There are different ways to describe anyons, including the Chern–Simons (CS) theory and lattice Hamiltonian approach [ 8,9 ]. For CS theory, it is well known that there is an additional gauge-invariant term that can be added to the Maxwell or Yang–Mills Lagrangian in (2 + 1) dimensions. This CS term is topological, as it does not depend on the metric [ 8,45 ]. At low temperatures, this term dominates. In the non-abelian case, the action is invariant under SU(2) =∼ Spin(3) and can be written as a Gauss constraint on a wave function of the gauge fields.

In the presence of sources (representations of a Lie algebra), anyonic behavior, such as fusion and braid dynamics, can be found with sufficient control of the low-temperature Hamiltonian, Lagrangian, or Gaussian constraints. The degenerate ground state of the effective theory is associated with the CS sources form the so-called fusion Hilbert space, which is proposed as a fault-tolerant topological quantum computing substrate. In the case of Fibonacci anyons, the sources can only be in the two lower-dimensional representations of SO(3), the spin-0 and spin-1 representations, with the fusion rules

If we have N spin-1 representations as sources and start to fuse them, they can build different fusion paths that can lead to either spin-1 or spin-0 representations with certain probabilities. The different paths to fuse the N spin-1 sources to only one spin-1 or spin-0 source can be seen as states in a fusion Hilbert space HN, where its dimension grows with the number of original spin-1 sources and is given by the Fibonacci sequence, ((0, 1, )1, 2, 3, 5, 8, 13, . . . , Fib(N + 1)) [ 46 ], i.e., HN = CFib(N+1), where Fib(N + 1) is the N + 1th Fibonacci number.

Rotating one physical source around the other is equivalent to an operation in the fusion Hilbert space described by the so-called braid operators (higher-dimensional representations of the braid group), which leads to non-trivial statistics given the necessary quantum evolution for topological quantum computation. The explicit construction of braid operators, B, is given as examples in ([ 46 ], Sections 2.4 and 2.5) through the so-called F-matrices and R-matrices operating in the fusion Hilbert space. For the case of fusing two anyons into a third one, this process is a five-dimensional space, and the explicit matrices in a suitable base can be given by

R = diag(e4πi/5, e−3πi/5, e−3πi/5, e4πi/5, e−3πi/5), with B = FRF−1 and φ = 2 cos( π5 ) ≈ 1.618, the golden ratio.

More details on Fibonacci anyons are well known and can be found in Ref. [ 46 ] and references therein. Less known is the isomorphism of the fusion Hilbert spaces with representations of certain C∗-algebras, in particular, the so-called Fibonacci C∗-algebra [ 12 ]. In [ 12 ], it is shown that the fusion rules determine the data of a Bratteli diagram [ 47 ], which specifies an approximately finite-dimensional (AF) C∗-algebra with a representation on a Hilbert space, which is isomorphic to the anyonic fusion Hilbert space. An AF C∗-algebra A is given by a direct limit A = l−i→mAn of a finite-dimensional C∗-algebra An, where An is a direct sum of matrix algebras over C, An = ⊕kN=n1Mrk (C). Similarly, a Hilbert-space representation of A, HA, is obtained as a direct limit of a system of finite-dimensional Hilbert spaces HnA, HnA = ⊕kN=n1Crk . A Bratteli diagram yields a unique C∗-algebra and allows for a simpler computation of the dimension of the Hilbert-space representations of this algebra by counting the number of paths to a certain node. For the Fibonacci C∗-algebra, see ([ 48 ], Example III.2.6) and ([ 12 ], Section 3.2), for the Bratteli diagram illustration and the dimension of the Hilbert-space computation. The isomorphism between the representations of Hilbert spaces and the anyonic-fusion Hilbert spaces is given in ([ 12 ], Lemma 3.3), where the dimensions of Fibonacci anyons and the Fibonacci C∗-algebra both grow with the Fibonacci sequence. (1) (2) (3) (4) (5)

In analogy with the anyonic case, we will provide a physical description of the anyonic behavior of quasicrystals to allow for concrete physical implementation and then the associated effective fusion Hilbert space to deal with topological quantum information processing. It is more common to deal with quasicrystals from the point of view of Bloch theory for periodic many-body atomic quantum systems, but even within this point of view there are different implementations. While the quasicrystal literature is fast growing, we mention the quasicrystalline extension of the Bloch theory in context of the gap-labelling theorem [ 16 ] and the discovery of a few exact solutions for quasicrystal Hamiltonians [ 17–19,25,28,32 ]. We also highlight more developments in terms of computations of the spectrum and band structure [ 20–24,26,27 ] and the study of topological properties [ 33–39 ]. Finally, quasicrystals have been actively studied in recent years [ 29–31,40–43,49 ]. From our understanding, the different approaches have convergent results, including the self-similar structure of the energy spectrum, band structure, and topological protected phases. The geometric cut-and-project method, or its more general form, called model sets, describes this structure.

The starting point is the periodic Bloch theory considering the Schrodinger equation for a particle over the atomic structure with a periodic potential V(r + R) = V(r) for all lattice vectors R of a given lattice L. With this setup, the Hamiltonian commutes with the translation operators, and the Bloch theory diagonalizes both simultaneously. For this, one introduces the reciprocal lattice L∗ with primitive translation vectors K, where the scalar product R · K is an integer multiple of 2π. The eigenfunctions are such that k exists as ψk+K(r + R) = eik·Rψk(r), in which ψk(r) the Bloch wavefunctions on Rn × Rn (r in the Voronoi cell V and k in its dual V∗, also called Brillouin zone). The curves of the spectrum are periodic in a dual reciprocal space, and the entire band structure is defined by the band structure inside the first Brillouin zone.

Our idea is to study the Hilbert space of ψ’s satisfying Bloch’s theorem, such that ||ψ||2 < ∞. We then introduce, for each k ∈ V∗, the Hilbert space Hk of the functions u on Rn, such that

u(r + R) = eik·Ru(r), and ||u||2 < ∞, with HL = ⊕Hk, and the dimension grows with the number of points on the lattice. The Hilbert spaces for a particle over an aperiodic potential from a quasicrystal will be seen as a subspace of the lattice Hilbert space HL, and we will need to review the cut-and-project method to obtain the quasicrystal from the lattice L.

We consider a cut-and-project scheme (CPS) to be a 3-tuplet G = Rd, Rd0 , L , where the parallel space Rd and the perpendicular space Rd0 are real euclidean spaces, L is the lattice in E = Rd × Rd0 , and is the embedding space with two natural projections π: Rd × Rd0 → Rd and π⊥: Rd × Rd0 → Rd0 subject to the conditions that π(L) is injective, and that π⊥(L) is dense in Rd0 . With L = π(L), this scheme has a well-defined map called the star map ? : L → Rd0 :

x 7−→ x? := π⊥(π−1(x)).

For a given CPS G and a window W, quasicrystal point sets (4λγ(W)) can be generated by setting two additional parameters: a shift γ ∈ Rd × Rd0 /L with γ⊥ = π⊥(γ) and a scale parameter λ ∈ R. The projected set (6) gives the quasicrystal point set.

Another important concept is the tiling of the Euclidean space from the point set. Consider that a pattern T in Rd (T @ Rd) is a non-empty set of non-empty subsets of Rd. The elements of T are the fragments of the pattern T . A tiling in Rd is a pattern T = {Ti | i ∈ I} @ Rd, where I is a countable index set, and the fragments Ti of T are non-empty closed sets in Rd subject to the conditions

∪i∈I Ti = Rd, int(Ti) ∩ int(Tj) = Ø for all i 6= j, and Ti is compact and equal to the closure of its interior Ti = int(Ti).

While this is trivial for lattices with unique unit cells, quasicrystals have more than one unit cell. Multiple quasicrystals with the same number of points N from L projected to the parallel space can lead to different tilings depending on the shift parameter γ.

The construction above identifies the quasicrystal point set as a subset of the original lattice in the embedding space and its Hilbert space H4 as a subspace of the lattice Hilbert space HL. An explicit example is given in ([ 16 ], Section 3.2) for the one-dimensional Fibonacci chain derived from the Z2 lattice. This provides access to the physical properties of quasicrystals, such as their electronic structure. However, the full tiling structure is not properly captured by these descriptions. To address the different tiling configurations of quasicrystals, it is standard to consider the associated C∗-algebra structures ([ 50 ], Sections II.3 and V.10) and the notion of tiling spaces [ 51 ]. A simple way to look at this is to decompose the quasicrystalline Hilbert space H4 further according to tile configurations. The one-dimensional Fibonacci chain and the two-dimensional Penrose tiling can be described by only two tiles. For the Fibonacci chain, they are called long (L) and short (S) edges. For the Penrose tiling, they can be given either by a fat rhombus (F) and a thin rhombus (T) or two quadrilaterals called kites and darts.

We can then consider the Hilbert spaces HL4,F and HS4,T associated with the two different tiles. The frequency of the appearance of these tiles in some tiling is constant and grows with the Fibonacci sequence, given, at some step, as F(N) for L or F to F(N − 1) for S or T. From the Bloch theory, the number of states depends on the number of points in the lattice, which translates to the number of tiles. A lattice trivially has only one tile. For quasicrystals, the number grows differently depending on the tiling considered. Both the Fibonacci chain and the Penrose tiling contain two fundamental tiles that grow with the Fibonacci 4 4 sequence. As such, the Hilbert spaces HL,F and HS,T subspaces of a quasicrystalline Hilbert space (which are subspaces of lattices Hilbert spaces) have dimensions that grow with the number of tiles added to the quasicrystal in the same way that the dimensions of the anyonic fusion Hilbert spaces grow with the addition of anyons. Following the discussion from the previous section, we conclude that these quasicrystalline subspaces are candidates for the implementation of representations of the Fibonacci C∗-algebra associated with Fibonacci anyons. We see the tiles emerging from the Bloch theory playing the same role of the non-abelian SO(3) sources in the Chern–Simons theory.

Another perspective is to consider the tiling space, which leads to Hilbert spaces that are isomorphic to the ones considered above with dimensions growing with the Fibonacci sequence. Basically, we start with a quasicrystal point set 4γ and associates a tiling with it. Then, we can shift the point set by shifting the window in perpendicular space using γ⊥. Each shift generates a new tiling with the same tiles but with a different configuration, where these tiles can be seen in both parallel and perpendicular spaces due to the star map. The difference is that, in parallel space, there is a growth of the quasicrystal with tiles of fixed length, while, in the perpendicular space, each point added rescales the tiles and reorganizes the configuration leading to a rescaling of the space, which is usually called inflation or deflation for the inverse process. Each tiling is a point in the so-called tiling space, which encodes all possible tilings that can be made with a fixed CPS and window. To encode this information, we can fix a point x inside the window in the perpendicular space. As the points are projected, with π⊥(L), we can track the tile type around x after a new point is projected. Then, we can generate different tilings from different shifts and track the sequence of tiles around that point x over the different sequence of projections.

Equivalently, one can use only one projection and track the evolution of different positions inside the window. Each tiling is described by a sequence that encodes the evolution of tiles around x in the perpendicular space as the quasicrystals grow in parallel space. By labelling the Fibonacci-chain and Penrose-tiling letters L or F as the symbols 1, and S or T as 0 we can associate different sequences (xi)n of 0s and 1s with x, where i indexes the different sequences of projections, and n ∈ N is the level in one sequence of projections. The only constraint on these sequences, which arises from the geometry of the CPS with fixed window, is that, if (xi)n = 0, then (xi)n+1 = 1. We illustrate this for the Fibonacci chain in Figure 1, where x1 = 1111101 . . ., and x2 = 011011 . . ., for example. (7) (8)

The Penrose tiling is shown in Figure 2, where x1 = 110 . . ., and x2 = 111 . . .

Additionally, an equivalence relation is defined on this space of sequences. Tilings Ti and Tj with some m, such that (xi)n = (xj)n for n ≥ m, are equivalent. This is presented in detail in ([ 50 ], Sections II.3 and V.10) for the tiling space of the Penrose tiling with the construction of a C∗-algebra A associated with this space. Remarkably, this algebra is the same Fibonacci C∗-algebra; the Hilbert-space representations are isomorphic to the anyonic fusion Hilbert spaces [ 12 ]. In the next section, we present detailed aspects of this algebra, quasicrystal physics interpretations, and topological quantum computation.

Let us consider a concrete solution of a Hamiltonian for a quasicrystal. Despite the difficulties with the generalization of the Bloch’s and Floquet’s theories, there are a few known exact solutions for quasicrystal Hamiltonians. Some of the state solutions of the so-called tight-binding model for the Fibonacci chain and the Penrose tiling are known [ 17–19,25,28,32 ]. These states include zero-energy degenerate states and have a similar form to the Bloch wave function, Equation (4), given by

ψ(i) = C(i)eκh(i) where κ ∈ R is a constant, C(i) are local site-dependent periodic functions given the local amplitudes and h(i) is a non-local height field dependent on the geometry of the specific tiling. For the Fibonacci chain in Equation (7), the zero-energy state takes the form ψ(2i) = (−1)ieκh(2i) with κ = ln φ, and the field h(2i) given by h(i) = ∑ B(2j → 2(j + 1)),

0≤j≤i where B(LS) = 1, B(SL) = −1, and B(LL) = 0. For the Penrose tiling, both κ and C(i) are computed numerically [ 28 ], but the ribbon description discussed above allows us to access the Fibonacci chain subspaces directly. Note that a flip LS → SL, such as the the one for the ribbon Rb in Figure 3, changes the state by a factor of φ−2, ψLS(i) = φ−2ψSL(i).

= = = Bn−1Bn,

BmBnBm, if |n − m| = 1 with φ = −A2 − A−2, where unitarity is guaranteed if the projections En are Hermitian. A contains four solutions, all with |A| = 1. The four solutions are A = e3πi/5, −e3πi/5, e2πi/5, and −e2πi/5. Note that the R-matrix for Fibonacci anyons in Equation (2) contains e3πi/5 on some of the diagonals. With the solution of A provided, one can verify that ρA(Bn)ρA(Bn−1) =

ρA(Bn−1)ρA(Bn) ρA(Bn)ρA(Bm)ρA(Bn) =

ρA(Bm)ρA(Bn)ρA(Bm) if |n − m| = 1 ρA(Bn)ρA(Bm) = ρA(Bm)ρA(Bn) if |n − m| > 1.

(15)

Therefore, the quasicrystal projection operators can be used to construct a representation of the braid group.

The usual step from quantum computation to topological quantum computation can now be performed with quasicrystals by finding an embedding e of an N-qubit space (C2)⊗N into a subspace of the tiling space. The embedding does not need to be efficient, because it is well known that the braid group can approximate any universal quantum gate to any desired precision. The computational subspace of the tiling space can be given by fixing one equivalence class (xi)n, n = 1, . . . , 2N + 1 and i = 1, . . . , d with d the number of sequences with (xi)2N+1 = 1. We represent this subspace using TN,1 = (xi)n. Finally, to simulate a quantum circuit, we can have

C2 ⊗N

U ↓ C2 ⊗N →e →e

TN,1 ↓ ρA(B) TN,1. (16) (17) (18)

Explicit matrix representations of ρA(B) can be obtained from the algebra AN(q) acting on the N-qubit Hilbert space (C2)⊗N, a subspace of the tiling space. Define E(q) acting on C2 ⊗ C2 as [ 55 ]

E(q) = [2]q−1 q−1e11 ⊗ e22 + qe22 ⊗ e11 + e12 ⊗ e21 + e21 ⊗ e12 with eij the two-dimensional matrix units and Ei(q) = I ⊗ . . . ⊗ I ⊗ E(q) ⊗ . . . ⊗ I, where E(q) acts on the positions i and i + 1 of the tensor product.

For TQC with a quantum–mechanical quasicrystal, suppose that researchers in the future could have complete control of how the quasicrystal is inflated or deflated. The number of possible inflation/deflation paths in the tiling space, which gives the Hilbertspace dimension, is tied to the number of physical tiles, analogous to how the number of physical anyons define the fusion Hilbert-space dimension. This allows us to obtain a dictionary between concepts related to Fibonacci anyons and TQC with a quantummechanical quasicrystal. For concreteness and simplicity, consider the Fibonacci chain, which has two inflation rules

Rule A: Rule B: {L → LS, S → L} {L → SL, S → L}.

To clarify, our conventions are that the inflation rules apply an inflation. It can be verified that the successive application of Rule A seeded by S leads to the reverse of the chain found by the successive application of Rule B. If n arbitrary combinations of Rule A and Rule B are applied from the seed, then 2n states can be found. However, these lead to various duplicate tilings, such that Fib(n + 2) unique tilings are found. For example, with seed L, for n = 2 we have {{L, SL, LSL}, {L, SL, LLS}, {L, LS, SLL}, and {L, LS, LSL}} resulting in three unique states {LSL, LLS, SLL}, or, in terms of the (xi), i = 1, 2, 3, describing the associated tiling space, we have {LSL, LLS, LLL}. The associated Bratteli diagram is shown in Figure 4, which is equivalent to the Fibonacci anyon diagram [ 12 ] and the AN(q) diagram for the Jones–Wenzl projections [ 53 ].

The analogue of an anyonic fusion process is given by the operation to forget the Nth step in (xi)n, n = 1, . . . , N, leaving the sequences (xi)n with n = 1, . . . , N − 1. This sends the system from level N to N − 1 or the Hilbert space of dimension from F(n) to F(n − 1) and is equivalent to a deflation of the physical quasicrystal. Since L is a fixed length, this operation acting on the Hilbert space associated with the two tiles LS would lead to L as a deflation, which decreases the length of the chain. When performing the analogue of braiding in the quasicrystal, one specifies a basis given by inflation/deflation paths (xi)n and decomposes the projection En in a direct sum of projections acting in lower-dimensional subspaces. From Equation (14), the subspace acted in by En reaches a different phase, which relates to A and a rescaling by φ. In usual anyonic systems, the braid operations involve a basis transformation. This selects two anyons to be fused and applies an operation to these two anyons, which gives a phase R and then applies an inverse basis transformation. In quasicrystals, the projection En directly selects the subspace to be acted on by a phase and rescaling. Table 1 summarizes a dictionary that compares the aspects of Fibonacci anyons and quantum–mechanical Fibonacci chains for TQC.

We have already noted that crystallographic theories, mainly Bloch’s and Floquet’s theories, do not extend directly to quasicrystals due to the lack of translational symmetry. We also discussed an isomorphism between anyonic and quasicrystalline Hilbert spaces. In this context, it is tempting to import well-developed techniques from anyonic systems for applications in quasicrystals to implement TQC. One example is the so-called golden chain [ 56 ], which models Fibonacci anyons in one dimension. The golden chain has a natural realization in terms of the Fibonacci-chain quasicrystal. The local Hamiltonian Hi acting on the ith Fibonacci anyon on the chain discussed in [ 56 ] is immediately identified with the projections En, acting on the inflation level n, (x)n of the Fibonacci-chain quasicrystal, allowing access to the quantum quasicrystal growth and shrinkage. A detailed analysis of this Hamiltonian (and other anyonic Hamiltonians) in the context of quasicrystals and their relationship with quasicrystal Hamiltonians could be discussed in future work.