3.7 Numerics
3.7.3 The toric code
As discussed in Section 3.6.3, for the toric code on the honeycomb lattice (see Fig. 3.1), the Hamiltonian of the model is Htop = −(P
pBp +P
vAv), where Bp = X⊗6 is a tensor product of Pauli-X operators on the six edges of the plaquette p, and Av = Z⊗3 is a tensor product of Pauli-Z operators on the three edges incident on the vertexv. We point out that the toric code on a honeycomb lattice has several differences compared to a toric code on a square lattice (which is often considered in the literature). Assuming that both lattices are defined with periodic boundary conditions,
(i) there are twice as many vertices compared to plaquettes on a honeycomb lattice (as opposed to the same number on a square lattice)
(ii) the vertex termsAv =Z⊗3 of the Hamiltonian have odd weights (as opposed to even weight for the square lattice)
(iii) the weight of a logical minimal ¯X-string operator (i.e. the number of spins it acts on) is roughly twice as large compared to the corresponding minimal ¯Z-string operator on the dual lattice (as opposed to the square lattice, where both operators have the same weight). For the 12-qubit code of Fig. 3.1, an example of such a pair ( ¯X,Z) of lowest-weight logical operators¯ is given below in Eq. (3.51).
Properties (i) and (ii) imply that the usual symmetries X ↔ Z and Z ↔ −Z of the toric code on the square lattice are not present in this case. The absence of these symmetries is reflected in our simulations. Property (iii) also directly affects the final state, as can be seen by the pertur-bative reasoning of Section 3.6.1: ¯Z-string operators appear in lower order in perturbation theory compared to ¯X-string operators.
(Non)-adiabaticity. We first present the adiabaticity erroradia(T) for the HamiltonianHtriv(θ) given by (3.46) (for different values of θ) as a function of the total evolution time T. Fig. 3.2 illlustrates the result. It shows that for sufficiently long total evolution times T, the Hamiltonian interpolation reaches the ground space of the toric code when the initial Hamiltonian is Htriv(θ= π) =−P
iZi; this is also the case forθ∈ {π/4, π/2,3π/4}. However, if the initial Hamiltonian isHtriv(θ= 0) =P
iZi, then the final state Ψ(T) is far from the ground space of the toric code Hamiltonian Htop. This phenomenon has a simple explanation along the lines of Observation 3.2.3. Indeed, ifθ= 0, then every vertex termsAv =Z⊗3commutes with both Htriv as well asHtop (and thus all intermediate Hamiltonians H(t)). In particular, the expectation value of the vertex terms remains constant throughout the whole evolution, and this leads to an adiabaticity error adia(T) of 1 in the case of Htriv(θ= 0) =P
iZi.
In Figs. 3.3a, 3.3b, we consider neighborhoods of Hamiltonians of the form (cf. (3.47)) Htriv+ (a, b) around Htriv+ (0,0) =Htriv(θ= 0) =X
j
Zj and Htriv− (a, b) around Htriv− (0,0) =Htriv(θ=π) =−X
j
Zj .
Total evolution time T
0 50 100 150
Adiabaticity error 0 adia(T) 10-5 10-5 10-4 10-3 10-2 10-1 100
3=0 3=:/4 3=:/2 3=3:/4 3=:
Figure 3.2: This figure gives the adiabaticity error adia(T) = 1− hΨ(T)|P0(T)|Ψ(T)i (cf. (3.49)) as a function of the total evolution time T and the initial Hamiltonian chosen. For the latter, we consider the one-parameter family Htriv(θ) given by (3.46). For θ = 0, the adiabatic evolution is not able to reach the final ground space because initially hAvi =−1 for every vertex operator Av =Z⊗3, and this quantity is conserved during the evolution. This is a feature of the honeycomb lattice because the vertex terms Av have odd weights. For other values of θ, the ground space is reached for sufficiently large total evolution times T.
The initial Hamiltonians Htriv(θ= 0) andHtriv(θ=π) correspond to the center points in Fig. 3.3a and 3.3b, respectively.
• In the first case (Fig. 3.3a), we observe that for all initial Hamiltonians of the formHtriv+ (a, b) in a small neighborhood ofHtriv+ (0,0), the adiabaticity error adia(T) is also large, but drops off quickly outside that neighborhood. This is consistent with the relevant level crossing(s) being avoided by introducing generic perturbations to the initial Hamiltonian.
• In contrast, almost all initial Hamiltonians in the familyHtriv− (a, b) (around the initial Hamil-tonian Htriv− (0,0)) lead to a small adiabaticity erroradia(T) (Fig. 3.3b), demonstrating the stability of the adiabatic preparation.
In a similar vein, Fig. 3.3c illustrates the non-adiabaticity for the family of Hamiltonian Htriv−X(a, b) =−(1−a2−b2)1/2X
j
Xj+bX
j
Yj+aX
j
Zj . (3.50)
The familyHtriv+X(a, b) (defined with a positive square root) would behave exactly the same due to the symmetry +X↔ −X.
Logical state. For the 12-qubit rhombic toric code (Fig. 3.1), logical observables associated with the two encoded logical qubits can be chosen as
X¯1 = X7X8X11X12
Z¯1 = Z10Z12 and X¯2 = X4X0X2X12
Z¯2 = Z1Z2 .
Because of the symmetry (3.7.2), however, these are not independent for a state Ψ(T) (or more precisely, its projectionP0(T)Ψ(T)) prepared by Hamiltonian interpolation from a product state:
their expectation values satisfy the identities
hZ¯1i=hZ¯2i and hX¯1i=hX¯2i. We will hence use the two (commuting) logical operators
X¯ = ¯X1=X7X8X11X12 and Z¯ = ¯Z2 =Z1Z2 (3.51)
3.7. NUMERICS 85
(a) The adiabaticity error adia(T) in the neighborhood around Htriv+ (0,0) = P
iZifor different HamiltoniansHtriv+ (a, b).
As explained, the evolution cannot reach the ground space of the toric code around (a, b) = (0,0) because the expecta-tion values of plaquette-operators are log-scale because the variation in values is small. The ground space of the toric code HamiltonianHtop is reached for almost the entire parameter region.
(c) The logarithm of adiabaticity error lnadia(T) in the neighborhood around Htriv−X(0,0) =−P
jXj for different HamiltoniansHtriv−X(a, b). Note that the resulting figure would look identical for the HamiltoniansHtriv+X(a, b) because of the−X ↔+X symmetry.
Figure 3.3: The adiabaticity error adia(T) = 1− hΨ(T)|P0(T)|Ψ(T)i, measuring how well the final state Ψ(T) overlaps with the ground space of the toric code. All three figures are for a total evolution time T = 120. In Fig. 3.3a, we consider the family of initial Hamiltonians Htriv+ (a, b) in the neighborhood of Htriv+ (0,0) =Htriv(θ= 0) =P
jZj. In contrast, Fig. 3.3b illustrates different choices of initial Hamiltonians Htriv− (a, b) aroundHtriv− (0,0) =Htriv(θ=π) =−P
jZj. The values (a, b)⊂R2are restricted to the unit disca2+b2 ≤1; the center points of the two figures correspond respectively to θ= 0 and θ=π in Fig. 3.2. Finally, Fig. 3.3c gives the non-adiabaticity error for initial Hamiltonians of the form Htriv−X(a, b) (as defined in Eq. (3.50)).
to describe the obtained logical state.
In Fig. 3.4, we plot the expectation values of ¯Z and ¯X in the final state Ψ(T) for initial Hamiltonians of the form (cf. (3.47) and (3.50))
Htriv− (a, b) around Htriv− (0,0) =−X
j
Zj
Htriv−X(a, b) around Htriv−X(0,0) =−X
j
Xj
We again discuss the center points in more detail. It is worth noting that the single-qubit {Zi} operators correspond to the local creation, hopping and annihilation of m anyons situated on plaquettes, whereas the operators{Xi}are associated with creation, hopping and annihilation ofe anyons situated on vertices. In particular, this means that the initial Hamiltonians associated with the center points in the two figures each generate processes involving only either type of anyon.
• ForHtriv− (0,0) =−P
iZi, we know that hZ¯i= 1 during the entire evolution because ¯Z com-mutes with the HamiltoniansH(t), and the initial ground state Ψ(0) is a +1 eigenstate of ¯Z.
In Figs. 3.4a and 3.4b, we can see that there is a large region of initial HamiltoniansHtriv− (a, b) around Htriv− (0,0) =−P
iZi which lead to approximately the same final state.
• On the other hand, as shown in Figs. 3.4c and 3.4d, the stable region of HamiltoniansHtriv−X(a, b) around the initial Hamiltonian Htriv−X(0,0) = −P
iXi is much smaller. This is due to the fact that the operator ¯X appears in higher order perturbation expansion compared to ¯Z, and the evolution time T is taken to be quite long. Given sufficiently large total evolution timeT, in the neighborhood of Htriv−X(0,0) =−P
iXi, the lower order term ¯Z in the effective Hamiltonian will dominate the term ¯X associated withV =−P
iXi.
However, in both cases considered in Fig. 3.4, we observe that one of two specific logical states is prepared with great precision within a significant fraction of the initial Hamiltonian parameter space.
3.7. NUMERICS 87
(a) The expectation valuehX¯iof the final state Ψ(T), for initial HamiltoniansHtriv− (a, b) in the neighborhood of Htriv− (0,0) = −P
jZj. Note that, as illustrated in Fig. 3.3b, the ground space of the toric code is reached for the whole parameter range; hence these values, to-gether with the expectation values shown in Fig. 3.4b uniquely determine the state Ψ(T). Hamiltoni-ansHtriv− (a, b) (we plot the logarithm because the vari-ation is small) as in Fig. 3.4a.
a
jXj. The corresponding adiabaticity error is shown in Fig. 3.3c.
(d) The quantity hZ¯i for initial Hamiltoni-ansHtriv−X(a, b) as in Fig. 3.4c.
Figure 3.4: These figures illustrate the expectation values hX¯i and hZ¯i of string-operators (cf. (3.51)) of the final state Ψ(T), for different choices of the initial Hamiltonian. The total evolution time is T = 120.
Total evolution time T
0 20 40 60 80 100
Adiabaticity error 0 adia(T) 10-6 10-4 10-2 100