Variational quantum compiling algorithms [1] aim to compress quantum circuits into parameterized ansatzes utilizing a hybrid quantum-classical approach. In this article a variant of VQC named the recursive variational quantum compiling algorithm (RVQC) is proposed. In contrast with existing VQC algorithms, where one is required to execute the full target circuit in order to compile it, RVQC works by dividing the target circuit into $N$ parts and recursively compresses one part at a time into the parameterized ansatz. Since smaller parts are being compressed at a time, the circuit depth required to implement the algorithm is not dependent on the depth of the target circuit, but on the depth of the ansatz and the depth of each of the $N$ parts. We are hence able to compress the target circuit by executing circuits of smaller depths than the target circuit itself. To show that this approach has an edge over the standard VQC method for sufficiently large circuit depths, RVQC was compared with VQC on a noise model of the IBM Santiago device with the goal of compressing a randomly generated five-qubit circuit of depth 5000. It was shown that VQC was not able to converge within 500 epochs of training. On the other hand, RVQC was able to converge to a fidelity ($0.88$) within a total of 500 epochs when splitting the target circuit into $N = 5$ parts.

§ INTRODUCTION

Quantum computers show potential in solving problems outside the reach of classical computers [Shor, 1994, Lloyd, 1996]. Unfortunately, today's noisy intermediate-scale quantum (NISQ) devices have quite limited coherence times and we are hence dependent on algorithms that are not only efficient, but also have a short enough circuit depth to respect the coherence times of the specific hardware (Quantum computing in the NISQ era and beyond). Hybrid quantum-classical algorithms are perfect candidates for NISQ devices, as they are built on the philosophy that even minimal quantum resources could be made useful when used in conjunction with classical routines [McClean et al., 2016]. Examples of such algorithms are the variational quantum eigensolvers (VQE) for finding the ground state of a Hamiltonian [Peruzzo et al., 2014], the quantum approximate optimization algorithm used for optimization problems [Farhi et al., 2014] and parameterized quantum circuits for machine learning problems [Benedetti et al., 2019]. Even though these algorithms are both ingenious and promising, could one utilize the hybrid quantum-classical approach to run existing deep quantum circuits on NISQ devices?

A general approach for making deep circuits more suitable for NISQ devices is to reduce their depth using Variational Quantum Compiling (VQC) [1]. This approach aims to compress a deep circuit $U$ using a shallow parameterized circuit $V(\boldsymbol{\theta})$, where $\boldsymbol{\theta}$ are the trainable parameters. The parameters are trained by minimizing a cost function in a hybrid quantum-classical fashion, where the cost usually is defined using some measure of distance between $U$ and $V(\boldsymbol{\theta})$. Sharma et al., 2020 showed that VQC exhibit a property called Optimal Parameter Resilience (OPR), which means that under typical incoherent noise models, such as decoherence processes and readout errors, the cost function has the same minimum as the cost function in the ideal scenario. Moreover, their numerical experiments also indicate that this property holds also for more realistic noise models. This suggests that it is possible to learn the optimal parameters for parameterized circuits used for variational quantum compiling, even on noisy quantum hardware.

While the cost functions may have the same minimum in the ideal and noisy case, Sharma et al., 2020 does not address the difficulty of finding this minimum, especially as the circuit depth and amount of noise increase. In this article, we illustrate that sufficiently deep circuits can be difficult to compress when noise is introduced, even though they are easily compressed in the ideal scenario. Afterwards, we introduce a recursive variational quantum compiling algorithm (RVQC) that is able to successfully compress such deep circuits despite of the noise. The algorithm works by dividing the target circuit $U$ into $N$ parts and recursively learns one part at a time until the whole of $U$ is compressed by a single ansatz $V(\boldsymbol{\theta})$. The circuit depth required to run the algorithm is not dependent on the depth of the target circuit, but on the depth of each of the $N$ parts and on the depth of the parameterized ansatzes. Thus, given a shallow ansatz and large enough $N$, the algorithm should be able to feasibly find optimal parameters when recursively compressing each part of the original circuit. To illustrate the algorithm, we will attempt to compress a five qubit random circuit using both ordinary VQC and RVQC on simulated noisy hardware. We then estimate the density matrices of the resulting optimized ansatzes and compare them with the density matrix of the ideal full circuit.

In section <ref>, we will explain VQC and RVQC in detail. We will then go through the target circuits, variational ansatzes and optimization methods used to test out the algorithm in section <ref>. We present and discuss the results in section <ref>, while we finally conclude the study and propose further work in section <ref>.

§ THEORY

§.§ Variational Quantum Compiling

Variational quantum compiling (VQC) is a method for compressing quantum circuits by training a variational ansatz $V(\boldsymbol{\theta})$ in order to approximate their effects. This can be achieved by minimizing following cost function with respect to the parameters $\boldsymbol{\theta}$ [1]:

\begin{align}
\label{eq:FUMC}
C_{FUMC}(\boldsymbol{\theta}) = 1 - |Tr(V^{\dagger}(\boldsymbol{\theta})U)|^2/d^2,
\end{align}

where the last term can be identified as the Hilbert-Schmidt inner product of $V(\boldsymbol{\theta})$ and $U$, and $d=2^n$ is the dimension of the Hilbert space for $n$ qubits. This particular loss yields a Full Unitary Matrix Compiling (FUMC), since the ansatz learns to approximate the complete effect of the target circuit. Alternatively, one can perform a Fixed Input State Compiling (FISC) where one is only interested in the effect of $U$ on a particular input state $\ket{\psi_0}$ [1]. For practical purposes, this input state can be taken to be the all-zero state, i.e., $\ket{\psi_0} = \ket{\boldsymbol{0}}$. FISC can be implemented by minimizing the following cost function with respect to the parameters $\boldsymbol{\theta}$:

\begin{align}
\label{eq:FISC}
C_{FISC}(\boldsymbol{\theta}) = 1 - |\bra{\boldsymbol{0}} U^{\dagger} V(\boldsymbol{\theta})\ket{\boldsymbol{0}}|^2,
\end{align}

where the last term can be identified as the Loschmidt Echo Test (LET) [Goussev et al., 2012]. For $C(\boldsymbol{\theta}) \approx 0$, we have that $V(\boldsymbol{\theta})\ket{\boldsymbol{0}} \approx U\ket{\boldsymbol{0}}$. This is a more restricted form of compiling compared to FUMC, but could in turn be more suitable for noisy quantum hardware since the trained anstazes generally consist of fewer gates. In this article, the focus will be on FISC because it is more suitable for noisy hardware. However, note that the following discussion can easily be extended to FUMC by introducing the appropriate cost function.

§.§ Recursive Variational Quantum Compiling

Since quantum circuits are sequences of quantum gates, they can always be written as a product of sub-sequences of gate, i.e., smaller circuits. Therefore, we can decompose a given quantum circuit $U$ as

\begin{align}
\label{eq:unitarymatrixproduct}
U = U_N U_{N-1} \cdots U_2 U_1,
\end{align}

where $N$ is the number of parts we divide the circuit into and each $U_k$ corresponds to one of these parts. See figure <ref> for an example with $N = 3$.

\[U \equiv \begin{array}{c}
\Qcircuit @C=1em @R=1em {
& \gate{H} & \ctrl{1} & \gate{X} & \qswap & \gate{R_y(\pi/3)} & \targ & \qw  \\
& \gate{H} & \targ & \gate{Y} & \qswap \qwx & \qw  & \ctrl{-1} & \qw \gategroup{1}{2}{2}{3}{.7em}{--} \gategroup{1}{4}{2}{5}{.7em}{--} \gategroup{1}{6}{2}{7}{.7em}{--} \\
\end{array}\]

$$\qquad \underbrace{\qquad \qquad}_{U_1}     \underbrace{\quad \qquad }_{U_2} \underbrace{\qquad \qquad \qquad \quad }_{U_3}$$

Quantum circuit divided into $N = 3$ different parts. Each part can be represented with each own unitary operator.

The Recursive Variational Quantum Compiling (RVQC) algorithm performs FISC, with $\ket{\boldsymbol{0}}$ as input state, on quantum circuits of the form <ref>. Rather than minimizing a single cost function to approximate the whole of $U$ at once, RVQC learns to approximate the circuit one part at a time. The first part $U_1$ of the circuit can be approximated by an ansatz $V(\boldsymbol{\theta}^{(1)})$ by minimizing the usual LET cost function

\begin{align}
\label{eq:CostFunction1}
C(\boldsymbol{\theta}^{(1)}) = 1 - |\bra{\boldsymbol{0}} U_1^{\dagger} V(\boldsymbol{\theta}^{(1)})\ket{\boldsymbol{0}}|^2,
\end{align}

where $\boldsymbol{\theta}^{(1)}$ are the trainable parameters of the ansatz. For $C(\boldsymbol{\theta}^{(1)}) \approx 0$, we have that $V(\boldsymbol{\theta}^{(1)})\ket{\bold{0}} \approx U_1\ket{\boldsymbol{0}}$. The next part of the circuit, $U_2$, can be incorporated by minimizing the cost function

\begin{align}
\label{eq:CostFunction2}
C(\boldsymbol{\theta}^{(2)}) = 1-|\bra{\boldsymbol{0}} V^{\dagger}(\boldsymbol{\theta}^{(1)})U_2^{\dagger}V(\boldsymbol{\theta}^{(2)}) \ket{\boldsymbol{0}}|^2,
\end{align}

where $\boldsymbol{\theta}^{(2)}$ is a new set of parameters independent of $\boldsymbol{\theta}^{(1)}$. Hence, the ansatz $V(\boldsymbol{\theta}^{(2)})$ is able to learn the product of $U_1$ and $U_2$ without having to coherently execute both parts. This gives rise to the RVQC algorithm, which involves minimizing the following cost functions for $k \in \{1, \cdots, N\}$:

\begin{align}
\label{eq:CostFunction}
C(\boldsymbol{\theta}^{(k)}) = 1-|\bra{\boldsymbol{0}} V^{\dagger}(\boldsymbol{\theta}^{(k-1)})U_k^{\dagger}V(\boldsymbol{\theta}^{(k)}) \ket{\boldsymbol{0}}|^2,
\end{align}

where $V(\boldsymbol{\theta}^{(0)}) = I$. This cost function can be evaluated on quantum hardware by implementing the circuit shown in <ref>. After minimizing all the cost functions, the final ansatz $V(\boldsymbol{\theta}^{(N)})$ will approximatly encode the effect of the whole circuit $U$ acting on $\ket{\boldsymbol{0}}$. This scheme bypasses the need to evaluate all of $U = U_N U_{N-1}\cdots U_1$ coherently during training, which can be potentially unfeasible if $U$ is a deep and the hardware used is noisy. Thus, we argue that RVQC can compress deep circuits using noisy quantum computers, where ordinary VQC would otherwise fail.

\[\begin{array}{c}
\Qcircuit @C=1em @R=1em {
& \multigate{3}{V(\boldsymbol{\theta}^{(k)})} & \qw & \qw & \multigate{3}{{U^\dagger_k}} & \multigate{3}{V^\dagger(\boldsymbol{\theta}^{(k-1)})} & \meter\\
& \ghost{V(\boldsymbol{\theta}^{(k)})} & \qw & \qw & \ghost{{U^\dagger_k}} & \ghost{V^\dagger(\boldsymbol{\theta}^{(k-1)})} & \meter\\
& \nghost{V(\boldsymbol{\theta}^{(k)})} & \vdots && \nghost{{U^\dagger_k}} & \nghost{V^\dagger(\boldsymbol{\theta}^{(k-1)})} &  \meter \\
& \ghost{V(\boldsymbol{\theta}^{(k)})} & \qw &\qw & \ghost{{U^\dagger_k}} & \ghost{V^\dagger(\boldsymbol{\theta}^{(k-1)})}  & \meter \\}
\end{array}\]

Quantum circuit to implement the cost function <ref>. If the unitary $V(\boldsymbol{\theta}^{(k)})$ is equal (up to a global phase) to $V(\boldsymbol{\theta}^{(k-1)})U_k$, all the qubits above should be in the $\ket{0}$ state. We can thus minimize the cost function in eq. <ref> by maximizing the probability of measuring the $\ket{\boldsymbol{0}}$ state in the above circuit.

The LET cost function <ref> can be evaluated on a quantum computer by implementing the circuit given by <ref> and calculating the probability for measuring all the qubits in the zero-state. Mathematically, this can be expressed as

\begin{equation}
\label{eq:LSET}
C(\boldsymbol{\theta}^{(k)}) = 1 - P(\ket{\boldsymbol{0}};\boldsymbol{\theta}^{(k)}) \approx 1 - \frac{1}{n_s}\sum_{i=1}^{n_s} p_i(\boldsymbol{\theta}^{(k)}),
\end{equation}

where $P(\ket{\boldsymbol{0}};\boldsymbol{\theta}^{(k)})$ is the probability of measuring $\ket{\boldsymbol{0}}$ for the circuit <ref>, $n_s$ is the number of shots used, and $p_i(\boldsymbol{\theta}^{(k)})$ indicates whether experiment $i$ was measured to be $\ket{\boldsymbol{0}}$ or not ($p_i(\boldsymbol{\theta}^{(k)}) = 1$ or $0$, respectively).

§ METHOD

§.§ Variational Ansatz

The ansatz used to compress the five qubit random circuit is defined as seen in figure <ref>, where the $R_yR_z$ blocks are independently parameterized Pauli rotations defined as in <ref>, and the blocks $ent$ establishes entanglement using CNOT-gates as showed in <ref>.

\[\begin{array}{c}
\Qcircuit @C=1em @R=1em {
& \multigate{4}{V(\boldsymbol{\theta})} & \qw\\
&        \ghost{V(\boldsymbol{\theta})} & \qw\\
&        \ghost{V(\boldsymbol{\theta})} & \qw\\
&        \ghost{V(\boldsymbol{\theta})} & \qw\\
&        \ghost{V(\boldsymbol{\theta})} & \qw\\}
\end{array}
\begin{array}{c}
\Qcircuit @C=1em @R=1em {
& \multigate{4}{R_y R_z} & \multigate{4}{ent} & \multigate{4}{R_y R_z} & \multigate{4}{ent} & \multigate{4}{R_y R_z} & \multigate{4}{ent} & \multigate{4}{R_y R_z} & \multigate{4}{ent} & \multigate{4}{R_y R_z}  & \qw\\
&        \ghost{R_y R_z} &        \ghost{ent} &        \ghost{R_y R_z} &        \ghost{ent} &        \ghost{R_y R_z} &        \ghost{ent} &        \ghost{R_y R_z} &        \ghost{ent} &        \ghost{R_y R_z}  & \qw\\
&        \ghost{R_y R_z} &        \ghost{ent} &        \ghost{R_y R_z} &        \ghost{ent} &        \ghost{R_y R_z} &        \ghost{ent} &        \ghost{R_y R_z} &        \ghost{ent} &        \ghost{R_y R_z}  & \qw\\
&        \ghost{R_y R_z} &        \ghost{ent} &        \ghost{R_y R_z} &        \ghost{ent} &        \ghost{R_y R_z} &        \ghost{ent} &        \ghost{R_y R_z} &        \ghost{ent} &        \ghost{R_y R_z}  & \qw\\
&        \ghost{R_y R_z} &        \ghost{ent} &        \ghost{R_y R_z} &        \ghost{ent} &        \ghost{R_y R_z} &        \ghost{ent} &        \ghost{R_y R_z} &        \ghost{ent} &        \ghost{R_y R_z}  & \qw \\}
\end{array}\]

Ansatz used for compressing the five-qubit random circuit. The $R_yR_z$ blocks are independently parameterized Pauli rotations defined as in <ref>, and the blocks $ent$ establishes entanglement using CNOT-gates as showed in <ref>.

\[\begin{array}{c}
\Qcircuit @C=1em @R=1em {
& \multigate{4}{R_y R_z} & \qw\\
&        \ghost{R_y R_z} & \qw\\
&        \ghost{R_y R_z} & \qw\\
&        \ghost{R_y R_z} & \qw\\
&        \ghost{R_y R_z} & \qw\\}
\end{array}=\begin{array}{c}
\Qcircuit @C=1em @R=1em {
& \gate{R_y(\boldsymbol{\theta}_1)} & \gate{R_z(\boldsymbol{\theta}_6)} & \qw\\
& \gate{R_y(\boldsymbol{\theta}_2)} & \gate{R_z(\boldsymbol{\theta}_7)} & \qw\\
& \gate{R_y(\boldsymbol{\theta}_3)} & \gate{R_z(\boldsymbol{\theta}_8)} &\qw \\
& \gate{R_y(\boldsymbol{\theta}_4)} & \gate{R_z(\boldsymbol{\theta}_9)} &\qw\\
& \gate{R_y(\boldsymbol{\theta}_5)} & \gate{R_z(\boldsymbol{\theta}_{10})} &\qw\\
\end{array}\]

Parameterized Pauli rotations acting on the qubits.

\[\begin{array}{c}
\Qcircuit @C=1em @R=1em {
& \multigate{4}{ent} & \qw\\
&        \ghost{ent} & \qw\\
&        \ghost{ent} & \qw\\
&        \ghost{ent} & \qw\\
&        \ghost{ent} & \qw\\}
\end{array}
\begin{array}{c}
\Qcircuit @C=1em @R=1em {
& \ctrl{1} & \qw      & \qw        & \qw      & \qw \\
& \targ    & \ctrl{1} & \qw        & \qw      & \qw \\
& \qw      & \targ    & \ctrl{1}   & \qw      & \qw \\
& \qw      & \qw      & \targ      & \ctrl{1} & \qw \\
& \qw      & \qw      & \qw        & \targ    & \qw \\
\end{array}\]

A ladder of linearly connected CNOT-gates that establishes entanglement in the ansatz.

The ansatz <ref> was chosen through trial and error, aiming for a shallow ansatz that is still able to successfully approximate the five-qubit random circuit. The parameters of the ansatz are initialized randomly as $\boldsymbol{\theta}_i \sim U[-\pi, \pi ]$, which has the benefit of coinciding with the period of Pauli rotations and hence enables us to uniformly sample models from model space.

§.§ Optimization

Since the variational ansatzes used in this work consists of only single-qubit Pauli rotations and non-parameterized multi-qubit gates, the gradients of the cost function with respect to the variational parameters are estimated using the parameter-shift rule [Schuld et al., 2019]. The parameters are then updated using the Adam gradient-based optimization method [Kingma and Ba, 2014].

§.§ Algorithm

In algorithm <ref> we can see how RVQC is implemented.

A circuit to compress, divided into $N$ parts (steps), represented by the unitary $U = U_N \cdots U_1$. Variational parameters $\boldsymbol{\theta}^{(k)}$, where $k \in \{1,2,\dots,N\}$. A variational ansatz $V(\boldsymbol{\theta})$. A tolerance $\epsilon$ for the cost function. A learning rate $\lambda$ for the gradient descent algorithm.
The unitary $V (\boldsymbol{\theta}^{(N)})$ such that $V (\boldsymbol{\theta}^{(N)})\ket{\boldsymbol{0}} \approx U_N \cdots U_1 \ket{\boldsymbol{0}}$.

initialize $\boldsymbol{\theta}^{(k)}$ for $k \in \{1,2,\dots,N\}$ with some proper initialization scheme
$k=1$ $N$

calculate $C(\boldsymbol{\theta}^{(k)})$ in <ref> with the circuit in <ref>.
$C(\boldsymbol{\theta}^{(k)}) > \epsilon$
$\boldsymbol{\theta}^{(k)} \gets \boldsymbol{\theta}^{(k)} - \lambda \nabla_{\boldsymbol{\theta}^{(k)}} C(\boldsymbol{\theta}^{(k)})$
calculate $C(\boldsymbol{\theta}^{(k)})$ in <ref> with the circuit in <ref>.

Circuit compression

§ RESULTS AND DISCUSSION

To test whether RVQC can provide improvements over ordinary VQC, a random five-qubit circuit of depth 5000 was generated and both methods was used to compress the circuit into the variational ansatz in <ref>. First, VQC was optimized by training it on Qiskits ideal statevector simulator, and then by training it on Qiskits noise model of the IBM Santiago five-qubit machine [Qis, 2019] with 8192 shots per circuit. In both cases we trained for 500 epochs using the ADAM optimizer with a learning rate of $\lambda = 0.1$. The results can be seen in <ref>.

Training loss (LET <ref>) as a function of epoch for VQC on a randomly generated five-qubit circuit of depth 5000. Ideal (noisy) loss means that the loss function was evaluated on a simulation of ideal (noisy) quantum hardware, while ideal (noisy) training means that the VQC training was done on a simulation of ideal (noisy) quantum hardware.

<ref> shows that when training on the ideal statevector simulator, VQC was able to converge to a satisfactory low loss value ($<10^{-2}$). However, it was not able to converge when trained on the noise model of the IBM Santiago device. Furthermore, the loss evaluated with the ideal and noisy simulation shows no obvious coinciding minima. Hence, the OPR property does not apply in practice for this example of a deep quantum circuit. In turn, this causes the optimal parameters to be hard to find through minimization. To see if the RVQC algorithm could successfully train in the noisy regime, the same random circuit was divided into $N = 5$ parts (steps) and a RVQC was trained for 100 epochs per part on the IBM Santiago noise model. We applied the same optimizer, learning rate, the same number of shots and the same ansatz as with VQC. On each part of the circuit, the parameters from the epoch with the lowest loss was chosen. The state fidelity between the RVQC at step $k$ and the first $k$ parts of the random circuit was calculated after each step. The results can be seen in <ref>.

State fidelity $F[\cdot]$ as a function of step number for the RVQC algorithm applied on a random circuit of depth 5000 split into 5 parts (steps). At each step $k$, we calculate the state fidelity $F[\cdot]$ between two density matrices. A density matrix $\rho$ with subscript $A$ and $R$ refers to the density matrix for the ansatz and random circuit, respectively. A density matrix $\rho$ with superscript $I$ and $N$ implies that the density matrix is ideal or estimated on the noise model, respectively. As an example, $\rho_R^I(k)$ is the ideal density matrix corresponding to the random circuit at step $k$, that is $\rho_R^I(k) = \ket{R_k}\bra{R_k}$, where $\ket{R_k} = U_k U_{k-1} \cdot U_1 \ket{\boldsymbol{0}}$. The density matrix $\rho_A^N(\boldsymbol{\theta}^{(k)})$ is the density matrix for the ansatz at step $k$, estimated on the noise model. For each step, the training of the ansatz is performed on the noise model for 100 epochs.

<ref> shows that the noisy execution of the full random circuit yielded a fidelity of $F = 0.03$, showing that some form of quantum compiling is necessary in order to achieve any meaningful results. Indeed, RVQC trained on the noisy device was able to compress the target circuit with an ideal and noisy fidelity of $F = 0.88$ and $F = 0.67$, respectivly. In contrast, ordinary VQC was seen to not converge in <ref>, making RVQC the preferred choice for this example. By splitting the full random circuit into smaller parts, RVQC is able to decrease the amount of noise. This ensures in turn that the optimization procedure is accomplishable in 500 epochs.

§ DISCUSSION

<ref> and <ref> shows that while VQC was unable to compress the target circuit, RVQC did so with a fidelity of over 0.95. This was done with a noise model of the IBM Santiago device,

Further, there are several factors that we believe will affect the fidelity. The first is the cost function tolerance $\epsilon$. It does not come as a surprise that a lower $\epsilon$ results in a higher fidelity given that the ansatzes are sufficiently expressive. This is because the tolerance $\epsilon$ controls how close the ansatz and target state needs to be at each step for the algorithm to move on. Further, it is also clear that the expressivenes of the ansatzes will affect the fidelity. This is because an ansatz that is not able to express a state close to some of the intermediate states $U_k U_{k-1}\cdots U_2 U_1 \ket{\boldsymbol{0}}$, will not be able to minimize the cost function to the appropriate level. We also believe that how the target circuit is split could affect the fidelity. This is because, while the final target state is indifferent to how the circuit is split, the splitting will dictate the pathway taken for the ansatz from the initial state to the target state. To see this, imagine that a two-qubit circuit represented by $U$ is split into two parts, that is $U = U_2 U_1$. The operator $U$ transforms the state $\ket{00}$ to $\ket{11}$, while the operator $U_1$ transforms the state $\ket{00}$ into $\frac{1}{\sqrt{2}} (\ket{00} + \ket{11})$. If our ansatz is not able to perform approximately the same transformation as $U_1$, the algorithm will fail. However, another split might lead to an intermediate state that our ansatz is able to approximate.

§ CONCLUSION

A random five-qubit circuit of depth 5000 was compressed into a parameterized ansatz utilizing RVQC and VQC. The circuit was split into $N=5$ parts when applying RVQC. It was shown that when optimizing for a total of 500 epochs on a simulation of a noisy quantum device, VQC was unable to converge to a satisfactory fidelity while RVQC was able to train an ansatz with fidelity of 0.88 with respect to the random circuit. The ansatz was sufficiently shallow such that its density matrix estimated on the noise model had a fidelity of 0.67 with respect to the actual density matrix of the random circuit. For comparison, the density matrix estimate of the random circuit on the noise model had a corresponding fidelity of only 0.03.

§.§ Further Work

The RVQC algorithm is dependent on the ansatz being expressive enough to successfully complete all of the $N$ steps. Meanwhile, it is expected that the required expressiveness will be varying with each step. An adaptive ansatz approach [Bilkis et al., 2022, Grimsley et al., 2019] thus seems like a natural extension of this work.

Research have shown that for a wide class of parameterized quantum circuits, the probability of non-zero gradients vanishes exponentially in the number of qubits [McClean et al., 2018], meaning that one has to chose ansatzes with care. Several ways to mitigate this problem have been proposed. For example, a cost function dependent on local observables will lead to only polynomially vanishing gradients if one uses an alternating layered ansatz composed of blocks forming local 2-designs, and the circuit depth does not exceed $\mathcal{O}(\log n)$ [Cerezo et al., 2021]. It would be interesting to find out if such ansatzes can be flexible enough to compress deep quantum circuits.

The RVQC algorithm aims to perform FISC, as opposed to FUMC. Note, however, that RVQC can easily be extended to the FUMC case by changing to the appropriate cost function.

Finally, although the RVQC algorithm was tested out on the IBM Santiago noise model, it has not yet been tested on any real quantum devices. Different methods of error mitigation, like Zero-Noise Extrapolation [Temme et al., 2017], can be explored in this case.

§ ACKNOWLEDGEMENTS

We would like to thank Morten Hjorth-Jensen for sharing his knowledge and engaging in helpful discussion.

[1]
Sumeet Khatri, Ryan LaRose, Alexander Poremba, Lukasz Cincio, Andrew Tyler
Sornborger, and Patrick Joseph Coles.
Quantum-assisted quantum compiling.

[Shor, 1994]
P.W. Shor.
Algorithms for quantum computation: discrete logarithms and
In Proceedings 35th Annual Symposium on Foundations of Computer
Science, pages 124–134, 1994.

[Lloyd, 1996]
Seth Lloyd.
Universal quantum simulators.
Science, 2730 (5278):0 1073–1078, 1996.

[McClean et al., 2016]
Jarrod R McClean, Jonathan Romero, Ryan Babbush, and Alán Aspuru-Guzik.
The theory of variational hybrid quantum-classical algorithms.
New Journal of Physics, 180 (2):0 023023, feb
URL <https://doi.org/10.1088/1367-2630/18/2/023023>.

[Peruzzo et al., 2014]
Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou,
Peter J. Love, Alán Aspuru-Guzik, and Jeremy L. O'Brien.
A variational eigenvalue solver on a photonic quantum processor.
Nature Communications, 50 (1):0 4213, 2014.
URL <https://doi.org/10.1038/ncomms5213>.

[Farhi et al., 2014]
Edward Farhi, Jeffrey Goldstone, and Sam Gutmann.
A quantum approximate optimization algorithm, 2014.

[Benedetti et al., 2019]
Marcello Benedetti, Erika Lloyd, Stefan Sack, and Mattia Fiorentini.
Parameterized quantum circuits as machine learning models.
Quantum Science and Technology, 40 (4):0
043001, nov 2019.
URL <https://doi.org/10.1088/2058-9565/ab4eb5>.

[Sharma et al., 2020]
Kunal Sharma, Sumeet Khatri, M Cerezo, and Patrick J Coles.
Noise resilience of variational quantum compiling.
New Journal of Physics, 220 (4):0 043006, apr
URL <https://doi.org/10.1088/1367-2630/ab784c>.

[Goussev et al., 2012]
A. Goussev, R. A. Jalabert, H. M. Pastawski, and D. Ariel Wisniacki.
Loschmidt echo.
Scholarpedia, 70 (8):0 11687, 2012.
revision #127578.

[Schuld et al., 2019]
Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan
Evaluating analytic gradients on quantum hardware.
Phys. Rev. A, 99:0 032331, Mar 2019.
URL <https://link.aps.org/doi/10.1103/PhysRevA.99.032331>.

[Kingma and Ba, 2014]
Diederik Kingma and Jimmy Ba.
Adam: A method for stochastic optimization.
International Conference on Learning Representations, 12 2014.

[Qis, 2019]
Qiskit: An open-source framework for quantum computing, 2019.

[Bilkis et al., 2022]
M. Bilkis, M. Cerezo, Guillaume Verdon, Patrick J. Coles, and Lukasz Cincio.
A semi-agnostic ansatz with variable structure for quantum machine
learning, 2022.

[Grimsley et al., 2019]
Harper R. Grimsley, Sophia E. Economou, Edwin Barnes, and Nicholas J. Mayhall.
An adaptive variational algorithm for exact molecular simulations on
a quantum computer.
Nature Communications, 100 (1):0 3007, 2019.
URL <https://doi.org/10.1038/s41467-019-10988-2>.

[McClean et al., 2018]
Jarrod R. McClean, Sergio Boixo, Vadim N. Smelyanskiy, Ryan Babbush, and
Hartmut Neven.
Barren plateaus in quantum neural network training landscapes.
Nature Communications, 90 (1):0 4812, 2018.
URL <https://doi.org/10.1038/s41467-018-07090-4>.

[Cerezo et al., 2021]
M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J. Coles.
Cost function dependent barren plateaus in shallow parametrized
quantum circuits.
Nature Communications, 120 (1):0 1791, 2021.
URL <https://doi.org/10.1038/s41467-021-21728-w>.

[Temme et al., 2017]
Kristan Temme, Sergey Bravyi, and Jay M. Gambetta.
Error mitigation for short-depth quantum circuits.
Physical Review Letters, 1190 (18), Nov 2017.
ISSN 1079-7114.
URL <http://dx.doi.org/10.1103/PhysRevLett.119.180509>.


# Well-posedness of non-autonomous transport equation on metric graphs

Christian Budde  University of the Free State, Department of Mathematics and
Applied Mathematics, P. O. Box 339, 9300 Bloemfontein, South Africa
<EMAIL_ADDRESS>and  Marjeta Kramar Fijavž  University of Ljubljana, Faculty
of Civil and Geodetic Engineering, Jamova 2, SI-1000 Ljubljana, Slovenia /
Institute of Mathematics, Physics, and Mechanics, Jadranska 19, SI-1000
Ljubljana, Slovenia<EMAIL_ADDRESS>
###### Abstract.

We consider transport processes on metric graphs with time-dependent
velocities and show that the corresponding non-autonomous abstract Cauchy
problem is well-posed by means of evolution families and evolution semigroups.
We discuss stability of the solutions in some special cases.

###### Key words and phrases:

non-autonomous problems, evolution families, evolution semigroups, metric
graphs, extrapolation spaces, transport equation

###### 2010 Mathematics Subject Classification:

37B55, 47D06, 35R02, 35F46

## Introduction

Consider a finite network (i.e., of pipelines) where some material is
transported along its branches (i.e., pipes). The velocity of the transport
depends on a given branch but may also change in time. We would like to know
under which condition such a system can be modelled in a way that for any
given initial distribution we are able to predict the state of the system in
any time. We would also like to obtain stable solutions, that continuously
depend on the initial state. In this case we will call our problem well-posed.

Such transport problems on networks have already been studied by several
authors. The operator theoretical approach by means of abstract Cauchy
problems on Banach spaces was initiated by the second author and E. Sikolya
[16], for an overview and further references we refer to the survey [17].
However, the majority of the publications concentrates on time-independent
transport and hence autonomous abstract Cauchy problems. A first attempt to
non-autonomous problems of this kind was performed by F. Bayazit et al. [8].
They considered transport on networks with boundary conditions changing in
time. The advantage of such approach is that the corresponding operator does
not change its action on the Banach space only its domain changes in time. Our
aim is to consider also the non-autonomous operator, that is, we study
transport problems on finite metric graphs with time-dependent velocities
along the edges. We use evolution families and evolution semigroups as studied
by G. Nickel [22] and show that the abstract Cauchy problem which can be
associated to the transport equation on these graphs is well-posed.

Let us also mention that diffusion and other processes on networks have also
been studied by semigroup techniques, see e.g. the monograph by D. Mugnolo
[18]. Similar non-autonomous problem as here was considered in [4] for time-
dependent diffusion on networks. There, a non-autonomous form method was used
that is only applicable in Hilbert spaces.

This paper is structured as follows. The first section consists of a reminder
on some notions from graph theory as well as of the definition and some
results on general non-autonomous abstract Cauchy problems and associated
evolution families. We also present our non-autonomous transport problem on a
metric graph and re-write it in an operator theoretical context. In section 2
we consider non-autonomous operators $A(t)$ with a common constant domain
$\mathrm{D}(A(t))\equiv\mathrm{D}$. Acquistapace and Terreni [1] considered
operators of this kind but their results are not applicable in our situation
since they assume the generation of analytic semigroups which cannot be the
case here. We can however apply the results by Kato [14]. We also state some
known stability results that apply for the periodic case. In section 3 we
treat the general case in which the operators $A(t)$ do not necessarily share
a common domain. We will obtain the aimed evolution family as a composition of
the translation and the multiplication semigroup, whereby the multiplication
semigroup will be constructed by the known solution semigroups of the
autonomous case using the results by T. Graser [13] on unbounded operator
multipliers. In order to determine the domain of the generator of the
evolution semigroup we will follow the approach via the extrapolation spaces
as presented by R. Nagel, G. Nickel and S. Romanelli [20]. Here we will make
use of the fact that in our case the extrapolated operators share a common
extrapolation space.

##  1\. Preliminaries

###  1.1. Metric graphs

We shall use the notation presented for example in [16, 6]. We take a simple,
connected, finite graph $(\mathrm{V},\mathrm{E})$ with the set of _vertices_
$\mathrm{V}=\\{\mathrm{v}_{1},\dots,\mathrm{v}_{n}\\}$ and the set of _edges_
$\mathrm{E}=\\{\mathrm{e}_{1},\dots,\mathrm{e}_{m}\\}\subseteq\mathrm{V}\times\mathrm{V}$.
We parametrize each edge on the interval $[0,1]$ and thus we obtain a metric
object called _metric graph_ (or network). By an abuse of notation we denote
the endpoints of the edge $\mathrm{e}=(\mathrm{v}_{i},\mathrm{v}_{j})$ as
$\mathrm{v}_{i}=\mathrm{e}(0)$ and $\mathrm{v}_{j}=\mathrm{e}(1)$. The
structure of a graph can be described by its incidence matrices as follows.
The _outgoing incidence matrix_ $\Phi^{-}:=(\phi^{-}_{ij})_{n\times m}$ and
_incoming incidence matrix_ $\Phi^{+}:=(\phi^{+}_{ij})_{n\times m}$ are
defined as

$\phi^{-}_{ij}:=\begin{cases}1,&\text{if }\mathrm{e}_{j}(1)=\mathrm{v}_{i},\\\
0,&\text{otherwise},\end{cases}\qquad\text{and}\qquad\phi^{+}_{ij}:=\begin{cases}1,&\text{if
}\mathrm{e}_{j}(0)=\mathrm{v}_{i},\\\ 0,&\text{otherwise}.\end{cases}$

Note, that given parametrisation implicitly orients our discrete graph whereby
for technical reasons we assume that the graph is oriented contrary to the
parametrisation of the edges. Matrix $\Phi:=\Phi^{+}-\Phi^{-}$ is the classic
incidence matrix of the obtained directed graph.

We will further assign some weights $0\leq w_{ij}\leq 1$ to the edges of our
metric graph such that

(1.1) $\sum_{j=1}^{m}{w_{ij}}=1\text{ for all }i=1,\dots,n.$

The so-called _weighted (transposed) adjacency matrix of the line graph_
$\mathbb{B}=(\mathbb{B}_{ij})_{m\times m}$ defined by

(1.2) $\mathbb{B}_{ij}:=\begin{cases}w_{ki},&\text{ if
}\mathrm{e}_{j}(0)=\mathrm{v}_{k}=\mathrm{e}_{i}(1),\\\ 0,&\text{
otherwise},\end{cases}$

contains all information on our weighted metric graph. By (1.1), the matrix
$\mathbb{B}$ is column stochastic, i.e., the sum of entries of each column is
$1$, and defines a bounded positive operator on $\mathbb{C}^{m}$ with
$r(\mathbb{B})=\|\mathbb{B}\|=1$. For additional terminology and properties we
refer to [6, Ch. 18].

###  1.2. Non-autonomous abstract Cauchy problems

The abstract operator-theoretical setting we will use are the so-called non-
autonomous abstract Cauchy problems. Let
$(A(t),\mathrm{D}(A(t)))_{t\in\mathbb{R}}$ be a family of (unbounded)
operators on a Banach space $X$. The associated _non-autonomous abstract
Cauchy problem_ is given by

(nACP) $\displaystyle\begin{cases}\dot{u}(t)=A(t)u(t),&\quad t\geq s,\\\
u(s)=x\in X.\end{cases}$

The _autonomous_ case, i.e., when the operators $A(t)\equiv A$ do not depend
on time, is well-understood by means of strongly continuous one-parameter
operator semigroups, or $C_{0}$-semigroups for short. There are several
monographs on this theory, we refer to K.-J. Engel and R. Nagel [12]. To some
extent this theory can be carried over to non-autonomous case, as described
below. First, let us recall the notion of (classical) solutions and well-
posedness of (nACP), see [12, Def. VI.9.1].

######  Definition 1.1.

Let $(A(t),\mathrm{D}(A(t)))$, $t\in\mathbb{R}$, be linear operators on a
Banach space $X$ and take $s\in\mathbb{R}$ and $x\in\mathrm{D}(A(s))$. A
_(classical) solution of ( nACP)_ is a function
$u(\cdot,s,x)=u\in\mathrm{C}^{1}\left(\left[s,\infty\right),X\right)$ such
that $u(t)\in\mathrm{D}(A(t))$ and $u$ satisfies (nACP) for $t\geq s$.

######  Definition 1.2.

The Cauchy problem (nACP) is called _well-posed_ (on spaces $Y_{t}$) if there
are dense subspaces $Y_{s}\subseteq\mathrm{D}(A(s))$, $s\in\mathbb{R}$, of $X$
such that for $s\in\mathbb{R}$ and $x\in Y_{s}$ there is a unique solution
$t\mapsto u(t,s,x)\in Y_{t}$ of (nACP). In addition, for $s_{n}\to s$ and
$Y_{s_{n}}\ni x_{n}\to x$, we have
$\widetilde{u}(t,s_{n},x_{n})\to\widetilde{u}(t,s,x)$ uniformly for $t$ in
compact intervals in $\mathbb{R}$, where we set
$\widetilde{u}(t,s,x):={u}(t,s,x)$ for $t\geq s$ and $\widetilde{u}(t,s,x):=x$
for $t<s$.

In the autonomous case, the solutions are represented by $C_{0}$-semigroups.
An appropriate analoge for the solutions of (nACP) are so-called evolution
families.

######  Definition 1.3.

[12, Def. VI.9.2] A family of bounded operators
$(U(t,s))_{t,s\in\mathbb{R},t\geq s}$ on a Banach space $X$ is called a
_(strongly continuous) evolution family_ if

  1. (i)

$U(t,s)=U(t,r)U(r,s)$ and $U(s,s)=\mathrm{I}$ for $t\geq r\geq$ s and
$t,r,s\in\mathbb{R}$ and

  2. (ii)

the mapping $\left\\{(\tau,\sigma)\in\mathbb{R}^{2}:\
\tau\geq\sigma\right\\}\ni(t,s)\mapsto U(t,s)$ is strongly continuous.

  3. (iii)

There exists $M\geq 1$ and $\omega\in\mathbb{R}$ such that
$\left\|U(t,s)\right\|\leq M\mathrm{e}^{\omega(t-s)}$ for all $t\geq s$.

We say that $(U(t,s))_{t\geq s}$ solves the Cauchy problem (nACP) (on space
$Y_{t}$) if there are dense subspaces $Y_{s}$, $s\in\mathbb{R}$, of $X$ such
that $U(t,s)Y_{s}\subseteq Y_{t}\subseteq\mathrm{D}(A(t))$ for $t\geq s$ and
the function $t\mapsto U(t,s)x$ is a solution of (nACP) for $s\in\mathbb{R}$
and $x\in Y_{s}$.

######  Proposition 1.4.

[22, Prop. 2.5] The Cauchy problem (nACP) is well-posed (on $Y_{t}$) if and
only if there is an evolution family solving (nACP) (on $Y_{t}$).

A uniform generation theorem in the style of Hille–Yosida is unfortunately not
known for non-autonomous Cauchy problems. In fact, the existence of solutions
of (nACP) is a priori not clear. However, there are several attempts to the
characterisation the well-posedness of certain classes of non-autonomous
abstract Cauchy problems, let us only mention the work by P. Acquistapace and
B. Terreni [2], T. Kato and H. Tanabe [15], as well as the survey by R.
Schnaubelt [24]. There is one possible approach to the well-posedness results
that applies operator semigroup theory by means of the so-called evolution
semigroups, as follows. To any evolution family $(U(t,s))_{t\geq s}$ on a
Banach space $X$ one can associate an operator semigroup
$(\mathcal{T}(t))_{t\geq 0}$ on the space $\mathrm{C}_{0}(\mathbb{R},X)$ by

$(\mathcal{T}(t)f)(s):=U(s,s-t)f(s-t),\quad t\geq 0,\
f\in\mathrm{C}_{0}(\mathbb{R},X),\ s\in\mathbb{R}.$

By [12, Chapter VI, Lemma 9.10] this semigroup is strongly continuous on
$\mathrm{C}_{0}(\mathbb{R},X)$ and is called the _evolution semigroup_. By
definition, every $C_{0}$-semigroup coming from an evolution family is called
an evolution semigroup. We denote its generator by $(G,\mathrm{D}(G))$. The
following result characterizes evolution semigroups. By
$(T_{\mathrm{r}}(t))_{t\geq 0}$ we denote the right translation semigroup on
$\mathrm{C}_{\mathrm{c}}(\mathbb{R})$ defined by
$(T_{\mathrm{r}}(t)\varphi)(s):=\varphi(s-t)$.

######  Theorem 1.5.

[12, Chapter VI, Thm. 9.14] Let $(T(t))_{t\geq 0}$ be a $C_{0}$-semigroup on
$\mathrm{C}_{0}(\mathbb{R},X)$ with generator $(G,\mathrm{D}(G))$. Then
following assertions are equivalent.

  1. (a)

$(T(t))_{t\geq 0}$ is an evolution semigroup.

  2. (b)

$T(t)(\varphi f)=(T_{\mathrm{r}}(t)\varphi)T(t)f$ for all
$\varphi\in\mathrm{C}_{\mathrm{c}}(\mathbb{R})$,
$f\in\mathrm{C}_{0}(\mathbb{R},X)$ and $t\geq 0$.

  3. (c)

For $f\in\mathrm{D}(G)$ and
$\varphi\in\mathrm{C}_{\mathrm{c}}^{1}(\mathbb{R})$ we have $\varphi
f\in\mathrm{D}(G)$ and $G(\varphi f)=\varphi Gf-\varphi^{\prime}f$.

Finally, one can characterize the well-posedness of (nACP) as follows.

######  Theorem 1.6.

[22, Thm. 2.9] Let $(A(t),\mathrm{D}(A(t)))_{t\in\mathbb{R}}$ be a family of
linear operators on a Banach space $X$. The following are equivalent.

  1. (a)

(nACP) is well-posed for the family of operators
$(A(t),\mathrm{D}(A(t)))_{t\in\mathbb{R}}$.

  2. (b)

There exists a unique evolution semigroup $(T(t))_{t\geq 0}$ with generator
$(G,\mathrm{D}(G))$ and an invariant core
$D\subseteq\mathrm{C}_{0}^{1}(\mathbb{R},X)\cap\mathrm{D}(G)$ such that
$Gf+f^{\prime}=A(\cdot)f$ for all $f\in D$.

The above result shows that determining the domain of the generator of the
evolution semigroup is the most important step on the way to solve given non-
autonomous abstract Cauchy problem.

###  1.3. The non-autonomous transport problem

Let us now consider the following non-autonomous dynamical system taking place
on our metric graph.

(nF) $\displaystyle\begin{cases}\frac{\partial}{\partial
t}u_{j}(x,t)=c_{j}(t)\frac{\partial}{\partial x}u_{j}(x,t),&\quad
x\in\left(0,1\right),\ t\geq s,\\\ u_{j}(x,s)=f_{j}(x),&\quad
x\in\left(0,1\right),\\\
\phi^{-}_{ij}c_{j}(t)u_{j}(1,t)=w_{ij}\sum_{k=1}^{m}{\phi^{+}_{ik}c_{k}(t)u_{k}(0,t)},&\quad
t\geq s,\end{cases}$

for $i=1,\dots,n$, $j=1,\dots,m$. The first equation models the transport
along the edge $\mathrm{e}_{j}$ where $c_{j}(t)$ is the time-variable velocity
coefficient along this edge. The second equation gives the initial mass
distribution along the edges while the third equation represent the boundary
conditions taking place in the vertices of the graph. Here, the graph
structure is encoded in terms of the incidence matrices. Observe, that the
weights $w_{ij}$ give proportions of the material arriving in the vertex
$\mathrm{v}_{i}$ that is distributed to the edge $\mathrm{e}_{j}$. By (1.1),
at all times the mass is conserved, i.e., the Kirchhoff law holds in all the
vertices.

We will show that there exists a solution of the non-autonomous set of
equations (nF) by means of evolution families. The equivalent problem in the
autonomous case, i.e., when all $c_{j}(t)\equiv c_{j}$ are independent of time
has already been considered by several authors, cf. [16, 17]. A first approach
to time-dependent transport equations on networks was done by F. Bayazit, B.
Dorn and M. Kramar Fijavž, cf. [8]. In particular, they studied transport
processes with time-dependent weights $\omega_{ij}(t)$ but constant velocities
$c_{j}(t)\equiv 1$. This can be interpreted as autonomous transport with time-
depending boundary conditions (i.e., the structure of the graph changes in
time). They were able to obtain well-posedness as well as some results on the
asymptotic behaviour. In our case, already the transport equation on each edge
is non-autonomous and a different approach is needed.

To introduce the operator theoretical approach to our problem we first define
Banach space

$X:=\mathrm{L}^{1}\left(\left[0,1\right],\mathbb{C}^{m}\right)\cong\mathrm{L}^{1}\left(\left[0,1\right],\mathbb{C}\right)^{m},$

with the usual norm

$\left\|f\right\|_{X}:=\sum_{j=1}^{m}\int_{0}^{1}{\left|f_{j}(t)\right|\
\mathrm{d}{t}}.$

On this Banach space $X$ we consider the family of operators
$(A(t),\mathrm{D}(A(t)))$ defined by

(1.3) $\displaystyle A(t)$
$\displaystyle:=\begin{pmatrix}c_{1}(t)\frac{\mathrm{d}}{\mathrm{d}{x}}&&0\\\
&\ddots&\\\ 0&&c_{m}(t)\frac{\mathrm{d}}{\mathrm{d}{x}}\end{pmatrix},$ (1.4)
$\displaystyle\mathrm{D}(A(t))$
$\displaystyle:=\left\\{v\in\mathrm{W}^{1,1}\left(\left[0,1\right],\mathbb{C}^{m}\right):\
v(1)=\mathbb{B}_{C}(t)v(0)\right\\}$

where the matrix $\mathbb{B}_{C}(t)$ is defined by
$\mathbb{B}_{C}(t):=C(t)^{-1}\mathbb{B}C(t)$ for
$C(t):=\mathrm{diag}(c_{j}(t))_{j=1}^{m}$ and $\mathbb{B}$ is the adjacency
matrix given in (1.2). We can now rewrite our transport problem (nF) in the
form of an abstract Cauchy problem as follows.

######  Lemma 1.

The abstract non-autonomous Cauchy problem

(1.5) $\displaystyle\begin{cases}\dot{u}(t)=A(t)u(t),&\quad t,s\in\mathbb{R},\
t\geq s,\\\ u(s)=(f_{j})_{j=1}^{m},\end{cases}$

associated to the family of operators $(A(t),\mathrm{D}(A(t)))$ defined in
(1.3)-(1.4) is equivalent to the transport problem stated in (nF).

###### Proof.

We only need to check that the condition in the domains $\mathrm{D}(A(t))$ is
equivalent to the boundary condition of (nF). This can be done the same way as
in the proof of [6, Prop. 18.2]. ∎

Our main goal is to show, that problem (1.5) is well-posed.

##  2\. Operators with constant domains

In the literature treating non-autonomous Cauchy problems it is often assumed
that operators $A(t)$ appearing in (nACP) have _constant domain,_ i.e.,
$\mathrm{D}:=\mathrm{D}(A(t))$ for all $t\in\mathbb{R}$. The following well-
posedness result, is a direct consequence of the results obtained by Kato
[14], see also [12, Chapter IV, Sect. 9.5].

######  Proposition 2.1.

Assume that $\mathrm{D}:=\mathrm{D}(A(t))$ holds for all $t\in\mathbb{R}$ and
that $c_{j}(\cdot)\in\mathrm{C}^{1}(\mathbb{R})$. Then (nF) is a well-posed
problem.

Let us state a simple structural condition that implies constant domains of
the operators $A(t)$ defined in (1.3)-(1.4). Note, that this condition is
reasonable in many situations. It means that whenever there is an inflow into
the $i$-th edge from the $j$-th edge, at all times the velocities of the flow
should stay in the same ratio which is, for example, determined by the radii
of the respective pipelines.

######  Lemma 2.

If for all $i,j$ such that $\mathbb{B}_{ij}\neq 0$ it holds

(2.1)
$\displaystyle\frac{c_{i}(t_{1})}{c_{j}(t_{1})}=\frac{c_{i}(t_{2})}{c_{j}(t_{2})}$

then for any $t_{1},t_{2}\in\mathbb{R}$ we have that
$\mathrm{D}(A(t_{1}))=\mathrm{D}(A(t_{2}))$.

###### Proof.

First observe that from the definition of $\mathbb{B}_{C}(t)$ and the
diagonality of $C(t)$ it follows that
$\left(\mathbb{B}_{C}(t)\right)_{ij}=c_{i}^{-1}(t)\mathbb{B}_{ij}c_{j}(t)$.
Hence, condition (2.1) clearly implies
$\mathbb{B}_{C}(t_{1})=\mathbb{B}_{C}(t_{2})$ an thus also
$\mathrm{D}(A(t_{1}))=\mathrm{D}(A(t_{2}))$. ∎

We continue with some results on the stability of the solutions in the
periodic case.

######  Definition 2.2.

Let $(A(t),\mathrm{D})_{t\in\mathbb{R}}$ be a family of linear operators on a
common domain $\mathrm{D}$. For $T>0$ we call the family of operators _$T$
-periodic_ if

(2.2) $\displaystyle A(t+T)=A(t),\quad t\in\mathbb{R}.$

Observe that the family of operators defined by (1.3) becomes $T$-periodic
whenever the functions $c_{j}(\cdot)$ are $T$-periodic for each $j$, i.e.,

(2.3) $\displaystyle c_{j}(t+T)=c_{j}(t),\quad t\in\mathbb{R},\
j\in\left\\{1,\ldots,m\right\\}.$

According to [11, Lemma 6.1], in this case there exists a $T$-periodic
evolution family $(U(t,s))_{t\geq s}$ in the sense that

(2.4) $\displaystyle U(t+T,s+T)=U(t,s),\quad t\geq s.$

One can associate the following important family of operators to each periodic
evolution family.

######  Definition 2.3.

Let $(U(t,s))_{t\geq s}$ be a $T$-periodic evolution family, i.e., the
evolution family satisfies (2.4). The _period-map_ is defined by

$M(t):=U(t+T,t),\quad t\in\mathbb{R}.$

The _monodromy operator_ is then defined by $M:=M(0)$.

By [11, Prop. 6.3] one has that either there exists $\omega_{0}\in\mathbb{R}$
with $r(M(t))=\mathrm{e}^{-T\omega_{0}}$ or $r(M(t))=0$ (in this case we set
$\omega_{0}=\infty$). Consequently, by [11, Thm. 6.6] we obtain the following
result.

######  Proposition 2.4.

If $(A(t),\mathrm{D})_{t\in\mathbb{R}}$ is $T$-periodic and
$\omega<\omega_{0}$, then there exists $M\geq 1$ such that

$\left\|U(t,s)\right\|\leq M\mathrm{e}^{-\omega(t-s)},\quad t\geq s.$

In particular, if $\omega_{0}>0$ the evolution family is exponentially stable.

##  3\. The General non-autonomous problem

We now study the general situation when the domains of operators $A(t)$ are
not neccessarilly constant. We use evolution families and evolution semigroups
approach to show the well-posedness of (nACP) for operators $A(t)$ given by
(1.3) with non-autonomous domains of the form (1.4).

###  3.1. The associated multiplication semigroup.

First, observe that by [6, Cor. 18.15], for each fixed $s\in\mathbb{R}$, the
abstract Cauchy problem corresponding to the single operator
$(A(s),\mathrm{D}(A(s)))$ is well-posed and the solution is given by a
positive contraction semigroup $(T_{s}(t))_{t\geq 0}$. From this we construct
a new operator semigroup $(\mathcal{S}(t))_{t\geq 0}$ on the vector-valued
function space $\mathrm{C}_{0}(\mathbb{R},X)$ by

(3.1) $(\mathcal{S}(t)f)(s):=T_{s}(t)f(s),\quad t\geq 0,\
f\in\mathrm{C}_{0}(\mathbb{R},X),\ s\in\mathbb{R}.$

Let us show that $(\mathcal{S}(t))_{t\geq 0}$ is a strongly continuous
semigroup on $\mathrm{C}_{0}(\mathbb{R},X)$ and give its generator.

######  Proposition 3.1.

Assume that $c_{j}(\cdot)\in\mathrm{C}(\mathbb{R})$ and that there exist
$m,M>0$ such that $m\leq c_{j}(t)\leq M$ for each $t\in\mathbb{R}$. Then, the
operator semigroup $(\mathcal{S}(t))_{t\geq 0}$ is strongly continuous on
$\mathrm{C}_{0}(\mathbb{R},X)$ and its generator
$(\mathcal{A},\mathrm{D}(\mathcal{A})$ is given by the multiplication operator
on $\mathrm{C}_{0}(\mathbb{R},X)$ induced by the family of differential
operators on $\mathrm{L}^{1}\left(\left[0,1\right],X\right)$ defined in
(1.3)-(1.4), i.e.,

$\displaystyle(\mathcal{A}f)(s)$ $\displaystyle=A(s)f(s),$
$\displaystyle\mathrm{D}(\mathcal{A})$
$\displaystyle=\left\\{f\in\mathrm{C}_{0}(\mathbb{R},X):\ \forall
s\in\mathbb{R}:\ f(s)\in\mathrm{D}(A(s)),\
A(\cdot)f(\cdot)\in\mathrm{C}_{0}(\mathbb{R},X)\right\\}.$

In addition, the set $\rho(\mathcal{A})\cap\rho(A(s))$ is nonempty for each
$s\in\mathbb{R}$ and the resolvent $R(\lambda,\mathcal{A})$ is a bounded
operator multiplier for all $\lambda\in\rho(\mathcal{A})$, given by

$\left(R(\lambda,\mathcal{A})f\right)(s)=R(\lambda,A(s))f(s),\quad
s\in\mathbb{R},f\in\mathrm{C}_{0}(\mathbb{R},X).$

###### Proof.

The assertion follows directly by [13, Thm. 3.4] once we show that the map
$\mathbb{R}\times\mathbb{R}_{\geq 0}\ni(s,t)\mapsto T_{s}(t)$ is strongly
continuous. To do so, we follow [13, Rem. 3.6] and prove that the mapping
$s\mapsto R(\lambda,A(s))$ is strongly continuous and uniformly bounded.

If for any fixed $s\in\mathbb{R}$ we take function $c_{k}(s)$ instead of
constant $c_{k}$ in [6, Prop. 18.12] we obtain an explicit formula for the
resolvent $R(\lambda,A(s))$. In particular, for $\mathrm{Re}\lambda>0$ we have

$R(\lambda,A(s))=\left(\mathrm{I}+E_{\lambda}(\cdot,s)(1-\mathbb{B}_{C,\lambda}(s))^{-1}\mathbb{B}_{C,\lambda}(s)\otimes\delta_{0}\right)R_{\lambda}(s),$

where

$E_{\lambda}(\tau,s)=\mathrm{diag}\left(\mathrm{e}^{(\lambda/c_{k}(s))\tau}\right),\quad\mathbb{B}_{C,\lambda}(s)=E_{\lambda}(-1,s)\mathbb{B}_{C}(s),$

$\delta_{0}$ is the point evaluation at $0$, and

$(R_{\lambda}(s)f)(\tau)=\int_{\tau}^{1}{E_{\lambda}(\tau-\xi,s)C^{-1}(s)f(\xi)\
\mathrm{d}\xi}.$

It is easy to see that by our assumptions on $c_{j}(\cdot)$, the mapping
$s\mapsto R(\lambda,A(s))f$ is continuous for every
$f\in\mathrm{C}_{0}(\mathbb{R},X)$. Moreover, uniform boundedness of the
resolvent follows from the contractive property of the semigroup. Now, the
Trotter–Kato theorem [23, Chapter III, Thm. 4.4] implies the strong continuity
of $(s,t)\mapsto T_{s}(t)$ and we are done. The addition on the resolvent
$R(\lambda,\mathcal{A})$ follows by [13, Lem. 3.5]. ∎

###  3.2. The associated evolution semigroup and its generator

Inspired by the work of R. Nagel, G. Nickel and S. Romanelli [20, Sect. 4] we
define another semigroup $(\mathcal{T}(t))_{t\geq 0}$ on
$\mathrm{C}_{0}(\mathbb{R},X)$ by

(3.2) $(\mathcal{T}(t)f)(s):=\mathcal{S}(t)f(s-t),\quad t\geq 0,\
f\in\mathrm{C}_{0}(\mathbb{R},X),\ s\in\mathbb{R},$

where $(\mathcal{S}(t))_{t\geq 0}$ is the multiplication semigroup given in
(3.1). Hence, $(\mathcal{T}(t))_{t\geq 0}$ is the composition of the
translation semigroup and the multiplication semigroup. An application of
Theorem 1.5 shows that $(\mathcal{T}(t))_{t\geq 0}$ is an evolution semigroup.
Our next goal is to determine the generator of this semigroup. This can be
done, as already mentioned in [20, Sect. 4], by means of extrapolation spaces.
Recall, that the first extrapolation space of a Banach space $X$ with respect
to a semigroup generator $(A,\mathrm{D}(A))$ is defined to be the completion
of $X$ with respect to a new norm $\left\|\cdot\right\|_{-1}$ on $X$ defined
by

$\left\|x\right\|_{-1}:=\left\|A^{-1}x\right\|,\quad x\in X$

(without loss of the generality we may assume that $0\in\rho(A)$). This
completion is denoted by $X_{-1}$ and is called the _first extrapolation
space_. If we want to stress the dependence on the operator
$(A,\mathrm{D}(A))$ we write $X_{-1}^{(A)}$. By continuity we can extend
$(T(t))_{t\geq 0}$ to a semigroup of linear operators on $X_{-1}$. This
semigroup is denoted by $(T_{-1}(t))_{t\geq 0}$. The first extrapolation space
and the corresponding extrapolated semigroup have the following properties
(see also [12, Chapter II, Thm. 5.5] and [19, Thm. 1.5]).

######  Proposition 3.2.

Let $(T(t))_{t\geq 0}$ be a strongly continuous semigroup on $X$ with
generator $(A,\mathrm{D}(A))$. The following assertions hold true.

  1. (i)

$X$ is dense in $X_{-1}^{(A)}$.

  2. (ii)

$(T_{-1}(t))_{t\geq 0}$ is a strongly continuous semigroup on $X_{-1}^{(A)}$.

  3. (iii)

The generator $A_{-1}$ of $(T_{-1}(t))_{t\geq 0}$ has domain
$\mathrm{D}(A_{-1})=X$ and is the unique extension of
$A\colon\mathrm{D}(A)\rightarrow X$ to an isometry $A_{-1}\colon X\rightarrow
X_{-1}^{(A)}$.

Typical examples for such extrapolation spaces are Sobolev spaces and weighted
$\mathrm{L}^{p}$-spaces. For more explicit examples we refer to [12, Chapter
II, Ex. 5.7,5.8], [20, Sect. 3] and [9, Sect. 5]. Recall, that by [13, Thm.
4.7] the first extrapolation space of $\mathrm{C}_{0}(\mathbb{R},X)$ with
respect to the multiplication operator $(\mathcal{A},\mathrm{D}(\mathcal{A}))$
looks like

$\left[\mathrm{C}_{0}(\mathbb{R},X)\right]_{-1}^{\mathcal{A}}=\left\\{f\in\prod_{s\in\mathbb{R}}{X_{-1,s}}:\
f\ \text{is fibre-continuous and vanishes at infinity}\right\\}.$

Similar results have been recently discovered for Bochner
$\mathrm{L}^{p}$-spaces as well, cf. [10]. Here, for fixed $s\in\mathbb{R}$,
$X_{-1}^{A(s)}$ denotes the first extrapolation space of $X$ with respect to
the operator $(A(s),\mathrm{D}(A(s)))$. Function
$f\in\prod_{s\in\mathbb{R}}{X_{-1}^{A(s)}}$ is called _fiber-continuous_ if
for any $s_{0}\in\mathbb{R}$ and $\varepsilon>0$ there exists $\delta>0$ such
that $\left|s_{0}-s\right|<\delta$ implies that

$\left\|f(s_{0})-x\right\|_{-1,s_{0}}+\left\|f(s)-x\right\|_{-1,s}<\varepsilon,$

see [13, Def. 4.4]. Moreover, such a function _vanishes at infinity_ if
$\lim_{\left|s\right|\to\infty}{\left\|f(s)\right\|_{-1,s}}=0$.

At the first glance, it seems that the elements of the family of extrapolation
spaces $(X_{-1}^{A(s)})_{s\in\mathbb{R}}$ may differ from each other and hence
the product of the extrapolated translation semigroup and the extrapolated
multiplication semigroups may not be well-defined. However, in our case the
operators $(A(s),\mathrm{D}(A(s)))$, $s\in\mathbb{R}$, have a special
appearance, that is, we can write $A(s)=C(s)\frac{\mathrm{d}}{\mathrm{d}{x}}$
for $C(s)\in\mathrm{C}\left(\mathbb{R},\mathscr{L}(X)\right)$ if we assume
that $c_{j}(\cdot)\in\mathrm{C}(\mathbb{R})$. We shall also assume that there
exist $m,M>0$ such that $m\leq c_{j}(t)\leq M$ for all $t\in\mathbb{R}$. By
[3, Rem. 6.6(b)], it follows, that all extrapolation spaces $X_{-1}^{A(s)}$
are equivalent, i.e., $X_{-1}^{A(s)}\cong X_{-1}^{A(0)}=:X_{-1}$ for all
$s\in\mathbb{R}$ and there exists $\eta>0$ such that
$\frac{1}{\eta}\left\|x\right\|_{X_{-1}}\leq\left\|x\right\|_{X_{-1}^{A(s)}}\leq\eta\left\|x\right\|_{X_{-1}}$
for all $s\in\mathbb{R}$. Hence, we have that
$\left[\mathrm{C}_{0}(\mathbb{R},X)\right]_{-1}^{\mathcal{A}}=\mathrm{C}_{0}(\mathbb{R},X_{-1})$.
Then, by following the procedure in [20, Sect. 4] we obtain a result similar
to [20, Prop. 4.3], i.e., we can describe the domain $\mathrm{D}(\mathcal{G})$
of the generator of the evolution semigroup $(\mathcal{T}(t))_{t\geq 0}$.

######  Proposition 3.3.

The generator $(\mathcal{G},\mathrm{D}(\mathcal{G}))$ of the $C_{0}$-semigroup
$(\mathcal{T}(t))_{t\geq 0}$ on $\mathrm{C}_{0}(\mathbb{R},X)$ defined in
(3.2) is given by

$\displaystyle(\mathcal{G}f)(s)$ $\displaystyle=A(s)f(s)-f^{\prime}(s),$
$\displaystyle\mathrm{D}(\mathcal{G})$
$\displaystyle=\left\\{f\in\mathrm{C}_{0}(\mathbb{R},X)\cap\mathrm{C}_{0}^{1}(\mathbb{R},X_{-1}):\
\mathcal{A}f-f^{\prime}\in\mathrm{C}_{0}(\mathbb{R},X)\right\\}.$

###### Proof.

As mentioned above, one has that
$\left[\mathrm{C}_{0}(\mathbb{R},X)\right]_{-1}^{\mathcal{A}}=\mathrm{C}_{0}(\mathbb{R},X_{-1})$.
On this space we consider the extrapolated multiplication semigroup
$(\mathcal{S}_{-1}(t))_{t\geq 0}$ which is defined by

$(\mathcal{S}_{-1}(t)f)(s):=T_{-1,s}(t)f(s),$

where $(T_{-1,s}(t))_{t\geq 0}$ is the extrapolated semigroup for
$(T_{s}(t))_{t\geq 0}$ on $X_{-1}$. Its generator is denoted by
$\mathcal{A}_{-1}$ and is by [13, Lem. 4.6] given by

$\left(\mathcal{A}_{-1}f\right)(s)=A_{-1}(s)f(s),\quad\mathrm{D}(\mathcal{A}_{-1})=\mathrm{C}_{0}(\mathbb{R},X).$

On the space $\mathrm{C}_{0}(\mathbb{R},X_{-1})$ we also consider the operator
$(\mathcal{B},\mathrm{D}(\mathcal{B}))$ defined by

$\mathcal{B}f:=f^{\prime},\quad\mathrm{D}(\mathcal{B}):=\mathrm{C}^{1}_{0}(\mathbb{R},X_{-1}),$

which generates the right translation semigroup on
$\mathrm{C}_{0}(\mathbb{R},X_{-1})$. Now define semigroup
$(\widetilde{\mathcal{T}}(t))_{t\geq 0}$ on
$\mathrm{C}_{0}(\mathbb{R},X_{-1})$ by

$(\widetilde{\mathcal{T}}(t)f)(s):=\mathcal{S}_{-1}(t)f(s-t),\quad t\geq
0,s\in\mathbb{R}.$

Following [5, Chapter A-I, Ex. 3.8] one obtains for the corresponding
generator $(\widetilde{\mathcal{G}},\mathrm{D}(\widetilde{\mathcal{G}}))$ that
for every $f\in\mathrm{D}(\mathcal{A}_{-1})\cap\mathrm{D}(\mathcal{B})$ one
has $f\in\mathrm{D}(\widetilde{\mathcal{G}})$ and

$(\widetilde{\mathcal{G}}f)(s)=-f^{\prime}(s)+(\mathcal{A}_{-1}f)(s)=-f^{\prime}(s)+A_{-1}(s)f(s),\quad
s\in\mathbb{R},$

see also [20, Lem. 4.1]. By observing that $(\mathcal{T}(t))_{t\geq 0}$ given
in (3.2) coincides with $(\widetilde{\mathcal{T}}(t))_{t\geq 0}$ on
$\mathrm{C}_{0}(\mathbb{R},X)$ one obtains that

$\mathrm{D}(\mathcal{G})=\\{f\in\mathrm{C}_{0}(\mathbb{R},X)\cap\mathrm{D}(\widetilde{\mathcal{G}}):\
\widetilde{\mathcal{G}}f\in\mathrm{C}_{0}(\mathbb{R},X)\\}$

and $\mathcal{G}f=\widetilde{\mathcal{G}}f$ for all
$f\in\mathrm{D}(\mathcal{G})$, see also [21, Lemma 2.6]. By the same reasoning
as in the proof of [20, Lem. 4.2] we finaly obtain

$\mathrm{C}_{0}(\mathbb{R},X)\cap\mathrm{D}(\widetilde{\mathcal{G}})=\mathrm{C}_{0}(\mathbb{R},X)\cap\mathrm{C}^{1}_{0}(\mathbb{R},X_{-1}).$

∎

We observe that
$\mathrm{D}(\mathcal{G})\cap\mathrm{C}_{0}^{1}(\mathbb{R},X)=\mathrm{D}(\mathcal{G})$
and that $\mathrm{D}(\mathcal{G})$ is a core. Hence, by applying Theorem 1.6,
we conclude that the problem (1.5) - and thus (nF) - is well-posed.

######  Corollary 1.

Assume that $c_{j}(\cdot)\in\mathrm{C}(\mathbb{R})$ and that there exist
$m,M>0$ such that $m\leq c_{j}(t)\leq M$ for each $t\in\mathbb{R}$. Then (nF)
is a well-posed problem.

Unfortunately, we can not give an explicit expression for the corresponding
evolution family due to the fact that there is no closed formula for
semigroups $(T_{s}(t))_{t\geq 0}$. Theoretically speaking, given an evolution
semigroup $(\mathcal{T}(t))_{t\geq 0}$ on $\mathrm{C}_{0}(\mathbb{R},X)$ one
defines the corresponding evolution family implicitly by

(3.3) $\displaystyle U(t,s)x:=(T_{\mathrm{r}}(s-t)\mathcal{T}(t-s)f)(s),\quad
s\in\mathbb{R},\ t\geq s,$

for $f\in\mathrm{C}_{0}(\mathbb{R},X)$ with $f(s)=x$ and
$(T_{\mathrm{r}}(t))_{t\geq 0}$ the right-translation semigroup on
$\mathrm{C}_{0}(\mathbb{R},X)$.

######  Remark 3.4.

  1. (a)

Following the construction of the evolution family given by (3.3) via the
contraction semigroups $(T_{s}(t))_{t\geq 0}$, $(\mathcal{S}(t))_{t\geq 0}$ ,
$(\mathcal{T}(t))_{t\geq 0}$, and translation semigroups we see that
$(U(t,s))_{t\geq s}$ actually becomes an evolution family of contractions,
i.e., $\left\|U(t,s)\right\|\leq 1$ for all $t\geq s$.

  2. (b)

Since each of the operators $(A(t),\mathrm{D}(A(t)))$, $t\in\mathbb{R}$,
generates a positive operator semigroup, our construction also yields that the
evolution family $(U(t,s))_{t\geq s}$ obtained in (3.3) consists of positive
operators.

  3. (c)

The analysis of the asymptotic behavior of the solutions to non-autonomous
problems is in general a very difficult task. Batty, Chill, and Tomilov
characterized strong stability of a bounded evolution family on a Banach space
$X$ by the stability of its associated evolution semigroup on
$\mathrm{L}^{p}(\mathbb{R}_{+},X)$ for some $1\leq p\leq\infty$ or,
equivalently, by the density of the range of the corresponding generator on
$\mathrm{L}^{1}(\mathbb{R}_{+},X)$ see [7, Thm. 2.2]. These conditions are
however not easy to verify in our case.

  4. (d)

In Section 2 we stated some stability results in the periodic situation.
Bayazit et al. [8] studied longterm behavior for the operators with periodic
non-autonomous domain. They used an explicit description of the evolution
family and the fact that in this case the period-map is a multiplication
operator. However, such results are not available for our general situation
so, a-priori, no statements regarding the asymptotics are possible.

###  3.3. Perturbations

Finally, we consider a slightly modified version of (nF) by adding a time-
varying absorption term. In particular, we consider the equation given by

(pnF) $\displaystyle\begin{cases}\frac{\partial}{\partial
t}u_{j}(t,x)=c_{j}(t)\frac{\partial}{\partial
x}u_{j}(t,x)+q_{j}(t,x)u_{j}(t,x),&\quad x\in\left(0,1\right),\ t\geq 0,\\\
u_{j}(t,0)=f_{j}(x),&\quad x\in\left(0,1\right),\\\
\phi^{-}_{ij}c_{j}(t)u_{j}(t,1)=\omega_{ij}\sum_{k\in\mathbb{N}}{\phi^{+}_{ik}c_{k}(t)u_{k}(t,1)},&\quad
t\geq 0.\end{cases}$

Then (pnF) can be written as an abstract Cauchy problem of the form

(3.4) $\displaystyle\begin{cases}\dot{u}(t)=(A(t)+B(t))u(t),&\quad
t,s\in\mathbb{R},\ t\geq s,\\\ u(s)=\mathsf{f},\end{cases}$

where the family of operators $(A(t),\mathrm{D}(A(t)))_{t\in\mathbb{R}}$ is
again defined by (1.3)–(1.4), $\mathsf{f}=(f_{j})_{j=1}^{m}$, and the family
of bounded linear operators $(B(t))_{t\in\mathbb{R}}$ is defined by

(3.5) $\displaystyle B(t):=\mathrm{diag}(M_{q_{j}(t)}),$

where $M_{q_{j}(t)}$ denotes the multiplication operator induced by the
function $q_{j}(t)$. Under certain regularity conditions, this leads to the
following result.

######  Theorem 3.5.

Let $c_{j}(\cdot)\in\mathrm{C}(\mathbb{R})$ such that there exist $m,M>0$ with
$m\leq c_{j}(t)\leq M$ for each $t\in\mathbb{R}$. Under the assumption that
$q_{j}(t,\cdot)\in\mathrm{L}^{\infty}\left(\left[0,1\right],\mathbb{C}^{m}\right)$
and $q_{j}(\cdot,x)\in\mathrm{C}_{\mathrm{b}}(\mathbb{R},X)$, there exists a
unique evolution family $(U_{B}(t,s))_{t\geq s}$, generated by the family of
operators $(A(t)+B(t),\mathrm{D}(A(t)))_{t\in\mathbb{R}}$.

###### Proof.

By Corollary 1, the abstract Cauchy problem (nF) is well-posed. By the
assumptions on the functions $q_{j}$ we obtain that
$B(\cdot)\in\mathrm{C}_{\mathrm{b}}(\mathbb{R},\mathscr{L}(X))$. The result
now follows by [12, Chapter VI, Cor. 9.20]. ∎

It is worth to mention, that Theorem 3.5 does not imply that (pnF) is well-
posed, see [12, Chapter VI, Ex. 9.21]. Nonetheless, for the evolution family
$(U_{B}(t,s))_{t\geq s}$ obtained in the theorem, the function
$U_{B}(\cdot,s)\mathsf{f}$ can be interpreted as a _mild solution_ of the
problem (3.4).

## Acknowledgement

C.B. acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) - 468736785. This article is based upon work from COST
Action CA18232 "Mathematical models for interacting dynamics on networks"
supported by COST (European Cooperation in Science and Technology).

## References

  * [1] P. Acquistapace and B. Terreni.  On the abstract Cauchy problem in the case of constant domains.  Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8), 76(1):7–13, 1984.
  * [2] P. Acquistapace and B. Terreni.  A unified approach to abstract linear nonautonomous parabolic equations.  Rend. Sem. Mat. Univ. Padova, 78:47–107, 1987.
  * [3] H. Amann.  Parabolic evolution equations in interpolation and extrapolation spaces.  J. Funct. Anal., 78(2):233–270, 1988.
  * [4] W. Arendt, D. Dier, and M. Kramar Fijavž.  Diffusion in networks with time-dependent transmission conditions.  Appl. Math. Optim., 69(2):315–336, 2014.
  * [5] W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander, and U. Schlotterbeck.  One-parameter semigroups of positive operators, volume 1184 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 1986.
  * [6] A. Bátkai, M. Kramar Fijavž, and A. Rhandi.  Positive operator semigroups, volume 257 of Operator Theory: Advances and Applications.  Birkhäuser/Springer, Cham, 2017.  From finite to infinite dimensions.
  * [7] C. J. K. Batty, R. Chill, and Y. Tomilov.  Strong stability of bounded evolution families and semigroups.  J. Funct. Anal., 193(1):116–139, 2002.
  * [8] F. Bayazit, B. Dorn, and M. Kramar Fijavž.  Asymptotic periodicity of flows in time-depending networks.  Netw. Heterog. Media, 8(4):843–855, 2013.
  * [9] C. Budde and B. Farkas.  Intermediate and extrapolated spaces for bi-continuous operator semigroups.  J. Evol. Equ., 19(2):321–359, 2019.
  * [10] C. Budde and R. Heymann.  Extrapolation of operator-valued multiplication operators.  Quaest. Math., 45(3):347–356, 2022.
  * [11] D. Daners and P. Koch Medina.  Abstract evolution equations, periodic problems and applications, volume 279 of Pitman Research Notes in Mathematics Series.  Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992.
  * [12] K.-J. Engel and R. Nagel.  One-parameter semigroups for linear evolution equations, volume 194 of Graduate Texts in Mathematics.  Springer-Verlag, New York, 2000.  With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.
  * [13] T. Graser.  Operator multipliers generating strongly continuous semigroups.  Semigroup Forum, 55(1):68–79, 1997.
  * [14] T. Kato.  Integration of the equation of evolution in a Banach space.  J. Math. Soc. Japan, 5:208–234, 1953.
  * [15] T. Kato and H. Tanabe.  On the abstract evolution equation.  Osaka Math. J., 14:107–133, 1962.
  * [16] M. Kramar and E. Sikolya.  Spectral properties and asymptotic periodicity of flows in networks.  Math. Z., 249(1):139–162, 2005.
  * [17] M. Kramar Fijavž and A. Puchalska.  Semigroups for dynamical processes on metric graphs.  Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 378(2185):20190619, 2020.
  * [18] D. Mugnolo.  Semigroup methods for evolution equations on networks.  Understanding Complex Systems. Springer, Cham, 2014.
  * [19] R. Nagel.  Extrapolation spaces for semigroups (nonlinear evolution equations and applications).  RIMS Kôkyûroku, 1009:181–191, 8 1997.
  * [20] R. Nagel, G. Nickel, and S. Romanelli.  Identification of extrapolation spaces for unbounded operators.  Quaestiones Math., 19(1-2):83–100, 1996.
  * [21] R. Nagel and E. Sinestrari.  Inhomogeneous Volterra integrodifferential equations for Hille-Yosida operators.  In Functional analysis (Essen, 1991), volume 150 of Lecture Notes in Pure and Appl. Math., pages 51–70. Dekker, New York, 1994.
  * [22] G. Nickel.  Evolution semigroups for nonautonomous Cauchy problems.  Abstr. Appl. Anal., 2(1-2):73–95, 1997.
  * [23] A. Pazy.  Semigroups of linear operators and applications to partial differential equations, volume 44 of Applied Mathematical Sciences.  Springer-Verlag, New York, 1983.
  * [24] R. Schnaubelt.  Well-posedness and asymptotic behaviour of non-autonomous linear evolution equations.  In Evolution equations, semigroups and functional analysis. In memory of Brunello Terreni. Containing papers of the conference, Milano, Italy, September 27-28, 2000, pages 311–338. Basel: Birkhäuser, 2002.