# Optimal parent Hamiltonians for time-dependent states

Davide Rattacaso  Dipartimento di Fisica, Università di Napoli “Federico II”,
Monte S. Angelo, I-80126 Napoli, Italy  Gianluca Passarelli  Dipartimento di
Fisica, Università di Napoli “Federico II”, Monte S. Angelo, I-80126 Napoli,
Italy  CNR-SPIN, c/o Complesso di Monte S. Angelo, via Cinthia - 80126 -
Napoli, Italy  Antonio Mezzacapo  IBM Quantum, IBM T.J. Watson Research
Center, Yorktown Heights, NY 10598, USA  Procolo Lucignano  Dipartimento di
Fisica, Università di Napoli “Federico II”, Monte S. Angelo, I-80126 Napoli,
Italy  Rosario Fazio  The Abdus Salam International Centre for Theoretical
Physics , Strada Costiera 11, I-34151 Trieste, Italy  Dipartimento di Fisica,
Università di Napoli “Federico II”, Monte S. Angelo, I-80126 Napoli, Italy

###### Abstract

Given a generic time-dependent many-body quantum state, we determine the
associated parent Hamiltonian. This procedure may require, in general,
interactions of any sort. Enforcing the requirement of a fixed set of
engineerable Hamiltonians, we find the optimal Hamiltonian once a set of
realistic elementary interactions is defined. We provide three examples of
this approach. We first apply the optimization protocol to the ground states
of the one-dimensional Ising model and a ferromagnetic $p$-spin model but with
time-dependent coefficients. We also consider a time-dependent state that
interpolates between a product state and the ground state of a $p$-spin model.
We determine the time-dependent optimal parent Hamiltonian for these states
and analyze the capability of this Hamiltonian of generating the state
evolution. Finally, we discuss the connections of our approach to shortcuts to
adiabaticity.

##  I Introduction

In the last decades, impressive progresses have been achieved in the
realization and manipulation of a vast variety of engineered quantum systems
zoller2005 . Solid-state quantum circuits, trapped ions, photonic or
atomic/molecular systems are among the most successful examples of this kind
buluta2011 . In these implementations, control has reached levels that allow
for an accurate driving of quantum states by means of a properly tailored
(time-dependent) Hamiltonian. The essence of most of quantum information
protocols consists in a well defined time modulation of a set of couplings
characterizing the Hamiltonian so to reach ultimately a final state that
encodes the solution to the quantum task NielsenChuang . The modulation may
break down into a set of elementary gates, evolve gently as in the adiabatic
computation or possibly evolve through some global controls as in quantum
simulators. In these situations, the Hamiltonian is known while the quantum
states are not (the final one embeds the desired solution).

Differently from these cases, there are some important instances in quantum
information processing where one is interested in the inverse problem. Given a
known quantum (many-body) state, one would like to determine the parent
Hamiltonian that generates it. What is the relevance of the inverse problem?
In a nutshell, there are several many-body states that one wants to realize
because of their properties, as for example Laughlin statesLaughlin , and
would like to understand how to generate them. The solution to this question
is not unique. In general, given the state of the system, several solutions to
the inverse problem exist in the space of Hermitian operators, but only a
limited number of these Hamiltonians can be realized in a laboratory either
due to fundamental constraints, like locality, or because of technological
limitations. For this reason, one usually considers solutions in a constrained
space of Hamiltonians.

In the time-independent case, the inverse problem consists in reconstructing a
stationary local Hamiltonian starting from a stationary state, for example an
eigenstate. Following the pioneering work by Chertkov and Clark Clark ,
several works have been devoted to this question Thomale ; Zeng ; Laflorencie
; Ranard ; lindner_stationary . A key role in the determination of the parent
Hamiltonian is played by the so-called _Quantum Covariance Matrix_ (QCM) of
the state. This matrix encodes the covariances of the local operators in the
stationary state and its kernel contains all the possible local parent
Hamiltonians. This approach has been extended in several ways, as the search
for a stationary parent Hamiltonian associated to a state after a quantum
quenchHsieh , to open quantum systems governed by a Lindblad dynamics
Lougovski ; Arad  and the determination of a parent Hamiltonian of a Matrix
Product StateMonthus . The non-uniqueness of the parent Hamiltonian is
reflected in the degeneracy of the kernel of the QCM. The characteristics of
this degeneracy have been investigated Ranard  showing under which conditions,
as a consequence of the analyticity of the matrix, the degeneracy is removed
and the reconstruction of the parent Hamiltonian from a unique eigenstate is
possible. The possibility to reconstruct the parent Hamiltonian from
measurements in a bounded region of space, and therefore from partial
knowledge of the correlation matrix, has been discussed in Refs. Ranard, ;
lindner_stationary, . A different approach to the inverse problem has been
proposed Dalmonte1  , based on the Bisognano-Wichmann theorem in quantum field
theory, which relates the parent Hamiltonian to the reduced density matrix of
the state. This method has been exploited to systematically reconstruct an
approximate local parent Hamiltonian for Jastrow-Gutzwiller wave functions
Dalmonte2 .

In this work, we would like to consider the inverse problem for a time-
dependent state. Given a quantum state $|\psi(t)\rangle$, we would like to
determine the Hamiltonian $\hat{H}(t)$ that has $|\psi\rangle$ as a solution
to the time-dependent Schrödinger evolution. The approach that we follow can
be thought as a time-dependent extension of the method proposed in Ref. Clark,
. In particular, we will see that the QCM has a central role also in the time-
dependent case. The reasons for considering this problem are multiple and, we
believe, relevant in quantum information processing. The most natural
application is quantum state preparation. Given a possibly complex target
state, it is possible to reach it by a time-path of quantum states once the
Hamiltonian that generates this sequence of states is found. An interesting
possibility of this strategy, beyond the scope of the present work, is to
design an inverse quantum adiabatic protocol for which the parent Hamiltonian
can be designed by a quantum computer. Also in this case, one needs to develop
an approach to finding the most suitable Hamiltonian to perform this task
taking into account the constraint on the available couplings present in the
Hamiltonian (for example the range or the many-body character of the
interactions). In these terms, the question is closely similar to a quantum
optimal control problem Gross ; opt_c_1 ; opt_c_2 ; opt_c_3 ; opt_c_4 ;
opt_c_5 ; opt_c_6  , with two fundamental differences: here the target state
is defined at each time and not only at the final time, and we consider an
arbitrary space of engineerable interactions. Moreover, given an effectively
realized state evolution, one could use this evolution to reconstruct the
interactions affecting the system, taking advantage of the knowledge of the
physical constraints affecting the Hamiltonian. This is time-dependent
Hamiltonian learning h_learning_1 ; h_learning_2 ; h_learning_3 ; h_learning_4
. Interesting connections of what we discuss here can also be found with
quantum verification. For a time-independent Hamiltonian, its relationship
with inverse problems has been outlined Zoller_verification ; verification_1 ;
verification_2 ; verification_3 ; verification_4 ; verification_5 ;
verification_6 . Time-dependent Hamiltonian learning, quantum verification and
optimal control are therefore different faces with a common ground, that is
the search for a parent Hamiltonian for the state in a space of constrained
Hermitian operators.

Since an exact parent Hamiltonian for a generic state usually does not satisfy
the unavoidable constraints dictated by fundamental principles or by
experimental requests, we look for an optimal parent Hamiltonian as the
minimum of a suitable cost functional, in the sense that the fidelity between
the target state and the state generated via the driving with a Hamiltonian
goes to one when the associated cost goes to zero. This ensures that a low-
cost solution can be useful both for quantum driving and for Hamiltonian
learning. A fundamental point of our analysis is the direct geometrical
meaning of the proposed cost functional, that reflects the geometry induced on
the space of states by a set of allowed interactions. Interpreting the
associated cost function as a geometrical object, we obtain the analytical
expression of the optimal parent Hamiltonian at each time as a function of the
state and its time derivative. The fundamental role of the QCM in this
expression also demonstrates the link between the several proposals for
stationary parent Hamiltonian reconstruction and the optimization methods
involved in optimal control. Moreover, as we show in the body of this work,
this geometrical point of view sets further links between quantum control and
the questions regarding the accessibility of the Hilbert space
illusion_hilbert  and the geometrization of quantum complexity Nielsen1 ;
Nielsen2 ; susskindcomplexity , allowing to design a geometrical picture of
the capability of reproducing a given evolution with a fixed set of allowed
interactions.

The paper is organized as follows. Section II is devoted to setting the stage
by defining the time-dependent inverse problem. In this section, we show how
to exploit the Euclidean structure of the space of Hermitian operators to find
all the exact parent Hamiltonians for a time-dependent state. Although the
inverse problems admit many solutions, engineering all the required
interactions can be an impossible task. One is left with the problem of
finding the parent Hamiltonian in a restricted parameter space. In Section
III, we include the constraints on the space of achievable interactions by
defining a cost functional that bounds the fidelity between the target time-
dependent state and the evolution generated by a trial Hamiltonian. The
minimization of this functional provides the optimal parent Hamiltonian. We
show that the cost functional is locally related to the infinitesimal Fubini-
Study length Wootters  of the target path of states by an _accessibility
angle_. The latter defines a geometrical interpretation of our ability to
generate a target evolution exploiting only some achievable interactions. In
Section IV, we exploit our method to define an optimal parent Hamiltonian for
several time-dependent states. As will be clear in the following, the search
for a parent Hamiltonian related to a time-dependent quantum state has also
tight links to shortcuts to adiabaticity shortcuts1 ; shortcuts2 ; shortcuts3
; shortcuts4 ; shortcuts5  performed via counterdiabatic drivingBerry ;
counter_2 ; counter_3 ; Polkovnikov . These links, as well as the
effectiveness of the optimal parent Hamiltonian as a method to generate a
shortcut to adiabaticity, are discussed in Section V. The last part of the
paper, Section VI, is devoted to conclusions and outlook.

##  II Time-independent inverse problem

The problem of determining the exact parent Hamiltonian associated to a time-
dependent quantum state can be easily posed. Given a pure time-dependent state
$\hat{\rho}(t)=|\psi(t)\rangle\langle\psi(t)|$ of a (possibly many-body)
quantum system in the time interval $[0,T]$, the evolution of this state is
related to the parent Hamiltonian $\hat{H}_{\text{p}}(t)$ via the Schrödinger
equation

$\partial_{t}\hat{\rho}(t)=-i[\hat{H}_{\text{p}}(t),\hat{\rho}(t)].$ (1)

The goal is to find $\hat{H}_{\text{p}}(t)$, given the state $\hat{\rho}(t)$.
The exact parent Hamiltonian is not unique. Hence, to find all the solution to
the inverse problem, we need to invert the Schrödinger equation. This task
becomes easier if we represent both the state and the unknown Hamiltonian as
vectors in the space of Hermitian operators. This space, endowed with the
Hilbert-Schmidt scalar product $\operatorname{Tr}(\hat{A}\hat{B})$, is a
Euclidean space, so we can choose an orthogonal basis
$\\{\hat{O}_{\alpha}\\}$. As an example, an orthogonal basis for the Hermitian
operators acting on an $N$-spins Hilbert space is given by all the tensor
product operators
$\hat{O}_{\alpha}=\hat{\sigma}_{1,\mu_{\alpha}}\otimes\hat{\sigma}_{2,\nu_{\alpha}}\otimes...\otimes\hat{\sigma}_{N,\tau_{\alpha}}$,
where $\hat{\sigma}_{i,\mu}$ is the $\mu^{\text{th}}$ Pauli matrix describing
the $i^{\text{th}}$ spin ($\hat{\sigma}_{i,0}$ is the normalized identity
$\hat{\mathds{1}}$ acting on the $i^{\text{th}}$ spin ).

Using this basis, the quantum state $\hat{\rho}$ can be written as

$\hat{\rho}(t)=\sum_{\alpha}o_{\alpha}(t)\hat{O}_{\alpha},$ (2)

where $o_{\alpha}(t)=\operatorname{Tr}(\hat{\rho}(t)\hat{O}_{\alpha})$, and a
generic Hamiltonian can be written as

$\hat{H}(t)=\sum_{\alpha}h_{\alpha}(t)\hat{O}_{\alpha},$ (3)

where $h_{\alpha}(t)=\operatorname{Tr}(\hat{H}(t)\hat{O}_{\alpha})$.

At this point, we can project Eq. (1) on each element of the basis
$\\{\hat{O}_{\alpha}\\}$ through the Hilbert-Schmidt scalar product, obtaining
a vector representation of the Schrödinger equation in the form

$\partial_{t}o_{\alpha}(t)=-\sum_{\beta}h_{\beta}K_{\alpha,\beta}(t),$ (4)

where
$K_{\alpha,\beta}=i\sum_{l}o_{\gamma}(t)\operatorname{Tr}([\hat{O}_{\beta},\hat{O}_{\gamma}]\hat{O}_{\alpha})$
is named _commutator matrix_. At each time $t$, each exact parent Hamiltonian
corresponds to a solution to this system of inhomogeneous equations for the
coefficients $h_{\alpha}(t)$. The kernel $\text{Ker}(K)$ of the commutator
matrix contains all the symmetries of the state $\hat{\rho}(t)$, that is, all
the Hamiltonians that do not generate any evolution acting on this state. As a
consequence of Eq. (4), the exact parent Hamiltonians, which form a vector
space, differ from each other in terms of the elements of $\text{Ker}(K)$. In
the stationary case, the kernel of $K_{\alpha,\beta}$ contains the solution to
the stationary inverse problem and corresponds to the kernel of the QCM, as
discussed in Ref. lindner_stationary, . In the next section, we will show how
this matrix also determines the behavior of the optimal solutions to the time-
dependent inverse problem. Here, instead, we just outline that the commutator
matrix and the QCM respectively express the dynamical and the static aspect of
an exact parent Hamiltonian for a stationary state: an operator lies in the
kernel of $K_{\alpha,\beta}$ if it does not generate any dynamics, and lies in
the kernel of the correlation matrix if it has zero variance.

As a simple illustration of what we discussed so far, it is useful to consider
the parent Hamiltonian reconstruction for a time-dependent spin-1/2 system. We
consider, for example, the state

$\hat{\rho}(t)=(\sin(\omega t)\hat{\sigma}_{x}+\cos(\omega
t)\hat{\sigma}_{y}+\hat{\mathds{1}})/2$

expressed in the orthogonal basis defined by the Pauli operators
$\\{\hat{\mathds{1}},\hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z}\\}$.
If we write the exact parent Hamiltonian as
$\hat{H}_{\text{p}}=h_{0}(t)\hat{\mathds{1}}+h_{x}(t)\hat{\sigma}_{x}+h_{y}(t)\hat{\sigma}_{y}+h_{z}(t)\hat{\sigma}_{z}$,
Eq. (4) reads

$\begin{pmatrix}0\\\ \omega\cos(\omega t)\\\ \\!-\omega\sin(\omega t)\\!\\\
0\\\ \end{pmatrix}\\!=\\!\begin{pmatrix}0&0&0&0\\\ 0&0&0&2\cos(\omega t)\\\
0&0&0&\\!-2\sin(\omega t)\\!\\\ 0&-\cos(\omega t)&\sin(\omega t)&0\\\
\end{pmatrix}\\!\\!\begin{pmatrix}h_{0}\\\ h_{x}\\\ h_{y}\\\ h_{z}\\\
\end{pmatrix}$

and the parent Hamiltonian for this evolution is

$\hat{H}_{\text{p}}(t)=\frac{\omega}{2}\hat{\sigma}_{z}+f(t)[\sin(\omega
t)\hat{\sigma}_{x}+\cos(\omega t)\hat{\sigma}_{y}]+g(t)\hat{\mathds{1}}$ (5)

with $f(t)$ and $g(t)$ arbitrary functions of time. In the previous equation,
the first term is the only one determining the evolution of the system. The
second term is the arbitrary element of the kernel of the commutator matrix,
which corresponds to the exact parent Hamiltonian for the steady state
$\hat{\rho}(t)$ at each time and does not generate any evolution. The above
example is also useful to understand the relationship between the inverse
problem and shortcuts to adiabaticity. From this perspective, the second term
is the adiabatic Hamiltonian for $\hat{\rho}(t)$ while the first one is the
counterdiabatic potential.

In general, for a many-body system the number of equations in Eq. (4) grows
exponentially with the size of the system. However, we will focus on realistic
Hamiltonians containing only a fraction of the operators $\hat{O}_{\alpha}$
implying few-body interactions, with well behaved couplings. That could be
well implemented in experimental systems.

The optimization method necessary to construct effective Hamiltonians with few
physically relevant interactions is illustrated in the next Section.

##  III Optimal parent Hamiltonian

In the previous section, we have shown that all the exact solution to the
inverse problem are solution to Eq. (4). Physically relevant systems can be
described by linear superpositions of a limited number “elementary”
interactions $\\{\hat{L}_{a}\\}$. Here, as in the rest of this paper, we use
the letter $L$ for operators in set of the _realistic_ elementary
interactions, while the letter $O$ is associated to operators that do not have
to satisfy any constraint. The form of these elementary interactions depends
on physical and technological constraints. Usually this subset includes a
polynomial (in the size of the system) number of operators. For instance, for
spin-1/2 lattices, a suitable basis of local interactions is the set
$\\{\hat{\sigma}_{i,\mu}\\}\cup\\{\hat{\sigma}_{i,\mu}\otimes\hat{\sigma}_{j,\nu}\\}\cup\\{\hat{\sigma}_{i,\mu}\otimes\hat{\sigma}_{j,\nu}\otimes\hat{\sigma}_{l,\tau}\\}$
of one-, two-, and three-body operators between the lattice sites $i,j,l$. The
Hamiltonian is the span of these operators and we define it as $k$-local, if
the maximum distance between interacting sites is
$k=\max\\{|i-j|,|j-l|,|i-l|\\}$. In the following, we will often suppose that
the constraint that determines this basis is locality, but our considerations
can be easily extended to arbitrary sets of allowed interactions.

We express a generic Hamiltonian with locality constraint as
$\hat{H}(t)=\sum_{i}h_{a}(t)\hat{L}_{a}$. The set of the possible
Hamiltonians, endowed with the Hilbert-Schmidt product defined in the previous
section, forms a Euclidean subspace of the space of Hermitian operators.
Therefore we can look for a realistic exact parent Hamiltonian by exploiting
again the method defined in the previous section, that is by looking for
solution to Eq. (4), where the commutator matrix has the form
$K_{a,\alpha}=i\sum_{\beta}o_{\beta}(t)\operatorname{Tr}([\hat{L}_{a},\hat{O}_{\beta}]\hat{O}_{\alpha})$.
Since the cardinality of the basis of local operators is polynomial in the
size of the system, while the dimension of the space of infinitesimal
evolutions for a quantum state is exponential in its size, for an arbitrary
time-dependent state we cannot expect that a realistic parent Hamiltonian
exists. Hence, we will seek for optimal parent Hamiltonians as approximate
solutions to the inverse problem by minimizing the cost functional

$\mathcal{F}[\hat{H}]={\int_{0}^{T}dtf(\hat{H},t)}=\int_{0}^{T}dt\lVert\partial_{t}\hat{\rho}(t)+i[\hat{H}(t),\hat{\rho}(t)]\rVert\;,$
(6)

where the argument of the last integral is the Frobenius norm
$\lVert\cdot\rVert=\sqrt{\operatorname{Tr}(A^{2})}$ of the difference between
the RHS and the LHS of the Schrödinger equation. Of course, this is a measure
of the error done in approximating the exact infinitesimal evolution
$\partial_{t}\hat{\rho}(t)$ of the state with the evolution
$-i[\hat{H}(t),\hat{\rho}(t)]$ generated by the candidate parent Hamiltonian
$\hat{H}(t)$. Moreover, as shown in Appendix A, $\mathcal{F}[\hat{H}]$ gives
an upper bound for the Hilbert-Schmidt distance between the target state
$\hat{\rho}(T)$ and the state

$\hat{\rho}_{\text{opt}}(T)=\left(\mathcal{T}e^{-i\int_{0}^{T}\hat{H}_{\text{opt}}(t)dt}\right)\hat{\rho}(0)\left(\mathcal{T}e^{-i\int_{0}^{T}\hat{H}_{\text{opt}}(t)dt}\right)^{\dagger}$
(7)

evolved with the optimal parent Hamiltonian. As illustrated in Figure 1, this
means that the state $\hat{\rho}_{\text{opt}}(t)$, during its evolution, has
to stay in a ball of radius
$\mathcal{F}[\hat{H},t]=\int_{0}^{t}f(\hat{H}_{\text{opt}},t^{\prime})dt^{\prime}$
with center in $\hat{\rho}(t)$. In terms of the squared fidelity
$F(\hat{\rho}(T),\hat{\rho}_{\text{opt}}(T))=\operatorname{Tr}(\hat{\rho}(T)\hat{\rho}_{\text{opt}}(T))$,
which measures how much the state $\hat{\rho}_{\text{opt}}$ is experimentally
indistinguishable from the state $\hat{\rho}$, inequality in Eq. (7) becomes

$1\geq F(\hat{\rho}(T),\hat{\rho}_{\text{opt}}(T))\geq
1-\frac{1}{2}\mathcal{F}[\hat{H},T]^{2}.$ (8)

Eq. (6) assures that the cost functional $\mathcal{F}[\hat{H}]$ is minimized
when $\hat{H}(t)$ locally generates an evolution as close as possible to
$\partial_{t}\hat{\rho}(t)$, while inequality in Eq. (8) assures that a
sufficiently low-cost minimum of $\mathcal{F}$ generates a high-fidelity
driving of the initial state. However, that does not imply that the minimum of
$\mathcal{F}$ is the best solution to the inverse problem in terms of
fidelity: some Hamiltonians may exist, with larger cost, generating the
driving with higher fidelity.

Figure 1: At each time $t$, the state $\hat{\rho}(t)$ generated by the
Hamiltonian $\hat{H}(t)$ lies inside a ball of radius $\mathcal{F}[\hat{H},t]$
centered on the target state $\hat{\rho}(t)$.

Once defined the optimal parent Hamiltonian as the minimum of the functional
$\mathcal{F}[\hat{H}]$ in Eq. (6), we can recast our problem from a global to
a local form by expressing the derivative of the Hamiltonian as a function of
the state evolution. The integrand of Eq. (6) does not depend on the
derivative of the Hamiltonian, therefore a continuous minimum for
$\mathcal{F}[\hat{H}]$ can be obtained as the time-dependent minimum of the
integrand $f(\hat{H},t)$. In other words, the time-dependent state determines
a time-dependent potential and the Hamiltonian evolves in time remaining in
the minimum of this potential, as in an adiabatic system.

In Figure 2, we show a geometrical interpretation of $f(\hat{H},t)$. The time-
derivative $\partial_{t}\hat{\rho}$ of the state $\hat{\rho}(t)$ is a vector
in the tangent space $T_{\hat{\rho}(t)}$ for the state $\hat{\rho}(t)$, that
is, the space of all the infinitesimal unitary evolutions and is spanned by
the vectors $\\{-i[\hat{O}_{\alpha},\hat{\rho}(t)]\\}$. On the other hand, the
term $i[\hat{H},\hat{\rho}(t)]$ is constrained in the subspace
$TL_{\hat{\rho}(t)}$ of $T_{\hat{\rho}(t)}$ spanned by the vectors
$\hat{l}_{a}(t)=-i[\hat{L}_{a},\hat{\rho}]$, that is, the space of evolutions
that can be generated by local operators. The cost $f(\hat{H},t)$ is the
Euclidean (Hilbert-Schmidt) distance between the vectors
$\partial_{t}\hat{\rho}(t)$ and $-i[\hat{H}_{\text{opt}}(t),\hat{\rho}(t)]$,
hence its minimum is given by the Hamiltonian $\hat{H}_{\text{opt}}(t)$ such
that $-i[\hat{H}_{\text{opt}}(t),\hat{\rho}(t)]$ is the projection of
$\partial_{t}\hat{\rho}(t)$ on $TL_{\hat{\rho}(t)}$.

Figure 2: The commutator $-i[\hat{H}_{\text{opt}}(t),\hat{\rho}(t)]$ has to be
the projection of $\partial_{t}\hat{\rho}(t)$ on the space spanned by the
vectors $\hat{l}_{a}(t)$.

When $i[\hat{H}_{\text{opt}}(t),\hat{\rho}(t)]$ is the projection of
$\partial_{t}\hat{\rho}(t)$, the projections of these two vectors on the
$\hat{l}_{a}(t)$ coincide. In formulas, this means that

$\operatorname{Tr}(\hat{l}_{a}(t)\partial_{t}\hat{\rho}(t))=-i\operatorname{Tr}(\hat{l}_{a}(t)[\hat{H}_{\text{opt}}(t),\hat{\rho}(t)])$
(9)

for each $\hat{l}_{a}$. If we replace the expression
$\hat{H}_{\text{opt}}(t)=\sum_{i}h_{\text{opt},a}(t)\hat{L}_{a}$ of the
optimal parent Hamiltonian in terms of the couplings in Eq. (9), it becomes

$\operatorname{Tr}(\hat{l}_{b}(t)\partial_{t}\hat{\rho}(t))=\sum_{a}\operatorname{Tr}(\hat{l}_{a}(t)\hat{l}_{b}(t))h_{\text{opt},b}(t)$
(10)

where $\operatorname{Tr}(\hat{l}_{a}(t)\hat{l}_{b}(t))$ corresponds to the QCM

$\displaystyle V_{ab}(t)$ $\displaystyle=$
$\displaystyle\langle\psi(t)|\\{\hat{L}_{a},\hat{L}_{b}\\}|\psi(t)\rangle$
$\displaystyle-$ $\displaystyle
2\langle\psi(t)|\hat{L}_{a}|\psi(t)\rangle\langle\psi(t)|\hat{L}_{b}|\psi(t)\rangle$

introduced by Chertkov and ClarkClark  in the context of the inverse problem
associated to a time-independent state. Since the vectors $\hat{l}_{a}(t)$ are
generally not linearly independent, the Gram matrix $V_{ab}(t)$ is not
invertible. In particular, as shown in Ref. Clark, , the kernel of this matrix
contains all the symmetries of the quantum state at a given time. As a
consequence, Eq. (10) represents a linear inhomogeneous system with several
solutions, that are the minima of the cost function $f(\hat{H},t)$ at time
$t$. Even if all these solutions have the same cost $f$, they will generate a
different time evolution with a different fidelity between the evolved state
and the target state. In the rest of this paper, we will focus our attention
on the simplest solution to Eq. (10), that is, the one that does not include
the instantaneous symmetries contained in the kernel of the QCM.

This particular choice has an important consequence on the fidelity between
the target time-dependent state $\hat{\rho}(t)$ and the state
$\hat{\rho}_{\text{opt}}(t)$ generated by the optimal parent Hamiltonian.
Indeed, if we define a new time-dependent state
$\hat{\phi}(\tau)=\hat{\rho}(t(\tau))$ evolving through the same path but at
different velocity, we have that

$F(\hat{\phi}(\tau(t)),\hat{\phi}_{\text{opt}}(\tau(t)))=F(\hat{\rho}(t),\hat{\rho}_{\text{opt}}(t)),$
(11)

where $\hat{\phi}_{\text{opt}}(\tau(t))$ is the evolution generated by the
optimal parent Hamiltonian associated to $\hat{\phi}(\tau)$. In words, with
this choice of the optimal parent Hamiltonian, the fidelity does not depend on
the velocity of the target evolution. This is a consequence of the fact that
the solution to Eq. (10) that does not include the instantaneous symmetries is
proportional to the time derivative of the state: an operator
$\hat{A}(\lambda)$ exists such that
$\hat{H}_{\text{opt}}(t)=\dot{\lambda}(t)\hat{A}(\lambda)$, hence the
generated unitary evolution
$\hat{U}(t)=\mathcal{T}\exp\left(-i\int_{0}^{t}\dot{\lambda}(t^{\prime})\hat{A}(\lambda)dt^{\prime}\right)$
only depends on $\lambda(0)$ and $\lambda(t)$. Note that if we include the
element of the kernel of the QCM in the optimal Hamiltonian, it is not anymore
proportional to the derivative of the state and the previous statement is not
valid.

We conclude this section with a comment on the physical meaning of the angle
between the vectors $\partial_{t}\hat{\rho}(t)$ and
$-i[\hat{H}_{\text{opt}}(t),\hat{\rho}(t)]$. In Figure 2 we can see that the
optimal cost is proportional to the Fubini-Study length of the evolution
$\partial_{t}\hat{\rho}(t)$ through the sine of this angle. Therefore, this
angle, that we call _accessibility angle_ , determines the capability of the
allowed interactions to generate the target time evolution. The more difficult
it is to access the target evolution $\hat{\rho}(t)$ using the available
interactions $\\{\hat{L}_{a}\\}$, the closer the accessibility angle is to
$\pi/2$ during the evolution. Since the difficulty to access quantum states
with a finite amount of resources and the time necessary to perform this task
are two sides of the same coin, we suggest that this angle could have a
central role in the geometrization of quantum complexity proposed by Nielsen
in the context of unitary operators and extended by Susskind to the space of
quantum states Nielsen1 ; Nielsen2 ; susskindcomplexity . Finally, this
relation between cost and length of the path helps us understand to what
extent it is possible to exploit the geodesics of the Fubini-Study metric to
select the most suitable path of states to prepare a given final quantum state
Brachistochrone ; AdiabBrachistochrone .

##  IV Examples

In this section, we apply the general approach illustrated so far to some
specific examples. We discuss three cases in detail. In the first two
examples, the time-dependent state $\hat{\rho}(t)$ coincides with the
instantaneous ground states of given many-body Hamiltonians: the one-
dimensional Ising model and the $p$-spin model in a time-dependent transverse
field. It is interesting to analyze this type of dynamics since it relates to
shortcuts to adiabaticity. In the third example, we consider a generic case in
which two ground states are connected through an arbitrary sequence of states.

###  IV.1 Time-dependent ground state of the Ising model in transverse field

We start from a paradigmatic example, and consider the state
$\hat{\rho}(t)=|\psi(t)\rangle\langle\psi(t)|$ as the instantaneous ground
state of the one-dimensional Ising model in a time-dependent transverse field,
i.e.,
$\hat{H}_{\text{I}}(\lambda(t))|\psi(t)\rangle=E_{\text{GS}}(t)|\psi(t)\rangle$
where $E_{\text{GS}}(t)$ is the energy of the instantaneous ground state and
the Ising Hamiltonian in a time-dependent transverse field
$\lambda(t)\in[0,3]$ is given by

$\hat{H}_{\text{I}}(\lambda(t))=-J\sum_{i=1}^{L}\hat{\sigma}_{i,x}\hat{\sigma}_{i+1,x}-\lambda(t)\sum_{i=1}^{L}\hat{\sigma}_{i,z}\;.$
(12)

We consider periodic boundary conditions
$\hat{\sigma}_{L+1,\mu}=\hat{\sigma}_{1,\mu}$ and an even number of sites $L$,
moreover we set $J=1$. It is important to remark that the Hamiltonian
$\hat{H}_{\text{I}}$ only determines the state $\hat{\rho}(t)$ for which we
will solve the inverse problem, while it is not involved in the definition of
the time-dependent parent Hamiltonian. Then, starting from $\hat{\rho}(t)$ we
obtain its optimal parent Hamiltonian with local interactions minimizing the
cost functional defined in Eq. (6).

First of all, we represent the Hamiltonian $\hat{H}_{\text{I}}(\lambda)$ in a
diagonal form. Since the ground states lie in the even parity sector of the
Hilbert space for the symmetry operator
$\hat{Q}=\prod_{i=1}^{L}{\hat{\sigma}_{i,z}}$, we will represent the
Hamiltonian only in this sector. Exploiting the equivalence of the Ising model
in transverse field with a chain of non-interacting Anderson pseudo-spins
pseudo_spins  $\tilde{\sigma}_{k}^{\mu}$ associated to the momenta in
$\mathcal{K}^{\pm}=\\{\pm\frac{(2i-1)\pi}{L}\text{, with }i=1,...,L/2\\}$ , we
find the following representation of $\hat{H}_{\text{I}}(\lambda)$ (see
Appendix C):

$\displaystyle\hat{H}_{\text{I}}$ $\displaystyle=$
$\displaystyle\sum_{k\in\mathcal{K}^{+}}\hat{H}_{k};$
$\displaystyle\hat{H}_{k}$ $\displaystyle=$
$\displaystyle\epsilon_{k}\left(\cos(\theta_{k})\tilde{\sigma}_{k}^{z}+\sin(\theta_{k})\tilde{\sigma}_{k}^{x}\right);$

where $\epsilon_{k}(\lambda)=2\sqrt{(\lambda-\cos(k))^{2}+\sin^{2}(k)}$ and
$\theta_{k}(\lambda)=-\arctan\left(\frac{\sin(k)}{\lambda-\cos(k)}\right)$. In
this representation, the ground state is the tensor product of single pseudo-
spin ground states of the Hamiltonians $\hat{H}_{k}$,

$|\psi(t)\rangle=\mathbin{\mathop{\otimes}\limits_{k\in\mathcal{K}^{+}}}\left(-\sin(\theta_{k}/2)|\uparrow_{k}\rangle+\cos(\theta_{k}/2)|\downarrow_{k}\rangle\right),$
(13)

that defines our path $\hat{\rho}(t)$. In Eq. (5), we have shown how to apply
our method to reconstruct the exact parent Hamiltonian for a single spin. We
can generalize that result to the state of Eq. (13) to find its exact parent
Hamiltonian in the form of a the sum of the corresponding rotations of single
pseudo-spins:

$\hat{H}_{\text{p}}(t)=\frac{1}{2}\dot{\lambda}\sum_{k\in\mathcal{K}^{+}}\left(\partial_{\lambda}\theta_{k}\right)\tilde{\sigma}_{k}^{y}.$
(14)

Remarkably, in this case, the exact parent Hamiltonian is also the
counterdiabatic potential for the Ising Hamiltonian
$\hat{H}_{\text{I}}(\lambda)$Zurek . This solution is not unique: there are
other exact solutions to the inverse problem for $\hat{\rho}(t)$. They can be
obtained from Eq. (14) by adding any term $\hat{H}_{S}(t)$ such that
$[\hat{H}_{S}(t),\hat{\rho}(t)]=0$.

The Hamiltonian $\hat{H}_{\text{p}}(t)$ drives the initial state through the
target evolution with fidelity one since it is an exact solution to the
inverse problem. Expressing Eq. (14) in terms of real-space Pauli matrices
$\hat{\sigma}_{i,\mu}$, we obtain

$\hat{H}_{\text{p}}(t)=\dot{\lambda}\sum_{j^{\prime}>j}\omega_{j,j^{\prime}}\mathcal{\hat{S}}_{jj^{\prime}}\;,$
(15)

where

$\omega_{j,j^{\prime}}=\frac{1}{2L}\sum_{k\in\mathcal{K}^{+}}\partial_{\lambda}\theta_{k}\sum_{j^{\prime}>j}\sin[k(j^{\prime}-j)]$

and

$\displaystyle\mathcal{\hat{S}}_{jj^{\prime}}$ $\displaystyle=$
$\displaystyle\hat{\sigma}_{j,x}\hat{\sigma}_{j+1,z}\dots\hat{\sigma}_{j^{\prime}-1,z}\hat{\sigma}_{j^{\prime},y}$
$\displaystyle+$
$\displaystyle\hat{\sigma}_{j,y}\hat{\sigma}_{j+1,z}\dots\hat{\sigma}_{j^{\prime}-1,z}\hat{\sigma}_{j^{\prime},x}$

is a string of Pauli operators acting on the spins from $i$ to $j$, in which
the dots represent a string of $\hat{\sigma}_{z}$ operators. Therefore,
$\hat{H}_{\text{p}}$ is nonlocal and involves interactions on a number of
spins that scales linearly with the size of the system.

So far, our approach allows for an exact solution in terms of arbitrary long
strings of spin operators. Now we look for an approximate solution to the
inverse problem. In particular, we look for an optimal parent Hamiltonian for
$\hat{\rho}(t)$ in the space spanned by single- and two-spin interactions in
the set
$B_{L}=\\{\hat{\sigma}_{i,\mu}\\}\cup\\{\hat{\sigma}_{i,\mu}\hat{\sigma}_{i+1,\nu}\\}$.
To this end, we replace the derivative of the state $\partial_{t}\hat{\rho}$
in Eq. (10) with $-i[\hat{H}_{\text{p}},\hat{\rho}]$. In this way, we obtain
the filter function that relates the couplings of the optimal parent
Hamiltonian to the couplings of the exact parent Hamiltonian. Here, it is
important to remark that, since the different exact solution to the inverse
problem generate the same infinitesimal evolution of the state, a different
choice of the exact parent Hamiltonian has no effect on the commutator
$[\hat{H}_{\text{p}},\hat{\rho}]$ in Eq. (10) and therefore it leads to the
same optimal parent Hamiltonian.

At this point, we use the symmetries of the problem to simplify the resolution
of Eq. (10) (see Appendix C). In particular, we exploit the axis reflection
symmetry of the Hamiltonian to show that the observables involving an odd
number of $\hat{\sigma}_{i,x}$ or $\hat{\sigma}_{i,y}$ have zero expectation
value. Moreover we exploit the symmetry of the ground state for a rotation
$i\rightarrow i+1$ and for the reflection $i\rightarrow-i$ of the real-space
axis. In this way, we prove that the optimal parent Hamiltonian has the form

$\hat{H}_{\text{opt}}=h(t)\sum_{i}(\hat{\sigma}_{i,x}\hat{\sigma}_{i+1,y}+\hat{\sigma}_{i,y}\hat{\sigma}_{i+1,x})$
(16)

where

$h(t)=\dot{\lambda}\sum_{l^{\prime}>l}\omega_{l,l^{\prime}}\frac{\langle\mathcal{\hat{S}}_{ll^{\prime}}\sum_{i}\mathcal{\hat{S}}_{ii+1}\rangle}{\langle\sum_{i}\mathcal{\hat{S}}_{ii+1}\sum_{j}\mathcal{\hat{S}}_{jj+1}\rangle}$
(17)

is the only nonzero coupling and $\langle\cdot\rangle$ is the expectation
value of an operator on the target state $\hat{\rho}(t)$. The correlation
functions in Eq. (17) have been previously computed Pfeuty ; Lieb  exploiting
the exact diagonalization of the Hamiltonian $\hat{H}_{\text{I}}(\lambda)$ and
Wick’s theorem. A compact expression of the optimal coupling is obtained
through the Anderson pseudo-spin representation:

$h(t)=-\dot{\lambda}\frac{\sum_{k,k^{\prime}\in\mathcal{K}^{+}}\left\langle\left(\partial_{\lambda}\theta_{k}\right)\tilde{\sigma}_{k}^{y}\sin(k^{\prime})\tilde{\sigma}_{k^{\prime}}^{y}\right\rangle}{8\sum_{k,k^{\prime}\in\mathcal{K}^{+}}\left\langle\sin(k^{\prime})\tilde{\sigma}_{k^{\prime}}^{y}\sin(k)\tilde{\sigma}_{k}^{y}\right\rangle}.$

Exploiting the orthogonality relation
$\left\langle\tilde{\sigma}_{k^{\prime}}^{y}\tilde{\sigma}_{k}^{y}\right\rangle=\delta_{k^{\prime}k}$,
this simplifies to

$h(t)=-\dot{\lambda}\frac{\sum_{k\in\mathcal{K}^{+}}\partial_{\lambda}\theta_{k}\sin(k)}{8\sum_{k\in\mathcal{K}^{+}}\sin(k)^{2}},$

which is the exact expression of the optimal coupling. This expression depends
on the time schedule only through $\lambda(t)$ and its time derivative. The
behavior of the renormalized coupling $h(t)/\dot{\lambda}$ for different
values of $\lambda$ is shown in Figure 3 for different system sizes. The
magnitude of the couplings does not scale with the size of the system,
allowing for a scalable implementation of the optimal Hamiltonian in synthetic
quantum systems.

Figure 3: Evolution of the (rescaled) optimal coupling $h/\dot{\lambda}$ for
different system sizes. $\lambda\in[0,3]$ is an arbitrary smooth function of
time.

The rescaled local cost $f(\hat{H}_{\text{opt}},t)/(\sqrt{L}\dot{\lambda})$ of
the optimal solution is shown in Figure 4 for different system sizes. For
noncritical states, the different curves converge for a sufficiently large
system, signaling that $f(\hat{H}_{\text{opt}},t)$ scales with $\sqrt{L}$. For
the critical states, instead, the cost scales faster than $\sqrt{L}$ and
finding a local optimal parent Hamiltonian becomes very hard. This behavior
for noncritical states is a direct consequence of the scaling of the Fubini-
Study length of the target path of states Zanardi1 ; Zanardi2 . This length
indeed, as we can see from Figure 2, upper bounds the cost of the optimal
Hamiltonian.

Figure 4: Evolution of the (rescaled) optimal cost
$f(\hat{H}_{\text{opt}})/(\sqrt{L}\dot{\lambda})$ for different system sizes.
$\lambda\in[0,3]$ is an arbitrary smooth function of time.

The fidelity $F(t)=\operatorname{Tr}(\hat{\rho}(t)\hat{\rho}_{p}(t))$ between
$\hat{\rho}(t)$ and the state $\hat{\rho}_{p}(t)$ driven by the optimal parent
Hamiltonian assumes a very simple form in the pseudo-spin representation of
the Hilbert space:

$F(t)=\prod_{k\in\mathcal{K}^{+}}|\langle\psi(t_{0})|e^{i\int_{t_{0}}^{t}dt^{\prime}(\frac{1}{2}\dot{\lambda}\partial_{\lambda}\theta_{k}+4h(t)\sin(k))\tilde{\sigma}_{k}^{y}}|\psi(t_{0})\rangle|^{2}.$

Defining the time-dependent angle
$\alpha_{k}(t)=4\sin(k)\int_{t_{0}}^{t}h(t^{\prime})dt^{\prime}+\frac{\theta_{k}(\lambda(t))-\theta_{k}(\lambda(t_{0}))}{2}$,
we obtain

$F(t)=\prod_{k\in\mathcal{K}^{+}}\cos(\alpha_{k}(t))^{2}.$

As anticipated in the previous section, the fidelity depends on time through
$\lambda$, but does not depend on the time schedule. In Figure 5, the fidelity
between the target state and the driven state is shown for different system
sizes. Away from the critical coupling $\lambda_{c}=1$, the optimal parent
Hamiltonian performs the driving of the state with high fidelity, while a
large amount of fidelity is irreversibly lost in correspondence of the
critical value of the control parameter. This behavior can be predicted from
the shape of the cost in Figure 4 by virtue of the inequality of Eq. (8). The
drastic drop of fidelity in correspondence of the critical state is a
consequence of Lieb-Robinson bounds LiebRobinson . These bounds state that the
local correlation length increases at a finite velocity under the evolution
generated by a local Hamiltonian. Since this velocity is proportional to the
magnitude of the interactions involved in the Hamiltonian, this means that if
we want the correlation length to diverge in the thermodynamic limit, as it
happens in the critical states, we need also diverging magnitude of the
couplings. As we see in Figure 3, here the couplings do not scale with the
size of the system, hence the critical state becomes inaccessible as predicted
by the Lieb-Robinson bounds and we see a drastic drop of the fidelity between
the target state and the state generated by the optimal Hamiltonian.

Figure 5: Evolution of the fidelity between the target state and the state
evolved through the optimal parent Hamiltonian for different system sizes.
$\lambda\in[0,3]$ is an arbitrary smooth function of time.

The problem of constructing the (exact or approximate) time-dependent
Hamiltonian generating the state considered here through shortcuts to
adiabaticity has been also discussed by del Campo, Rams and Zurek Zurek  as an
application of the counterdiabatic driving Berry . Their aim was to understand
the performance of a local counterdiabatic potential for the Ising model as a
shortcut to adiabaticity. To this end, the authors have exploited the Ising
diagonal form of the Hamiltonian to determine the counterdiabtic potential in
Eq. (15), and then have filtered out nonlocal interactions. This yields our
Eq. (16). This situation rises an important question: does our optimal parent
Hamiltonian for time-dependent ground states generally correspond to the exact
counterdiabtic potential in which nonlocal interactions have been removed? Our
answer to this question is negative: the filter function that our optimization
method implicitly defines is analyzed in Appendix D where, from Eq. (26), we
can see that the filter depends on the effect of the interactions only on the
target state. On the other hand the procedure followed in Ref. Zurek,  does
not depend on the target states and therefore takes into account the effect of
the interactions in the whole Hilbert space. This may become more evident in
the following subsection.

###  IV.2 Time-dependent ground state of the $p$-spin model

Another paradigmatic case is the inverse problem for the ground state of a
time-dependent $p$-spin Hamiltonian in transverse field. This Hamiltonian,
originally introduced in derrida  and mezard  to model spin glasses, consists
in a spin system in which the elementary interactions connect each set of $p$
spins regardless of their distance. From the computational point of view, the
nonlocal interactions involved in this model are required for the resolution
of some computationally hard problems like the Grover’s searchgrover . Hence,
the possibility of generating the dynamics induced by this kind of
Hamiltonians exploiting realistic resources can give an interesting boost in
the direction of quantum algorithm implementation. For this reason, several
works have been dedicated in recent years to propose techniques to implement
the ground states of the $p$-spin Hamiltonian, with a particular focus on
techniques that involve shortcuts to adiabaticity Passarelli1 ; Passarelli2 ;
Passarelli3 ; Passarelli4 ; Nishimori ; Santoro1 ; Santoro2 ; Santoro3 . The
search for an optimal parent Hamiltonian can be considered a further
contribution to the debate on this topic.

The Hamiltonian of the $p$-spin model, and consequently its ground state, can
be easily represented with an amount of resources that is linear in the size
of the system, allowing for the resolution of the inverse problem of Eq. (10)
also for a large number of spins.

Let us consider the $p$-spin Hamiltonian in transverse field

$\hat{H}_{\text{$p$-spin}}(\lambda(t))=-\Gamma(1-\lambda(t))\hat{\Sigma}_{x}-\lambda(t)\frac{J}{n^{p-1}}(\hat{\Sigma}_{z})^{p}$
(18)

where $\lambda(t)\in[0,1]$ is a time-dependent parameter and

$\hat{\Sigma}_{\mu}=\sum_{i}\hat{\sigma}_{i,\mu}.$

We set $J=\Gamma=1$ and $p=3$. For this choice of $p$, the system exhibits a
first-order phase transition and the ground state is non-degenerate
Passarelli4 . The non-degenerate ground state $\hat{\rho}(t)$ of
$\hat{H}_{\text{$p$-spin}}(\lambda(t))$ is the state for which we solve the
inverse problem.

The model is nonlocal since the interactions in the term
$(\hat{\Sigma}_{z})^{p}$ connect sites regardless of their mutual distance in
the spin chain. We look for an optimal parent Hamiltonian for $\hat{\rho}(t)$
that involves nonlocal interactions connecting a finite number of sites. The
maximum number of sites connected by the elementary interactions is called
_weight_. We look for a parent Hamiltonian spanned by all the interactions of
weight one, two and three. Because of the symmetries inherited by the state
$\hat{\rho}(t)$, these sets must contain only interactions that are invariant
for an arbitrary permutation of the chain sites and are the product of an odd
number of $\hat{\Sigma}_{y}$ operators. Therefore we consider the following
sets of elementary interactions

$\displaystyle B_{1}$ $\displaystyle=$ $\displaystyle\\{\hat{\Sigma}_{y}\\},$
$\displaystyle B_{2}$ $\displaystyle=$ $\displaystyle
B_{1}\cup\\{\hat{\Sigma}_{xy},\hat{\Sigma}_{yz}\\},$ $\displaystyle B_{3}$
$\displaystyle=$ $\displaystyle
B_{2}\cup\\{\hat{\Sigma}_{yyy},\hat{\Sigma}_{xyx},\hat{\Sigma}_{zyz},\hat{\Sigma}_{xxy},\hat{\Sigma}_{yzz},\hat{\Gamma}_{xxy},\hat{\Gamma}_{yzz},$
$\displaystyle\hat{\Sigma}_{xyz},\hat{\Sigma}_{xzy},\hat{\Sigma}_{yxz},\hat{\Gamma}_{xyz},\hat{\Gamma}_{xzy},\hat{\Gamma}_{yxz}\\},$

where
$\hat{\Sigma}_{\mu_{1},..,\mu_{n}}=(\hat{\Sigma}_{\mu_{1}}\dots\hat{\Sigma}_{\mu_{n}}+\text{h.\,c.})/2$
and
$\hat{\Gamma}_{\mu_{1},..,\mu_{n}}=i(\hat{\Sigma}_{\mu_{1}}\dots\hat{\Sigma}_{\mu_{n}}-\text{h.\,c.})/2$.

We calculate the couplings of an optimal parent Hamiltonian that is the span
of each of the previous sets of interactions by numerically solving Eq. (10).
In Figure 6, these coupling are shown for different system sizes when the
chosen interactions have weight $w=2$. We observe that, in correspondence of
noncritical states, the optimal coupling of each interaction converges in the
thermodynamic limit, facilitating a scalable implementation of the optimal
parent Hamiltonian. Instead, in correspondence of the critical state, the
magnitude of the coupling exhibits a large peak whose height increases with
the system size. Analogous features are shared by the optimal coupling for
weight $w=1$ and $w=2$.

Figure 6: Evolution of the (rescaled) optimal coupling $h_{i}/\dot{\lambda}$
for different system sizes when the available interactions have maximum weight
$w=2$. The couplings shown in figure respectively correspond to the
interactions $\hat{\Sigma}_{y}$, $\hat{\Sigma}_{xy}$ and $\hat{\Sigma}_{yz}$.
$\lambda\in[0,1]$ is an arbitrary smooth function of time.

The local cost associated to the optimal solution $f(\hat{H}_{\text{opt}},t)$
of weight $w=2$ is shown in Figure 7. Here, we can observe that the cost
decreases and converges to a minimum value for noncritical states when the
system size is increased: a better parent Hamiltonian for a $p$-spin model
ground state exists when the size of the system is enlarged. As we can see in
Figure 8 (a), this feature is inherited by the fidelity between the target
state and the driven one: larger $p$-spin noncritical states are easier to be
driven. The situation is reversed in correspondence of the critical states,
where the cost of the optimal solution drastically increases with the system
size, generating a sharp drop of the fidelity in Figure 8: as in the Ising
model, the optimal parent Hamiltonian fails to generate a quantum phase
transition. Analogous features are shared by the cost function and the
fidelity for weight $w=1$ and weight $w=3$.

Figure 7: Evolution of the (rescaled) optimal local cost
$f(\hat{H}_{\text{opt}})/\dot{\lambda}$ for different system sizes when the
available interactions have maximum weight $w=2$. $\lambda\in[0,1]$ is an
arbitrary smooth function of time. Figure 8: In panel (a) evolution of the
fidelity between the target state and the state evolved through the optimal
parent Hamiltonian of weight $w=2$ for different system sizes, where
$\lambda\in[0,1]$ is an arbitrary smooth function of time. In panel (b),
fidelity between the target state and the state generated by the optimal
parent Hamiltonian at time $T$ as a function of the system size and for
different weights of the available interactions.

The final fidelity between the target state and the driven one for
$\lambda(t)=1$ is shown in Figure 8 (b) for different sizes of the spin chain
and for different weights of the interactions. As expected, the fidelity is
drastically improved using operators of larger weights, since the basis
$B_{i}$, and therefore the space $TL_{\hat{\rho}}$ on which the exact
evolution is projected, is expanded.

As anticipated, the search for an optimal Hamiltonian capable of generating
the path of ground states for the $p$-spin model in exam has been largely
investigated in the recent literature in the context of shortcuts to
adiabaticity. In these works, the authors look for an optimal countediabatic
Hamiltonian by minimizing the cost functional proposed by Sels and Polkovnikov
Polkovnikov . The differences between the results of these papers and our
results can be easily understood by investigating the difference between the
minimization of Eq. (6) and the counterdiabatic cost proposed in Ref.
Polkovnikov, . Section V of this paper is devoted to this aim.

###  IV.3 Linear interpolation of ground states

In the previous examples, we have solved the inverse problem for a path of
states that have been explicitly built as ground states of a given time-
dependent Hamiltonian. However, our method is suited to the search of an
optimal parent Hamiltonian for a generic time-dependent state. Here we attack
the inverse problem for a time-dependent linear interpolation of a pair of
states. The initial and final states are ground states of the $p$-spin
Hamiltonian $\hat{H}_{\text{$p$-spin}}(\lambda)$ at $\lambda=0$ and
$\lambda=1$. In this way, we can compare the performances of our method for
different paths of states with the same extremal points.

The target time-dependent state is

$|\psi(t)\rangle=Z(t)\left[\cos\left(2\pi\lambda(t)\right)|\psi_{0}\rangle+\sin\left(2\pi\lambda(t)\right)|\psi_{1}\rangle\right]$

where $|\psi_{0}\rangle$ and $|\psi_{1}\rangle$ are the ground states of the
Hamiltonian in Eq. (18) with $\lambda=0,1$, respectively, and $Z(t)$ is a
normalization factor such that $\lVert|\psi(t)\rangle\rVert=1$ at each time
$t$. $\lambda(t)$ is an arbitrary smooth function of time such that
$\lambda(0)=0$ and $\lambda(T)=1$. We represent this state through the density
matrix $\hat{\rho}(t)=|\psi(t)\rangle\langle\psi(t)|$. We construct an optimal
parent Hamiltonian with the operators of weight one, two and three of the sets
$B_{1}$, $B_{2}$ and $B_{3}$.

The solution to Eq. (10) in the space spanned by the operators of weight two
leads to the optimal couplings shown in Figure 9. Here, we can observe that
the magnitude of the coupling decreases with the system size and is maximized
at the initial and final times. A similar behavior is shared by the local cost
associated to the optimal Hamiltonian, which is shown in Figure 10 for
different numbers of spins. We can see that the local cost increases with the
system size and is maximized at the initial and final times. The behavior of
the couplings and of the total cost at the extremal times seems to reflect the
fact that constructing an interpolation of ground states with the available
interactions is very hard.

Figure 9: Evolution of the (rescaled) optimal coupling $h_{i}/\dot{\lambda}$
for different system sizes when the available interactions have maximum weight
$w=2$. $\lambda\in[0,1]$ is an arbitrary smooth function of time. The
couplings shown in figure respectively correspond to the interactions
$\hat{\Sigma}_{y}$, $\hat{\Sigma}_{xy}$ and $\hat{\Sigma}_{yz}$. Figure 10:
Evolution of the (rescaled) optimal local cost
$f(\hat{H}_{\text{opt}})/\dot{\lambda}$ for different system sizes when the
available interactions have maximum weight $w=2$. $\lambda\in[0,1]$ is an
arbitrary smooth function of time.

The behavior of the optimal cost function is reflected in the fidelity between
the target state and the optimally driven one in Figure 11 (a), where larger
drops in fidelity correspond to larger values of the local cost function.

Figure 11: Panel (a): evolution of the fidelity between the target state and
the state evolved through the optimal parent Hamiltonian of weight $w=2$ for
different system sizes, where $\lambda\in[0,1]$ is an arbitrary smooth
function of the time. Panel (b): fidelity between the target state and the
state generated by the optimal parent Hamiltonian at time $T$ for different
sizes of the system and for different weights of the available interactions.

The final fidelity between the target state and the driven one at time $T$ is
shown in Figure 11 (b) for different system sizes and different weights of the
allowed interactions. As expected, we can see that a larger weight corresponds
to a larger final fidelity.

Both the time-dependent state analyzed here and the one analyzed previously
start from and end at the same points, but, for fixed number of spins, the
cost associated to the time-dependent state studied here is larger and the
final fidelity is dramatically lower. We can state that the path of states of
the previous section is more accessible with the interactions in exam. The
investigation of the concept of “accessibility” and the search for a more
accessible path of states are stimulating questions but are beyond the aims of
this work. Here, driven by the observation that building interpolations of
states seems more difficult than building ground states, we just conjecture a
direct relationship between the evolution of the entanglement and the
accessibility of the time-dependent state.

##  V OPTIMAL DRIVING AND SHORTCUT TO ADIABATICITY

Two examples from the previous section have been devoted to the search for an
optimal parent Hamiltonian for the time-dependent ground state
$\text{GS}(\lambda(t))$ of an Hamiltonian $\hat{H}_{\text{a}}(\lambda(t))$. In
this context, the inverse problem is exploited to perform a shortcut to
adiabaticity. A shortcut to adiabaticity is a protocol to drive a quantum
state to the ground state of a target Hamiltonian in a shorter time than that
required by an adiabatic process shortcuts1 ; shortcuts2 ; shortcuts3 ;
shortcuts4 ; shortcuts5 . To reach this goal one needs to suppress Landau-
Zener transitions to the excited eigenstates, which naturally arise when the
control parameters of the system rapidly change. One of the most exploited
shortcuts to adiabaticity consists in suppressing these transition through the
so-called counterdiabatic potential associated to
$\hat{H}_{\text{a}}(\lambda(t))$Berry . This potential also is one of the
exact parent Hamiltonians for $\text{GS}(\lambda(t))$. However, since the
counterdiabatic potential is the span of interactions that are very hard to
engineer in an experiment, one looks for an approximate potential that only
involves a given set of interactions $B_{L}=\\{L_{i}\\}$. Sels and Polkovnikov
Polkovnikov  proposed a local approximation of the counterdiabatic potential
through the minimization of a cost functional depending on the adiabatic
Hamiltonian. This local counterdiabatic potential can be exploited to
optimally control a time-dependent state, so we need to understand which is
the relationship between the local counterdiabatic potential and our optimal
solution to the inverse problem. This section is devoted to the investigation
of this analogy.

There are two fundamental differences between the counterdiabatic potential
and the application of the exact parent Hamiltonian as a shortcut to
adiabaticity. First of all, in order to construct a counterdiabatic potential,
one needs to know the adiabatic Hamiltonian, but the reconstruction of the
latter from a time-dependent state $\hat{\rho}(t)$ is not a trivial problem
and could be considered a particular solution to the optimal inverse problem.
Moreover, the optimal counterdiabatic Hamiltonian is suited to the
simultaneous driving of all the eigenvalues of the adiabatic Hamiltonian. This
can be easily shown if one consider that the potential in exam is the minimum
of the functional
$S_{\hat{H}_{\text{a}}}(\hat{A}^{*})=\int_{0}^{T}\operatorname{Tr}\left[(\partial_{t}\hat{H}_{\text{a}}+i[\hat{A}^{*},\hat{H}_{\text{a}}])^{2}\right]dt$
with respect to the candidate counterdiabatic potential $\hat{A}^{*}$, where
$\hat{H}_{\text{a}}(t)$ is the adiabatic Hamiltonian. One can show (see
Appendix E) that the minimum of $S_{\hat{H}_{\text{a}}}$ is also the minimum
of

$S_{\hat{H}_{\text{a}}}^{\prime}(\hat{A}^{*})=\int_{0}^{T}\lVert\sum_{i}E_{i}(\partial_{t}\hat{\rho}_{i}+i[\hat{A}^{*},\hat{\rho}_{i}])\rVert
dt,$ (19)

where the $E_{i}$ and the $\hat{\rho}_{i}$ are the eigenvalues and the
eigenvectors of $\hat{H}_{\text{a}}$ respectively. This last functional is
related to our cost functionals $\mathcal{F}_{\hat{\rho}_{i}}$ associated to
the inverse problems for the eigenstates $\hat{\rho}_{i}(t)$ as follows:

$S_{\hat{H}_{\text{a}}}^{\prime}\leq\sum_{i}\lvert
E_{i}\rvert\mathcal{F}_{\hat{\rho}_{i}}$ (20)

Therefore, the minimum of $S_{\hat{H}_{\text{a}}}$ optimally drives all the
eigenstates of the adiabatic Hamiltonian, and the driving of each of these
states $\hat{\rho}_{i}$ is obtained by minimizing the inverse problem cost
functional $\mathcal{F}_{\hat{\rho}_{i}}$ and the corresponding energy has the
role of a penalty. As a consequence, a minimum for the potential
$S_{\hat{H}_{\text{a}}}$ is generally different from a minimum of
$\mathcal{F}$. As an example, let us consider the transformation
$\hat{H}_{\text{a}}\rightarrow\hat{H}_{\text{a}}+d\hat{H}_{\text{a}}$ for the
Hamiltonian $\hat{H}_{\text{a}}$, where
$\hat{H}_{\text{a}}=\text{diag}(E_{1},E_{2},E_{3})$ and

$d\hat{H}_{\text{a}}=dt\dot{\lambda}\begin{pmatrix}0&(E_{2}-E_{1})&0\\\
(E_{1}-E_{2})&0&2(E_{3}-E_{2})\\\ 0&2(E_{2}-E_{3})&0\\\ \end{pmatrix}i,$

in which $i$ is the imaginary unit. If we look for the minimum of the cost
function $S_{\hat{H}_{\text{a}}}$ in the space spanned by the operator

$\hat{P}=\begin{pmatrix}0&1&0\\\ 1&0&1\\\ 0&1&0\\\ \end{pmatrix},$

we obtain the counterdiabatic potential

$\hat{H}_{\text{CD}}=-\dot{\lambda}\frac{(E_{1}-E_{2})^{2}+2(E_{3}-E_{2})^{2}}{(E_{1}-E_{2})^{2}+(E_{3}-E_{2})^{2}}\hat{P}.$

The optimal parent Hamiltonian for the ground state of $\hat{H}_{\text{a}}$
instead corresponds to the minimum of the associated cost functional
$\mathcal{F}$. This can be obtained by replacing $E_{2}$ and $E_{3}$ with zero
in the last expression. Now the minimization leads to

$\hat{H}_{\text{opt}}=-\dot{\lambda}\hat{P}.$

The difference between these Hamiltonians mirrors the different goals for
which the corresponding potential are designed: while the first one is suited
to simultaneously drive all the eigenstates of $\hat{H}_{\text{a}}$, the
second one exploits all the available resources to drive only its ground
state.

##  VI CONCLUSIONS AND OUTLOOK

In this paper, we have illustrated the exact solutions to the quantum time-
dependent inverse problem. When the space of Hamiltonians is restricted to a
subset of realistic interactions, we have also defined an engineerable optimal
solution to this problem. Beyond the reconstruction of the time-dependent
parent Hamiltonian, an optimal solution with a sufficiently low cost also
drives the initial state along the target evolution with a high fidelity. As a
consequence, the applications of this methods include both the manipulation of
quantum states and the investigation of the interactions affecting the system.
Exploiting the geometrical interpretation of the cost functional, we have
found the analytical expression of the optimal Hamiltonian in terms of the
state and its derivative at each time. Finally, we have demonstrated the
performance of our method in driving a time-dependent ground state of the
Ising model in transverse field and of the $p$-spin model.

Beyond the direct extension to time-dependent quantum verification and
Hamiltonian learning, an interesting application of our method consists in
calculating the correlations in Eq. (10) using a quantum device. In this way,
one could use our equation to look for a simplified version of the driving
protocol that the device exploits to implement the state $\hat{\rho}(t)$.

In this paper, we have focused our attention to a particular optimal solution
to the inverse problem, whose performance in driving the state does not depend
on the total time of the process. A different approach consists in selecting
the adiabatic solution to the inverse problem, defined as the Hamiltonians
capable of generating the driving with high fidelity for a sufficiently slow
evolution. Such an adiabatic solution has to simultaneously satisfy two
constraints: it has to be a symmetry for the state at any time, and the state
must belong to a nondegenerate eigenspace of the Hamiltonian at any time. As
previously shown in the literature, local symmetries can be easily found as
the kernel of the matrix $V$, but until now there is no systematic way to
select the local Hamiltonians that also obey the second constraint. The design
of a cost functional capable of this selection is one of the most interesting
natural prosecution of this work.

In Section III, we have also defined a geometrical measure of the difficulty
of generating a target time-dependent state exploiting the local interaction,
that is the _accessibility angle_. This geometrical notion could be used to
select the best path to connect quantum states by looking for geodesics of a
related metric. Moreover, the physical meaning of this object has a strong
analogy with the idea of a geometry of quantum complexityNielsen1 ; Nielsen2 ;
susskindcomplexity . This insight could be consolidated by showing a
functional relation between the cost associated to the optimal parent
Hamiltonian and the complexity of the best locally driven approximation of
$\hat{\rho}(t)$. Once this relation has been proven, the optimal cost function
can be associated to an _accessibility metric_ on the manifold of quantum
states, and its relation to the complexity metric proposed by NielsenNielsen1
; Nielsen2 ; susskindcomplexity  could be clarified.

Finally, the central role of the density matrix in our approach suggests the
possibility of an extension to open quantum systems and the better performance
demonstrated in driving states with exponential decaying correlations is a
good reason to support the application to Matrix Product States.

## ACKNOWLEDGEMENTS

P. L. acknowledges Università di Napoli Federico II financial support under
the initiative _Programma di scambi internazionali per la mobilita’ di breve
durata_ D.R. n. 2243 del 12/06/2017.

R. F. acknowledges partial financial support from the Google Quantum Research
Award.

R. F. research has been conducted within the framework of the Trieste
Institute for Theoretical Quantum Technologies (TQT).

##  Appendix A Relationship between cost and fidelity

In this section, we prove the inequality in Eq. (8) of Section III. Given a
trial Hamiltonian $\hat{H}(t)$, the associated cost functional is

$\mathcal{F}[\hat{H}]=\int_{0}^{T}dtf(\hat{H},t),$

where

$f(\hat{H},t)=\lVert\partial_{t}\hat{\rho}(t)+i[\hat{H}(t),\hat{\rho}(t)]\rVert.$

Here we show that, at the final time $T$, the Hilbert-Schmidt distance between
the target state $\hat{\rho}(T)$ and the state

$\hat{\rho}_{\text{opt}}(t)=\left(\mathcal{T}e^{-i\int_{0}^{t}\hat{H}(t^{\prime})dt^{\prime}}\right)\hat{\rho}(0)\left(\mathcal{T}e^{-i\int_{0}^{t}\hat{H}(t^{\prime})dt^{\prime}}\right)^{\dagger}$

evolved with the trial Hamiltonian, is upper bounded by the cost
$\mathcal{F}[\hat{H}]$.

We start from the inequality

$\lVert\hat{\rho}(T)-\hat{\rho}_{\text{opt}}(T)\rVert\leq\int_{0}^{T}dt\,\big{\lvert}\partial_{t}{\lVert\hat{\rho}(t)-\hat{\rho}_{\text{opt}}(t)\rVert}\big{\rvert}.$
(21)

The derivative of the squared norm in the last equation is

$\displaystyle\partial_{t}\lVert\hat{\rho}-\hat{\rho}_{\text{opt}}\rVert^{2}$
$\displaystyle=2\operatorname{Tr}\bigl{[}(\hat{\rho}-\hat{\rho}_{\text{opt}})\partial_{t}(\hat{\rho}-\hat{\rho}_{\text{opt}})\bigr{]}$
$\displaystyle=2\operatorname{Tr}\bigl{[}(\hat{\rho}\\!-\\!\hat{\rho}_{\text{opt}})(\partial_{t}\hat{\rho}+i[\hat{H},\hat{\rho}]\\!-\\!i[\hat{H},\hat{\rho}\\!-\\!\hat{\rho}_{\text{opt}}])],$

where we have exploited the Leibniz rule and we have written
$\partial_{t}\hat{\rho}_{\text{opt}}$ as
$-i[\hat{H},\hat{\rho}_{\text{opt}}]$. Taking into account the Hilbert-Schmidt
orthogonality between $\hat{A}$ and $[\hat{A},\hat{B}]$ for any pair of
operators $\hat{A}$ and $\hat{B}$, the latter becomes

$\partial_{t}\lVert\hat{\rho}-\hat{\rho}_{\text{opt}}\rVert^{2}=2\operatorname{Tr}\bigl{[}(\hat{\rho}-\hat{\rho}_{\text{opt}})(\partial_{t}\hat{\rho}+i[\hat{H},\hat{\rho}])\bigr{]}.$

Finally, we apply the Cauchy-Schwartz inequality to obtain

$\big{\lvert}\partial_{t}{\lVert\hat{\rho}-\hat{\rho}_{\text{opt}}\rVert}^{2}\big{\lvert}\leq
2\lVert\hat{\rho}-\hat{\rho}_{\text{opt}}\rVert\,f(\hat{H},t).$

On the other hand, the Leibniz rule also holds in the following form:

$\partial_{t}{\lVert\hat{\rho}-\hat{\rho}_{\text{opt}}\rVert}^{2}=2\lVert\hat{\rho}-\hat{\rho}_{\text{opt}}\rVert\,\partial_{t}\lVert\hat{\rho}-\hat{\rho}_{\text{opt}}\rVert,$

hence

$\big{\lvert}\partial_{t}\lVert\hat{\rho}-\hat{\rho}_{\text{opt}}\rVert\big{\rvert}\leq
f(\hat{H},t).$

Replacing the latter in Eq. (21), we obtain the upper bound

$\lVert\hat{\rho}(T)-\hat{\rho}_{\text{opt}}(T)\rVert\leq\int_{0}^{T}dt\,f(\hat{H},t)=\mathcal{F}[\hat{H}].$
(22)

For pure states, we have
$\lVert\hat{\rho}(T)-\hat{\rho}_{\text{opt}}(T)\rVert=2(1-F(\hat{\rho}(T),\hat{\rho}_{\text{opt}}(T)))$,
where $F$ is the squared fidelity, therefore the latter equation corresponds
to the inequality in Eq. (8).

##  Appendix B Fubini-Study and Hilbert-Schmidt distance

Here we demonstrate the relationship between the Hilbert-Schmidt metric for
pure states and the Fubini-Study metric. Let $\\{\hat{T}_{n}\\}$ be a set of
Hermitian generators for a basis $\\{|\partial_{n}\psi\rangle\\}$ of a tangent
space for $|\psi\rangle$, that is,
$|\partial_{n}\psi\rangle=i\,\hat{T}_{n}|\psi\rangle$ for each
$|\partial_{n}\psi\rangle$. The derivative of a vector in an arbitrary
direction can be written as

$|d\psi\rangle=i\,\delta^{n}|\partial_{n}\psi\rangle=i\,\delta^{n}\hat{T}_{n}|\psi\rangle$

where the $\delta^{n}$ are coordinates. The Fubini-Study metric is defined as

$g(d\psi,d\psi)=\text{Re}(\langle
d\psi|d\psi\rangle-\langle\psi|d\psi\rangle\langle d\psi|\psi\rangle).$

In the basis defined above, the coordinate representation of the Fubini-Study
metric is the matrix $g_{ij}$ such that

$g(d\psi,d\psi)=\delta^{n}\delta^{m}g_{nm}.$

We show that this matrix is proportional to the correlation matrix $V_{nm}$ of
the operators $\\{\hat{T}_{n}\\}$:

$\displaystyle\delta^{n}\delta^{m}g_{nm}$ $\displaystyle=\text{Re}(\langle
d\psi|d\psi\rangle-\langle\psi|d\psi\rangle\langle d\psi|\psi\rangle)$
$\displaystyle=\delta^{n}\delta^{m}\text{Re}(\langle\psi|\hat{T}_{n}\hat{T}_{m}|\psi\rangle-\langle\psi|\hat{T}_{n}|\psi\rangle\langle\psi|\hat{T}_{m}|\psi\rangle)$
$\displaystyle=\frac{\delta^{n}\delta^{m}}{2}\text{Re}\left(\langle\psi|\\{\hat{T}_{n},\hat{T}_{m}\\}|\psi\rangle-\langle\psi|\hat{T}_{n}|\psi\rangle\langle\psi|\hat{T}_{m}|\psi\rangle\right)$
$\displaystyle=\frac{1}{2}\delta^{n}\delta^{m}V_{nm}.$

The correlation matrix $V_{nm}$ is also the metric associated to the Hilbert-
Schmidt length of tangent vectors:

$\displaystyle{\lVert d\psi\rVert}^{2}$
$\displaystyle=\operatorname{Tr}\bigl{[}(|d\psi\rangle\langle\psi|+\text{h.\,c.})^{2}\bigr{]}$
$\displaystyle=\operatorname{Tr}\bigl{[}(i\,\delta^{n}[\hat{T}_{n},\hat{\rho}])^{2}\bigr{]}$
$\displaystyle=-\delta^{n}\delta^{m}\operatorname{Tr}\bigl{[}(\hat{T}_{n}\hat{\rho}-\hat{\rho}\hat{T}_{n})(\hat{T}_{m}\hat{\rho}\\!-\\!\hat{\rho}\hat{T}_{m})\bigr{]}$
$\displaystyle=\delta^{n}\delta^{m}V_{nm}.$

Therefore, the Hilbert-Schmidt length of tangent vectors is twice the Fubini-
Study length, and the correlation matrix associated to the operators
$\hat{T}_{n}$ is twice the Fubini-Study metric on the tangent space with basis
$\\{i\,\hat{T}_{n}|\psi\rangle\\}$.

##  Appendix C Ground states of the Ising model

Here we show some details of the reconstruction of the exact and optimal
parent Hamiltonians for the ground states of the Ising model that have been
omitted from the main text.

###  C.1 Pseudo-spins representation of the Hamiltonian

The Hamiltonian of the Ising model is

$\hat{H}_{\text{I}}(\lambda)=-\sum_{i=1}^{N}\hat{\sigma}_{i,x}\hat{\sigma}_{i+1,x}-\lambda\sum_{i=1}^{N}\hat{\sigma}_{i,z},$

where, for simplicity, we suppose that the number of spins $N$ is even and
that there are periodic boundary conditions
$\hat{\sigma}_{N+1,\mu}=\hat{\sigma}_{1,\mu}$.

Exploiting the Jordan-Wigner transformation jordan_wigner

$\displaystyle\hat{K}_{j}$
$\displaystyle=\prod_{1}^{j-1}(\hat{\mathds{1}}-2\hat{c}_{j}^{\dagger}\hat{c}_{j})$
$\displaystyle\hat{\sigma}_{j,z}$
$\displaystyle=(\hat{\mathds{1}}-2\hat{c}_{j}^{\dagger}\hat{c}_{j})$
$\displaystyle\hat{\sigma}_{j,x}$
$\displaystyle=\hat{K}_{j}(\hat{c}_{j}^{\dagger}+\hat{c}_{j})$
$\displaystyle\hat{\sigma}_{j,y}$
$\displaystyle=i\,\hat{K}_{j}(\hat{c}_{j}^{\dagger}-\hat{c}_{j}),$

$\hat{H}_{\text{I}}$ can be written as

$\displaystyle\hat{H}_{\text{I}}$
$\displaystyle=-\sum_{1}^{L-1}\left(\hat{c}_{j}^{\dagger}\hat{c}_{j+1}^{\dagger}+\hat{c}_{j}^{\dagger}\hat{c}_{j+1}+\text{h.\,c.}\right)+\lambda\sum_{j=1}^{L}(2\hat{c}_{j}^{\dagger}\hat{c}_{j}-\hat{\mathds{1}})$
$\displaystyle+\left(\hat{c}_{L}^{\dagger}\hat{c}_{L+1}^{\dagger}+\hat{c}_{L}^{\dagger}\hat{c}_{L+1}+\text{h.\,c.}\right)\hat{Q},$

where we have defined the parity operator
$\hat{Q}=(-\hat{\mathds{1}})^{\sum_{j}\hat{c}_{j}^{\dagger}\hat{c}_{j}}$. This
operator is a symmetry for the Hamiltonian and divides the Hilbert space in
two parity sectors. The ground state of the Ising model lies in the even
sector where $Q=1$. Therefore, from now, with a little abuse of notation we
represent the Hamiltonian only in this sector. Once defined the Fourier’s
operators

$\displaystyle\hat{c}_{k}$
$\displaystyle=\frac{e^{-i\pi/4}}{\sqrt{L}}\sum_{j=1}^{L}e^{-ikj}\hat{c}_{j},$
$\displaystyle\hat{c}_{j}$
$\displaystyle=\frac{e^{i\pi/4}}{\sqrt{L}}\sum_{k\in\mathcal{K}}e^{ikj}\hat{c}_{k},$

in the even parity sector the Hamiltonian is

$\displaystyle\hat{H}_{\text{I}}$
$\displaystyle=\sum_{k\in\mathcal{K}^{+}}\hat{H}_{k}$
$\displaystyle\hat{H}_{k}$
$\displaystyle=\epsilon_{k}\bigl{[}\cos(\theta_{k})(\hat{c}_{k}^{\dagger}\hat{c}_{k}-\hat{c}_{-k}\hat{c}_{-k}^{\dagger})$
$\displaystyle+\sin(\theta_{k})(\hat{c}_{k}^{\dagger}\hat{c}_{-k}^{\dagger}+\hat{c}_{-k}\hat{c}_{k})\bigr{]},$

with

$\displaystyle\epsilon_{k}(\lambda)$
$\displaystyle=2\sqrt{(h-J\cos(k))^{2}+(J\sin(k))^{2}},$
$\displaystyle\theta_{k}(\lambda)$
$\displaystyle=-\arctan\left(\frac{J\sin(k)}{h-J\cos(k)}\right),$

and $\mathcal{K}=\\{\pm\frac{(2n-1)\pi}{L}\text{, with }n=1,...,L/2\\}$.

Finally, we can introduce the pseudo-spins operators

$\displaystyle\tilde{\sigma}_{k}^{x}$
$\displaystyle=\hat{c}_{k}^{\dagger}\hat{c}_{-k}^{\dagger}+\hat{c}_{-k}\hat{c}_{k}$
$\displaystyle\tilde{\sigma}_{k}^{y}$
$\displaystyle=-i(\hat{c}_{k}^{\dagger}\hat{c}_{-k}^{\dagger}-\hat{c}_{-k}\hat{c}_{k})$
$\displaystyle\tilde{\sigma}_{k}^{z}$
$\displaystyle=\hat{c}_{k}^{\dagger}\hat{c}_{k}-\hat{c}_{-k}\hat{c}_{-k}^{\dagger}$

and write the Hamiltonians $\hat{H}_{k}$ as

$\hat{H}_{k}=\epsilon_{k}\left[\cos(\theta_{k})\tilde{\sigma}_{k}^{z}+\sin(\theta_{k})\tilde{\sigma}_{k}^{x}\right].$

Since the operators $\tilde{\sigma}_{k}^{\mu}$ obey spin commutation rules, in
this representation the Ising Hamiltonian is a Hamiltonian of noninteracting
spins in an inhomogeneous transverse field, with trivial eigenvalues and
eigenvectors.

The pseudo-spin operators can be represented as a function of the real-space
spins by inverting all the previous transformations as follows:

$\displaystyle\tilde{\sigma}_{k}^{y}$
$\displaystyle=-i(\hat{c}_{k}^{\dagger}\hat{c}_{-k}^{\dagger}-\hat{c}_{-k}\hat{c}_{k})$
$\displaystyle=\frac{1}{L}\sum_{j,j^{\prime}}e^{ik(j-j^{\prime})}(\hat{c}_{j}^{\dagger}\hat{c}_{j^{\prime}}^{\dagger}+\hat{c}_{j}\hat{c}_{j^{\prime}})$
$\displaystyle=-\frac{i}{L}\sum_{j,j^{\prime}}\sin[k(j^{\prime}-j)](\hat{c}_{j}^{\dagger}\hat{c}_{j^{\prime}}^{\dagger}+\hat{c}_{j}\hat{c}_{j^{\prime}})$
$\displaystyle=\frac{1}{L}\sum_{j^{\prime}>j}\sin[k(j^{\prime}-j)]\hat{\mathcal{S}}_{jj^{\prime}},$

where we have defined the spin string operators

$\displaystyle\hat{\mathcal{S}}_{jj^{\prime}}$ $\displaystyle=$
$\displaystyle\hat{\sigma}_{j,x}\hat{\sigma}_{j+1,z}\dots\hat{\sigma}_{j^{\prime}-1,z}\hat{\sigma}_{j^{\prime},y}$
$\displaystyle+$
$\displaystyle\hat{\sigma}_{j,y}\hat{\sigma}_{j+1,z}\dots\hat{\sigma}_{j^{\prime}-1,z}\hat{\sigma}_{j^{\prime},x}.$

###  C.2 Exploiting symmetries

We show how to exploit the symmetries of the Ising model to find a simple
expression for the optimal parent Hamiltonian. The coefficients of the optimal
parent Hamiltonian are solution to Eq. (10). If we replace the time derivative
$\partial_{t}\hat{\rho}$ with $-i[\hat{H}_{\text{p}},\hat{\rho}]$ where
$\hat{H}_{\text{p}}$ is the exact parent Hamiltonian, this equation becomes

$\langle\hat{H}_{\text{p}},\hat{L}_{a}\rangle=\sum_{b}h_{b}\langle\hat{L}_{b},\hat{L}_{a}\rangle,$
(23)

where the $\hat{L}_{a}\in
B_{L}=\\{\hat{\sigma}_{i,\mu}\\}\cup\\{\hat{\sigma}_{i,\mu}\hat{\sigma}_{i+1,\nu}\\}$
are Pauli operators acting on one spin and two adjacent spins and
$\langle\hat{A},\hat{B}\rangle=\operatorname{Tr}[\hat{\rho}(\hat{A}\hat{B}+\hat{B}\hat{A})]-2\operatorname{Tr}(\hat{\rho}\hat{A})\operatorname{Tr}(\hat{\rho}\hat{B})$
is a connected correlation.

First of all, we note that the system is symmetric for a reflection of the
$x$-axis ($\hat{\sigma}_{i,x}\rightarrow-\hat{\sigma}_{i,x}$) or of the
$y$-axis ($\hat{\sigma}_{i,y}\rightarrow-\hat{\sigma}_{i,y}$). As a
consequence, all the expectation values of an odd number of Pauli operators on
these axis are null and Eq. (23) can be represented in a block-matrix form as
follows

$\begin{pmatrix}\vec{0}\\\ \langle\hat{H}_{\text{p}},\vec{XY}\rangle\\\
\langle\hat{H}_{\text{p}},\vec{YX}\rangle\\\
\end{pmatrix}=\begin{pmatrix}A&0&0\\\
0&\langle\vec{XY},\vec{XY}\rangle&\langle\vec{YX},\vec{XY}\rangle\\\
0&\langle\vec{XY},\vec{YX}\rangle&\langle\vec{YX},\vec{YX}\rangle\\\
\end{pmatrix}\begin{pmatrix}\vec{h}_{0}\\\ \vec{h}_{xy}\\\ \vec{h}_{yx}\\\
\end{pmatrix},$

where
$\vec{XY}=(\hat{\sigma}_{1,x}\hat{\sigma}_{2,y},\hat{\sigma}_{2,x}\hat{\sigma}_{3,y},...,\hat{\sigma}_{L,x}\hat{\sigma}_{1,y})^{T}$
and
$\vec{YX}=(\hat{\sigma}_{1,y}\hat{\sigma}_{2,x},\hat{\sigma}_{2,y}\hat{\sigma}_{3,x},...,\hat{\sigma}_{L,y}\hat{\sigma}_{1,x})^{T}$.
It follows that the optimal parent Hamiltonian is

$\hat{H}_{\text{opt}}=\sum_{i}\left[(\vec{h}_{xy})_{i}\hat{\sigma}_{i,x}\hat{\sigma}_{i+1,y}+(\vec{h}_{yx})_{i}\hat{\sigma}_{i,y}\hat{\sigma}_{i+1,x}\right].$

The symmetry of the Ising Hamiltonian for a translation or an inversion of the
spin chain, that is, $\hat{\sigma}_{i,\mu}\rightarrow\hat{\sigma}_{i+1,\mu}$
and $\hat{\sigma}_{i,\mu}\rightarrow\hat{\sigma}_{-i,\mu}$, is inherited by
the optimal parent Hamiltonian, which becomes

$\hat{H}_{\text{opt}}(t)=h(t)\sum_{i}\left[\hat{\sigma}_{i,x}\hat{\sigma}_{i+1,y}+\hat{\sigma}_{i,y}\hat{\sigma}_{i+1,x}\right].$

We have shown that the unique nonzero optimal couplings are associated to the
interactions $\hat{\sigma}_{i,x}\hat{\sigma}_{i+1,y}$ and
$\hat{\sigma}_{i,y}\hat{\sigma}_{i+1,x}$ and are equal to $h$. In this way,
taking into account that the ground state expectation values of Pauli
operators with an odd number of $\hat{\sigma}_{i,x}$ or $\hat{\sigma}_{i,y}$
interactions vanish, Eq. (23) becomes

$h=\frac{\langle\hat{H}_{\text{p}}\sum_{n}(\hat{\sigma}_{i,x}\hat{\sigma}_{i+1,y}+\hat{\sigma}_{i,y}\hat{\sigma}_{i+1,x})\rangle}{\langle\left(\sum_{i}(\hat{\sigma}_{i,x}\hat{\sigma}_{i+1,y}+\hat{\sigma}_{i,y}\hat{\sigma}_{i+1,x})\right)^{2}\rangle},$
(24)

where $\langle\cdot\rangle$ is the expectation value of an operator on the
target state $\hat{\rho}(t)$.

###  C.3 Pseudo-spin representation of the optimal coupling formula

Equation (24) can be expressed using the pseudo-spin operators. Indeed,

$\displaystyle\sum_{i}(\hat{\sigma}_{i,x}\hat{\sigma}_{i+1,y}+\hat{\sigma}_{i,y}\hat{\sigma}_{i+1,x})$
$\displaystyle\quad=2i\sum_{i}(\hat{c}_{i}^{\dagger}\hat{c}_{i+1}^{\dagger}+\hat{c}_{i}\hat{c}_{i+1})$
$\displaystyle\quad=2\sum_{k\in\mathcal{K}}(e^{ik}\hat{c}_{k}^{\dagger}\hat{c}_{-k}^{\dagger}-e^{-ik}\hat{c}_{k}\hat{c}_{-k})$
$\displaystyle\quad=4\sum_{k\in\mathcal{K}^{+}}\sin(k)i(\hat{c}_{k}^{\dagger}\hat{c}_{-k}^{\dagger}-\hat{c}_{-k}\hat{c}_{k})$
$\displaystyle\quad=-4\sum_{k\in\mathcal{K}^{+}}\sin(k)\tilde{\sigma}_{k}^{y},$

and, as shown in the main text,

$\hat{H}_{\text{p}}(t)=\frac{1}{2}\dot{\lambda}\sum_{k\in\mathcal{K}^{+}}\left(\partial_{\lambda}\theta_{k}\right)\tilde{\sigma}_{k}^{y}.$
(25)

Replacing these expressions in Eq. (24), we obtain

$h=-\frac{\dot{\lambda}}{8}\frac{\sum_{k,k^{\prime}\in\mathcal{K}^{+}}\langle\left(\partial_{\lambda}\theta_{k}\right)\tilde{\sigma}_{k}^{y}\sin(k^{\prime})\tilde{\sigma}_{k^{\prime}}^{y}\rangle}{\sum_{k,k^{\prime}\in\mathcal{K}^{+}}\langle\sin(k)\tilde{\sigma}_{k}^{y}\sin(k^{\prime})\tilde{\sigma}_{k^{\prime}}^{y}\rangle}.$

Finally, if we take into account the orthogonality relation
$\langle\tilde{\sigma}_{k}^{y}\tilde{\sigma}_{k^{\prime}}^{y}\rangle=\delta_{kk^{\prime}}$,
we obtain the final expression of the optimal coupling:

$h=-\frac{\dot{\lambda}}{8}\frac{\sum_{k\in\mathcal{K}^{+}}\left(\partial_{\lambda}\theta_{k}\right)\sin(k)}{\sum_{k\in\mathcal{K}^{+}}\sin(k)^{2}}.$

##  Appendix D Filtering out not allowed interactions

In this Appendix, we show that, given the time-dependent state $\hat{\rho}(t)$
and an exact parent Hamiltonian
$\hat{H}_{\text{p}}(t):=\sum_{k}f_{\alpha}(t)\hat{O}_{\alpha}$ for this state,
Eq. (10) systematically defines a filter function $M_{a}^{\alpha}(t)$ such
that the optimal parent Hamiltonian is

$\hat{H}_{\text{opt}}=\sum_{a}h_{\text{opt},a}(t)\hat{L}_{a},$

where $h_{\text{opt},a}(t)=\sum_{\alpha}M^{\alpha}_{a}(t)f_{\alpha}(t)$. This
filter function defines new couplings for the allowed interaction in order to
counterbalance, when possible, part of the effect of eliminating not allowed
interactions.

We can derive the filter $M^{\alpha}_{a}(t)$ from Eq. (10) by replacing the
derivative of the state $\partial_{t}\hat{\rho}(t)$ with
$-i[\sum_{\alpha}f_{\alpha}(t)\hat{O}_{\alpha},\hat{\rho}(t)]$. In this way,
we obtain

$h_{\text{opt},a}(t)=-\sum_{\alpha,a^{\prime}}f_{\alpha}(V_{aa^{\prime}}(t))^{-1}\operatorname{Tr}([\hat{L}_{a^{\prime}},\hat{\rho}(t)][\hat{O}_{\alpha},\hat{\rho}(t)]),$

hence

$M^{\alpha}_{a}(t)=-\sum_{a^{\prime}}(V_{aa^{\prime}}(t))^{-1}\operatorname{Tr}([\hat{L}_{a^{\prime}},\hat{\rho}(t)][\hat{O}_{\alpha},\hat{\rho}(t)]).$
(26)

These last equations reflect the fact that the optimal parent Hamiltonian
$\hat{H}_{\text{opt}}$ is the projection of an arbitrary exact parent
Hamiltonian $\hat{H}_{\text{p}}$ on the space of the allowed interactions
through the degenerate scalar product $(\cdot,\cdot)$ defined as
$(\hat{A},\hat{B})=-\operatorname{Tr}([\hat{A},\hat{\rho}][\hat{B},\hat{\rho}])$.
In the context of shortcuts to adiabaticity, this filter can also be used to
systematically generate a local counterdiabatic potential starting from the
exact one.

##  Appendix E Cost functional and counterdiabatic potential

Here, we show that the counterdiabatic potential defined in Ref. Polkovnikov,
is equal to the potential that minimizes the functional in Eq. (6) of the main
text.

In Ref. Polkovnikov, , the counterdiabatic potential is defined as the
operator $\hat{A}^{*}$ that minimizes

$S_{\hat{H}_{\text{a}}}(\hat{A}^{*})=\int_{0}^{T}\operatorname{Tr}\left[\bigl{(}\partial_{t}\hat{H}_{\text{a}}+i[\hat{A}^{*},\hat{H}_{\text{a}}]\bigr{)}^{2}\right]dt,$

where $\hat{H}_{\text{a}}(t)$ is the adiabatic Hamiltonian. The authors also
show that the minima of the latter potential are also the minima of

$S_{\hat{H}_{\text{a}}}^{\prime\prime}(\hat{A}^{*})=\int_{0}^{T}\operatorname{Tr}\biggl{[}\Bigl{(}\partial_{t}\hat{H}_{\text{a}}+i[\hat{A}^{*},\hat{H}_{\text{a}}]-\sum_{i}\partial_{t}E_{i}\hat{\rho}_{i}\Bigr{)}^{2}\biggr{]}dt,$

where the $E_{i}$ and the $\hat{\rho}_{i}$ are the eigenvalues and the
eigenvectors of $\hat{H}_{\text{a}}$ respectively. Exploiting the spectral
decomposition of $\hat{H}_{\text{a}}$, this becomes

$S_{\hat{H}_{\text{a}}}^{\prime\prime}(\hat{A}^{*})=\int_{0}^{T}\operatorname{Tr}\biggl{[}\Bigl{(}\sum_{i}E_{i}\bigl{(}\partial_{t}\hat{\rho}_{i}+i[\hat{A}^{*},\hat{\rho}_{i}]\bigr{)}\Bigr{)}^{2}\biggr{]}dt.$

Since this functional does not depend on the derivative of $\hat{A}^{*}$, its
minimum is the minimum of the integrand function. Therefore, taking into
account that the square root is a monotonic function, we can replace the
integrand with its square root. We finally obtain

$S_{\hat{H}_{\text{a}}}^{\prime}(\hat{A}^{*})=\int_{0}^{T}\Bigl{\lVert}\sum_{i}E_{i}\bigl{(}\partial_{t}\hat{\rho}_{i}+i[\hat{A}^{*},\hat{\rho}_{i}]\bigr{)}\Bigr{\rVert}dt,$

where $\lVert\cdot\rVert$ is the Frobenius norm.

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