and subsequently, Song et al.66 have expanded the method in a three-level system based on two nitrogen-vacancy-center ensembles coupled to a transmission line resonator. Moreover, without directly using the counterdiabatic Hamiltonian, Torrontegui et al.67 have used the dynamical symmetry of the Hamiltonian to find alternative Hamiltonians that achieved the same goals as speed-up schemes with Lie transforms. Chen et al.70 have proposed a method for constructing shortcuts to adia-baticity by a substitute of counterdiabatic driving terms.

In this paper, inspired by TQD and the previous schemes61-70, a new scheme for reverse engineering of a Hamiltonian by designing the evolution operators is proposed for eliminating the terms of Hamiltonian which are hard to be realized in practice. The present scheme is focus on only one or parts of moving states in a D-dimension (D > 3) system, that is different from TQD with which all instantaneous eigenstates evolve parallel. According to the numerical simulation, the present scheme not only contains the results of TQD, but also has more free parameters, which make this scheme more flexible. Moreover, the problematic terms of Hamiltonian may be eliminated by suitably choosing these new free parameters. For the sake of clearness, an example is given to realize the population transfer for a Rydberg atom, where numerical simulation shows the scheme is effective. Therefore, this scheme may be used to construct a Hamiltonian which can be realized in experiments.

The article is organized as follows. In the section of "Reverse engineering of a Hamiltonian", we will introduce the basic principle of the scheme for reverse engineering of a Hamiltonian by designing the evolution operators. In the section of "The population transfer for a Rydberg atom", we will show an example using the present scheme to realize the population transfer for a Rydberg atom. Finally, conclusions will be given in the section of "Conclusion".

Reverse engineering of a Hamiltonian

We begin to introduce the basic method of the scheme for reverse engineering of a Hamiltonian by designing the evolution operators. Firstly, we suppose that the system evolves along the state | ^(t)) and the initial state of the system is |V>(0)). So, the condition |^1(0)) = |V>(0)) should be satisfied. We can obtain a complete orthogonal basis {|^„(i})} through a process of completion and orthogonalization. Therefore, the vectors in basis ||^„(i))} satisfy the orthogonality condition {^>m(t)|^>n(t)) = Smn and the completeness condition 4>n(t)){4>„(t) | = 1. Since the system evolves along |^1(t)), the evolution operator can be designed as

U(t) = №(t))(^(0)| + £ A„„(t)(t))(0)|,

m ,n (1)

where parameters \mn(t) (m, n ^ 1) are chosen to satisfy the unitary condition UU = UtU = 1. Submitting the unitary condition into Eq. (1), we obtain

(t) A„k (t) = ¿mn^ n ^ 1). k*1 (2)

Secondly, according to Schrödinger equation (h = 1), we have

¿dt| v (t)) = h (t) (t)),

idtU(t) |t(0)) = H(t) U(t) |t(0)). (3)

On account of the arbitrariness of | V>(0)), Eq. (3) can be written by

idtU(t) = H(t) U(t). (4)

The Hamiltonian can be formally solved from Eq. (4), and be given as

H (t) = i (dtU (t))Uf (t)

= i|&(t)) (&(t)| + i £ Xml (t) A nl (t) [4>m (t)) (4>n (t)|

+ i £ A m, (t) A li (t) (t)) №n(t) |.

l ,m ,n*1 (5)

By submitting Eq. (2) into Eq. (5), the Hamiltonian in Eq. (5) can be described as

H (t) = i£l h(t)) {№ I + i E -Ut) A n*l (t) l^m (t)) (tn (t) |. , x

k l,m,n*1 (6)

Different from TQD, which gives Hamiltonian in the following from

h (t) = i£l iß k(t)) <&(t)|,

Figure 1. Energy levels of the three-energy-level Rydberg atom.

the present scheme has more free parameters Xmn(t). Therefore, this scheme may construct some new and different Hamiltonians. Moreover, when parameters Xmn (m, n ^ 1) are independent of time, Eq. (6) will degenerate into Eq. (7), which shows that the present scheme contains the results of TQD. On the other hand, once the unitary condition UU = UU = 1 for evolution operator is satisfied, the Hamiltonian given in Eq. (6) should be a Hermitian operator, because

H (t) = i (dtU (t))U (t)

= idt(U(t)Uf(t)) - iU(t)dtU (t)) = -iU (t) dt U (t)) = Ht (t).

As an extension, for a N-dimension system (N > 4), the evolution operator can be designed as U (t) = £ (t) (0) |+ £ Xmn (t) |0m (t) )($« (0) |,(1 < 5 < N - 2) .

j = 1,2,-,s

Then, the initial state |V>(0)) of the system can be expressed by the superposition of {|0.(O))} (j =1,2,..., 5). Thus, the system can evolve along more than one moving states in this case. This might sometimes help us to simplify the design of the system's Hamiltonian.

The population transfer for a Rydberg atom

For the sake of clearness, we give an example to emphasize the advantages of the scheme. Here, we consider a Rydberg atom with the energy levels shown in Fig. 1. The transition between |1) and |3) is hard to realize. So, the Hamiltonian of the Rydberg atom is usually written as the following form

H (t) = fti2(f)|1)(2| + ^23(0 (0|2)(3| + H.c.,

where, H12 and H23 are the Rabi frequencies of laser pulses, which drive the transitions |1) ^ |2) and |2) ^ |3), respectively, and they are ^-dephased from each other. Suppose the initial state of the three-energy-level Rydberg atom is |1), the target state is |^far) = cos ^|1) + sin fj]3). We choose a complete orthogonal basis as below

|^(t) ) = cos a cos (311) + sin (3\2) + sin a cos (3\3), |02(t) ) = cos a sin ^1) — cos ^2) + sin a sin ^3), |03(t) ) = sin a|1) — cos a|3).

With the unitary condition in Eq. (2), the evolution operator can take this form

U (t) = )(^(0) |+ cos A (t)(\ï2(t) )(02(O) | + |&(t) )(&(0) I) + sin A (t)(ew (t) 102 (t ) )(03(O) I - e-0 (t)^(t) )(^(0) |). According to Eq. (6), the evolution operator in Eq. (12) gives the following Hamiltonian

H (t) = l El k (t)) (t)| + lÀ (e W\Ut)) {Ht)\ - e lU(t )> (02(t)|)

- 6 sin À cos À (e >2(t)> (№t)\ + e-WO) (t2(t)\)

- 6 sin2À(|02(t))($2(t)\ - |&(t))(03(t)|).

For simplicity, we set d = 0 here, the Hamiltonian in Eq. (13) can be written by

H(t) = i(A sin3 + a)(|3){1| - |1)<3|)

+ i (3 cos a - A cos 3 sin a)(|2)(1| - |1)(2|)

+ i (3 sin a + A cos a cos 3)(|2)(3| - |3)(2|). (14)

Here, the Hamiltonian in Eq. (14) is already a Hermitian operator. To eliminate the terms with |1) (3| and |3) (1|, which are difficult to realize for the three-energy-level Rydberg atom, we set A sin 3 + a = 0. Eq. (14) will be changed into

H(t) = i01(t)(|^(1 - |1)(2|)

+ in2(t)(|2)(3| - |3)(2|),

01(t) = 3 cos a + a cot 3 sin a,

02(t) = 3 sin a — a cos a cot 3. (15)

For simplicity, we suppose the initial time is t = 0 and the final time is tf = T, so T is the total interaction time. To satisfy the boundary conditions a(0) = 0,a(T) = j, a(0) = a(T) = 0, ,0(0) = 3(T) = 0, /3(0) = ¡3(T) = 0 and avoid the singularity of Hamiltonian, we choose the parameters as

f3 (t) =

£(t) =

a (t) =

a (t) = m

8m _ 3T ' t T

where A is an arbitrary constant. Then, the Hamiltonian in Eq. (15) can be written by H(t) = i01(t)(|2)(1| - |1)<2|) + ifi2(t)(|2)(3| - |3)(2|),

^(t) = nA T . (2nt) si^ - cos a - I T ) 3T i 2nt) 1 — cos - I T ) 2 sin a cot 3,

fi2(t) = nA T . (2nt) . si^ - sin a - I T ) 2^ 3T (2nt) 1 — cos - I T ) 2 cos a cot 3. (17)

For the sake of obtaining a relatively high speed, the values of H1T and H2T in Eq. (17) should not be too large. Noticing that, with A increasing, nA increases while cot 0 decreases. Therefore, to obtain a relatively small |H1 T| and |H2T|, A should be neither too large nor too small. Therefore, we choose A = 1 here. However, we can see from Eq. (17) that the functions of Rabi frequencies H1(t) and H2(t) are too complex for experimental realization. Fortunately, we can solve the problem by using simple functions to make a curve fitting for the H1(t) and H2(t). As an example, ^ = n is taken here. We use fi'(t) and Q,'2(t) in the following, which are linear superposition of the Gaussian or trigonometric functions, to make a curve fitting for the H1(t) and H2(t),

in;(t)(|2)(1| - |1)(2|) + ini(t)(|2)(3| - |3)(2|) 3.154

o; (t) =

o2(t) =

T 1.686

sin(5.939t/T - 0.02523), 0 < t < 0.534T, sin(6.531t/T - 0.3177), 0.534T < t < T,

i t-0.3185T\ 0.9443e ( 0.1848T )

/ t-0.7233T\

2.95e ( 0.2004T )

In this case, we have |fi{T| < 3.154 and|fi2T| < 2.96.

To compare the values of H1(t) and fi( (t), H2(t) and 02(t), we plot H1T and fi( (t) versus t/T with j = n/4 and A = 1 in Fig. 2(a) and plot H2T and Q,'2(t) versus t/T with j = n/4 and A = 1 in Fig. 2(b). From Fig. 2(a,b), one can find that the curves of H1(t) and fi( (t) (H2(t) and 02(t)) are well matched with each other. Therefore, we may use fi'(t) (02(t)) instead of H1(t) (H2(t)) to obtain the same effect. To test the effectiveness of the approximation by using fi((t) (02(t)) instead of H1(t) (H2(t)), a simulation for the varies of populations of states |1), |2) and |3) when the Rydberg atom is driven by laser pulses with Rabi frequencies H1(t) and H2(t) with parameters j = n/4 and A = 1, is shown in Fig. 3(a). We can see from Fig. 3(a) that the evolution is consonant with the expectation coming from the evolution operator in Eq. (12). As a comparison, a simulation for the varies of populations of states |1), |2) and |3) when the Rydberg atom is driven by laser pulses with Rabi frequencies H'1(t) and H'2(t) with parameters j = n/4 and A = 1, is shown in Fig. 3(b). As shown in Fig. 3(a,b), we can conclude that the

0.2 0.4 0.6

0.2 0.4 0.6 0.8

Figure 2. (a) H1T and tt[T versus t/T with f = n/4. (b) H2T and H2,T versus t/T with f = n/4 and A = 1.

approximation by using (t) (^2(t)) instead of H1(t) (H2(t)) is effective here. In addition, seen from Fig. 3, the population of intermediate state |2) reaches a peak value about 0.72, because the system does not evolve along the dark state of the Hamiltonian of the system but a nonadiabatic shortcut, which greatly reduces the total evolution time.

Since most of the parameters are hard to faultlessly achieve in experiment, that require us to investigate the variations in the parameters caused by the experimental imperfection. We would like to discuss the fidelity F = K^J^T))!2 with the deviations ST, and 6Q'2 of total interaction time T, Rabi frequencies of laser pulses

and 02 being considered.

Firstly, we plot F versus ¿O'/O' and 6U'2/U'2 with parameters f = n/4 and A = 1in Fig. 4 (a). Moreover, we calculate the exact values of the fidelities F at some boundary points of Fig. 4 (a) and show the results in Table 1. According to Table 1 and Fig. 4 (a), we find that the final fidelity F is still higher than 0.9822 even when the deviation = |^2/n2| = 10%. Therefore, the realizing of the population transfer for a Rydberg atom given in this paper is robust against deviations ¿O' and 6U'2 of Rabi frequencies Q' and U'2 for laser pulses.

Secondly, we plot F versus SQ[/Q[ and ST/T with parameters f = n/4 and A = 1 in Fig. 4 (b). Moreover, /Q[ and ST/T with corresponding fidelity F are shown in Table 2. Seen from Table 2 and Fig. 4 (b), we obtain that the fidelity F is still high than 0.9729 even when the deviation |£Q}/Q1| = \ST /T| = 10%. So, the scheme is insensitive to deviations and ST.

Thirdly, F versus 6Q'2/^'2 and ST/T with parameters f = n/4 and A = 1 is plotted in Fig. 4 (c). And 6Q'2/^'2 and ST/T with corresponding fidelity F are given in Table 3. As indicated in Table 3 and Fig. 4 (c), the fidelity F is still high than 0.9588 even when the deviation \SQ'2/Q'2\ = \ST/T| = 10%. Moreover, when deviations of 5^2 and ST have the different signs (one negative and one positive), the fidelity F can still keep in a high level. Hence, we can say the scheme suffers little from deviations 6Q'2 and ST.

Fourthly, we discuss the fidelity F when <50|, 6Q'2 and ST are all considered. Some samples are given in Table 4. Table 4 shows that the fidelity F is still with a high level when the three deviations <SQ', 6Q'2 and ST are all considered. Moreover, in the worst case, when /0' = 6Q,'2№'2 = ST/T = -10%, the fidelity F is still higher than 0.9469.

According to the analysis above, we summarize that, the scheme to realize the population transfer for a Rydberg atom is robust against operational imperfection.

To prove that the present scheme can be used to speed up the system's evolution and construct the shortcut to adiabatic passages, we make a comparison between the present scheme and the fractional stimulated Raman

adiabatic passage (STIRAP) method via dark state \^dark(t)) =

Hamiltonian shown in Eq. (10). According to STIRAP method, by setting boundary condition lim = 12 = 0, lim = 12 =-tanß = -1,

t^-œ 023(t) t^+ж 023(t)

one can design the Rabi frequencies H12(t) and H23(t) as following

(«23<t)|1) -ад0|3)) of

Qi2(t) = —Ц) exp ^(t) = exp

t - 10 - T/2

t + 10 - T/2

sin ß,

П0 exp

t - 10 - T/2

cos ß,

where H0 denotes the pulse amplitude, tc and t0 are some related parameters. Setting tc = 0.19tf and t0 = 0.14f, Rabi frequencies H12(t) and H23(t) can well satisfy the boundary condition in Eq. (19). We plot Fig. 5 to show the

Figure 3. (a) Populations of states |1), |2) and |3) versus t/Twhen the Rydberg atom is driven by laser pulses with Rabi frequencies H1 and H2. (b) Populations of states |1), |2) and |3) versus t/T when the Rydberg atom is driven by laser pulses with Rabi frequencies and O^. Here we set the parameters j = n/4 and A = 1.

rflia/iii

Figure 4. (a) Fidelity F of the target state versus ¿O'/O' and 6Q,2/Q,2. (b) Fidelity F of the target state versus ¿0'/0' and 6T/T. (c) Fidelity F of the target state versus 8Q,/2/Q/2 and 6T/T. Here we set the parameters j = n/4 and A = 1.

fidelity F when the Rydberg atom is driven by laser pulses with Rabi frequencies H12(t) and H23(t) shown in Eq. (20) versus H0T. And a series of samples of H0T and corresponding fidelity F are shown in Table 5. From Fig. 5 and Table 5, we can see that, to meet the adiabatic condition and obtain a relatively high fidelity by using STIRAP method, one should take H0T about 30. Moreover, when H0T = 3.154, the adiabatic condition is badly violated and the fidelity is only 0.5538 for STIRAP method. But for the present scheme, we can obtain F = 1.000 while |n|T| < 3.154 and\Qr2T\ < 2.96. Therefore, the evolution speed with the present scheme is faster a lot comparing with that using STIRAp method. It confirms that the present scheme can be used to speed up the system's evolution and construct the shortcut to adiabatic passages. Therefore, we conclude that the present scheme can construct a Hamiltonian with both fast evolution process and robustness against operational imperfection.

än'j/n'j sn'2/n'2 F

10% 10% 0.9835

10% 0 0.9951

0 10% 0.9916

0 0 1.0000

-10% 0 0.9938

0 -10% 0.9902

-10% -10% 0.9822

10% -10% 0.9875

-10% 10% 0.9887

Table 1. <5Oj/O' and /n2 with corresponding fidelity F.

5nJ/n' ST/T F

10% 10% 0.9855

10% 0 0.9951

0 10% 0.9942

0 0 1.0000

-10% 0 0.9938

0 -10% 0.9855

-10% -10% 0.9729

10% -10% 0.9879

-10% 10% 0.9915

Table 2. <5Oj/O' and ST/T with corresponding fidelity F.

50'2/n'2 ST/T F

10% 10% 0.9688

10% 0 0.9916

0 10% 0.9942

0 0 1.0000

-10% 0 0.9902

0 -10% 0.9855

-10% -10% 0.9588

10% -10% 0.9974

-10% 10% 0.9994

Table 3. Sfi'/fi' and ST/T with corresponding fidelity F.

5n''/n' 50'2/n'2 ST/T F

-10% -10% -10% 0.9469

10% -10% -10% 0.9607

-10% 10% -10% 0.9853

-10% -10% 10% 0.9926

10% 10% -10% 0.9990

10% -10% 10% 0.9956

-10% 10% 10% 0.9713

10% 10% 10% 0.9531

Table 4. <5n1/n(, Sty/«2 and ST/T with corresponding fidelity F.

In the end, we discuss the fidelity F is robust to the decoherence mechanisms. In this scheme, the atomic spontaneous emission plays the major role. The evolution of the system can be described by a master equation in Lindblad form as following

p = i [p, Hi] + Y^UpL} - 1 (!/Lp + pljli) 1,

I 2 (21)

Figure 5. Fidelity F of the target state versus O0T with the STIRAP method.

3.154 0.5538

5 0.6263

10 0.8516

15 0.9604

20 0.9898

25 0.9960

30 0.9992

Table 5. O0T for STIRAP and corresponding fidelity F.

Figure 6. Fidelity F of the target state versus rj/O0 and r2/00.

where, Ll is the Lindblad operator. There are two Lindblad operators here. They are L1 = ,^r1|1)(2| and L2 = ^Jt2\2) (3|, in which, r1 and r2 are the atomic spontaneous emission coefficients for |2) ^ 11) and |3) ^ |2), respectively. Fidelity F versus r1 T and r2T is plotted in Fig. 6. From Fig. 6, we can see that the fidelity F decreases when r1 and r2 increase. When in the case of strong coupling 01, > r1, r2, the influence caused by atomic spontaneous emission is little. For example, if r1 = r2 = 0.01 x 3.154/T, the fidelity is 0.9901. Even when r1 = r2 = 0.1 x 3.154/T, the fidelity is 0.9101, still higher than 0.9. With current experimental technology, it is easy to obtain a laser pulse with Rabi frequency much larger than the atomic spontaneous emission coefficients. Therefore, the population transfer for a Rydberg atom with the reverse engineering scheme given here can be robustly realized.

Conclusion

In conclusion, we have proposed an effective and flexible scheme for reverse engineering of a Hamiltonian by designing the evolution operators. Different from TQD, the present scheme is focus on only one or parts of moving states in a D-dimension (D > 3) system. The numerical simulation has indicated that the present scheme not only contains the results of TQD, but also has more free parameters, which make this scheme more flexible.

Moreover, the new free parameters may help to eliminate the terms of Hamiltonian which are hard to be realized practically. Furthermore, owing to suitable choice of boundary conditions for parameters, by making a curve fitting, the complex Rabi frequencies ^ and ^ of laser pulses can be respective superseded by Rabi frequencies O' and 02 expressed by the superpositions of the Gaussian or trigonometric functions, which can be realized with current experimental technology. The example given in Sec. III has shown that the present scheme can design a Hamiltonian to realize the population transfer for a Rydberg atom successfully and the numerical simulation has shown that the scheme is fast and robustness against the operational imperfection and the decoherence mechanisms. Therefore, the present scheme may be used to construct a Hamiltonian which can be realized in experiments.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grants No. 11575045 and No. 11374054, and the Major State Basic Research Development Program of China under Grant No. 2012CB921601.

Author Contributions

Y.X. and Y.-H.K. came up with the initial idea for the work and performed the simulations for the model. Y.-H.C., Q.-C.W., J.S. and B.-H.H. performed the calculations for the model. Y.X., Y.-H.K. and Y.-H.C. performed all the data analysis and the initial draft of the manuscript. All authors participated in the writing and revising of the text.

Additional Information

Competing financial interests: The authors declare no competing financial interests.

How to cite this article: Kang, Y.-H. et al. Reverse engineering of a Hamiltonian by designing the evolution operators. Sci. Rep. 6, 30151; doi: 10.1038/srep30151 (2016).

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