*3.4. Illustration*

We illustrate our results in a simple two-qubit model, where the system and and the environment are each described by a single qubit. Note that the environment can live in a larger Hilbert space while still being characterized as a virtual qubit [44]. The aforementioned virtual qubit notion simply means that the state of the environment is of rank equal to two.

We choose a driving Hamiltonian *<sup>H</sup>*0(*t*), such that

$$H\_0(t) = \frac{B}{2}\sigma\_\ddagger + \frac{J(t)}{2}\sigma\_\ddagger,\tag{18}$$

where *J*(*t*) is the driving/control field, *B* is a constant, and *σz* and *σx* are Pauli matrices. Depending on the physical context, *B* and *J*(*t*) can be interpreted in various ways. In particular, as noted in ref. [45], in some contexts, the constant *B* can be regarded as the energy splitting between the two levels [46–48], and in others, the driving *J*(*t*) can be

interpreted as a time-varying energy splitting between the states [49–52]. To illustrate our results we choose the following:

$$(\forall \ t \in [0, \ \tau]); \ J(t) = B \cos^2 \left(\frac{\pi t}{2\tau}\right). \tag{19}$$

The above driving evolves the system beyond the adiabatic manifold, and we quantify this by plotting, in Figure 2, the overlap between the evolved state |*φn*(*t*)- ≡ *<sup>U</sup>*(*t*)|*n*(0)- and the instantaneous eigenstate |*n*(*t*)- of the Hamiltonian *<sup>H</sup>*0(*t*), for *n* ∈ {*g*,*<sup>e</sup>*}. To illustrate our main result, we also plot the overlap between the states resulting from the two shortcut schemes (illustrated in Figure 1): the first scheme is the usual counterdiabatic (CD) driving, where we add a counterdiabatic field *H*CD to the system of interest, and we note the resulting composite state as "|*ψ*CDSE -". The second scheme is the environment-assisted shortcut to adiabaticity (EASTA), and we note the resulting composite state as "|*ψ*EASTA SE -". Confirming our analytic results, the local driving on the environment ensures that the system evolves through the adiabatic manifold at all times since the state overlap is equal to one for all *t* ∈ [0, *τ*].

Finally, we compute and plot the cost of both shortcut schemes and verify that they are both equal to each other for all times "*t*" (cf. Figure 2b), and for all "*τ*" (cf. Figure 2c).

**Figure 2.** *Cont.*

**Figure 2.** In panel (**a**), the blue curve illustrates the overlap between the *n*th evolved state |*φn*(*t*)- and the *n*th instantaneous eigenstate |*n*(*t*)- of the Hamiltonian *<sup>H</sup>*0(*t*). This curve shows that the driving *<sup>H</sup>*0(*t*) evolves the system beyond the adiabatic manifold. The red curve illustrates that EASTA guarantees an exact evolution through the adiabatic manifold. In panel (**b**), we illustrate the cost of both the EASTA and the CD schemes, and numerically verify that they are equal C = *C*CD = *C*env for all *t*/*τ* ∈ [0, 1], and *τ* = 1. Note that in the illustrations we pick the driving field *J*(*t*) = *B* cos<sup>2</sup> *πt* 2*τ* and *B* = 1. In panel (**c**), we illustrate the costs for different values of *τ*. For infinitely fast processes (*τ* → 0) the cost diverges and it tends to zero for infinitely slow processes (*τ* → ∞).

#### **4. Concluding Remarks**
