*4.1. Summary*

In the present manuscript, we considered branching states {|*ψ*SE -}, on a Hilbert space HS ⊗ HE of arbitrary dimension, and we derived the general analytic form of the timedependent driving on HE , which guarantees that the system S evolves through the adiabatic manifold at all times. Through this environment-assisted shortcuts to adiabaticity scheme, we explicitly showed that the environment can act as a proxy to control the dynamics of the system of interest. Moreover, for branching states |*ψ*SE - with equal branch probabilities, we further proved that the cost associated with the EASTA technique is exactly equal to that of counterdiabatic driving. We illustrated our results in a simple two-qubit model, where the system and the environment are each described by a single qubit.

It is interesting to note that while we focused in the present manuscript on counterdiabatic driving, the technique can readily be generalized to any type of control unitary map "*U*control", resulting in a desired evolved state |*<sup>κ</sup>n*(*t*)- ≡ *<sup>U</sup>*control|*n*(0)-. The corresponding unitary *U* on HE has then the following form:

$$(\forall \ (m, n) \in [0, N - 1]^2); \ l\!\!/^{\prime}\_{m, n} = \langle n(0) | l \!\!/^{\dagger}\_{m} | \kappa\_{m}(t) \rangle. \tag{20}$$

In the special case, for which the evolved state is equal to the *n*th instantaneous eigenstate of *<sup>H</sup>*0(*t*) (with a phase factor),

$$|\kappa\_n(t)\rangle = e^{-\frac{i}{\hbar}f\_n(t)}|n(t)\rangle,\tag{21}$$

we recover the main result of the manuscript. The above generalization illustrates the broad scope of our results. Any control unitary on the system S can be realized solely by acting on the environment E, without altering the dynamics of the system of interest S (i.e., for any arbitrary driving *<sup>H</sup>*0(*t*) and thus, any driving rate).

#### *4.2. Envariance and Pointer States*

In the present work, we leveraged the presence of an environment to induce the desired dynamics in a quantum system. Interestingly, our novel method for shortcuts to adiabaticity relies on branching states, which play an essential role in decoherence theory and in the framework of quantum Darwinism.

In open system dynamics [53–55], the interaction between system and environment superselects states that survive the decoherence process, also known as the pointer

states [56,57]. It is exactly these pointer states that are the starting point of our analysis, and for which EASTA is designed. While previous studies [58–60] have explored STA methods for open quantum systems, to the best of our understanding, the environment was only considered a passive source of additional noise described by quantum master equations. In our paradigm, we recognize the active role that an environment plays in quantum dynamics, which is inspired by envariance and reminiscent of the mindset of quantum Darwinism. In this framework [44,61–77], the environment is understood as a communication channel through which we learn about the world around us, i.e., we learn about the state of systems of interest by eavesdropping on environmental degrees of freedom [44].

Thus, in true spirit of the teachings by Wojciech H. Zurek, we have understood the agency of quantum environments and the useful role they can assume. To this end, we have applied a small part of the many lessons we learned from working with Wojciech, to connect and merge tools from seemingly different areas of physics to gain a deeper and more fundamental understanding of nature.

**Author Contributions:** Conceptualization, A.T. and S.D.; Formal analysis, A.T.; Funding acquisition, S.D.; Supervision, S.D.; Writing—original draft, A.T. and S.D.; Writing—review & editing, A.T. and S.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** S.D. acknowledges support from the U.S. National Science Foundation under Grant No. DMR-2010127. This research was supported by gran<sup>t</sup> number FQXi-RFP-1808 from the Foundational Questions Institute and Fetzer Franklin Fund, a donor advised fund of Silicon Valley Community Foundation (SD).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** We would like to thank Wojciech H. Zurek for many years of mentorship and his unwavering patience and willingness to teach us how to think about the mysteries of the quantum universe. Enlightening discussions with Agniva Roychowdhury are gratefully acknowledged.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. Cost of Environment-Assisted Shortcuts to Adiabaticity**

In this appendix, we show that CD and EASTA have the same cost. Generally, we have the following:

$$\begin{split} H\_{\text{env}}(t) &= i\hbar \frac{d\mathcal{U}'(t)}{dt} \mathcal{U}'^{\dagger}(t), \\ &= i\hbar \sum\_{i,j} \sum\_{k} \frac{d\mathcal{U}'\_{i,k}}{dt} (\mathcal{U}'\_{j,k})^\* |\mathcal{E}\_i(0)\rangle\langle\mathcal{E}\_j(0)|. \end{split} \tag{A1}$$

From the main result in Equation (17), we obtain the following:

$$\begin{split} H\_{\text{env}}(t) &= \sum\_{i,j} \left( \sum\_{k} \left( \langle k(0)|i\hbar \partial\_{t} \mathcal{U}^{\dagger}|\psi\_{i}(t)\rangle \langle \mathcal{U}\_{j,k}^{\prime}\rangle^{\*} + i\hbar \langle k(0)|\mathcal{U}^{\dagger}|\partial\_{t}\psi\_{i}(t)\rangle \langle \mathcal{U}\_{j,k}^{\prime}\rangle^{\*} \right) \right) \\ &\times |\mathcal{E}\_{i}(0)\rangle\langle\mathcal{E}\_{j}(0)|. \end{split} \tag{A2}$$

Given that *<sup>H</sup>*0(*t*) = *ih*¯ *dU*(*t*) *dt <sup>U</sup>*†(*t*), we also have

$$\begin{split} H\_{\text{env}}(t) &= \sum\_{i,j} \left( \sum\_{k} \left( \langle k(0) | (-\mathcal{U}^{\dagger} H\_0) | \psi\_i(t) \rangle (\mathcal{U}\_{j,k}')^\* + i\hbar \langle k(0) | \mathcal{U}^{\dagger} | \partial\_i \psi\_i(t) \rangle (\mathcal{U}\_{j,k}')^\* \right) \right) \\ &\times |\mathcal{E}\_i(0)\rangle \langle \mathcal{E}\_j(0)| , \end{split} \tag{A3}$$

which implies

$$\begin{split} H\_{\text{env}}(t) &= \sum\_{i,j} \left( \sum\_{k} \left( \langle k(0) | (-\mathcal{U}^{\dagger} H\_0) | \psi\_i(t) \rangle (\mathcal{U}\_{j,k}')^\* + \langle k(0) | \mathcal{U}^{\dagger} H | \psi\_i(t) \rangle (\mathcal{U}\_{j,k}')^\* \right) \right) \\ &\times |\mathcal{E}\_i(0) \rangle \langle \mathcal{E}\_j(0) |, \end{split} \tag{A4}$$

and hence,

$$H\_{\rm env}(t) = \sum\_{i,j} \left( \sum\_{k} \langle k(0) | \mathcal{U}^{\dagger} H\_{\rm cp} | \psi\_{i}(t) \rangle \langle \mathcal{U}\_{j,k}' | \* \right) | \mathcal{E}\_{i}(0) \rangle \langle \mathcal{E}\_{j}(0) |. \tag{A5}$$

Using |*φk*(*t*)- ≡ *U*(*t*)|*k*(0)- we can write the following:

$$H\_{\rm env}(t) = \sum\_{i,j} \left( \sum\_{k} \langle \psi\_{\rangle}(t) | \phi\_{k}(t) \rangle \langle \phi\_{k}(t) | H\_{\rm cp} | \psi\_{i}(t) \rangle \right) | \mathcal{E}\_{i}(0) \rangle \langle \mathcal{E}\_{j}(0) |. \tag{A6}$$

Therefore, the following holds:

$$H\_{\rm env}(t) = \sum\_{i,j} \left( \langle \Psi\_{\rangle}(t) | H\_{\rm cp} | \Psi\_{\bar{i}}(t) \rangle \right) | \mathcal{E}\_{\bar{i}}(0) \rangle \langle \mathcal{E}\_{\bar{j}}(0) |. \tag{A7}$$

By definition, we also have the following:

$$H\_{\rm cp} = \sum\_{i,j} \left( \langle \psi\_i(t) | H\_{\rm cp} | \psi\_j(t) \rangle \right) |\psi\_i(t)\rangle \langle \psi\_j(t)|,\tag{A8}$$

and hence,

$$H\_{\rm cp}^T = H\_{\rm cp}^\* = \sum\_{i,j} \left( \langle \psi\_j(t) | H\_{\rm cp} | \psi\_i(t) \rangle \right) |\psi\_i(t)\rangle \langle \psi\_j(t)|. \tag{A9}$$

Thus, there exists a similarity transformation between *H*<sup>∗</sup>CD and *H*env, and *C*CD = *C*env for any arbitrary driving *<sup>H</sup>*0(*t*). The similarity transformation is given by the matrix *S* = ∑*j* |E*j*(0)*ψj*(*t*)|, such that *SH*<sup>∗</sup>CD*S*−<sup>1</sup> = *H*env. Since we proved that the Hamiltonians *H*CD and *H*env have the same eigenvalues, our result can be valid for other definitions of the cost function C, which might involve other norms (e.g., the Frobenius norm).

It is noteworthy that in the above analysis, we do not consider the effect of quantum fluctuations [78] in the control fields, since their energetic contribution to the cost function is negligible in our context.

#### **Appendix B. Generalization to Arbitrary Branching Probabilities**

Finally, we briefly inspect the case of non-even branching states. We begin by noting the consequences of our assumptions. In particular, we have assumed that the state of system+environment evolves unitarly. Thus, consider a joint map of the form *U* ⊗ *M*, where *U* is a unitary on S. Then, it is a simple exercise to show that the map *M*, on E, is also unitary, *MM*† = *M*†*M* = I. In what follows, we prove by contradiction that there exists no unitary map *M* that suppresses transitions in S, for branching states with arbitrary probabilities.

Consider the following:

$$|\psi\_{\mathcal{SE}}(0)\rangle = \sum\_{n=0}^{N-1} \sqrt{p\_n} |n(0)\rangle \bigotimes\_{l=1}^{N\_{\mathcal{E}}} |\mathcal{E}\_n^l(0)\rangle,\tag{A10}$$

and assume that there exists a unitary map *M* on E that suppresses transitions in S, i.e.,

$$\sum\_{n=0}^{N-1} \sqrt{p\_n} \mathcal{U}|n(0)\rangle \otimes \left(M \bigotimes\_{l=1}^{N\_{\mathcal{E}}} |\mathcal{E}\_n^l(0)\rangle\right) = \sum\_{n=0}^{N-1} \sqrt{p\_n} \mathcal{e}^{-\frac{i}{\hbar}f\_n(t)} |n(t)\rangle \otimes \left(\bigotimes\_{l=1}^{N\_{\mathcal{E}}} |\mathcal{E}\_n^l(0)\rangle\right). \tag{A11}$$

Following the same steps of Section 3, we obtain the following:

$$(\forall \ (m, n) \in \left[0, N - 1\right]^2); \ M\_{m, n} = \sqrt{\frac{p\_m}{p\_n}} e^{-\frac{i}{\hbar} f\_m(t)} \langle n(0) | \mathcal{U}^\dagger | m(t) \rangle. \tag{A12}$$

Comparing the above map with our main result in Equation (17), we conclude that the additional factor *pmpn* violates unitarity, and hence we conclude that EASTA cannot work for non-even branching states (A10).

This can be seen more explicitly from the form of the matrices *MM*† and *M*†*M*. Generally, and by dropping the superscript in environmental states *<sup>N</sup>*E *l*=1 |E*ln*(0)- ≡ |E*n*(0)-, we have the following:

$$MM^\dagger = \sum\_{i,j,k} M\_{i,k} M^\*\_{j,k} |\mathcal{E}\_i(0)\rangle\langle\mathcal{E}\_j(0)|,\tag{A13}$$

from the expression of the elements of *M* (cf. Equation (A12)), and by adopting the notation |*φn*- ≡ *<sup>U</sup>*(*t*)|*n*(0)-, we obtain the following:

$$MM^\dagger = \sum\_{i,j,k} \frac{\sqrt{p\_i p\_j}}{p\_k} \langle \phi\_k | \psi\_i \rangle \langle \psi\_j | \phi\_k \rangle \langle \mathcal{E}\_i(0) \rangle \langle \mathcal{E}\_j(0) |, \tag{A14}$$

which implies

$$MM^\dagger = \mathbb{I} + \sum\_{i,j} \sqrt{\frac{p\_j}{p\_i}} \langle \psi\_j | D\_{(i)} | \psi\_i \rangle | \mathcal{E}\_i(0) \rangle \langle \mathcal{E}\_j(0) |, \tag{A15}$$

such that the matrix

$$D\_{(i)} = \sum\_{k} \frac{p\_i}{p\_k} |\phi\_k\rangle\langle\phi\_k| - \mathbb{I} \tag{A16}$$

is diagonal in the basis spanned by the orthonormal vectors {|*φk*-}*<sup>k</sup>*. This matrix is generally (for any choice of *<sup>H</sup>*0(*t*) and initial state of S) different from the null matrix for non-equal branch probabilities. A similar decomposition can be made for the matrix *<sup>M</sup>*†*M*, such that

$$M^\dagger M = \mathbb{I} + \sum\_{i,j} \sqrt{\frac{p\_i}{p\_j}} \langle \phi\_j | \mathcal{D}\_{(i)} | \phi\_i \rangle | \mathcal{E}\_i(0) \rangle \langle \mathcal{E}\_j(0) |, \tag{A17}$$

where

$$\mathcal{D}\_{(i)} = \sum\_{k} \frac{p\_k}{p\_i} |\psi\_k\rangle\langle\psi\_k| - \mathbb{I}. \tag{A18}$$

In conclusion, for branching states with non-equal probabilities, there is no unitary map that guarantees that the system evolves through the adiabatic manifold at all times and for any arbitrary driving *<sup>H</sup>*0(*t*). Hence, we can realize the EASTA technique only for a system maximally entangled with its environment (cf. Equation (A10) with √*pn* = 1/√*N* for all *n* ∈ -0, *N* − 1), or in the general case (non-equal branch probabilities) when we can access an extended Hilbert space.
