What remains open is to implement this system at a small scale, using both simulation and physical systems. Initial effort has been made in this direction, but more work is needed to have a complete proof of concept. Already, an implementation using quantum network simulator QuNetSim [53] and Deltaflow together has shown distributed quantum computing can be simulated. More, Deltaflow has been shown to work with distributed circuit boards which leads to the first steps of simulating distributed quantum algorithms. The methods for distributing the Ansatz states could consider more parameters for further optimization, such as the coherence times of the qubits and location of the control gates in the circuits. We aim to explore this in more detail in future work.

As discussed, distributed quantum computing is a promising path to developing large scale quantum computers. Much effort has gone into this in the classical computing domain, and the overlap between the fields is high. We can use this knowledge to design robust and secure distributed quantum computers, and as quantum technologies improve, this will surely become a reality.

\clearpage
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\appendix
\section{Additional Material}
\subsection{Algorithms}

\begin{algo}[H]
\small
\caption{Does Not Fit}
\textbf{Input:}
\begin{compactitem}
\item $Q = [q_1, ..., q_n]$ the collection of QPUs in the distributed system, non-increasingly sorted by number of available qubits
\item $A$ the size of the Ansatz
\end{compactitem}
\textbf{Output:} If the Ansatz can fit in the distributed QPU $Q$\\
\textbf{DoesNotFit}$(Q, A)$
\begin{algorithmic}
\If{$Q$ is empty}
\State \textbf{return} $true$
\EndIf
\For{$q_i \in Q$}
\State $curAllocation \gets [0,..., 0]$ \Comment{Allocate $n$ element array of $0$s}
\State $possibleQPUs \gets \{q_1,...,q_i\} \subseteq Q$
\If {$i == 1$}
\State $k \gets \textbf{QPUNumber}(possibleQPUs[1])$ \Comment{The QPU index}
\State $curAllocation[k] \gets q_1 - 1$
\Else
\State State $k \gets \textbf{QPUNumber}(possibleQPUs[1])$ \Comment{The QPU index}
\State $curAllocation[k] \gets q_1 - 3$
\For{$q_j \in \{q_2, ..., q_i\}$}
\State  $k \gets \textbf{QPUNumber}(possibleQPUs[j])$ \Comment{The QPU index}
\State $curAllocation[k] \gets q_j - 2$
\EndFor
\EndIf
\If{$\textbf{sum}(curAllocation) \geq A$}
\State \textbf{return} $false$ \Comment{The Ansatz does fit}
\EndIf
\EndFor
\State \textbf{return} $true$
\end{algorithmic}
\label{algo:does_not_fit}
\end{algo}

\subsection{Control System Commands}
\begin{table}[H]
\centering
\begin{tabularx}{\textwidth}{|s|L|}
\hline
\bf{Command}  &\bf{Description}\\
\hline
$TWO\_QUBIT[G, \text{qID}_1, \text{qID}_2]$ & Two qubit gate $G$ on qubits with memory IDs $\text{qID}_1, \text{qID}_2$, where $\text{qID}_1$ is the control if needed. $\text{qID}_1$ is potentially a classical register.  \\
\hline
$SINGLE[G, \text{qID}]$ & A single qubit gate $G$ applied to qubit with memory ID qID\\
\hline
$GEN\_ENT[\text{qID}_1, \text{qID}_2]$ & Generate entanglement and store it in the qIDs $\text{qID}_1, \text{qID}_2$\\
\hline
$REC\_ENT[QPU, \text{qID}]$ & Receive entanglement half and store it in at memory qID\\
\hline
$SEND\_ENT[QPU, \text{qID}]$ & Send entanglement half and keep other half in qubit memory qID \\
\hline
$CLASSICAL[\text{cID}]$ & Transmission of classical bit in classical memory with ID cID\\
\hline
$SEND\_CLA[QPU, \text{cID}]$ &  Send classical bit in classical memory with ID cID \\
\hline
$REC\_CLA[QPU, \text{cID}]$ &  Receive classical bit and store in classical memory with ID cID \\
\hline
\end{tabularx}
\caption{Commands for QPU schedule}
\label{tab:controls}
\end{table}

\end{document}


Introduction sec-intro

A long-run average objective for a Markov decision process (MDP) $D$ is a
property depending on the proportion of time (frequency) spent in the
individual states of $D$. Typical examples of such properties include

  * •

the total frequency of visits to “bad” states is ${\leq}0.05$;

  * •

the state frequency vector is equal to a given vector $\nu$.

The existing works on long-run average optimization (see Related Work)
concentrate on constructing a strategy $\sigma$ such that the Markov chain
$D^{\sigma}$ obtained by applying $\sigma$ to $D$ is irreducible and the
_invariant_ (also called _steady-state_) distribution $\Inv_{\sigma}$ achieves
the objective111Recall that by ergodic theorem [Norris:book], the invariant
distribution is the vector of limit state frequencies computed for longer and
longer prefixes of runs.. Unfortunately, the existing algorithms cannot
influence the _local stability_ of the invariant distribution along a run.
More concretely, for a given time horizon $n$, consider the _local frequency_
$\Freq_{n}$ of states sampled from $n$ consecutive states along a run,
starting at a randomly chosen _pivot position_ (we refer to Section LABEL:sec-
model for precise definitions). The local stability of the invariant
distribution is the probability that $\Freq_{n}$ stays “close” to
$\Inv_{\sigma}$. If the local stability is low, then the probability of
achieving the considered objective _locally_ (i.e., within the prescribed time
horizon) is also low, and this may lead to severe problems in many application
scenarios. Consider a system of Fig. LABEL:fig-RM-system(a) that can be either
in the running (R) or maintenance (M) state. A long-run sustainability of the
system requires that the system is running for $90\%$ of time and the
remaining $10\%$ is spent on maintenance. Hence, we aim at constructing a
strategy $\sigma$ such that $\Inv_{\sigma}=\nu$, where $\nu(R)=0.9$ and
$\nu(M)=0.1$. Ideally, the maintenance should be performed _regularly_ , i.e.,
the state $M$ should be visited once in $10$ consecutive states. That is,
$\Freq_{10}$ should be equal to $\nu$ with high probability. For every
$y\in[0,1)$, the memoryless strategy $\sigma_{y}$ of Fig. LABEL:fig-RM-
system(b) satisfies $\Inv_{\sigma_{y}}=\nu$. However, the probability of
$\Prob^{\sigma_{y}}[\Freq_{10}{=}\nu]$ approaches _zero_ as $y\to 1$. The best
result is achieved for $y=0$, where this probability is ${\approx}0.43$.
Hence, even the best memoryless strategy may considerably degrade the
reliability of the system. The simple deterministic strategy $\pi$ of Fig.
LABEL:fig-RM-system(c) satisfies $\Inv_{\pi}=\nu$ and
$\Prob^{\pi}[\Freq_{10}{=}\nu]=1$. Note that $\pi$ needs $9$ memory states to
“count” the repeated visits to $R$ before visiting $M$. A “tradeoff” between
memory size and the local satisfaction of the sustainability objective is
achieved by the strategy $\eta$ of Fig. LABEL:fig-RM-system(d) where
$\Inv_{\eta}=\nu$ and $\Prob^{\eta}[\Freq_{10}{=}\nu]\approx 0.74$. Other
examples of long-run average objectives where the local satisfaction/stability
requirements rise naturally are

  * •

_critical supply delivery_ (see, e.g., [Skwirzynski-multiuser-comm,
Lazar:optimal-flow-queues]), where a bundle of items with limited lifespan
should be delivered with a given frequency $f$. A high level of local
instability of the frequency causes a high probability of early/late
deliveries that are both undesirable (early deliveries lead to wasting the
items that are not consumed before expiration, and late deliveries lead to a
shortage of items).

  * •

_dependability_ , i.e., an upper bound on failure frequency (see, e.g.,
[BL:MDP-failure-rate, BLF:non-ergodic-MDP-failure-rate]). If this bound is
locally violated with considerable probability, a user may interpret this as a
violation of the dependability guarantee. For example, consider a device
supposed to fail at most once in a month _on average_ during the device
lifetime. If the device fails twice in two weeks with probability $0.2$ (which
is possible _without_ violating the guarantee on the long-run average failure
frequency), the device is likely to be perceived as _unreliable_.

The above list of examples is not exhaustive. Scenarios documenting the
importance of local satisfaction/stability can be found in every application
area involving long-run average objectives.

[x=4.2cm, y=2.5cm, ¿=stealth’, scale=0.48,font=] in 0,-1[min] (R) at (0,) $R$;
[min] (M) at (1,) $M$; [-¿,thick] (R) to [in=170,out=10] node[above]