Methodology for bus layout for topological quantum error correcting codes
 Martin Wosnitzka^{1},
 Fabio L Pedrocchi^{1}Email author and
 David P DiVincenzo^{1}
Received: 2 November 2015
Accepted: 24 February 2016
Published: 9 March 2016
Abstract
Most quantum computing architectures can be realized as twodimensional lattices of qubits that interact with each other. We take transmon qubits and transmission line resonators as promising candidates for qubits and couplers; we use them as basic building elements of a quantum code. We then propose a simple framework to determine the optimal experimental layout to realize quantum codes. We show that this engineering optimization problem can be reduced to the solution of standard binary linear programs. While solving such programs is a NPhard problem, we propose a way to find scalable optimal architectures that require solving the linear program for a restricted number of qubits and couplers. We apply our methods to two celebrated quantum codes, namely the surface code and the Fibonacci code.
Keywords
1 Introduction
Since the theoretical demonstration of faulttolerant quantum information processing, a holy grail of modern physics has been to realize faulttolerant quantum computing architectures in the lab. While this still remains a very challenging task, many experimental advances have been achieved. Arguably, one can hope to see the first smallsize implementations in a near future.
Among the most promising quantum computing platforms, one finds so called topological quantum codes [1]. The main idea is to encode quantum information (in the form of logical qubits) using a large number of physical qubits. The additional degrees of freedom introduced in the Hilbert space then allow the extraction of some information about the errors induced by the environment (the error syndrome) and to correct them without collapsing the stored logical qubit. Furthermore, topological codes are, by definition, immune to local and static perturbations [1].
Most of the topological quantum codes are realizable as a lattice of qubits (some of them might require qudits instead) that are coupled to each other. Depending on the specifics of the quantum code, one qubit might be coupled to several other qubits in its neighborood. In this work, we present a general framework to determine the optimal architecture to couple the qubits of a quantum code. Here we assume that couplers can be introduced between qubits and we identify the coupling architecture that minimizes the total length of the couplers, rendering the physical implementation more practical. Our analysis is valid for any quantum code and we show that this set of optimization problems are identical to wellknown binary linear programs. We would like to point out that our choice of metric defining what we call an optimal scheme might turn out not to be the one that experimentalists will eventually consider relevant. Experimental and theoretical works with several qubits per transmission lines have just started, [2–5] and the results obtained from experiments will be crucial to determine which optimality criteria should be satisfied in practice. In this context, our work builds a methodology where a huge number of layouts can be studied and from which an optimal layout can be determined; this methodolody applies to a variety of metrics and not only to the precise choice we have made here. We thus believe that it represents a useful tool.
We apply our formalism to two celebrated quantum codes, namely the surface code [6, 7] and the Fibonacci LevinWen code [8, 9]. The former one is a planar version of Kitaev’s toric code [10] that is among the most promising quantum computing platforms because of its simplicity and its surprisingly high error threshold of about 1%. The latter code is more involved but supports Fibonacci anyons that are universal for topological quantum computation; in other terms every quantum gate can be approximated to any accuracy by braiding Fibonacci anyons. Since the Fibonacci model is universal, it can hardly be simulated on a classical computer. In order to determine the error threshold of the Fibonacci LevinWen model, one would probably need to perform a full quantum simulation and thus have a quantum computer at hand. However, what precise algorithm should be run on the hypothetical quantum computer is still open and is a very interesting problem. Recently, some specific limiting cases where the Fibonacci model can be simulated classically have been investigated where the error threshold is approximatively 12.5 % [11].
We think that our work on the surface code is particularly timely since the first set of experiments to build small fragments of surface code (with 9 data qubits) have now started [12]. It is thus interesting to understand what architecture is optimal and could be realized in the lab. Finally we compare our results for the surface code with previously suggested architectures [13, 14].
The paper is organized as follows. In Section 2.1 we present the physical model under consideration for a generic quantum code as well as the formalization of the optimization problem. In particular, we show that the optimal architecture is found by solving binary linear programs. In Section 3.1 we apply the formalism developed in Section 2.1 in order to find an optimal architecture for the Fibonacci code. In particular, we present a methodology to find scalable architectures by solving tractable binary linear programs. Section 3.2 finally contains our results for the surface code.
2 Connecting qubits optimally
In this work, we consider Transmon Qubits (TQs) and Transmission Line Resonators (TLRs) as the prototypical examples of physical qubits and moderate distance couplers [15, 16]. However, it is worth pointing out that our approach does not depend on the technological details of the implementation but can be applied to any kinds of qubits and couplers [17–23].
2.1 Model
However, using a new TLR for each pair of qubits that should be coupled to participate in twoqubit gate operations might not be the most optimal approach to this engineering problem. In fact TLRs are able to couple to more than two TQs and we assume that m individual TQs can reside inside the resonant cavity provided by the TLR. Each of the TQs can be controlled separately and coupled to any of the other \(m1\) TQs through the TLR. Following recent experimental progress [2], we find that \(m\leqslant5\) is a realistic upper bound. Informative micrographs of TLRs hosting several TQs can be found in [2] and on page 104 of [5] or in Figure 1(a) of [3]. Also, it seems natural to restrict the number p of TLRs that are connected to a single TQ; here we choose \(p=5\) [5]. We would like to mention here that Xmon qubits, a type of transmon qubit, are specifically designed so that they can couple to several TLRs, see Figure 1 in [12] for example where each branch of the Xmon can be coupled to a different TLR.
 1.For all \(i_{j}\in\{1,2,\ldots,N\}\) ,$$ \sum_{\mathcal{S}\in\mathfrak{S}_{m}  i_{j}\in \mathcal{S}}\kappa_{\mathcal{S}}\leqslant p. $$(3)
 2.
\(W_{m}\) realizes every twoqubit coupling of \(\mathcal{P}\) .
It is now clear why we call this a binary linear program; every component of the vector \(W_{m}\) is either 0 or 1. Solving such a binary linear program is generally very difficult and is in fact an NPhard problem. However, specific instances of such problems can be tractable, and we give explicit examples below. As a side remark, note that when all the numbers in the program are allowed to be real, then the situation is dramatically simplified and the optimization problem can be solved in polynomial time.
In this work, we use the free software lpsolve, available at http://lpsolve.sourceforge.net/5.5/, to find the optimal solution to the binary linear program defined above. In order to simplify the program, we leave out all the superfluous TLRs. We call a TLR superfluous if it can be replaced by two (or more) TLRs that host no common qubits such that the same set of required twoqubit couplings is realized; one can thus always replace a superfluous TLR by two TLRs that will have a lower overall cost.
As mentioned in the Introduction, we aim to find the optimal architectures for two important quantum error correcting codes, namely the surface code and the LevinWen model. We find interesting that such quantum technological problems can be turned into standard optimization problems.
3 Application to Quantum error correcting codes
3.1 Fibonacci LevinWen model
On a surface with nontrivial topology this groundstate subspace of Hamiltonian (5) is degenerate and one uses this set of states to encode logical qubits. A nontrivial operation (a logical error) applied to the logical qubit is implemented by creating pairs of τexcitations, braiding them, and annihilating them. The logical operation does not depend on the details of the braiding process, but only on its topology; this is in fact the main idea of topological quantum computation [27]. Importantly, Fibonacci anyons are universal for quantum computation and any quantum gate can thus be performed in a topologically protected fashion.
We note here that ancillary qubits are needed to perform nondemolition measurements of \(Q_{v}\) and \(B_{p}\). According to the plaquette reduction method of Ref. [9], in Figure 2 the ancillary qubit \(\alpha_{0}\) is used to measure \(B_{p}\), while the ancillary qubits \(\alpha_{16}\) are used to measure the six vertex operators \(Q_{v}\).
Said differently, \(\mathcal{C}_{\mathcal{S}}\) is the geometric length of the shortest path going through all the TQs specified in the string \(\mathcal{S}\). In this work we thus look for the TLR scheme that minimizes the total length of the TLR wires.
Primary qubit  Qubits to which the primary qubit couples 

\(\alpha_{0}\)  \(i_{6}\) 
\(a_{1}\)  \(a_{6}\), \(i_{1}\), \(i_{2}\), \(i_{6}\), \(\alpha_{1}\) 
\(a_{2}\)  \(i_{1}\), \(i_{2}\), \(i_{3}\), \(i_{6}\), \(\alpha_{2}\) 
\(a_{3}\)  \(i_{2}\), \(i_{3}\), \(i_{4}\), \(i_{6}\), \(\alpha_{3}\) 
\(a_{4}\)  \(i_{3}\), \(i_{4}\), \(i_{5}\), \(i_{6}\), \(\alpha_{4}\) 
\(a_{5}\)  \(i_{4}\), \(i_{5}\), \(i_{6}\), \(\alpha_{5}\) 
\(a_{6}\)  \(a_{1}\), \(i_{1}\), \(i_{2}\), \(i_{6}\), \(\alpha_{6}\) 
\(i_{1}\)  \(a_{1}\), \(a_{2}\), \(a_{6}\), \(i_{2}\), \(i_{3}\), \(i_{6}\), \(\alpha_{1}\), \(\alpha_{6}\) 
\(i_{2}\)  \(a_{1}\), \(a_{2}\), \(a_{6}\), \(i_{1}\), \(i_{3}\), \(i_{4}\), \(i_{6}\), \(\alpha_{1}\), \(\alpha_{2}\) 
\(i_{3}\)  \(a_{2}\), \(a_{3}\), \(a_{4}\), \(i_{1}\), \(i_{2}\), \(i_{4}\), \(i_{5}\), \(i_{6}\), \(\alpha_{2}\), \(\alpha_{3}\) 
\(i_{4}\)  \(a_{3}\), \(a_{4}\), \(a_{5}\), \(i_{2}\), \(i_{3}\), \(i_{5}\), \(i_{6}\), \(\alpha_{3}\), \(\alpha_{4}\) 
\(i_{5}\)  \(a_{4}\), \(a_{5}\), \(i_{3}\), \(i_{4}\), \(i_{6}\), \(\alpha_{4}\), \(\alpha_{5}\) 
\(i_{6}\)  \(\alpha_{0}\), \(a_{1}\), \(a_{2}\), \(a_{3}\), \(a_{4}\), \(a_{5}\), \(a_{6}\), \(i_{1}\), \(i_{2}\), \(i_{3}\), \(i_{4}\), \(i_{5}\), \(\alpha_{5}\), \(\alpha_{6}\) 
\(\alpha_{1}\)  \(a_{1}\), \(i_{1}\), \(i_{2}\) 
\(\alpha_{2}\)  \(a_{2}\), \(i_{2}\), \(i_{3}\) 
\(\alpha_{3}\)  \(a_{3}\), \(i_{3}\), \(i_{4}\) 
\(\alpha_{4}\)  \(a_{4}\), \(i_{4}\), \(i_{5}\) 
\(\alpha_{5}\)  \(a_{5}\), \(i_{5}\), \(i_{6}\) 
\(\alpha_{6}\)  \(a_{6}\), \(i_{1}\), \(i_{6}\) 
Length of the TLR wire  Qubits contained inside the TLR wire 

2.52 a.u.  \(a_{6}\), \(\alpha_{0}\), \(i_{6}\), \(\alpha_{6}\), \(i_{1}\) 
4.73  \(i_{6}\), \(a_{6}\), \(a_{1}\), \(i_{2}\), \(a_{2}\) 
3.15  \(i_{1}\), \(i_{2}\), \(a_{2}\), \(\alpha_{2}\), \(i_{3}\) 
4.73  \(i_{2}\), \(i_{3}\), \(\alpha_{3}\), \(i_{4}\), \(a_{3}\) 
3.15  \(i_{3}\), \(a_{3}\), \(a_{4}\), \(i_{5}\), \(i_{6}\) 
3.15  \(i_{6}\), \(\alpha_{5}\), \(a_{5}\), \(i_{5}\), \(i_{4}\) 
1.15  \(i_{1}\), \(\alpha_{1}\), \(a_{1}\) 
1.15  \(i_{4}\), \(\alpha_{4}\), \(a_{4}\) 
0.58  \(i_{2}\), \(\alpha_{1}\) 
0.58  \(i_{5}\), \(\alpha_{4}\) 
Primary Qubit  Qubits to which the primary qubit couples 

\(\alpha_{0}\)  \(\alpha_{9}\) 
\(a_{1}\)  \(i_{1}\), \(i_{2}\), \(\alpha_{1}\), \(\alpha_{7}\), \(\alpha_{9}\) 
\(a_{2}\)  \(i_{2}\), \(i_{3}\), \(\alpha_{2}\), \(\alpha_{7}\), \(\alpha_{9}\) 
\(a_{3}\)  \(i_{3}\), \(i_{4}\), \(\alpha_{3}\), \(\alpha_{7}\), \(\alpha_{9}\) 
\(a_{4}\)  \(i_{4}\), \(i_{5}\), \(\alpha_{4}\), \(\alpha_{7}\), \(\alpha_{9}\) 
\(a_{5}\)  \(i_{5}\), \(i_{6}\), \(\alpha_{5}\), \(\alpha_{7}\), \(\alpha_{9}\) 
\(a_{6}\)  \(i_{1}\), \(i_{6}\), \(\alpha_{6}\), \(\alpha_{7}\), \(\alpha_{8}\), \(\alpha_{9}\) 
\(i_{1}\)  \(a_{1}\), \(a_{6}\), \(i_{2}\), \(i_{6}\), \(\alpha_{1}\), \(\alpha_{6}\), \(\alpha_{7}\), \(\alpha_{8}\), \(\alpha_{9}\) 
\(i_{2}\)  \(a_{1}\), \(a_{2}\), \(i_{1}\), \(i_{3}\), \(\alpha_{1}\), \(\alpha_{2}\), \(\alpha_{7}\), \(\alpha_{9}\) 
\(i_{3}\)  \(a_{2}\), \(a_{3}\), \(i_{2}\), \(i_{4}\), \(\alpha_{2}\), \(\alpha_{3}\), \(\alpha_{7}\), \(\alpha_{9}\) 
\(i_{4}\)  \(a_{3}\), \(a_{4}\), \(i_{3}\), \(i_{5}\), \(\alpha_{3}\), \(\alpha_{4}\), \(\alpha_{7}\), \(\alpha_{9}\) 
\(i_{5}\)  \(a_{4}\), \(a_{5}\), \(i_{4}\), \(i_{6}\), \(\alpha_{4}\), \(\alpha_{5}\), \(\alpha_{7}\), \(\alpha_{9}\) 
\(i_{6}\)  \(a_{5}\), \(a_{6}\), \(i_{1}\), \(i_{5}\), \(\alpha_{5}\), \(\alpha_{6}\), \(\alpha_{7}\), \(\alpha_{8}\), \(\alpha_{9}\) 
\(\alpha_{1}\)  \(a_{1}\), \(i_{1}\), \(i_{2}\) 
\(\alpha_{2}\)  \(a_{2}\), \(i_{2}\), \(i_{3}\) 
\(\alpha_{3}\)  \(a_{3}\), \(i_{3}\), \(i_{4}\) 
\(\alpha_{4}\)  \(a_{4}\), \(i_{4}\), \(i_{5}\) 
\(\alpha_{5}\)  \(a_{5}\), \(i_{5}\), \(i_{6}\) 
\(\alpha_{6}\)  \(a_{6}\), \(i_{1}\), \(i_{6}\) 
\(\alpha_{7}\)  \(a_{1}\), \(a_{2}\), \(a_{3}\), \(a_{4}\), \(a_{5}\), \(a_{6}\), \(i_{1}\), \(i_{2}\), \(i_{3}\), \(i_{4}\), \(i_{5}\), \(i_{6}\), \(\alpha_{8}\), \(\alpha_{9}\) 
\(\alpha_{8}\)  \(a_{6}\), \(i_{1}\), \(i_{6}\), \(\alpha_{7}\), \(\alpha_{9}\) 
\(\alpha_{9}\)  \(\alpha_{0}\), \(a_{1}\), \(a_{2}\), \(a_{3}\), \(a_{4}\), \(a_{5}\), \(a_{6}\), \(i_{1}\), \(i_{2}\), \(i_{3}\), \(i_{4}\), \(i_{5}\), \(i_{6}\), \(\alpha_{7}\), \(\alpha_{8}\) 
Length of the TLR wire  Qubits contained inside the TLR wire 

2.52 a.u.  \(\alpha_{9}\), \(\alpha_{8}\), \(i_{6}\), \(\alpha_{0}\), \(a_{6}\) 
2.84  \(a_{1}\), \(i_{1}\), \(\alpha_{7}\), \(\alpha_{8}\), \(\alpha_{9}\) 
4.04  \(\alpha_{7}\), \(\alpha_{9}\), \(i_{4}\), \(i_{3}\), \(a_{3}\) 
4.30  \(\alpha_{9}\), \(\alpha_{7}\), \(a_{5}\), \(i_{5}\), \(a_{4}\) 
2.44  \(i_{1}\), \(\alpha_{7}\), \(\alpha_{6}\), \(i_{6}\), \(a_{6}\) 
2.15  \(i_{2}\), \(\alpha_{1}\), \(i_{1}\), \(a_{1}\) 
2.15  \(i_{3}\), \(\alpha_{2}\), \(i_{2}\), \(a_{2}\) 
3.04  \(\alpha_{7}\), \(\alpha_{9}\), \(i_{2}\), \(a_{2}\) 
2.15  \(i_{5}\), \(\alpha_{4}\), \(i_{4}\), \(a_{4}\) 
1.63  \(i_{5}\), \(\alpha_{5}\), \(i_{6}\), \(a_{5}\) 
1.15  \(i_{4}\), \(\alpha_{3}\), \(a_{3}\) 
0.58  \(i_{3}\), \(\alpha_{3}\) 
3.1.1 Scaling
Qubit q in unit cell 0  Qubits to which q couples 

\(i_{0,1}\)  \(\alpha_{0,7}\), \(i_{0,6}\), \(\alpha_{0,8}\), \(\alpha _{0,9}\), \(i_{1,5}\), \(\alpha_{1,4}\), \(i_{1,6}\), \(\alpha_{1,9}\), \(\alpha _{2,7}\), \(i_{2,5}\), \(\alpha_{2,5}\), \(i_{2,6}\), \(\alpha_{2,9}\), \(i_{6,6}\), \(\alpha_{6,9}\) 
\(i_{0,5}\)  \(\alpha_{0,7}\), \(\alpha_{0,5}\), \(\alpha_{0,4}\), \(i_{0,6}\), \(\alpha _{0,9}\), \(\alpha_{4,7}\), \(i_{4,1}\), \(i_{4,6}\), \(\alpha_{4,7}\), \(\alpha _{4,9}\), \(i_{5,1}\), \(i_{5,6}\), \(\alpha_{5,9}\), \(i_{6,6}\), \(\alpha_{6,9}\) 
\(i_{0,6}\)  \(\alpha_{0,7}\), \(i_{0,1}\), \(i_{0,5}\), \(\alpha_{0,8}\), \(\alpha _{0,9}\), \(i_{1,5}\), \(\alpha_{2,7}\), \(i_{2,5}\), \(\alpha_{3,7}\), \(i_{3,1}\), \(i_{3,5}\), \(\alpha _{4,7}\), \(i_{4,1}\), \(i_{5,1}\) 
\(\alpha_{0,0}\)  \(\alpha_{0,9}\) 
\(\alpha_{0,4}\)  \(i_{0,5}\), \(\alpha_{4,7}\), \(i_{4,1}\) 
\(\alpha_{0,5}\)  \(\alpha_{0,7}\), \(i_{0,5}\), \(i_{5,1}\) 
\(\alpha_{0,7}\)  \(i_{0,1}\), \(i_{0,5}\), \(\alpha_{0,5}\), \(i_{0,6}\), \(\alpha _{0,8}\), \(\alpha_{0,9}\), \(i_{1,5}\), \(\alpha_{1,4}\), \(i_{1,6}\), \(\alpha _{1,9}\), \(i_{5,1}\), \(i_{5,6}\), \(\alpha_{5,9}\), \(i_{6,6}\), \(\alpha_{6,9}\) 
\(\alpha_{0,8}\)  \(\alpha_{0,7}\), \(i_{0,1}\), \(i_{0,6}\), \(\alpha_{0,9}\), \(i_{1,5}\), 
\(\alpha_{0,9}\)  \(\alpha_{0,7}\), \(i_{0,1}\), \(i_{0,5}\), \(\alpha _{0,0}\), \(i_{0,6}\), \(\alpha_{0,8}\), \(i_{1,5}\), \(\alpha_{2,7}\), \(i_{2,5}\), \(\alpha _{3,7}\), \(i_{3,1}\), \(i_{3,5}\), \(\alpha_{4,7}\), \(i_{4,1}\), \(i_{5,1}\) 
4  \(\alpha_{0,7}\), \(i_{0,1}\), \(\alpha_{0,8}\), \(\alpha_{1,4}\) 
3  \(i_{0,5}\), \(\alpha_{0,5}\), \(\alpha_{0,4}\) 

Consider the set \(\mathcal{P}^{0}_{\mathrm{swapping}}\) of twoqubit couplings that contains at least one qubit in the unit cell 0.

Define \(\mathfrak{W}\) as the set of TLR schemes that include all TLRs that are not superfluous with respect to \(\mathcal{P}_{\mathrm{swapping}}^{0}\) and all their equivalent TLRs.

Out of every set of equivalent TLRs, choose one unique representative TLR. For each TLR scheme \(W_{m}\in\mathfrak{W}\), define an associated TLR scheme \(B_{m}\). This scheme \(B_{m}\) includes the same TLRs as \(W_{m}\) but contains the following TLRs: All TLRs that do not have an equivalent TLR and are contained in \(W_{m}\) as well as all representatives for which \(W_{m}\) contains at least one equivalent TLR. We call this new set of TLR schemes \(\mathfrak{B}\).

For each TLR scheme \(B_{m}\in\mathfrak{B}\), define a new TLR scheme \(V_{m}\) that includes the same TLRs as \(B_{m}\) and contains all the TLRs that are contained in \(B_{m}\) as well as all equivalent TLRs.

Perform the linear optimization over \(\mathfrak{B}\) to find a TLR scheme \(B_{m}\) that minimizes the cost \(C(B_{m})^{t}\cdot B_{m}\) such that
 1.
\(V_{m}\) realizes every twoqubit coupling of \(\mathcal {P}^{0}_{\mathrm{swapping}}\).
 2.For all \(i_{j}\in\{1,2,\ldots,N\}\),$$ \sum_{\mathcal{S}\in\mathfrak{S}_{m}  i_{j}\in \mathcal{S}}\kappa_{\mathcal{S}}\leqslant p. $$(15)
 1.
Optimal TLR scheme for the set of couplings \(\pmb{\mathcal{P}_{\mathrm{swpapping}}^{0}}\) , i.e., for couplings between qubits in the unit cell 0 and the rest of the lattice
Length of the TLR  Qubits contained inside the TLR wire 

3.57 a.u.  \(i_{0,6}\), \(\alpha_{0,9}\), \(i_{2,5}\), \(\alpha_{3,7}\), \(i_{3,1}\) 
3.15  \(\alpha_{1,9}\), \(\alpha_{1,7}\), \(i_{1,6}\), \(i_{1,5}\), \(\alpha_{0,7}\) 
3.48  \(\alpha_{2,7}\), \(\alpha_{2,5}\), \(i_{0,1}\), \(\alpha_{0,9}\), \(i_{0,6}\) 
2.52  \(i_{0,5}\), \(\alpha_{4,0}\), \(i_{4,6}\), \(\alpha_{4,8}\), \(\alpha_{4,9}\) 
3.80  \(i_{17,1}\), \(i_{6,6}\), \(\alpha_{6,9}\), \(i_{5,1}\), \(i_{0,5}\) 
1.44  \(\alpha_{0,8}\), \(\alpha_{0,7}\), \(\alpha_{1,4}\), \(i_{0,1}\) 
1.15  \(\alpha_{0,4}\), \(i_{0,5}\), \(\alpha_{0,5}\) 
3.2 Surface code
The surface code [1] is a planar version of Kitaev’s toric code [10] and represents arguably the most promising quantum computing architecture. It is thus justified to determine its optimal architecture using the simple formalism developed in this work, in particular because experimental groups are nowadays starting to build small fragments of the surface code.
The set \(\pmb{\mathcal{P}^{0}_{\mathrm{surface}}}\) of couplings between qubits of the unit cell 0 and the rest of the lattice to measure plaquette and star operators of the surface code, see Figure 9
Qubit q in unit cell 0  Qubits to which q couples 

\(i_{0,1}\)  \(\alpha_{0,1}\), \(\alpha_{0,2}\), \(\alpha_{3,1}\), \(\alpha_{2,2}\) 
\(\alpha_{0,1}\)  \(i_{0,1}\), \(i_{0,2}\), \(i_{1,1}\), \(i_{2,2}\) 
\(i_{0,2}\)  \(\alpha_{0,1}\), \(\alpha_{0,2}\), \(\alpha_{1,2}\), \(\alpha_{4,1}\) 
\(\alpha_{0,2}\)  \(i_{0,1}\), \(i_{0,2}\), \(i_{4,1}\), \(i_{3,2}\) 
3.2.1 Distance 5 surface code
4 Conclusions
In this work we have developed a methodology to find optimal architectures for quantum codes. Our starting point is to consider a twodimensional lattice of transmon qubits that interact with each others over moderate distances by coupling them to transmission line resonators. For each layout, we define a cost that allows to designate an optimal scheme. We show that finding such optimal scheme reduces to solve standard binary programs. What optimal means here depends obviously on the choice of a cost function. While we decided to choose to optimize over the total length of transmission line resonators for the Fibonacci and surface codes, our formalism is general enough to be straightforwardly applicable to many other codes and cost functions. While further experimental and theoretical studies will be necessary to determine which cost function will turn out to be most relevant in practice, we believe that our work represents a useful methodology to investigate a large number of layouts and find an optimal one according to some metric. In particular, we show how to apply our method to a restricted set of qubit and couplers that can be scaled up to a large twodimensional structure.
Declarations
Acknowledgements
We are happy to thank N Breuckmann and B Criger for useful discussions. We are grateful for support from the Alexander von Humboldt foundation, ScaleQIT, and QALGO.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Terhal BM. Quantum error correction for quantum memories. Rev Mod Phys. 2015;87:307. ADSMathSciNetView ArticleGoogle Scholar
 Chow JM. Quantum information processing with superconducting qubits. PhD dissertation, Yale University. 2010. http://www.eng.yale.edu/rslab/papers/jmcthesis.pdf.
 Ristè D, Poletto S, Huang MZ, Bruno A, Vesterinen V, Saira OP, DiCarlo L. Detecting bitflip errors in a logical qubit using stabilizer measurements. Nat Commun. 2015;6:6983. ADSView ArticleGoogle Scholar
 Yang CP, Su QP, Zheng SB, Nori F. Entangling superconducting qubits in a multicavity system. New J Phys. 2016;18:013025. View ArticleGoogle Scholar
 Ristè D. Feedback control of superconducting quantum circuits. PhD dissertation, TU Delft University of Technology. 2014. http://repository.tudelft.nl/view/ir/uuid%3A787bc48896b24e03abbe0a9fba2c4d75/.
 Bravyi SB, Kitaev AY. Quantum codes on a lattice with boundary. arXiv:quantph/9811052 (1998).
 Freedman MH, Meyer DA. Projective plane and planar quantum codes. Found Comput Math. 2001;1:32532. MathSciNetView ArticleMATHGoogle Scholar
 Koenig R, Kuperberg G, Reichardt BW. Quantum computation with TuraevViro codes. Ann Phys. 2010;325:270749. ADSMathSciNetView ArticleMATHGoogle Scholar
 Bonesteel NE, DiVincenzo DP. Quantum circuits for measuring LevinWen operators. Phys Rev B. 2012;86:165113. ADSView ArticleGoogle Scholar
 Kitaev AY. Faulttolerant quantum computation by anyons. Ann Phys. 2003;303:230. ADSMathSciNetView ArticleMATHGoogle Scholar
 Burton S, Brell CG, Flammia ST. Classical simulation of quantum error correction in a Fibonacci anyon code. arXiv:1506.03815 (2015).
 Kelly J, et al.. State preservation by repetitive error detection in a superconducting quantum circuit. Nature. 2015;519:669. ADSView ArticleGoogle Scholar
 DiVincenzo DP. Faulttolerant architectures for superconducting qubits. Phys Scr T. 2009;137:014020. ADSView ArticleGoogle Scholar
 Ghosh J, Fowler AG, Geller MR. Surface code with decoherence: an analysis of three superconducting architectures. Phys Rev A. 2012;86:062318. ADSView ArticleGoogle Scholar
 Blais A, Gambetta J, Wallraff A, Schuster DI, Girvin SM, Devoret MH, Schoelkopf RJ. Phys Rev A. 2007;75:032329. ADSView ArticleGoogle Scholar
 DiCarlo L, Chow JM, Gambetta JM, Bishop LS, Johnson BR, Schuster DI, Majer J, Blais A, Frunzio L, Girvin SM, Schoelkopf RJ. Demonstration of twoqubit algorithms with a superconducting quantum processor. Nature. 2009;460:2404. ADSView ArticleGoogle Scholar
 Childress L, Sørensen AS, Lukin MD. Mesoscopic cavity quantum electrodynamics with quantum dots. Phys Rev A. 2004;69:042302. ADSView ArticleGoogle Scholar
 Burkard G, Imamoglu A. Ultralongdistance interaction between spin qubits. Phys Rev B. 2006;74:041307(R). ADSView ArticleGoogle Scholar
 Trif M, Golovach VN, Loss D. Spin dynamics in InAs nanowire quantum dots coupled to a transmission line. Phys Rev B. 2008;77:045434. ADSView ArticleGoogle Scholar
 Trifunovic L, Dial O, Trif M, Wootton JR, Abebe R, Yacoby A, Loss D. Longdistance spinspin coupling via floating gates. Phys Rev X. 2012;2:011006. Google Scholar
 Shulman MD, Dial OE, Harvey SP, Bluhm H, Umansky V, Yacoby A. Demonstration of entanglement of electrostatically coupled singlettripel qubits. Science. 2012;336:2025. ADSView ArticleGoogle Scholar
 Trifunovic L, Pedrocchi FL, Loss D. Longdistance entanglement of spin qubits via ferromagnet. Phys Rev X. 2013;3:042023. Google Scholar
 Hassler F, Catelani G, Bluhm H. Exchangeinteraction of two spin qubits mediated by a superconductor. arXiv:1509.0638 (2015).
 Nielsen MA, Chuang IL. Quantum computation and quantum information. Cambridge: Cambridge University Press; 2000. MATHGoogle Scholar
 Levin MA, Wen XG. Stringnet condensation: a physical mechanism for topological phase. Phys Rev B. 2005;71:045110. ADSView ArticleGoogle Scholar
 Bonderson PH. NonAbelian anyons and interferometry. PhD dissertation, California Institute of Technology. 2007. http://resolver.caltech.edu/CaltechETD:etd06042007101617.
 Nayak C, Simon SH, Stern A, Freedman M, Das Sarma S. NonAbelian anyons and topological quantum computation. Rev Mod Phys. 2008;80:1083159. ADSMathSciNetView ArticleMATHGoogle Scholar
 Feng W, Bonesteel NE, DiVincenzo DP. NonAbelian errors in the Fibonacci LevinWen model. In preparation. Google Scholar
 Dennis E, Kitaev A, Landahl A, Preskill J. Topological quantum memory. J Math Phys. 2002;43:4452505. ADSMathSciNetView ArticleMATHGoogle Scholar
 Horsman C, Fowler AG, Devitt S, Van Meter R. Surface code quantum computing by lattice surgery. New J Phys. 2012;14:123011. MathSciNetView ArticleGoogle Scholar