The Small Stellated Dodecahedron Code and Friends
Jonathan Conrad, Christopher Chamberland, Nikolas P. Breuckmann, Barbara M. Terhal
rrspa.royalsocietypublishing.org
Research
Article submitted to journal
Subject Areas: quantum information
Keywords: quantum error correction,fault-tolerance, homological quantumcodes
Author for correspondence:
Barbara M. Terhale-mail: [email protected]
The Small StellatedDodecahedron Code andFriends
J. Conrad , C. Chamberland ,N. P. Breuckmann , B. M. Terhal , JARA Institute for Quantum Information, RWTHAachen University, Aachen 52056, Germany Institute for Quantum Computing and Department ofPhysics and Astronomy, University of Waterloo,Waterloo, Ontario, N2L 3G1, Canada Department of Physics and Astronomy, UniversityCollege London, London WC1E 6BT, UK QuTech, Delft University of Technology, P.O. Box5046, 2600 GA Delft, The Netherlands Institute for Theoretical Nanoelectronics,Forschungszentrum Juelich, D-52425 Juelich,Germany
We explore a distance-3 homological CSS quantumcode, namely the small stellated dodecahedron code,for dense storage of quantum information and wecompare its performance with the distance-3 surfacecode. The data and ancilla qubits of the small stellateddodecahedron code can be located on the edges resp.vertices of a small stellated dodecahedron, makingthis code suitable for 3D connectivity. This codeencodes 8 logical qubits into 30 physical qubits (plus22 ancilla qubits for parity check measurements) ascompared to 1 logical qubit into 9 physical qubits (plus8 ancilla qubits) for the surface code. We develop fault-tolerant parity check circuits and a decoder for thiscode, allowing us to numerically assess the circuit-based pseudo-threshold. c (cid:13) The Authors. Published by the Royal Society under the terms of theCreative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author andsource are credited. a r X i v : . [ qu a n t - ph ] M a r r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... Contents (a) 2D Hyperbolic Surface Codes and Star Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . 3(b) Some 3D and 4D Examples Based on Regular Tessellations . . . . . . . . . . . . . . . . . . 4 (a) Decoder for [[30 , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1. Introduction
The popular toric or surface codes are members of a family of topological codes calledhomological CSS codes [1–3] which can be obtained from tessellations of D -dimensionalmanifolds. Curvature and topology of these manifolds determine features of these codes.Although a code does not specify a specific physical lay-out or physical distance betweenqubits, its prescription of which parity checks need to be measured, dictates what high-precisioninteractions need to be engineered between the physical qubits and ancilla qubits for measuringparity checks. As such, a code based on a tessellation of a 2D flat manifold suits a planar 2Dconnectivity between qubits, while a three-dimensional representation of a code in terms of apolyhedron could be used as a template of how physical qubits could be placed and connectedup in 3D.In this paper we continue the exploration of so-called hyperbolic surface codes [4,5] todetermine whether such codes, being block codes with high rate, have advantages over thesurface code. The work in [4] constructed various classes of hyperbolic surface codes based onregular tessellations and numerically examined noise thresholds of these codes when subjectedto depolarizing noise (assuming noiseless parity checks). The work in [5] went one step furtherby including effective noise in the parity check measurements themselves, focusing uniquely on { , } -hyperbolic surface codes. Ref. [5] also showed how to do read/write operations using Dehntwists if these block codes are used as a quantum memory. In this paper we focus on one of thesmallest and simplest members of the hyperbolic surface code family, namely a code which has arepresentation as a small stellated dodecahedron. Going beyond the previous work, we examinethe performance of the code when all elementary gates and operations, including those in theparity check circuits, are noisy (more details of the circuit level noise model are given in Section 5).The interest in the small stellated dodecahedron code is that it can pack logical qubits verydensely while, like the [[9 , , surface code, still allowing for plain fault-tolerant parity checkmeasurements in combination with a look-up table decoder. Even denser packings of logicalqubits in block stabilizer codes are certainly feasible: there are non-CSS codes such as [[8 , , , [[10 , , , [[11 , , , [[13 , , , and [[14 , , codes listed in [6]. However, one may expect thatthe construction of fault-tolerant parity check circuits for such codes requires resource-intensemethods such as Steane, Shor or Knill error correction, or flag-fault-tolerance methods [7,8]. r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... Ref. [7] also proposed fault-tolerant circuits for a non-topological [[15 , , Hamming code, usingonly 17 physical qubits in total: a disadvantage of this code is that the weight of the parity checksis high, namely 8, and in the tally of 17 qubits all parity checks are done using the same ancillaqubit.We find that for a depolarizing circuit-level noise model the stellated dodecahedron code paysfor its dense storage with a pseudo-threshold which is a factor 19 lower than that of the Surface-17 code. Despite this somewhat negative message, the methods developed in this paper lay thegroundwork for further exploration of these families of codes.We start the paper by recalling the notion of homological CSS codes, illustrating this codeconstruction by a variety of examples in 2D, representable as star polyhedra, as well as a few 3Dand 4D codes. In Section 3 we zoom in on the small stellated dodecahedron code, while we zoomout again in Section 4 by formalizing the problem of optimally scheduling the entangling gates ofparity check circuits of LPDC codes or more specifically hyperbolic surface codes. We apply thesetechniques in Section 5 to the dodecahedron code obtaining fault-tolerant circuits and describingthe decoding method. In Section 6 we report on the results of our numerical implementation,which includes a direct comparison with the Surface-17 code. We end the paper with a Discussion.
2. Homological CSS Codes
Here we briefly review the definition of homological CSS codes. We start with a regulartessellation of a D -dimensional closed manifold. This defines a cell complex composed of i-cells ,with i = 0 , , .., D referring to the cell dimension. The i -cells span a vector space C i = Z dim( C i )2 whose elements will be called i-chains . Given such a cell complex, one can define a CSS codeby associating the i -cells with physical qubits, the ( i + 1) -cells with Z -checks (i.e. generatorsof elements in the stabilizer group which only involve Pauli Z operators) and the ( i − -cellswith X -checks (i.e. generators of elements in the stabilizer group which only involve Pauli X operators). The number of physical qubits of the code is n = dim( C i ) . A Z -parity check isassociated with each ( i + 1) -cell and it takes the parity of the qubits/ i -cells which form theboundary of the ( i + 1) -cell. Formally, the boundary operator ∂ i +1 is defined as ∂ i +1 : C i +1 → C i .Similarly, a X -parity check is associated with each ( i − -cell so that it takes the X -parity of allqubits/ i -cells which are the co-boundary of the ( i − -cell (that is, which have the ( i − -cellas their boundary). Formally, the coboundary operator δ i − is defined as δ i − = ∂ Ti : C i − → C i .The X - and Z -parity checks commute since the boundary of any ( i + 1) -cell and the co-boundaryof any ( i − -cell overlap on an even number of i -cells/qubits.By the parity check weight of a X - or Z -parity check we mean the number of qubits on whichthis parity check acts non-trivially. The logical Z operators (denoted as Z ) of the code are closed i -chains which are not the boundary of any collections of ( i + 1) -cells. Similarly, the logical X operators (denoted as X ) are closed i -cochains which are not the co-boundary of any collection of ( i − -cells. The number of logical qubits of the code is given by k = dim( H i ( Z )) where H i ( Z ) is the i -th homology group over Z , that is H i ( Z ) = Ker( ∂ i ) / Im( ∂ i +1 ) . In the next sections wediscuss some concrete code families. (a) 2D Hyperbolic Surface Codes and Star Polyhedra Taking a surface ( D = 2 ), the only choice is for qubits to be associated with -cells or edges so that n = | E | . We only consider regular tilings of the surface. Such tilings can be denoted by the Schläflisymbol { r, s } , meaning that each face is a regular r -gon and s of such r -gons meet at each vertex.When { r, s } is such that r + s < , the surface is negatively curved or hyperbolic. For r + s = ,it is flat, and for r + s > it is positively curved, providing a regular tiling of the sphere. Thelast choice for { r, s } gives us all the Platonic solids (e.g. the dodecahedron { , } ) with trivialtopology of the sphere, hence not interesting for encoding quantum information using topologysince every closed loop can be contracted to a point. r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... In order to make a code out of a hyperbolic surface, one needs to close the surface so it istopologically equivalent to a many-handled torus. The Euler characteristic χ of such tessellatedclosed surface equals χ = 2 − g = | V | − | E | + | F | where | V | , | E | and | F | are the number ofvertices, edges and faces and g is the genus of the surface. The surface encodes k = 2 g logicalqubits. As was argued and reviewed in [1,4], hyperbolic surface codes based on an { r, s } -tilinghave an encoding rate kn = 1 + n − (cid:0) r + s (cid:1) and distance d ≥ c r,s log n with some constant c r,s which depends on the tessellation.Some of the smallest codes that one obtains from this construction can be representedas uniform star polyhedra, see Table 1. Examples are the dodedecadodecahedron based onclosing a { , } -tiling of the hyperbolic plane [5] with 60 qubits and the small stellateddodecahedron obtained from closing a { , } -tiling of the hyperbolic plane, depicted in Fig. 1.In its representation as star polyhedron, a regular p -gon can be represented as a star- pk -gon ( k and p mutually prime) whose vertices are generated by rotating by an angle πnpk with integer n [9].The Schläfli-notation for a star polygon is { pk } , that is, the pentagram is represented as { } so thatthe small stellated dodecahedron is denoted as { , } . n k = 2 − χ wt Z (wt X ) d Z ( d X )Tetrahemihexahedron U (a projective plane code) 12 1 3,4 (4) 3 (4)Octahemioctahedron U (a toric code) 24 2 3,6 (4) 4 (5)Cubohemioctahedron U (N.O.) 24 4 4,6 (4) 3 (4)Small stellated dodecahedron U (hyperbolic { , } ) 30 8 5 (5) 3 (3)Great dodecahedron U (dual to U ) 30 8 5 (5) 3 (3)Small rhombihexahedron U
48 8 4,8 (4) 3 (4)Small cubicuboctahedron U
48 6 3,4,8 (4) 4 (4)Great cubicuboctahedron U
48 6 3,4,8 (4) 4 (4)Great rhombihexahedron U (N.O.) 48 8 4,8 (4) 3 (4)Ditrigonal dodecadodecahedron U (hyperbolic { , } ) 60 18 5 (6) 3 (4) [4]Small ditrigonal icosidodecahedron U
60 10 3,5 (6) 4 (4)Great ditrigonal icosidodecahedron U
60 10 3,5 (6) 4 (4)Great dodecahemicosahedron U
60 10 5,6 (4) 5 (4)Small dodecahemicosahedron U (N.O.) 60 10 5,6 (4) 5 (4)Dodecadodecahedron U (hyperbolic { , } ) 60 8 5 (4) 6 (4) [5]Cubitruncated cuboctahedron U
72 6 6,8 (3) 8 (4)Table 1: Some small uniform star polyhedra with | E | = n physical qubits, k = 2 g = 2 − χ logicalqubits, Z - (resp. X ) parity check weight wt Z (wt X ) and Z - (resp. X ) distance d Z and d X . Thedistances d Z and d X were determined using the algorithm described in [5]. A full list of uniformstar polyhedra can be found at https://en.wikipedia.org/wiki/List_of_uniform_polyhedra . We omit all uniform polyhedra with χ = 2 , all polyhedra with faces with 10 edges( Z -parity check weight 10) and all polyhedra with 120 physical qubits or more. N.O. indicatesthat the surface represented by the polyhedron is not orientable. Since each vertex looks the same(vertex-transitivity) in the polyhedron, all X -checks have the same, fairly low, weight and act thesame. Except for the small stellated dodecahedron, all dual polyhedra have faces which are notregular polygons (they can be, say, arbitrary quadrilaterals), hence they are not star polyhedra.The many polyhedra with more than one type of polygonal face can also be viewed as quotientspaces of the uniformly-tiled hyperbolic plane. (b) Some 3D and 4D Examples Based on Regular Tessellations If we consider regular tessellations of three-dimensional manifolds, we have the option of placingqubits on edges or faces. Since these are dual to each other, one can only construct one code from r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... a given cell complex, so let’s imagine that we associate qubits with faces. Since 3D manifoldsare complicated mathematical objects, it is best to restrict any discussion to concrete three-dimensional cell complexes. A honeycomb is a set of polyhedra filling space such that each face isshared by two polyhedra. We can use the Schläfli-symbol { p, q, r } to denote a regular honeycomb,meaning that r regular polyhedra, each of type { p, q } , meet at a vertex. There is only one regularhoneycomb, namely { , , } , a tiling by cubes, which fills flat 3D space and can be wrapped intoa -torus, hence leading to the 3D toric code.The three-dimensional versions of the Platonic solids are 6 convex 4-polytopes: examples are { , , } (tesseract) and { , , } (120-cell) and its dual { , , } (600-cell). Instead of filling a flatspace, these tile a sphere. In other words, similar as the dodecahedron is a regular tiling of the2-sphere, one can view these cells as regular tilings (by volumes) of the 3-sphere S . This impliesthat dim( H ) = dim( H ) = 0 , or no qubits are encoded in such objects. The Euler characteristic ofthese convex polytopes is χ ( S ) = | V | − | E | + | F | − | C | = 0 (with | C | the number of 3-cells). Forexample, the 120-cell { , , } has | V | = 600 , | E | = 1200 , | F | = 720 and | C | = 120 .Similar to the stellation of a dodecahedron, one can also stellate or greaten a 120-cell or a600-cell to obtain so-called star polychora with non-trivial topology. An example is the smallstellated 120-cell { , , } which has | F | = 720 qubits, | C | = 120 Z -check cells, | E | = 1200 X -checkedges and | V | = 120 vertices, hence its Euler characteristic is χ = 120 − −
120 = − .Since χ = (cid:80) di =0 ( − i dim( H i ) and dim( H ) = dim( H ) = 1 , it follows that −
480 = dim( H ) − dim( H ) , so allowing for the encoding of logical information.In 4D, a natural choice is to put qubits on 2-cells, so that one associates a Z -check with each3-cell and an X -check with each edge. Beyond the 4D toric code which corresponds to a filling offlat 4D space, namely the honeycomb { , , , } [10,11], generalizations of the hyperbolic surfacecodes to 4D are known to exist as well [12,13]. These codes have a number of logical qubits k which scales linearly with the number of physical qubits n , just like the hyperbolic surface codes.Unlike the 2D hyperbolic codes, the distance of these codes has been shown to scale polynomiallywith the number of physical qubits d ∈ O ( n (cid:15) ) with < (cid:15) < . , see [13]. In principle one couldcreate a code starting with a regular tessellation of 4D hyperbolic space by 4-polytopes. In orderto have a closed 4D hyperbolic manifold one needs to find certain normal, torsion-free subgroupsof the Coxeter group [4,14] such that 4-cells related by generators of this group can be identified.One known example is the orientable closed Davis manifold obtained from identifying opposingdodecahedra in the 120-cell, viewed as a 4-polytope [15]. It encodes dim( H ) = 72 logical qubitsand n = 144 physical qubits (and dim( H ) = dim( H ) = 24 ) [16].In [14] an exhaustive search for finding normal torsion-free subgroups of the { , , , } tessellation of a 4D hyperbolic space is reported. In this tessellation qubits are associated withpentagons and dodecahedral cells act on 12 qubits. The X -checks correspond to tetrahedra in thedual lattice (with Schläfli-symbol { , } ), having weight-4. Each qubit is acted on by 5 X-checks( deg X = 5 ) and 3 Z-checks ( deg Z = 3 ). Unfortunately, running MAGMA to find an exhaustivelist of small subgroups of this { , , , } Coxeter group returns only one quantum code whichencodes k = 197 logical qubits ( dim( H ) = 197 ) into n = 16320 physical qubits. For { , , , } it isthe only example which has less than × physical qubits.
3. Features of The Small Stellated Dodecahedron Code
Some of the features of the small stellated dodecahedron code are summarized in Fig. 1. The codeencodes logical qubits (genus 4) into 30 and has distance 3. The Z -checks of the code are givenby the pentagrammic faces, that is, a Z -check acts on the five edges of each pentagrammic face.The X -checks are located at the vertices, i.e. an X -check acts on each of the five edges that meet ata vertex. There are thus 12 X - and Z -checks each of weight wt ( S ) = 5 . Since the product of all X -checks is I , the number of independent X -checks is 11 (and similarly there are 11 independent Z -checks). The small stellated docecahedron is obtained by stellating the dodecahedron as in Fig. 2, r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... (a) (b) (c)Figure 1: The small stellated dodecahedron as a [[30 , , code (Fig. 1a) and its dual polyhedron(Fig. 1b) which is called the great dodecahedron. Both polyhedra have the same number offaces, vertices and edges. The vertices of the small stellated dodecahedron lie at the starswhere edges meet. With the qubits placed on the edges, the Z-checks of the small stellateddodecahedron are described by intersecting pentagrammic faces, denoted as { } . By computingthe Euler characteristic χ using the parameters in Fig. 1c, it can be seen that the small stellateddodecahedron surface is topologically equivalent to the surface with genus g = 4 [17].Figure 2: Illustration of the small stellated dodecahedron construction by extending or stellatingthe edges of the (colored) regular dodecahedron until they intersect. The labeling of each vertexwill be used to identify data qubits as well as the check and logical operators. For example, thecheck operator localized at vertex is S X = X (0 , X (0 , X (0 , X (0 , X (0 , .that is, we extend the edges until they meet at new vertices. One can understand the emergenceof logical operators due to stellation for this specific polyhedron .The dodecahedron itself does not encode qubits but this trivial dodecahedron code has qubitson its 30 edges and weight-3 X checks and weight-5 Z -checks which commute. Now we extendthe edges, creating new vertices at which these edges meet. For this new code we keep the weight-5 dodecahedral Z -checks and add the weight-5 X -checks located at the 12 vertices where the For polyhedra one can also extend faces instead of edges, this is called greatening. An example is the greatening of theoctahedron into the stella octangula. Since qubits are not defined on faces, this process does not create an interesting code.One can stellate the icosahedron into the small triambic icosahedron (which is dual to U , see Table 1), but there does notseem to be an interpretation of stellation as creating non-trivial topology in this case. r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... extended edges meet. The 20 weight-3 X -checks of the dodecahedron still commute with the Z -checks and become possible logical operators.In addition, the stellation process creates new weight-3 loops running along a triangleconnecting three vertices and these loops cannot be the product of dodecahedral faces sincethe edges of the triangle lie in a single plane. When we take the 12 vertices and only representthe edges of the small stellated dodecahedron as a graph, one obtains an icosahedron, Fig. 3,which allows one to see the linear dependencies between the 20 Z -loops. In Fig. 3, the trianglelogical Z -operators are represented by the highlighted green edges (any weight-two Z operatorwould have odd support on at least one X -check). Around each vertex the product of 5 ofthese triangular Z -loops is a Z -check, hence the number of independent logical Z -operators is −
12 = 8 . Similarly, the 20 vertices of the dodecahedron are logical X operators, but productsof 5 of them around a dodecahedral face are identical to one weight-5 X -check, so there are −
12 = 8 linearly-independent logical operators. A possible basis for the logical operators isgiven in Table 2.Figure 3: Construction of the logical Z operators ( Z ) of the small stellated dodecahedron codefrom its disentangled graph, the icosahedron, where each blue or red vertex corresponds to an X -check acting on all incident edges. Each green triangle is a logical Z operator commuting withall X -checks. In the right figure one sees how a product of 5 logical Z operators, listed in Table2, equals the Z -check S Z . S Z is the Z -check associated with the face located above the vertexlabeled in Fig. 2.Logical Z s Logical X s Z = Z (0 , Z (0 , Z (6 , X = X (0 , X (2 , X (3 , Z = Z (0 , Z (0 , Z (7 , X = X (0 , X (1 , X (3 , Z = Z (0 , Z (0 , Z (8 , X = X (0 , X (1 , X (2 , Z = Z (1 , Z (1 , Z (7 , X = X (1 , X (4 , X (5 , Z = Z (2 , Z (2 , Z (5 , X = X (2 , X (3 , X (4 , Z = Z (3 , Z (3 , Z (7 , X = X (2 , X (3 , X (3 , X (4 , Z = Z (5 , Z (5 , Z (6 , X = X (1 , X (2 , X (5 , Z = Z (0 , Z (0 , Z (6 , Z (6 , X = X (0 , X (0 , X (1 , X (1 , X (3 , X (5 , X (6 , X (8 , Table 2: Set of independent logical X and Z operators of the small stellated dodecahedron codeobeying X i Z j = ( − δ ij Z j X i . Each qubit is labeled by the edge ( u, v ) with vertices u, v rangingfrom 0 to 11 as in Fig. 2. r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... The three-dimensional representation of this code as a small stellated dodecahedronimmediately suggests (but does not necessitate) a placement and connectivity of qubits in 3Dspace. We have also argued in [5] that any hyperbolic surface code can be implemented in abilayer of qubits with CNOTs required between the two layers while the connectivity betweenqubits in each layer is planar. Recent experiments on superconducting qubits also demonstratethe feasibility of variable-range planar (hyperbolic) connectivity between qubits [18].
4. Parity Check Scheduling for LDPC CSS Codes
In general, fault-tolerant error correction protocols for LDPC codes are implemented usingentangling gates between ancilla and data qubits in order to measure the parity checks. In thissection we assume that parity X -checks (resp. Z -checks) are measured via the interaction of asingle ancilla qubit with wt ( X ) data qubits via CNOT gates (resp. wt ( Z ) data qubits via CNOTgates). Thus, at any point in time an ancilla qubit can interact via a CNOT with at most one dataqubit. Similarly, any data qubit can interact with at most one ancilla qubit. A relevant problem isto find a scheduling of the CNOT gates which minimizes the number of time steps to measure allparity checks (so as to suppress the occurrence of errors).This scheduling problem for homological codes based on non-flat geometries is not as trivial asit is for a surface code (or a 4D tesseract code [19]) where a local orientation and order in terms ofNorth, East, South, West can be parallel transported over the whole lattice [20]. This idea does nottranslate to hyperbolic surface codes since the parallel transport of a vector around a closed curvedoes not bring it back to itself (in other words, the parallel transported vector depends on thepath that one takes) capturing the local curvature. Hence we formulate the scheduling problemas an optimization problem which can be attacked numerically.For starters, let’s imagine that we consider a CSS LDPC (low-density-parity-check) code andwe wish to do all X -check measurements with maximal parallelism followed by an optimizedschedule for the Z -check measurements. In such a scenario, the optimization of the numberof time-steps in the measurement of all, say X -checks, corresponds to a graph vertex-coloringproblem in a graph (to be defined) associated with the LDPC code. This graph and its coloringproblem is (non-uniquely) obtained as follows. Each data qubit q in the code is replaced by deg X ( q ) vertices, together forming the vertex set V X of the X -check scheduling graph G X . Here deg X ( q ) is the number of X -checks that the qubit participates in, hence the number of CNOTgates that it has to undergo. The edges of G X are taken as follows. For each qubit q we makea clique (complete subgraph) on its deg X ( q ) vertices, capturing the constraint that none of theCNOT gates are simultaneous. For example, for homological surface codes, we replace each qubitby two connected vertices. Secondly, for each X -check of weight wt ( X ) we create a completegraph K wt ( X ) between the vertices which represent the qubits on which the parity check acts,capturing the constraint that the CNOTs on the X -ancilla qubit cannot act simultaneously. Notethat this choice is not unique as each qubit has deg X ( q ) possible representatives. For homologicalsurface codes, a natural choice which is the same for every edge and face is shown in Fig. 4. Thisgives the edge set E X of the scheduling graph G X = ( V X , E X ) .Any vertex-coloring with m colors of the graph G X gives a schedule which requires T = m + 2 time-steps for the X -parity check measurements. In other words, the chromatic number χ ( G X ) ofthe graph G X determines the number of required time steps. In the first time step, ancilla qubitsare prepared in | + (cid:105) . In the subsequent m time steps, CNOT gates are performed between dataand ancilla qubits with the colors of vertices represented by the data qubits labeling the time stepat which the CNOT is performed. Note that the coloring assignment only prescribes a temporalordering up to permutations of time slots. In the last time step, the ancilla qubits are measured. Agraph G with maximum degree ∆ ( G ) always admits a vertex coloring, i.e. χ ( G ) ≤ ∆ + 1 [21]. Thedegree of G X is ∆ ( G X ) = deg X + wt ( X ) − when all qubits have the same degree deg X and allparity checks have weight wt ( X ) . An example for a planar { , } -tiling is shown in Fig. 4. Notethat for { r, s } -hyperbolic surfaces codes, the degrees of these scheduling graphs are ∆ ( G X ) = s and ∆ ( G Z ) = r . r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... (a) (b)Figure 4: Separate X and Z -scheduling graphs for the { , } -tiling. In Fig. 4a the scheduling graphfor Z -checks is shown whereas in Fig. 4b the scheduling graph for X -checks is shown. In Fig. 4aeach qubit is replaced by two vertices connected by a blue edge. The vertices of all qubits whichparticipate in a hexagonal face are connected by red edges.Figure 5: The vertices and edges of the interleaved scheduling graph G for a code based ona { , } -tiling: one takes the union of the vertices and edges in the graphs G X and G Z andadds additional green edges so that each qubit is represented by a clique of four vertices. In thehighlighted "interleaving" box, an X and a Z -check act on the same pair of data qubits.However, in order to minimize the total number of time steps, it is advantageous tosimultaneously apply CNOTs for X - and Z -check measurements instead of scheduling X and Z -check measurements sequentially. Such an interleaved schedule has been worked out for thesurface code [20,22] leading to a minimal schedule which requires T = 4 + 2 time steps (includingancilla preparation and measurement).Determining an optimal interleaved schedule can again be mapped onto a graph coloringproblem obeying an additional constraint which ensures that there is no interference betweenthe two types of measurements. To construct the interleaved scheduling graph G , we replaceeach qubit q by deg q vertices, where deg q is now the total number of parity checks that the qubitparticipates in. This constitutes the set of vertices V . As to the edges, we again make each clusterof deg q vertices into a clique. Then we add both the edges of the X - as well as the Z -checks as wedid separately in the graph G X and G Z . Fig. 5 shows the example of the { , } -tiling. For codeswith qubit degree deg , X -parity checks of weight wt ( X ) and Z -parity checks of weight wt ( Z ) ,the degree of this interleaved scheduling graph equals ∆ ( G ) = deg − wt ( X ) , wt ( Z )) .For { r, s } -surface codes, this results in ∆ ( G ) = 2 + max( r, s ) so that χ ( G ) ≤ r, s ) since r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... deg = 4 . At the same time, the chromatic number χ ( G ) ≥ max( r, s ) since the graph containscliques of size r and s .However, these coloring-based schedules may not be achievable since the CNOT order isadditionally constrained due to the noncommutativity of Pauli X and Z . In order to capture thisconstraint in the coloring problem, one can focus on homological surface codes in which X -checksand Z -checks have common support on either two or zero qubits (see also another expression ofthe constraints in [23]).Consider a pair of qubits, let’s call them a and b, on which an X - and a Z -check both havesupport, see Fig. 6. Both these qubits have to undergo CNOTs with an X -ancilla, as well asCNOTs with a Z -ancilla. Irrespective of what other data qubits are involved in the parity checkmeasurements, the outcomes of the two measurements are proper when, either both qubits firstinteract with the X -ancilla and then with the Z -ancilla or vice versa. In these cases one can deduce(by propagating Pauli operators through the CNOT gates) that the measurement of X x of the | + (cid:105) x ancilla equals the measurement of the observable X x X a X b I z . Since the X -ancilla is prepared in | + (cid:105) x , this is equivalent to X a X b I z . Similarly, a proper schedule shows that measurement of Z z is equivalent to measuring I x Z a Z b Z z , which is equivalent to I x Z a Z b due to the Z -ancilla beingprepared in | (cid:105) z . For an improper circuit shown in Fig. 6b the outcome of the parity checks israndomized since the X -check measurement depends on the expectation value of X on | (cid:105) z (andthe Z -check depends on Z on | + (cid:105) x ).(a) Proper circuit (b) Improper circuitFigure 6: The parity check circuits are proper when for each pair of qubits a and b which areinvolved in a X - and a Z -check, both qubits first interact with the X -ancilla and then the Z -ancillaor both first with the Z -ancilla and then the X -ancilla.There is no general efficient algorithm to find the chromatic number of a graph since theproblem is NP-complete. However, for sparse graphs Ref. [24] (e.g. Theorem 5) discusses anefficient algorithm under some assumptions. However, our problem is compounded by theadditional constraint that the schedule has to be proper. This means that a schedule of roundsof CNOT for the small stellated dodecahedron code might not be achievable, at least we havenot found it. In addition, the schedule is required to be fault-tolerant which puts additionalconstraints on the schedule. For the small stellated dodecahedron code we have numericallyobtained a sequential non-interleaved X and Z -parity check schedule which is automaticallyproper.We leave the existence of an efficient algorithm for determining a minimal-time parity checkschedule for LDPC codes (with sufficiently large distance) as an open question.
5. Fault-Tolerant Circuits for The Small Stellated DodecahedronCode
In this section we present the fault-tolerant methods that will be used to analyze the performanceof the small stellated dodecahedron code. The first step is to find a scheduling of the CNOTs usedto measure the checks as discussed in the previous Section. By applying a degree of saturation r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... (greedy) algorithm [25], a separate schedule with 5 colors for both G X and G Z could be found(see Figs. 13 and 14). Consequently, all checks can be measured in
10 + 2 time steps, Fig. 14. Forthis schedule we have verified that faults occurring during CNOT gates meet the requirements offault-tolerance (see the discussion in Section (a)).We consider the following circuit-level depolarizing noise model for our analysis:(i) With probability p , each two-qubit gate is followed by a two-qubit Pauli error drawnuniformly and independently from { I, X, Y, Z } ⊗ \ { I ⊗ I } .(ii) With probability p , the preparation of the | (cid:105) state is replaced by | (cid:105) = X | (cid:105) . Similarly,with probability p , the preparation of the | + (cid:105) state is replaced by |−(cid:105) = Z | + (cid:105) .(iii) With probability p , any single qubit measurement has its outcome flipped.(iv) Lastly, with probability p , each resting qubit location is followed by a Pauli error drawnuniformly and independently from { X, Y, Z } .The reason to choose the idling location to have a lower error probability of p is that it is areasonable assumption for actual qubits (such as trapped-ion qubits [26] or nuclear spin qubitsaround a diamond NV center [27]) and it brings out more clearly the effect of CNOT errors whichdominate the logical failure rate. Taking the idling location to have the same error probability p asall other locations would give a disadvantage to the dodecahedron code versus the surface codesince the parity check schedule for the dodecahedron code has more qubit idling.As was shown in [28] (see also the concise description in [8]), a d = 3 code should obey thefollowing fault-tolerance criteria for an error correction (EC) unit in order that the logical errorprobability is possibly below the physical error probability p : Condition 5.1. (Fault-Tolerant Criteria for an EC unit of a distance-3 code) (i) If the input state has r errors and the EC unit has s faults with r + s ≤ , then idealdecoding of the output state of the EC will result in the same codeword as ideal decodingof the input state.(ii) Regardless of the number of errors in the input state, if there are s faults during the ECunit with s ≤ , the output state can differ from a valid codeword by an error of at mostweight s .Here, ideal decoding means a round of fault-free error correction. Furthermore, by a fault wemean any gate, state-preparation, measurement or idle qubit failing according to the noise modeldescribed above. The second criteria states that if E | ψ (cid:105) is the input state with codeword | ψ (cid:105) andwt ( E ) is arbitrary, the output state must be written as E (cid:48) | φ (cid:105) where | φ (cid:105) is a codeword and wt ( E (cid:48) ) ≤ s ≤ . Note that it is not required that | ψ (cid:105) = | φ (cid:105) .The second condition is particularly important in order to guarantee that errors won’taccumulate during multiple rounds of EC resulting in a logical fault. It was shown in [28] (andapplied in e.g. [29]) that it is the logical failure probability of an exRec, see Fig. 7, instead ofthe failure probability of a single EC unit that should be compared to the bare qubit failureprobability p in order to determine whether the lifetime of an encoded qubit will be longer thanthat of an unencoded qubit. The reason is that single faults in each consecutive EC unit can leadto logical failure since an incoming error and a fault in the unit can combine together. In theliterature, pseudo-thresholds for small distance codes are often computed using a single EC unit.The pseudo-threshold is thus set by the total logical failure probability (probability of a logical X , Y or Z error) of the exRec being equal to p . In Section 6 we explicitly show that the logical failurerate of a single EC cannot be used to estimate the encoded qubit life-time. r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... Figure 7: Illustration of an extended rectangle (exRec). The EC unit consists of performing around of fault-tolerant error correction (in our case, three rounds of syndrome measurementsfollowed by the decoding protocol described in Section (a)). The exRec consists of performingtwo consecutive EC’s and its logical failure rate is determined by the occurrence of two malignantfaults which lead to logical failure on output.Figure 8: Circuit for measuring a weight-five X -check. A single X fault occurring after the thirdCNOT gate can propagate to two data qubits resulting in two outgoing X errors. (a) Decoder for [[30 , , Since the small stellated dodecahedron code is a distance-3 code, any single data qubit error in theEC unit will be corrected. However, the stabilizer checks are weight five which implies, as shownin Fig. 8, that a single fault occurring on some of the CNOT gates can lead to potentially dangerouserrors of weight two. Note that for any check P with wt ( P ) = 5 , a single fault occurring duringits measurement can lead to a data error E with weight at most 2 since min ( wt ( E ) , wt ( EP )) ≤ .Therefore, we need to ensure that any weight-two errors E and E (cid:48) arising from a single faultduring the measurement of an X or Z check either have a unique syndrome compared to eachother ( s ( E ) (cid:54) = s ( E (cid:48) ) ) and compared to single faults which lead to an outgoing weight-one error, orif s ( E ) = s ( E (cid:48) ) , then they must be logically equivalent ( EE (cid:48) ∈ S where S is the stabilizer group).A useful feature of the code is that the triangular logical Z operators in Fig. 3 overlap with anyweight-5 Z -checks on at most 0 or 1 qubit: a triangular logical Z lies in a plane which intersectsthe pentagrammic planes on at most one edge. However, an example of a problematic scenarioinvolving a product of these logical operators is shown in Fig. 9. In this scenario, both pairs ofqubits 1,4 and 2,3 could undergo Z errors arising from a single fault during the measurement ofthe checks, and notice that both pairs of errors have the same error syndrome but are not logicallyequivalent. Thus, correcting the wrong error would lead to a logical fault. To resolve this issue,for the parity check schedule given in Fig. 13, it was verified that every weight two X or Z -errorarising from a single fault during a stabilizer measurement has a unique syndrome.With the above considerations, we now describe a decoding protocol which satisfies the fault-tolerant criteria outlined in Condition 5.1. Given the size of the small stellated dodecahedron code,it is possible to decode X and Z errors separately using full lookup tables (since each containsonly = 2048 syndromes). For a given syndrome s , the lookup table chooses the lowest weighterror E that corresponds to the measured syndrome. However, note that there can be weight-twoerrors E and E (cid:48) such that s ( E ) = s ( E (cid:48) ) with EE (cid:48) ∈ N ( S ) \ S where N ( S ) is the normalizer ofthe stabilizer group. Thus when constructing the lookup table, the corrections associated to allsyndromes s ( E ) where E is a weight-2 error that can arise from a single fault during a stabilizer r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... Figure 9: Two pentagonal Z -checks, illustrated in green and blue, which overlap on two Z logicaloperators (in red). The pairs of qubits, 1,4 and 2,3 are data qubits which can have Z errors arisingfrom a single fault occurring during the measurement of the two checks. The syndromes areindicated by the red exclamation marks.measurement should be E and not some other weight-two error with the same syndrome as E and which is not logically equivalent to E . Note that from the above discussion, this lookup tableconstruction is possible.As with other distance-3 codes, a single round of syndrome measurement isn’t sufficient todistinguish measurement errors from data qubit errors and would thus not be fault-tolerant. Tomake our decoding scheme fault-tolerant, we use the following protocol: Fault tolerant EC unit (for either X or Z errors): Perform three rounds of syndrome measurements resulting in syndromes s , s and s . Note that each round uses the CNOT scheduling depicted in Fig. 13 andFig. 14.(i) If at least two syndromes are trivial, apply no correction.(ii) If at least two syndromes s are identical, apply the correctioncorresponding to s using the lookup table.(iii) If the first two conditions are not satisfied, apply the correctioncorresponding to s (the last measured syndrome) using the lookuptable.Note that this procedure could be implemented to fault-tolerantly decode any distance-3 code,as long as one can pick a scheduling of the CNOTs (or other entangling gates) that guarantees thatall errors arising from a single fault have unique syndromes (and those with the same syndromeare logically equivalent) and one uses these particular errors as the minimum weight correctionsin the lookup table.We now give a rigorous proof that the above procedure satisfies the fault-tolerance criteria ofCondition 5.1.First, if there is an input error E with wt ( E ) = 1 and no faults during the EC rounds, thenall three rounds will report the syndrome s ( E ) and the error will be corrected. Now supposethere are no input errors but a single fault occurs during the EC. If the fault occurs during thefirst round, then rounds two and three will produce the same syndrome and the resulting error r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... will be corrected. If the fault occurs during the third round, then the first two rounds will yielda trivial syndrome and no correction will be applied. However, the output error must then be acorrectable error. Thus ideally decoding the output would result in the input state. Now if thefault occurs during the second round, then all three syndromes could be different (dependingat which time step the error occurred). There is also the possibility that s = s . In both cases, acorrection corresponding to s would be applied removing all errors on the data. Hence the firstcriterion will be satisfied.Lastly, we need to show that the second criterion is satisfied. In fact, we modify the secondcriterion and demand that the output state differs from a valid codeword by an error which iscorrectable by our ideal decoder (the ideal decoder is our Look-Up Table Decoder assuming nofurther errors). As discussed, this could be an error of weight-2. This modification does not alterthe use of this condition in deriving fault-tolerance [28].In what follows, we will consider the case where the input error E has arbitrary weight. Ifthere are no faults during the EC, then all three syndromes will be equal to s ( E ) . Hence applyinga correction E (cid:48) based on this syndrome will always project the code back to the code space (i.e. EE (cid:48) ∈ N ( S ) ). Now suppose there’s a single fault during the first round of the EC. Then thesyndromes s = s will be the syndromes for the combined error E and the resulting errors fromthe single fault during the first round. Thus correcting using s will always project the code backto the code space. If the fault occurs during the second round, then, as in the previous paragraph,the correction will correspond to the last syndrome s which includes both the input error andthe error arising from the fault. Thus correcting using s will always project the code back to thecode space. Lastly, if the fault occurs during the third round, then the first two syndromes s = s will correspond to the input error E . Let E (cid:48) be the resulting data qubit error from the third round.Then correcting using the recovery ˜ E where E ˜ E ∈ N ( S ) , the output state will differ from a validcodeword by an error of weight wt(E’) ≤ which is correctable using our look-up table decoder.
6. Numerical Results
In this section we present numerical results for the pseudo-thresholds of the surface-17 codeof [22] and the small stellated dodecahedron code using the fault-tolerant decoding schemesand circuit-level noise model presented in Section 5. To provide a fair comparison, we choosea sequential X and Z -check schedule for the surface-17 code as well (such sequential schedulemay be a necessity in some architectures anyhow, see e.g. the schedule in [30]). Some of the codecan be found at https://github.com/einsteinchris .To obtain the average lifetime of a physical qubit, suppose that an error is introduced withprobability p at any given time step. The probability that an error is introduced after exactly t time steps is given by f p ( t ) = (1 − p ) t − p . Thus the mean time before a failure occurs is given by (cid:80) ∞ t =1 tf p ( t ) = 1 /p . To obtain a lower bound of the lifetime of an encoded qubit, we can simplyreplace p by the logical failure rate curve p L ( p ) of the exRec (see [28]). For a distance-3 code, p L ( p ) = cp + O ( p ) since the code can correct any single qubit error.In Fig. 10, plots illustrating the pseudo-threshold of the Surface-17 and the small stellateddodecahedron code are shown. In Fig. 11, the circular dots show the average number of ECrounds before failure of encoded qubits for both a single qubit encoded in Surface-17 and 8qubits encoded in the small stellated dodecahedron code (in the simulation, we decoded everythree rounds and propagated residual errors into the next EC unit). Unfortunately, the Surface-17 code has a pseudo-threshold which is about 19 times larger than the dodecahedron code( (3 . ± . × − compared to (1 . ± . × − ). Note that the pseudo-threshold valueswere obtained by the intersection between the curve p (since we are considering a noise modelwhere idle qubits fail with probability p and are concerned about quantum memories) and thelogical failure rate curve of the exRec. The differences in pseudo-thresholds are primarily due tothe larger number of locations in the fault-tolerant circuits of the dodecahedron code comparedto the surface-17 code circuits as well as the fact that both codes have the same distance. In fact,just by counting the number of pairs of CNOT gates in an EC unit one can get an indication of r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... (a) (b)Figure 10: Fig. 10a and Fig. 10b show the total logical failure rate (probability of either an X , Y or Z logical fault) curve for the exRec of the Surface-17 ( p L ( p ) ≈ p ) code and the exRec ofthe small stellated dodecahedron code ( p L ( p ) ≈ p ). The intersection between these curvesand the curve f ( p ) = p gives, in principle, the pseudo-threshold of the codes. Note however thatsince idle qubits fail with probability p , for quantum memories, the relevant cross-over point isgiven by the intersection with the curve p and not p . We find that it is (3 . ± . × − forSurface-17 and (1 . ± . × − for the small stellated dodecahedron code.(a) (b)Figure 11: Fig. 11a shows the average number of EC rounds before failure of an encoded qubitin the surface-17 code while Fig. 11b shows the average number of EC rounds before failure of8 encoded qubits in the small stellated dodecahedron code. Solid lines show /p L where p L isthe logical failure rate (as a function of p ) obtained for both the exRec circuit and the single ECunit circuit. The data clearly shows that the lifetime is lower bounded by /p L obtained from theexRec circuit and not a single EC unit.the pseudothreshold. For the small stellated dodecahedron code the number of CNOT gates is × ×
22 = 330 so that (cid:0) (cid:1) = 54285 while for Surface-17 the number of CNOT gates in an EC-unit is × × so that (cid:0) (cid:1) = 4560 , in rough correspondence with the c ’s in P L ( p ) = cp observed in Fig. 10.Note that since the dodecahedron code encodes 8 logical qubits, a fairer comparison would beto compare the logical failure rate of the dodecahedron code with that of 8 qubits encoded in thesurface-17 code. In general, if the logical failure rate of an extended rectangle of the code is givenby p L ( p ) , the logical failure rate of m copies of the code is given by p ( m ) L ( p ) = 1 − (1 − p L ( p )) m = mp L ( p ) + O (( p L ( p )) ) . r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... Figure 12: Comparison of eight logical qubits encoded in the surface-17 code, with a total logicalfailure rate given by − (1 − p L ( p )) where p L ( p ) ≈ p , with 8 logical qubits encoded in thedodecahedron code with p L ( p ) ≈ p . It can be seen that the surface-17 still outperforms thedodecahedron code since × (cid:28) .In Fig. 12, we compare the logical failure rate of eight qubits encoded in the dodecahedroncode with eight qubits encoded in the surface-17 code. It can be seen that the surface-17 code stillachieves a lower logical failure rate compared to the dodecahedron code.
7. Discussion
The fault-tolerance analysis for the small stellated dodecahedron code has shown the difficultyof getting a block code with high pseudo-threshold when we include noise in the parity checkcircuits themselves. The EC unit of this code is simply larger since many more checks need to bemeasured and the pseudo-threshold is determined by pairs of malignant locations in this largeunit. In contrast, separate copies of the surface code, each with a much smaller EC unit benefitfrom having "room for each logical operator". One might expect that this problem becomes lesssevere for larger hyperbolic codes which have shown lower but still good performance versussurface codes for a phenomenological noise model [5].One could consider how Steane error correction can improve the performance of the smallstellated dodecahedron code: we expect that the qubit overhead will be larger (mainly due to therequirement for preparing four logical | (cid:105) and four logical | + (cid:105) ancillas) but the pseudo-thresholdwould be quite better. The tetrahemihexahedron code [[12 , , (see Table 1) with some weight-3checks might be an interesting variation on the × rotated surface code (with d Z = 3 , d X = 4 ).Lastly, we also tried to use only four of the eight logical qubits of the small stellateddodecahedron code for encoding logical information in order to see if significant improvementsin the pseudo-threshold could be obtained. However, our numerical simulations showed that forvarious choices of the logical qubits, the pseudo-threshold improved by less than a factor of two.The primary reason is that in most cases where a failure occurred, several logical qubits wereafflicted.A goal for future work would be to compare the performance of the small stellateddodecahedron code with the surface code for a physically-motivated noise model in an optically-linked ion-trap architecture [31] or an optically-linked NV-center in diamond architecture [32].Funding. We acknowledge support through the EU via the ERC GRANT EQEC No. 682726. This researchwas supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute issupported by the Government of Canada through Industry Canada and by the Province of Ontario through r s pa . r o y a l s o c i e t y pub li s h i ng . o r g P r o c R S o c A .......................................................... the Ministry of Economic Development & Innovation. C.C. acknowledges the support of NSERC through thePGS D scholarship.
Acknowledgements.
We would like to thank Kasper Duivenvoorden and Christophe Vuillot for usefuldiscussions. We acknowledge the use of valuable computing time on the RWTH Aachen Compute Cluster.We thank Koen Bertels for quick access to the 4 machines at Computer Engineering TU Delft, and Steve Weissfor the use of computing clusters at IQC Waterloo. C.C. would like to acknowledge TU Delft for its hospitalitywhere the work was completed.
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