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Methodology for bus layout for topological quantum error correcting codes
EPJ Quantum Technology volume 3, Article number: 4 (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.
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
Consider a set of N TQs \(q_{1},\ldots,q_{N}\) that lie on a twodimensional plane at positions \(\mathbf{x}_{1},\ldots, \mathbf{x}_{N}\). Depending on the specific quantum codes that one wants to realize, see Sections 3 for examples, several TQs must interact with each other and thus be coupled through TLRs. As any quantum circuit can be reduced to a succession of single and twoqubit operations [24], the most straightforward approach is to introduce TLRs containing each exactly two TQs; in this way TLRs realize the set \(\mathcal{P}\) of all twoqubit couplings necessary to implement a given quantum circuit, see Figure 1.
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.
We call an unordered sequence of sites \(i_{k}\in\{1,\ldots,N\}\) a string \(\mathcal{S}=\{i_{1},i_{2},\ldots, i_{m}\}\). The length \( \mathcal{S}\) of a string is defined by the number of sites it contains. To each string \(\mathcal{S}\), we associate a number \(\kappa _{\mathcal{S}}=0,1\); if \(\kappa_{\mathcal{S}}=1\), then a TLR is present and hosts the m TQs \(q_{i_{1}},\ldots,q_{i_{m}}\), otherwise no single TLR hosts all those specific m qubits. We denote by \(\mathfrak {S}_{m}\) the set of all strings \(\mathcal{S}\) with \(\mathcal {S}\leqslant m\). We call the vector
a TLR scheme. We say that a TLR corresponding to a string \(\mathcal{S}\) is included in a TLR scheme \(W_{m}\) if \(\kappa_{\mathcal{S}}\) is one of the elements of the vector \(W_{m}\). We say that a TLR scheme \(W_{m}\) contains a TLR associated with string \(\mathcal{S}\) if it is included and \(\kappa_{\mathcal{S}}=1\).
In order to formalize the concept of optimal TLR scheme, we introduce a cost \(\mathcal{C}_{\mathcal{S}}\in\mathbb{R}\) associated with each string \(\mathcal{S}\in\mathfrak{S}_{m}\). The cost vector of the scheme \(W_{m}\) is then
Using this notation, the total cost of a given TLR scheme \(W_{m}\) is \(\mathcal{C}(W_{m})^{T}\cdot W_{m}\). The goal of this work is to determine one TLR scheme \(W_{m}\) that minimizes the cost and realizes the set \(\mathcal{P}\) of twoqubit couplings. In Sections 3 we present concrete examples of cost functions for the surface code and for the LevinWen model.
The problem of finding the TLR scheme \(W_{m}\) that has minimal cost is solved by using standard binary linear optimization methods. The problem is formalized as follows: Given a set of twoqubit connections \(\mathcal{P}\) , and given two integers m and p, find the TLR scheme \(W_{m}\) that minimizes the cost \(\mathcal{C}(W_{m})^{T}\cdot W_{m}\) such that

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 worth pointing out again that the maximal number m of qubits per TLR, as well as the maximal number p of TLR per qubit, is fixed.
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
LevinWen models are a class of spin systems defined on trivalent lattices whose excitations realize any consistent (Abelian or nonAbelian) anyonic theory [25]. Here we focus on a particular LevinWen model, namely the Fibonacci LevinWen model [8, 9]. Its name takes its origin in the nature of the excitations above the ground states; indeed they are Fibonacci anyons with topological charge τ and fusion rules
Here 1 represents the vacuum topological charge.
Considering a trivalent lattice with each edge carrying a spin\(1/2\) particle, we define the Fibonacci LevinWen Hamiltonian [8, 9],
where \(Q_{v}\) and \(B_{p}\) are operators that are respectively associated with vertex v and plaquette p of the lattice, see Figure 2.
The vertex operator \(Q_{v}\) acts on the three qubits residing on the edges that meet at vertex v. If the states of the three qubits on theses edges are \( i\rangle\), \( j\rangle\), and \( k\rangle \), then we have
with
The plaquette operators are more complicated and involve 12qubit interactions. Consider the twelve qubits \(a_{16}\) and \(i_{16}\) around a given plaquette p, see Figure 2(b). The plaquette operators are then defined through
with \(\phi=\frac{1+\sqrt{5}}{2}\) the golden ratio and
where \(s=1,\tau\) and
For the Fibonacci theory we have [26]
and all other F’s are trivial. One can then show that the LevinWen plaquette and star operators satisfy \([B_{p},Q_{v} ]=[B_{p},B_{p^{\prime }}]=[Q_{v},Q_{v^{\prime}}]=0\), for all v, \(v^{\prime}\), p, \(p^{\prime}\) [8, 9].
We define the Fibonacci code [8] \(\mathcal{F}\) (an example of a stabilizer code) as the groundstate subspace of Hamiltonian (5), namely
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.
Recently, Ref. [9] has shown how to explicitly construct quantum circuits that measure plaquette and vertex operators of the Fibonacci LevinWen model; this is required to measure the error syndrome of \(\mathcal{F}\) and to decide how to perform error correction. Here we go one step further and determine the optimal qubitcoupler architecture to realize those quantum circuits. It is not the goal of the present work to review in detail how vertex and plaquette quantum circuits are constructed. But these circuits indicate which qubits must be coupled and this indicates the binary linear program of Section 2 that is to be solved to obtain the optimal architecture. For the sake of completeness in Figure 3 we reproduce the circuit of Ref. [9] for the plaquette reduction method.
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 \(Q_{v}\).
Here we choose the cost function \(\mathcal{C}_{\mathcal{S}}\) that measures the geometric length of the TLR corresponding to \(\mathcal {S}=\{i_{1}, i_{2},\ldots, i_{m}\}\),
where σ is a permutation of m elements.
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.
Following the plaquette reduction method of Ref. [9] and using the notation of Figure 2, we present in Table 1 the set \(\mathcal{P}_{\mathrm{reduction}}\) of twoqubit couplings that are necessary to measure the six vertex operators and the single plaquette operator of Figure 2.
Having in hand \(\mathcal{P}_{\mathrm{reduction}}\), we can solve the binary linear program of Section 2 and determine the optimal TLR scheme. The result is summarized in Table 2 and a pictorial representation is given in Figure 4.
For completeness, we also investigate the plaquette swapping method of Ref. [28] to measure plaquette operators. In this case, more ancillary qubits are required, see Figure 5. For the sake of completeness, in Figure 6 we reproduce the plaquette swapping circuit of Ref. [28]. The set \(\mathcal{P}_{\mathrm{swapping}}\) of twoqubit couplings required by the plaquette swapping method is summarized in Table 3. Again, we solve the binary linear program and find the optimal architecture of Table 4; here we have again chosen \(m=p=5\). As the pictorial representation would be too crowded, we refrain from drawing the TLRs corresponding to Table 4.
3.1.1 Scaling
While binary linear programs can be solved rapidly for a small number of qubits, as is the case for the 12 data qubits of Figures 2 and 5, the problem becomes rapidly unsolvable when we increase the number of qubits. This seems to be problematic as one wants to find the optimal architecture for a large LevinWen model and not only for a single plaquette. Fortunately, most of the time there is a lot of redundancy in the problem in the sense that a fundamental circuit unit can be identified and translated over the whole lattice. In fact, if one wants for example to measure all the vertex and plaquette operators of a large Fibonacci LevinWen model, the circuit will look the same around any plaquette of the lattice. In such a scenario, it is possible to identify a small number of qubits that we couple optimally and that we translate to cover the whole lattice. The aim of this section is thus to introduce a simple method to optimally solve a given unit cell of the model that can be scaled up by simple translation to build a large twodimensional lattice, see Figure 7. For the sake of simplicity, we just focus here on the plaquette swapping method of Ref. [28]. In Table 5 we present the set \(\mathcal{P}_{\mathrm{swapping}}^{0}\) of twoqubit couplings that are required between the qubits of unit cell 0, see Figure 7, and the remaining qubits of the lattice. If one would now straightforwardly solve the binary linear program for the unit cell, as we did in Section 3.1, then one would encounter the problem of equivalent connections, i.e., connections that are doubled due to the shifting of the unit cell. As an explicit example, let us consider the connection between qubits \(i_{0,6}\) and \(i_{2,5}\) as well as the connection between qubits \(i_{0,5}\) and \(i_{5,6}\), see Figure 7. It is straightforward to see that after translating unit cell 0 onto unit cell 5, a TLR will be doubled. In order to avoid such doublings, one needs to slightly modify the algorithm as follows.
Consider a given unit cell 0 and two distinct strings \(\mathcal {S}_{1}\) and \(\mathcal{S}_{2}\) that each contains at least one site inside unit cell 0. We say that \(\mathcal{S}_{1}=\{ i_{1},i_{2},\ldots, i_{m}\}\) and \(\mathcal{S}_{2}=\{j_{1},j_{2},\ldots, j_{m}\}\) are equivalent if \(\forall k\in[1,m]\) \(\exists \ell\in[1,m]\) such that
where \(\lambda_{1,2}\in\mathbb{Z}\) and \(\mathbf{v}_{1,2}\) are basis vectors of the lattice, see Figure 7. If a TLR scheme possesses a TLR hosting the qubits along \(\mathcal{S}_{1}\) and another TLR hosting the qubits along \(\mathcal{S}_{2}\), it is clear that this will not be optimal. Indeed, when we translate unit cell 0 by the vector \(\lambda_{1}\mathbf{v}_{1}+\lambda_{2}\mathbf{v}_{2}\) and the associated TLRs, to cover the whole lattice, then some TLRs will be doubled. Having set these definitions, we present the steps that we follow to find the optimal scalable TLR without doubled TLRs.

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.
Following the above algorithm, we find the scalable optimal architectures presented in Table 6 and Figure 8.
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.
Consider a square lattice with a spin\(1/2\) particle on each vertex, see Figure 9. We define the star operators \(A_{s}\) and plaquette operators \(B_{p}\) of the surface code as
where s and p label respectively light and dark squares of the lattice, see Figure 9. Note that plaquette and star operators at the boundaries are products of three qubit operators and not four as is the case in the bulk. Similar to the Fibonacci code, we define the surface code \(\mathcal{S}\) as
With the boundary conditions represented in Figure 9, the surface code is twofold degenerate and can thus encode a logical qubit [1]. Similar to the Fibonacci code, the surface code is a topological code and it is thus protected against local (static) perturbations. Its most striking property is its surprisingly high error threshold of about 1%, see Ref. [1] for a detailed review on this subject.
Here we do not review the construction of quantum circuits to measure star and plaquette operators of the surface code, rather, using the notation of Figure 9, in Table 7 we show the set \(\mathcal{P}_{\mathrm{surface}}^{0}\) of twoqubit couplings required to measure the eigenvalues of \(A_{s}\) and \(B_{p}\) in a scalable manner [29]. As was the case for the Fibonacci code, we need to introduce ancillary qubits to measure plaquette and star operators nondestructively.
Solving the binary linear optimization problem with \(m=5\), we find the scalable optimal TLR scheme reported in Figure 10. The result is that each bulk TLR hosts four TQs and each bulk TQ is hosted by two TLRs. Interestingly, this result is the one originally proposed in Ref. [13], see also Ref. [14]. We point out that the optimal architecture of Figure 10 is clearly valid for the smallest possible surface code (consisting of 13 data qubits and 12 ancillary qubits) able to detect and correct a single physical qubit error. In fact, we have solved our optimization problem for this small surface code directly and obtained the solution of Figure 10 with two and threequbit TLRs at the boundaries.
3.2.1 Distance 5 surface code
As a final relevant explicit example, we consider a surface code that can correct two physical errors. While the surface code depicted in Figure 9 would contain 41 data qubits and 40 ancillary qubits in order to detect and cure two physical errors, there are simple methods to reduce the number of qubits while keeping the distance the same [30]. Such modifications are important for smallscale implementations of surface codes in a near future; indeed there is clearly an intention to realize a quantum code that requires the smallest possible amount of resource. Here we follow the approach of Ref. [30] and consider the rotated surface code of Figure 11. The qubits that are part of the rotated code reside inside the black square and some of the boundary ancillary qubits are also incorporated to measure the boundary stabilizers. As was shown in Ref. [30], such rotated surface code can correct two physical qubit errors although it possesses many fewer than 41 data qubits, in fact it consists only of 25 data qubits and 24 ancillary qubits. Furthermore, one can do slightly better by requiring not each stabilizer to have its individual ancillary qubit but instead by reusing an ancillary qubit to measure several stabilizer operators. We thus remove the 14 ancillary qubits with a (yellow) cross in Figure 11.
We can now solve the linear binary program and find the optimal architecture given in Figure 12.
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.
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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.
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MS carried out the numerical simulations. MS, FLP, and DPD contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Wosnitzka, M., Pedrocchi, F.L. & DiVincenzo, D.P. Methodology for bus layout for topological quantum error correcting codes. EPJ Quantum Technol. 3, 4 (2016). https://doi.org/10.1140/epjqt/s4050701600428
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DOI: https://doi.org/10.1140/epjqt/s4050701600428