Current address: IBM Quantum, IBM Research Europe – Zurich
E-Mail: dmi(at)zurich.ibm.com

Fathoming out quantum state space is a challenging endeavor due to its exponentially growing dimensionality.
At the expense of being bound in its expressiveness, the discrete and finite subspace of graph states is easier to investigate
via a pictorial framework accompanied with a theoretical toolkit from the stabilizer formalism.
Analyzing hand-drawn graphs is a tedious and time-consuming task and imposes limitations to the problem sizes that can be addressed.
Similarly, algorithmic studies using adjacency matrices alone lack the benefit of a visual representation of the states.
We argue that applying visual analytics to investigate graph states can be advantageous.
To this end, we introduce , a web-based application for the visual analysis of qubit graph states and their stabilizer groups.
Our tool facilitates the interactive construction of a graph through multiple components supported by linking and brushing.
The user can explore graph-state-specific properties, including the Pauli-weight distribution of its stabilizer operators and noise thresholds for entanglement criteria.
We propose a use case in the context of near-term quantum algorithms to illustrate the capabilities of our prototype.
We provide access to  as an open-source project and invite the broader quantum computing and engineering communities to take advantage of this tool and further boost its development.

§ INTRODUCTION

The advent of quantum computers increasingly stimulates interdisciplinary research and development between computer and quantum information science <cit.>.
Visual analytics is a growing subfield of computer science, which integrates the user into the analysis process to better understand complex relationships in data [6].
For instance, visual analytics has already been successfully applied to investigate graph structures in social networks [7].
We argue that the quantum information community can also benefit from using visual analytics  because the state space of a quantum computer is  high-dimensional and complex.

While a register of $n$ classical bits can be simply described by bit strings $\mathbf{b}=(b_1,\ldots,b_n)\in \FF_2^n$,
an $n$-qubit register can also be in a quantum superposition state $\ket{\psi}= \sum_{\mathbf{b}\in \FF_2^n} z_\mathbf{b} \ket{\mathbf{b}}$.
Hereby, the only constraint on the probability amplitudes $z_\mathbf{b}\in \CC$ is the normalization condition $\sum_{\mathbf{b}} \vert z_{\mathbf{b}}\vert^2=1$.
This flexibility enables novel quantum information protocols having no analog in the classical computing paradigm [1].

When approaching a given problem, quantum information scientists often first investigate graph states, which allow for visual illustration [8].
For an $n$-qubit register, one considers undirected, simple graphs $G=(V,E)$, where $V=\{1,\ldots, n\}$ is the set of vertices.
Every choice of edge set $E$ leads to a different graph state $\ket{G}$, which is defined as follows.
First, every qubit is initialized in state $\ket{+}= \frac{1}{\sqrt{2}}(\ket{0}+\ket{1})$.
Then, for every edge $\{i,j\}\in E$ one applies a controlled-$Z$ gate, $\CZ = \diag(1,1,1,-1)$, between qubit $i$ and qubit $j$ (see <ref>).

A ring graph with six vertices (left) and the quantum circuit, which prepares the corresponding 6-qubit ring graph state (right).
The layer of Hadamard gates prepares every qubit in state $H\ket{0}=\ket{+}$.
The order of the subsequent two-qubit gates  (one for each edge in the graph) is irrelevant because $\CZ$-gates commute with each other.

The resulting state can be written as

\begin{align} \label{eq:graph_state}
\ket{G} = \frac{1}{\sqrt{2^n}} \sum_{\mathbf{b}\in \FF_2^n} (-1)^{\sum\limits_{j=1}^n\sum\limits_{k=j+1}^n b_j \gamma_{j,k} b_k} \ket{\mathbf{b}},
\end{align}

where $\gamma_{j,k}=1$ if there is an edge $\{j,k\}\in E$, and $\gamma_{j,k}=0$ otherwise.
The binary matrix $\Gamma=(\gamma_{j,k})_{j,k}\in \FF_2^{n\times n}$ is called adjacency matrix of the graph $G=(V,E)$.

Graph states play an essential role in the stabilizer formalism [9], which relies on the concept of Pauli operators

\begin{align}
X= \begin{pmatrix}
\end{pmatrix}, \
Y = \begin{pmatrix}
0&-i \\i&0
\end{pmatrix}, \text{ and }\
Z= \begin{pmatrix}
1&0 \\ 0&-1
\end{pmatrix}.
\end{align}

Since $Y = iXZ$, any $n$-qubit Pauli operator can be written as

\begin{align} \label{eq:Pauli}
i^qX^\mathbf{r}Z^\mathbf{s} = i^q\bigotimes_{k=1}^n X^{r_k}Z^{s_k},
\end{align}

for some bit strings $\mathbf{r},\mathbf{s}\in \FF_2^n$ and an integer $q$ modulo $4$.
A central result of the stabilizer formalism is that, given a group $\mathcal{S}$ of $2^n$ commuting Pauli operators with $-\mathbbm{1}\not\in\mathcal{S}$, there is a unique $n$-qubit state $\ket{\psi}$ with the property $S\ket{\psi}= \ket{\psi}$ for all $S\in\mathcal{S}$.
One calls $\ket{\psi}$ a stabilizer state and $\mathcal{S}$ its stabilizer group.
Every graph state $\ket{G}$, as defined in Eq. (<ref>), is a stabilizer state  with stabilizer group  $\mathcal{S}=\{ \pm X^\mathbf{r}Z^{\Gamma \mathbf{r}} \ \vert \ \mathbf{r}\in \FF_2^n \}$ [8, 10].
Vice versa,  every stabilizer state $\ket{\psi}$ can be transformed into a graph state $\ket{G}$ by applying suitable single-qubit gates, i.e., $\ket{\psi}$ and $\ket{G}$ are local-unitary (LU) equivalent [11].
Since many quantum state properties do not change under such local transformations, the graph state formalism provides a valuable paradigm for studying quantum entanglement and multipartite correlation in general [8, 12].
Further applications of graph states contain, but are not limited to,
quantum error correction [2], simultaneous measurements of commuting Pauli operators [12], and measurement-based quantum computation [13].

In this paper, we contribute , a web-based interactive prototype for visualizing graph states and their stabilizer groups.
This application enabled us to discover many results in Ref. [12].
As we expect our prototype to be stimulating for the broader quantum computing and engineering community, we make it publicly available as an open source project [14].

§ PROTOTYPE DESIGN

The interface of  comprises six integrated components facilitating the exploration and analysis of graph states as illustrated in <ref>.
Our application allows the user to construct graphs and investigate the corresponding graph states through linking and brushing [15].

§.§ Constructing graphs

When  is launched, a Pusteblume graph with $n=8$ vertices is initialized as a starting graph, which can be modified through mouse interactions in the Graph State Visualization widget .
It is also possible to start with a different graph by choosing from a list

in the Load a Predefined Graph widget .
Furthermore, the user can create a random graph for which the edge-creation probability $p$ can be  adjusted using a slider, which appears in widget  once this option is selected.
For example, setting $p=0$ and $p=1$ leads to the edgeless and complete graph, respectively.

To create or remove vertices and edges, the user can follow the graph manipulation workflow as described in <ref>.

Typical workflow of manipulating a graph in our application.
A right-click on the background opens a context menu facilitating the creation of a new isolated vertex .
With another right-click on , the user can add a new edge to an available (green) vertex;
vertices can be unavailable (red) because simple graphs cannot contain loops or multiple edges.
Similarly, the edge — can be deleted via the edge context menu.

Whenever the graph is changed, its adjacency matrix  is updated accordingly.
Vice versa, it is also possible to create (and delete) edges by flipping the respective entries of the adjacency matrix from 0 to 1 (and back).
The rows and columns are explicitly enumerated, and a right-click on one of these header cells opens the same vertex context menu as in <ref>.
Besides Add New Edge and Delete Vertex, there is a third option in this menu: Local Complementation [16].
Clicking this button will invert the neighborhood of vertex $i$; i.e., for each pair of vertices $j,k$ with $\gamma_{i,j}=\gamma_{i,k}=1$, the value of $\gamma_{j,k}$ is flipped.

If a sequence of local complementations relates two graphs, then the respective graph states only differ by the application of certain single-qubit Clifford gates; in particular, they are LU equivalent [8].

By default, the prototype reduces overlapping of vertices, leading to balanced graph representations by exploiting the force-directed graph layout algorithm that is available through the D3 library [17]  .
The automatic layout animation can be toggled by using the Lock / Unlock Force button  in the graph state visualization widget .
In particular, the user can deactivate the repelling forces between vertices to create a manual layout tailored to his or her preference.

Triggering the button Zoom to Fit  above will automatically adjust the view to exploit the available space displaying the whole graph.
The user can flexibly navigate the graph visualization via zoom and pan functionality.

The user can copy the graph ID  or a specific URL  into the clipboard to share a configured graph or save it for later reference .

The graph ID consists of the number of vertices followed by the upper half of the adjacency matrix converted into hexadecimal representation.
This unambiguous mapping allows a graph to be created based on its graph ID, which
consists of approximately $\frac{n(n-1)}{8}$ characters for a graph with $n$ vertices.
In our GitHub repository [14], we provide python functions that convert a Graph ID into the corresponding adjacency matrix and vice versa.
We expect this graph ID conversion functionality accompanied by the URL loading and saving feature to be beneficial to those researchers who prefer to interactively debug and configure graphs instead of mere coding.

§.§ Analyzing the graph state and its stabilizer group

A core functionality of  is to facilitate the investigation of the stabilizer group of a given $n$-qubit graph state $\ket{G}$ with adjacency matrix $\Gamma\in \FF_2^{n\times n}$.
As it is not feasible to depict all $2^n$ stabilizers individually, the prototype instead displays the sector length distribution  (SLD)
$\mathbf{A}=(A_0, \ldots, A_n)$

of the graph state [18];
the $k$-body sector length (SL) of a general $n$-qubit state $\rho \in \CC^{2^n\times 2^n}$ is defined as

\begin{align}
A_k [ \rho] = \sum_{\substack{P \in \{\mathbbm 1,X,Y,Z\}^{\otimes n}\\ \wt(P)=k}}
\Tr[\rho P]^2 ,
\end{align}

where  $k\in\{0,\ldots,n\}$ and $\wt(P)$ denotes the weight of a Pauli operator $P$,
i.e., the number of its non-identity tensor factors.
It is well known that for graph states (and, more generally, for stabilizer states), the $k$-body SL coincides with the number of weight-$k$ stabilizer operators $S\in \mathcal{S}=\{(-1)^{\sum_{i<j} r_i  \gamma_{i,j}r_j } X^\mathbf{r}Z^{\Gamma \mathbf{r}} \ \vert\  \mathbf{r}\in \FF_2^n\}$ [19].[ For example, the three-qubit star-graph state .9—— has one weight-0 stabilizer ($\mathbbm1\mathbbm1\mathbbm1$), no weight-1 stabilizers, three weight-2 stabilizers ($XZ\mathbbm1$, $\mathbbm1ZX$, $X\mathbbm1 X$) and four weight-3 stabilizers ($ZXZ$, $ZYY$, $YYZ$, $-YXY$). Therefore, its SLD is given by $\mathbf{A}=(1,0,3,4)$.]
Thus, we can write

\begin{align} \label{eq:SLD_graphstate}
A_k [ \ket{G}\bra{G}] = \# \{ \mathbf{r} \in \FF_2 ^n \ \vert \ \swt(\mathbf{r},\Gamma \mathbf{r})=k \},
\end{align}

where the symplectic weight
of a pair of  vectors $\mathbf{r},\mathbf{s}\in~\FF_2^n$ is defined as

\begin{align}
\swt(\mathbf{r},\mathbf{s}) = \# \{i\in\{1,\ldots, n\}\ \vert \ r_i=1 \text{ or } s_i=1\}.
\end{align}

<ref> gives rise to a brute force algorithm, which we have implemented in  to compute the SLDs of those graph states, which correspond to the connectedness components of the graph built by the user.
These partial SLDs are then recursively combined using the formula

\begin{align} \label{eq:SLD_combination}
A_k[\rho\otimes \rho'] = \sum_{j=0}^k A_j[\rho]A_{k-j}[\rho'],
\end{align}

which holds for arbitrary multi-qubit states $\rho$ and $\rho'$ [20].
The SLD-computation runtime is therefore limited by the size of the largest connected subgraph. It doubles with every new vertex in this subgraph; taking hours for a connected graph with $n\approx 30$ verices.
For this reason, the SLD histogram  is automatically updated only for connected graphs with $n\le 16 $ vertices (and an equivalent for disconnected graphs).
If the SLD is not provided automatically, the user can still enforce its calculation by clicking the button Compute Sector Length Distribution, appearing in the SLD widget, if needed.
Whenever the SLD of a new graph is calculated, it is cached; the next time the same graph is created, the SLD is immediately displayed without the need to wait.
As the cache is stored on a remote server, the user can also access  SLDs that have been previously computed by other users.

To facilitate reading, the SLD histogram bars representing $A_k$ are displayed in blue and red color for even and odd $k$, respectively.
For every stabilizer state, the SLD falls into one of two categories [21]:
If every vertex of the graph has an odd number of neighbors, it holds $A_k = 0$ for all odd $k$ (type II).
Otherwise, it holds $\sum_{k\text{ odd}} A_k = \sum_{k\text{ even}} A_k$ (type I).
In , an SLD of type II is quickly recognized by viewing only blue bars in component , while for an SLD of type I the blue and red bars cover the same area (see <ref>).

By clicking the button Parity Coloring , every vertex with an even and odd number of neighbors is colored red and blue, respectively .
Thus, the SLD is of type II if and only if all vertices are colored in blue.
This can only happen if $n$ is even.
The parity coloring feature facilitates the identification of a near-minimal number of graph modifications, which can turn an SLD of type I into type II and vice versa.

Real quantum hardware is always subject to noise.
A simple noise model is given by the depolarization channel

\begin{align}
\mathcal{E}_p: \rho \longmapsto p\frac{ \mathbbm1}{2} + (1-p) \rho,
\end{align}

which replaces a single-qubit state $\rho$ by a maximally mixed state $\mathbbm 1/2$ with probability $p$ [1, 2].
The slider in widget  allows the user to adapt $p$ and to visualize how $n$ depolarizing channels (one for each qubit) cause the SLD to decay (see panel (c) of <ref>).

The visualization is based on the formula

\begin{align}
A_k\left[\mathcal{E}_{p}^{\otimes n}[\rho]\right] = (1-p)^{2k} A_k[\rho],
\end{align}

which is derived in Ref. [12].
In the absence of noise, every graph state $\ket{G}$ is entangled, except if $G$ is the edgeless graph.
With gradual increase of noise levels, quantum entanglement is eventually lost.
This phenomenon is known as sudden death of entanglement [22].
While the exact threshold value $p_\mathrm{thr}$ at which this transition occurs is difficult to determine, entanglement criteria can provide lower bounds on $p_\mathrm{thr}$.
For example, every $n$-qubit state $\rho$ with $A_n[\rho]>1$ is entangled [23] (but there are also highly entangled states $\rho$ with $A_n[\rho]=0$ [24]).
In the noise robustness widget , we refer to this sufficient (but not necessary) entanglement criterion as  criterion.
Similarly, the value of  is a lower bound on $p_\mathrm{thr}$ based on the majorization criterion [25], which implies that every $n$-qubit state $\rho$ with
$\sum_k(2k-n)A_k[\rho]  > 0$ is entangled [12].
The third and final lower bound on $p_\mathrm{thr}$, which is displayed in the noise robustness widget , is based on an entanglement distillation protocol [26].
Its value is given by

\begin{align} \label{eq:loc_thresh_dist}
p_\texttt{Distill.}  = 1- {2^{-2/ \left(2+ \max\limits_{\{i,j\}\in E} \deg(i)+\deg(j)\right) }},
\end{align}

where $\deg(i)$ denotes the degree of vertex $i\in V$.
In  it is possible to highlight a pair of neighboring vertices $i,j\in V$, which maximizes the expression $\deg(i)+\deg(j)$, by clicking the Distillation button  in the graph visualization widget .
This option is activated in <ref> , coloring vertex 0.8$\vertOne$ and 0.8$\vertFour$ in yellow.

§ USE-CASE PROPOSAL: HARDWARE-EFFICIENT MEASUREMENTS OF COMMUTING PAULI OPERATORS

The variational quantum eigensolver (VQE) algorithm is a new quantum algorithm that is hoped will outperform the best classical simulation algorithms in the near future by exploiting noisy intermediate-scale quantum computers [27, 28].
The goal of VQE is to minimize the expectation value

\begin{align}
\langle H \rangle _{\boldsymbol{\theta}} =
\bra{\Psi(\boldsymbol{\theta})}  H \psiTheta
\end{align}
of a problem Hamiltonian $H$ by optimizing parameters $\boldsymbol{\theta}$ in the preparation circuit of a trial quantum state $\ket{\Psi(\boldsymbol \theta)}$.
To estimate this expectation value, the Hamiltonian is expanded into a linear combination of Pauli operators

\begin{align}
H = \sum_{i=1}^m \lambda_i P_i,
\end{align}

where $\lambda_i\in \RR$ and $P_i\in\{\mathbbm1, X,Y,Z\}^{\otimes n}$ for all $i\in\{1,\ldots,m\}$.
This allows us to obtain  $\langle H \rangle _{\boldsymbol{\theta}}$  as a linear combination of individually measured expectation values $\langle P_i\rangle _{\boldsymbol \theta}$.
Using simultaneous measurements of commuting Pauli operators, it is possible to reduce the total runtime of the VQE algorithm [29, 30, 31].
The partitioning of the set  $\mathcal{M}=\{P_1,\ldots, P_m\}$ into commuting subsets, however, is not unique and comes with a trade-off between the number and depth of  measurement circuits (see Table I of Ref. [32] for an overview of available partitioning methods).

From the stabilizer formalism, it is well known that $2^n$ is the maximum number of commuting $n$-qubit Pauli operators $S$  (up to a global phase).
After possibly replacing some of these operators by $-S$, the resulting set $\mathcal{S}$ is the stabilizer group of a stabilizer state $\ket{\psi}$.
Since $\{Z^\mathbf{r} \ \vert \ \mathbf{r}\in \FF_2^n\}$ is the stabilizer group of $\ket{0}^{\otimes n}$, we can write $\mathcal{S}=\{UZ^\mathbf{r} U^\dagger \}$, where $U$ is a Clifford circuit fulfilling $\ket{\psi}= U \ket{0}^{\otimes n}$.
For all operators in $\mathcal{M}$, which are of the form $P_i=\pm UZ^\mathbf{r}U^\dagger$, we can simultaneously measure the desired expectation values

\begin{align}
\langle  P_i \rangle_{\boldsymbol{\theta}}
=  \pm  \bra{ \Psi(\boldsymbol{\theta})}  U
U^\dagger \psiTheta
\end{align}

by preparing the VQE trial state $\psiTheta$ on a quantum computer, followed by the application of $U^\dagger$ (we refer to this step as uncomputing the stabilizer state $\ket{\psi}$), followed by a qubit-wise readout in the computational basis.
Note that the latter is nothing else but the simultaneous measurement of  $Z^\mathbf{r}$ for all $\mathbf{r}\in \FF_2^n$.

Since state-of-the-art quantum computers are still very noisy, it can be beneficial to partition  $\mathcal{M}$ into (subsets of) stabilizer groups $\mathcal{S}$, for which the corresponding measurement circuits $U^\dagger$ are tailored to the hardware.
The least noisy measurement circuits consist of a single layer of single-qubit gates.
They measure the stabilizer group of a product state, which is also known as a tensor product basis (TPB) in this context [29].
The Pauli-weight distribution of the operators in a TPB is given by the SLD of a fully separable state, $A_k=\binom{n}{k}$ (see panel (a) of Fig. <ref>).
In the full Pauli group, however, there are $4\binom{n}{k}3^{k}$  operators having weight $k$, recall <ref>.
Therefore, the probability of drawing a weight-$k$ operator from the uniform distribution over a TPB and over the full Pauli group, respectively, is given by $p_\mathrm{TPB} = \binom{n}{k}2^{-n}$ and $p_\mathrm{PG}=\binom{n}{k}3^k4^{-n}$.
Although $\mathcal{M}$ does not arise as the sample set of a uniform distribution over the Pauli group (but from encoding schemes such as Jordan–Wigner [28]),
a similar mismatch occurs for typical VQE instances [29]:
If the operators $P_i\in \mathcal{M}$ are assigned to TPBs, those with a low Pauli weight run out first. The remaining (large-weight) operators are not likely to fit into joint TPBs.
Thus, they require are a large number of additional measurement circuits to complete the partitioning of $\mathcal{M}$ into TPBs.
One step in the direction of overcoming this problem (without sacrificing hardware efficiency) is to complement the layer of single-qubit gates in the measurement circuit of a TPB with a single layer of two-qubit gates [30].
Up to LU-equivalence, the TPB-only approach corresponds to the measurement of the edgeless graph's stabilizer group, while the second approach also allows for graphs where every vertex is either isolated or a leaf.

The connectivity graph $\Gcon$ of a quantum computer is the graph whose vertices correspond to qubits, while its edges display to which pairs of qubits physical two-qubit gates can be implemented.
For ion trap quantum computers, $\Gcon$ typically is the complete graph [33].
For super- or semiconductor based quantum computers, however,
the vertices of $\Gcon$ have a limited degree, causing the number of edges to scale linearly in the number of qubits.
In this case, the preparation and, more importantly (for our purposes), the uncomputation of the graph state, which corresponds to $\Gcon$ or one of its subgraphs, is  a hardware efficient quantum circuit.
As we explained above, uncomputing a graph state facilitates the measurement of its stabilizer group.
By adding single qubit Clifford gates at the beginning of the measurement circuit, one can access a multitude of stabilizer groups without altering the Pauli-weight distribution.
This approach significantly enlarges the number of efficiently-measurable stabilizer groups and includes the  approaches of Refs. [29] and [30] as a special case.
Whether relevant instances of the VQE algorithm can benefit from our idea, however, requires further investigation.

We argue that  can support such an investigation.
As discussed in <ref>, the user can construct the connectivity graph $\Gcon$ and interactively explore its subgraphs by deleting some of its edges.
Simultaneously, the SLD histogram will display the Pauli-weight distribution of the operators, which can be measured simultaneously by uncomputing the corresponding graph state.
In <ref>, we depict the connectivity graph $\Gcon$ of [IBM, the IBM logo, and IBM Q are trademarks or registered trademarks of IBM Corp., in the U.S. and/or other countries.] (a quantum computer with 27 superconducting qubits [34]), one of its subgraphs, and their SLDs.
As it is typically observed for graph states with $I$ isolated vertices, their $k$-body SLs  are approximately given by $A_k \approx 2^n\binom{n}{k}p^k(1-p)^{n-k}$, where $p=(3n-I)/4n$ [12].
Depending on the Pauli-weight distribution of the operators in $\mathcal{M}$ (which are not yet assigned to a measurement circuit), one should explore graph states with an appropriate value of $I$.
In the special case where $\mathcal{M}$ is the sample set of the unit distribution over the Pauli group, one should start with $I=0$.

Greenberger-Horne-Zeilinger (GHZ) states

$\GHZn = \frac{1}{\sqrt{2}} (\ket{0}^{\otimes n}+\ket{1}^{\otimes n})$

lie in the same LU-class as star- and complete-graph states [8].
Since the star graph does not match the connectivity of super- or semiconducting quantum computers, GHZ states are most efficiently prepared (and uncomputed) recursively:
Since $\ket{\mathrm{GHZ}^{n-1}}$ is symmetrical under qubit exchange, any qubit can serve as control for a CNOT gate targeting the $n$-th qubit initialized in state $\ket{0}$, i.e.,

\begin{align}
\mathrm{CNOT}_{i,n} \ket{\mathrm{GHZ}^{n-1}}\otimes \ket{0}= \GHZn
\end{align}

for all $i\in\{0,\ldots,n-1\}$.
In this way, the limited connectivity of a quantum chip is leveraged.
From the theory of SLDs it is known that there is no quantum state having more weight-$n$ stabilizers than $\GHZn$ and its LU-equivalents [23, 35].
Their $k$-body SLs are given by

\begin{align} \label{eq:SLD_GHZ}
A_k = \binom{n}{k}\delta_{k,\mathrm{even}} + 2^{n-1}\delta_{k,n},
\end{align}

where $\delta_{k,n}$ is the Kronecker symbol and similarly for $\delta_{k,\mathrm{even}}$ [10, 35].
Thus, these states are of high importance if only weight-$n$ operators from $\mathcal{M}$ remain to be assigned to measurement circuits.
Similarly, if mostly large-weight operators remain, one can investigate tensor products of $\ket{\mathrm{GHZ}^{n_i}}$ for different choices of $n_i$ (and their LU-equivalents).
By inserting <ref> into <ref>, we find that the $k$-body SL of $\ket{\mathrm{GHZ}^{m}}\otimes \ket{0}^{\otimes n- m}$ is given by

\begin{align} \label{eq:SLD_GHZ_and_product}
A_k=2^{m-1}\binom{n-m}{k-m}+ \sum_{j=0}^{\lfloor k/2\rfloor} \binom{m}{2j} \binom{n-m}{k-2j},
\end{align}

where we define $\binom{n}{k}=0$ if $k<0$.
Besides a broad peak at $k=n/2$, these SLDs have a second, sharp peak at $k=n-m/2$ (see <ref>).
Therefore, these states can serve as starting points to design hardware efficient measurement circuits for large-weight operators, which remain after all the low-weight operators in $\mathcal{M}$ have been partitioned into e.g. TPBs.

These examples illustrate how  can support the design of new VQE measurement strategies.
When pursuing our use case proposal, researchers can flexibly explore the space of available SLDs using our prototype.
We anticipate that this will help them to identify states whose stabilizer groups match the Pauli operators to be measured.

§ CONCLUSION AND OUTLOOK

In this paper, we presented , an open-source interactive application for the visual analysis of  graph states, which is ready to use [14].

We illustrated how our prototype could be applied when designing new hardware-efficient measurements for variational quantum algorithms.
In addition, we believe that our application can support the identification of local-unitary invariants of graph states and their meaning in terms of graph invariants.

We consider our approach to be only a first step in applying visual analytics in the field of quantum information theory.
Therefore, we invite intrigued researchers to tailor and extend the features of  as desired.
For example, it could be modified to support the design of new $k$-uniform states by visually highlighting subgraphs that pose an obstruction to $k$-uniformity [36].

Moreover, the sector-length-decay visualization could be extended to a larger class of error channels by adding noise probability sliders to analyze their combined effects.

Directions for major extensions of our tool include the visualization of hypergraph states or qudit graph states [37, 38].

Finally, visual analytics could also be applied to investigate how the sector length distribution of a quantum state changes during a noisy quantum computation.
Insights about such dynamical effects are essential for the design of error mitigation techniques.
A good starting point is the investigation of Clifford circuits with Pauli noise since theoretical tools are available in this case [39].

§ ACKNOWLEDGMENTS

The authors are thankful for stimulating discussions with Ivano Tavernelli (IBM Research Europe - Zurich) and Nikolai Wyderka.
Felix Huber suggested the term Pusteblume graph.

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# Superconducting RF cavity R&D for future accelerators

C. M. Ginsburg  Fermilab, Batavia, IL 60510 USA

###### Abstract

High-beta superconducting radiofrequency (SRF) elliptical cavities are being
developed for several accelerator projects including Project X, the European
XFEL, and the International Linear Collider (ILC). Fermilab has recently
established an extensive infrastructure for SRF cavity R&D for future
accelerators, including cavity surface processing and testing and cavity
assembly into cryomodules. Some highlights of the global effort in SRF R&D
toward improving cavity performance, and Fermilab SRF cavity R&D in the
context of global projects are reviewed.

##  I Introduction

Substantial effort is being expended in the quest for high gradients in
superconducting radiofrequency (SRF) cavities for ongoing and proposed
projects: (1) the test/user facilities STF (KEK), NML (Fermilab), and FLASH
(DESY), (2) the European XFEL currently under construction, (3) Project X at
Fermilab, and (4) the International Linear Collider (ILC). The common
requirements or choices for the high-gradient cavities in these projects are
gradients of at least 23 MV/m, $\beta$=1, elliptical shape and an accelerating
mode frequency of 1.3 GHz. Recent SRF R&D highlights in the pursuit of high
cavity gradients and the status of Fermilab SRF infrastructure development are
reviewed.

##  II Achieving High Gradient

Achieving high gradient in SRF cavities for current projects requires pure
niobium, an excellent surface quality, and a good geometry.

For the ILC ilc_rdr , the niobium sheets from which cavities are formed must
have a residual resistivity ratio (RRR) of at least 300, consistent with very
high purity niobium, to ensure good superconducting properties.

Because RF fields occupy the first $\sim$40 nm of the inner cavity surface for
these cavities, as shown in Fig. 1, the quality of the innermost surface is
critical and must be carefully controlled both during cavity fabrication and
test preparation. To avoid the inclusion of contaminants or defects apparent
after the sheet manufacturing, eddy current scanning (ECS) of all sheets is
performed before cavity fabrication. ECS has proven to be a very useful
feedback mechanism for material vendors, constributing substantially to an
overall material quality improvement in recent years. After fabrication, the
cavity inner surface must be very smooth, with no inclusion of foreign
particles, or topological defects such as bumps or pits or sharp niobium grain
boundaries, which could increase surface resistance locally. In addition to
the strict quality assurance for the material, the introduction of dust or
other microscopic contaminants must be avoided after the final surface
preparation, to prevent field emission which can become dark current during
machine operation and cause undesirable heating in superconducting elements
and damage to accelerator components.

Figure 1: Simplified drawing of an SRF cavity surface. [Courtesy of Hui Tian.]

In selecting a cavity geometry for optimum performance, cavity cell shapes are
typically optimized for low peak surface magnetic field (Hpeak) and low peak
surface electric field (Epeak) relative to the gradient (Eacc), as well as
ease of surface processing and fabrication.

Many cavities have reached 35 MV/m or more in the last decade SRF2007_Saito ;
MAC2007_Lilje , particularly single-cell cavities of varying elliptical shapes
and 9-cell Tesla-shape cavities, as shown in Fig. 2.

Figure 2: Maximum cavity gradient as a function of time for 1-cell cavities of
non-standard shapes (top, courtesy of Kenji Saito) and of 9-cell cavities of
Tesla shape (bottom, courtesy of Lutz Lilje). All tests were performed at 2K.
(Data are from KEK, Cornell, JLab, and DESY.)

##  III Surface Processing

The surface treatment intended to maximize cavity performance, developed for
the ILC TTC_surfaceprep , includes initial preparation steps to remove
$\sim$$150\ \mu$m from the inner surface using electropolishing (EP). This
initial removal step may be performed with centrifugal barrel polishing (CBP)
or buffered chemical polishing (BCP) at some labs; however, the maximum
gradients reached with BCP as the initial preparation step are lower than
those achieved using the other methods. Cavities then undergo an 800 C
annealing step, to drive hydrogen from the surface. The final preparation
steps include degreasing with detergent, another light electropolishing
($\sim$$20\ \mu$m), a high pressure rinse (HPR) with ultrapure water, drying
in a class-10 cleanroom, and then evacuation and low-temperature baking (120
C) for about 48 hours after the final assembly with couplers. Additional
surface treatments to address field emission, which occur after the final EP,
will be described later. The primary methods for material removal during
surface preparation are CBP, EP and BCP.

CBP is a standard technique developed for cavities at KEK in which abrasive
small stones are placed into a cavity with water to form a slurry and the
cavity is rotated. The KEK CBP machine and a diagram showing the CBP process
are shown in Fig. 3. As a centrifugal process, material is preferentially
removed from the equator region. Since standard cavities have an equator weld,
CBP is very effective in smoothing the weld.

Figure 3: The KEK CBP machine (top) and a diagram showing the CBP process
(bottom). [Courtesy of Kenji Saito.]

EP is an electrolytic current-supported material removal, and has been
developed for use on cavities by KEK in collaboration with industrial
partners, and adopted at most labs. The EP process is complementary to CBP
because the material removal is preferentially on the iris. The ANL EP machine
and a diagram showing the EP process are shown in Fig. 4. In this case, the
niobium cavity functions as an anode, and an aluminum cathode is inserted on
the cavity axis. The electrolyte is HF(40%):H2SO4 in a ratio of 1:9 by volume.
Some sulfur remains on the surface after EP and will cause field emission
unless it is removed.

Figure 4: The ANL EP machine (top, courtesy of Michael Kelly), and a diagram
showing the EP process (bottom, courtesy of Lutz Lilje).

BCP involves filling a cavity with a combination of
HF(40%):HNO3(65%):H3PO4(85%) in a 1:1:2 ratio by volume. BCP has been shown to
cause hydrogen contamination at the surface; this problem can be mitigated by
using the appropriate proportion of acid and buffer and by keeping the
temperature below about 15 C. BCP is rather less expensive than EP and is
often sufficient to produce cavities attaining gradients as high as about 25
MV/m. However, it tends to enhance grain boundaries which may degrade the
performance of standard fine-grain cavities.

An improvement to the average maximum gradient was seen after the previous BCP
standard was replaced with EP, as shown in Fig. 2. The comparison of best test
results of DESY cavities treated with BCP and EP is shown in Fig. 5, revealing
that the maximum gradient achieved with BCP is $\sim$25-30 MV/m, whereas the
maximum gradient achieved with EP is rather higher at $\sim$35-40 MV/m. A
comparison of the resulting surface smoothness is shown in Fig. 6. Although it
is possible now to achieve gradients in 9-cell cavities higher than 35 MV/m,
there are several factors which limit cavity performance.

Figure 5: Comparison of cavities processed with BCP (top) and EP (bottom). The
EP’d cavities reach noticeably higher gradients. [Courtesy of DESY.] Figure 6:
Comparison of a sample surface after BCP (left) or EP (right). The EP’d
surface is noticeably smoother than the BCP’d surface. [Courtesy of DESY.]

##  IV Reducing Field Emission

One of the factors which limits cavity performance is the presence of field
emission. Four recent field emission studies have been performed: flash EP
(also known as fresh EP), dry ice cleaning, degreasing, and ethanol rinsing.
In tests at KEK using six single-cell Ichiro-shape cavities, a 3 $\mu$m EP
using fresh acid after the final EP was studied flashEP . A gradient
improvement to both average and rms was observed. Furthermore, the treatment
was found to increase the gradient at which field emission turns on. Dry ice
cleaning is a less developed but promising alternative being developed at DESY
dry_ice . It reduces or eliminates contaminating particles in several ways:
they become brittle upon rapid cooling, they encounter pressure and shearing
forces as CO2 crystals hit the surface, and they are rinsed due to the 500
times increased volume after sublimation. Also, LCO2 is a good solvent and
detergent for hydrocarbons and silicones, etc. Dry ice cleaning is a dry
process which leaves no residues, because the loosened contaminants are blown
out the ends of the cavities by the positive pressure, and can be performed in
a horizontal orientation. The DESY dry ice nozzle is shown in Fig. 7. Dry ice
cavity cleaning might be possible after coupler installation, a procedure
which risks introducing field emission. Improved field emission
characteristics have been seen in single-cell cavity tests, and an extension
of the system to 9-cell cavities is planned.

Figure 7: The DESY dry ice nozzle. [Courtesy of DESY.]

One of several degreasing R&D studies is that of a KEK 9-cell Ichiro-shape
cavity which was processed and tested at KEK and JLab geng_detergent . In an
initial test at JLab, the cavity produced substantial field emission which was
observed at gradients above 15 MV/m. After the cavity was ultrasonically
cleaned with a 2% Micro-90 solution and standard HPR, the field emission was
substantially reduced or even eliminated.

An extensive study of ethanol rinsing after the final EP was carried out at
DESY DESY_ethanol . Out of 33 cavities used in the study, 20 were treated
using the standard procedure and 13 with an additional ethanol rinse after the
final EP. The number of tests in which field emission was observed was
substantially reduced. In addition, the maximum gradient was somewhat
improved, from $27\pm 4$ MV/m without ethanol to $31\pm 5$ MV/m with ethanol
rinse. Since field emission was found to be reduced so substantially, the
ethanol rinse is now part of the standard DESY cavity treatment.

##  V Manufacturing and Shape Studies

The vast majority of cavities which have been processed and tested in the last
decade are Tesla-shape fine-grain cavities TESLA_cavities . Some fundamental
changes to the standard Tesla-shape fine-grain cavities which have been under
investigation include changes to the fabrication techniques, material
composition and cavity shape.

Because electron-beam welding is a substantial portion of the cavity
fabrication cost, and may be a source of surface defects which cause premature
quenches (described later), reducing the number of equator welds required in
cavity fabrication would be advantageous. Recently, a 9-cell cavity was
fabricated at DESY using a hydroforming technique Singer_hydroforming . The
cavity was built from three 3-cell hydroformed units, so it had only two iris
welds and two beampipe welds. The surface was treated with the standard
procedure including ethanol rinse. The cavity performance was comparable to
cavities built using standard techniques.

Material is wasted, and a lot of time is required, to roll sheets and stamp
out the disks from which fine-grain cavities are made. In principle, it should
be possible to save manufacturing costs by slicing an ingot directly into
large-grain sheets. It may also be possible to achieve high gradients using
BCP only, since the performance of large-grain cavities should be less
sensitive to the grain boundary enhancement seen in fine-grain cavities, as
long as the grain boundaries are strategically located in low surface-field
regions. Recent experience at DESY with large-grain cavities shows their
performance is comparable to fine-grain cavities. It is still unclear whether
BCP will be sufficient or whether EP will be necessary. Many 1-cell tests have
shown high gradients, in a range comparable to fine-grain cavities. Recent
tests on three 9-cell large-grain cavities Singer_LG  showed gradients
comparable to fine-grain cavities as well. Effective large-grain ingot cutting
methods are being pursued in industry. Single-crystal niobium cavities have
been difficult to produce, because it is difficult to produce large diameter
single-grain ingots. Six single-cell single-grain cavities of varying shape
and fabrication technique were fabricated, processed and tested recently by a
JLab/DESY collaboration Kneisel_singlecrystal , with performance found to be
comparable to that of fine-grain cavities. Unless substantial performance
improvement is seen, the difficulty of producing single-grain cavities may not
justify the effort required. In both the large-grain and single-grain cases,
further study of crystal orientation effects is needed. For a summary of
recent large-grain and single-grain cavity work, see Ref. Kneisel_LG .

It may be possible to increase the RF breakdown magnetic field of
superconducting cavities by creating a multilayer coating of alternating
insulating layers and thin superconducting layers Gurevich . By using
multilayers thinner than the RF penetration depth, the critical magnetic field
can be increased from niobium Hc1 to one similar to high-Tc superconductors,
thereby significantly improving the maximum gradient achievable. Recently, a
cavity was prepared with such a composite surface and tested Proslier . A 10
nm layer of Al2O3 was chemically bonded to the niobium surface of an existing
single-cell cavity using atomic layer deposition. The surface was then covered
by a 3 nm layer of Nb2O5. The cavity showed promising early results, and
further study is underway.

Cell accelerating length and equator diameter are fixed by $\beta$ and
frequency respectively. However, the details of the shape may be optimized for
low field emission (low Epeak/Eacc) and reduced sensitivity to the fundamental
maximum surface magnetic field (low Hpeak/Eacc); see, e.g., Ref. Jacek_shapes
. Excellent results on single-cell elliptical cavities have recently been
obtained. A comparison of three shapes of elliptical single-cell cavities is
shown in Fig. 8. A re-entrant shape cavity built and tested by Cornell
University recently reached 59 MV/m Cornell_record , as shown in Fig. 9,
setting a world record for the type of cavities described in this paper.
Excellent results have also been achieved with an Ichiro shape cavity at KEK,
with a record of 53.5 MV/m SRF2007_Saito . Furthermore, $46.7\pm 1.9$ MV/m was
reached on six single-cell cavities with optimized surface treatment
parameters flashEP . Another low-loss shape single-cell cavity which was
processed and tested by a DESY/KEK collaboration reached 47.3 MV/m
Furuta_EPAC06 . It is rather more difficult to manufacture an excellent multi-
cell cavity than an excellent single-cell cavity, and the very high gradients
seen in single-cell alternative-shape cavities have not been achieved in
9-cell versions yet. One 9-cell Ichiro-shape cavity, without endgroups, which
was processed and tested by a KEK/JLab collaboration reached up to 36 MV/m
Geng_LINAC08 .

Figure 8: Single-cell 1.3 GHz elliptical cavities: Cornell re-entrant (left),
Tesla (center) and Ichiro (right). Figure 9: The record SRF 1-cell cavity
gradient. [Courtesy of Cornell University.]

##  VI Understanding Cavity Behavior

Quenches and field emission appear as hot spots on the outer cavity surface.
Temperature mapping (T-map) systems have been used at nearly all of the labs
active in this field for many years to study these phenomena Padamsee . The
existing T-map systems primarily use Allen-Bradley carbon resistors as thermal
sensors, as developed at Cornell Cornell_thermometry . T-map systems commonly
in use for many years at DESY include a fixed type for single-cell cavities
with 768 sensors DESY_fixed_thermometry , and a rotating system for 9-cell
cavities with 128 sensors DESY_rotating_thermometry . A fixed thermometry
system for 9-cell cavities with four sensors around each equator and a few
sensors on the endgroups has been developed at KEK and was used for tests of
STF Tesla-like cavities KEK_diagnostics . Second sound detection has been used
at ANL for quench location on split-ring resonators Shepard_secondsound . A
selection of new hot spot detection systems are described; these systems vary
in coverage, flexibility, and in the number of cavity tests required to
extract useful T-map information.

A system of Cernox temperature sensors has been developed at Fermilab
fast_thermometry  and used for quench location. Up to 32 sensors may be
attached as needed to suspect locations; therefore the system is very flexible
but also time consuming to install, typically requiring several cavity test
cycles to conclusively locate quenches. It is highly portable and suitable for
any cavity shape. An example installation of this system on a 9-cell cavity is
shown in Fig. 10, in which a hard quench at 16 MV/m was observed, accompanied
by a temperature rise of about 100 mK, above the 2K operation, over about 2
sec in sensors #3 and #4 before the quench. The quench was seen on all
sensors.

Figure 10: Fermilab Cernox thermometry system installed on a 9-cell cavity in
the region of a pre-determined hot spot.

New multi-cell T-map systems have also been developed. A new 2-cell system at
JLab was recently commissioned JLab_2-cell_thermometry , with 160 Allen-
Bradley sensors installed around each of the two equators. Understanding
cavity behavior requires two cooldowns: one to measure the TM010 passband
modes and determine within two the limiting cells, then a second after the
T-map installation to measure the temperature distribution of the limiting
cell. At LANL, a new 9-cell T-map system has been developed
LANL_9-cell_thermometry  which employs 4608 Allen-Bradley sensors and a
multiplexing scheme to map an entire 9-cell cavity in a single test yet limit
the number of cables leaving the cryostat. The preliminary results are very
promising. A 9-cell T-map system at Fermilab using 8640 diodes as a
multiplexed thermometer array is under development diode_thermometry . All of
these multi-cell T-maps are specifically designed for the Tesla cavity shape.

Cornell has developed a new quench location system using second sound sensors
Cornell_secondsound . Second sound is a thermal wave which can propagate only
in superfluid helium. It is generated when a heat pulse is transmitted from a
heat source, such as a quench, through superfluid helium. In this system,
eight sensors detect the arrival of the temperature oscillation, and the
location is determined from the relative timing of the arrival of the
oscillation from different sensors. This simple system is suitable for any
cavity shape or number of cells and can locate quenches in a single cavity
test. The detector is shown in Fig. 11, and an example of the data used to
triangulate the location of the second sound source is shown in Fig. 12.

Figure 11: The Cornell second sound sensor (left, courtesy of Cornell
University) and a diagram showing the function of the sensor (right, courtesy
of Genfa Wu). Figure 12: Cornell data showing the arrival of second sound
waves at three different sensors. [Courtesy of Zachary Conway.]

A resurgence of interest in optical inspection occurred recently when several
cavities with hard quench limitation in the 15-20 MV/m range, were observed to
have surface defects correlated with hot spots, using a new optical inspection
system developed by Kyoto University/KEK with a clever lighting technique and
excellent resolution. This optical system Kyoto_camera  consists of an
integrated camera, mirror, and lighting system on a fixed rod; the cavity is
moved longitudinally and axially with respect to the camera system. The
lighting is provided by a series of electroluminescent strips, which provide
lighting with low glare on the highly polished surface. These strips can be
turned off and on, and the resulting shadows studied for an effective 3D image
of the depth or height of the defect. The camera resolution is about 7 $\mu$m.
A photograph of the Kyoto/KEK camera system is shown in Fig. 13. An optical
inspection system employed regularly at JLab Geng_LINAC08  includes a Questar
long-distance microscope and mirror to inspect any cavity inner surface. This
system uses electroluminescent lighting. Additional optical inspection systems
are in use or under development at other labs.

Figure 13: The KEK/Kyoto University cavity inspection camera system. [Courtesy
of Yoshihisa Iwashita.]

Because many of the cavity-limiting surface defects found by thermometry and
optical inspection are located in the heat-affected zone of the electron-beam
welds, weld properties are under intense scrutiny. Understanding which weld
parameters might contribute to creating such defects, and ultimately
eliminating the defect occurrence, is critical to improving the yield of high-
gradient cavities. Note that cavities with such defects are largely, but not
exclusively, from new cavity manufacturers. One group has attempted to
reproduce these features on samples Cooley , to permit their systematic study.
In time, study of cavity surface quality may improve cavity quality in a
manner similar to the improvement seen in niobium sheet quality resulting from
regular eddy current scanning.

##  VII Fermilab Infrastructure for SRF Development

At Fermilab, the SRF infrastructure has been substantially built up in recent
years. The key program goals are the achievement of high gradients with high
yields for future accelerators, the testing of cavities, and the ramp up of
the production rates needed to construct cryomodules for Project X and ILC
R&D. The infrastructure which has been built to achieve these goals includes
facilities for single-cell cavity R&D, an ANL/FNAL joint cavity processing
facility, a vertical cavity test facility, a horizontal test stand, a
cryomodule assembly facility, and a cryomodule test facility.

The ANL/FNAL cavity processing facility provides surface processing and
assembly of cavities for vertical test, for both 1-cell and 9-cell elliptical
cavities. Recent photographs of the primary facility components: EP,
ultrasonic degreasing, high-pressure ultrapure water rinsing, and assembly
support and vacuum leak testing, are shown in Fig. 14. All of these components
are now in routine operation.

Figure 14: ANL/FNAL join facility infrastructure: (clockwise from top left)
EP, ultrasonic degreasing, HPR and assembly stands. [Courtesy of Michael Kelly
and Damon Bice.]

At the FNAL vertical cavity test facility (VCTF), more than 40 cavity tests
have been performed in FY08/FY09, where a test is defined as a cryogenic
cycle. The test cavities have been primarily 9-cell and single-cell 1.3 GHz
elliptical cavities, with two 325 MHz single-spoke resonators included. The
tests have primarily been for instrumentation development, e.g., variable
input coupler, thermometry, cavity vacuum pumping system, and for cavity
diagnostic tests for vendor development. Upgrades are planned to increase the
cavity test throughput to more than 200 cavity tests per year by October 2011,
to support Project X and ILC R&D. Some of the cavities recently tested at the
VCTF are shown in Fig. 15. A summary of the vertical test results of 9-cell
cavities is shown in Fig. 16.

Figure 15: Some of the cavities recently tested at the Fermilab VCTF:
(clockwise from top left) SSR1, 9-cell Tesla-shape, and 1-cell Tesla-shape
(with diode thermometry) cavities. [Courtesy of Bill Mumper.] Figure 16: A
summary of maximum gradients from 9-cell vertical tests in the ILC Americas
region. The cavities tested at the VCTF are shown by the black histogram.

The horizontal test stand (HTS), which is used to test 9-cell cavities which
have been welded into a helium tank, i.e., dressed, using pulsed high power,
is now operational. It has been commissioned for both 1.3 and 3.9 GHz
cavities. Four 3.9 GHz cavities were tested in 2008; they have been installed
in a cryomodule which is currently at DESY. Upgrades are planned to expand the
capacity with the addition of a second cryostat by 2012. A photograph of the
HTS is shown in Fig. 17.

Figure 17: A photograph of the HTS, showing a 9-cell 1.3 GHz cavity being
installed. [Courtesy of Fermilab Visual Media Services.]

Two cryomodule assembly facilities (CAF’s) have been built at Fermilab, CAF-
MP9 and CAF-ICB. The CAF-MP9 facility receives dressed cavities and their
peripheral parts, and assembles these into a cavity string in a class-10
cleanroom. Then, the string assembly is installed onto the cold mass and
transported to CAF-ICB. At CAF-ICB, the cavity string is aligned and assembled
into the vacuum vessel. After the cryomodule is complete, it is shipped to the
cryomodule test facility (NML) for testing. One cryomodule, CM1, built with
cavities from DESY, has been assembled in the CAF facilities. Recent
photographs of the CAF-MP9 and CAF-ICB, showing steps in the assembly of CM1,
are shown in Figs. 18 and 19, respectively.

Figure 18: Recent photographs of the CAF-MP9 facility, showing stages in the
assembly of the first cryomodule, CM1. [Courtesy of Tug Arkan.]

Figure 19: Recent photographs of the CAF-ICB facility, showing stages in the
assembly of the first cryomodule, CM1. [Courtesy of Tug Arkan.]

The cryomodule test facility, NML, will test one RF unit, i.e., 3 cryomodules,
using a 10 MW RF system and an electron beam with ILC parameters. Various
Project-X parameters will also be tested with beam in this facility. NML will
provide the capability to conduct advanced accelerator R&D for future
accelerator components. Through the end of FY09, the facility is being
prepared to test the first Fermilab-assembled cryomodule, CM1, without beam,
requiring completion of the RF system and the cryogenic system. A recent
photograph of the NML facility is shown in Fig. 20.

Figure 20: A recent photograph of the NML facility, indicating where the first
cryomodule, CM1, has been installed. [Courtesy of Jerry Leibfritz.]

##  VIII Conclusions and Outlook

Highlights of the rich R&D activity in the quest for high gradient and reduced
cost have been described. Very high gradients have been measured in niobium
SRF cavities: more than 50 MV/m in single-cell cavities of various shapes, and
more than 35 MV/m in several 9-cell Tesla-shape cavities. Several promising
new results from large-grain 9-cell cavities and hydroformed cavities address
the cost and reliability issues of these cavities. Several cavities which have
been limited to 15-20 MV/m by hard quench have been studied using various
techniques, and the outlook is good that useful information can be fed back to
cavity manufacturers to improve the yield of high-gradient cavities over time.
Surface treatment is crucial for optimum performance, and several promising
studies on final preparation methods have been found to sharply reduce field
emission. Fundamental SRF studies are also underway, many showing promising
results.

An overview of Fermilab infrastructure in pursuit of SRF R&D has been given.
Among the recent accomplishments, one 1.3 GHz cryomodule using DESY dressed
cavities has been built and transported to the cryomodule test facility for
testing. Most key infrastructure components are now in place, with final
commissioning underway.

###### Acknowledgements.

I am grateful to my colleagues for their contributions to the content for this
talk, as referenced. I especially thank those who graciously permitted me to
show their unpublished data. This manuscript has been authored by Fermi
Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S.
Department of Energy. The United States Government retains and the publisher,
by accepting the article for publication, acknowledges that the United States
Government retains a nonexclusive, paid-up, irrevocable, worldwide license to
publish or reproduce the published form of this manuscript, or allow others to
do so, for United States Government purposes.

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