An optimal dissipative encoder for the toric code

Dissipation, while generally seen as detrimental for quantum computers, can nevertheless be a useful resource if suitably engineered. Appropriately chosen system–bath couplings can result in non-equilibrium dynamics where initial states converge toward some dynamical steady state. This kind of 'quantum reservoir engineering' has been proposed as a viable approach to the experimental preparation of interesting many-body states [11, 18]. Remarkable examples include the preparation of pure states with long-range order in Bose–Einstein condensates [6], photonic arrays [17], as well as topologically ordered states [2, 18]. More generally, Verstraete et al [18] argued that, at least in principle, an arbitrary quantum computation can be realized by dissipation. The corresponding process is similar to Feynman's clock construction and has the final state of the computation as its steady state. Subsequent work [8] following this program proposed dissipative gadgets allowing the realization of different dissipative dynamics during subsequent time intervals.

Here we examine the dissipative preparation of specific topologically ordered states. While not realizing a fully dissipative computation, this basic primitive could act as a building block in a hybrid scheme where initial states for quantum computation are prepared by thermalization and subsequent computations are performed in the usual framework of topological quantum computation. We ask whether dissipative processes can be used to realize an encoder, i.e. a map which turns states on individual physical qubits into encoded (many-qubit) states. In contrast, previous work only considered the dissipative preparation of some ground state without guarantees on the logical information.

It is worth mentioning that various unitary encoders are known for topologically ordered systems. For the toric code [9] on an L × L lattice, Dennis et al [5] gave a unitary circuit with two-local controlled-not (CNOT) gates of depth Θ(L²) acting as an encoder. Bravyi et al [3] showed for any evolution under a local time-dependent Hamiltonian acting as an encoder requires time at least Ω(L). In turn, Brown et al [4] present a duality transformation from a two-dimensional (2D) cluster state to a topologically ordered state, which can be interpreted as a geometrically local quantum circuit of matching depth. Dropping the requirement of locality, Aguado and Vidal [1] gave an encoder with depth O(logL) with geometrically non-local two-qubit gates.

The dissipative encoder considered here may be realized by designing suitable system–environment interactions. This is to be contrasted with schemes involving error correction, which generally consist of syndrome extraction by measurement and associated correction operations. For the toric code, an encoding procedure of this form was given [7]. It involves active error correction operations similar to the minimal matching technique used in [5].

To define the notion of an encoder in more detail, consider a quantum error-correcting code $\mathcal {Q}\cong (\mathbb {C}^2)^{\otimes k}\subset (\mathbb {C}^2)^{\otimes n}$ encoding k logical qubits into n physical qubits. Informally, an encoder is a map taking any state $|\Psi \rangle \in (\mathbb {C}^2)^{\otimes k}$ into its encoded version $|\overline {\Psi }\rangle \in \mathcal {Q}\subset (\mathbb {C}^2)^{\otimes n}$. (This notion implicitly assumes a choice of basis of $\mathcal {Q}.$) Since we are interested in a physical system of n qubits, we will require the encoder to convert a 'simple' unencoded initial state into an encoded logical state. That is, we ask that for a fixed subset A₁,...,A_k of qubits and a fixed product state |φ〉_Ak+1⊗···⊗|φ〉_An on the remaining qubits, the encoder maps

$$\begin{equation} |\Psi\rangle_{A_1\cdots A_k}\otimes |\varphi\rangle_{A_{k+1}}\otimes\cdots\otimes |\varphi\rangle_{A_n}\,\mapsto \, |\overline{\Psi}\rangle\in\mathcal{Q} \quad{\rm for \ all} \ |\Psi\rangle\in(\mathbb{C}^2)^{\otimes k}. \end{equation} \tag{ 1 }$$

We are interested in encoders realized by evolution under a Markovian master equation

$$\begin{eqnarray*} &&\frac{\rm d}{{\rm d}t}\rho=\mathcal{L}(\rho). \end{eqnarray*}$$

Here the Liouvillian has Lindblad form

$$\begin{eqnarray*} \mathcal{L}(\rho) &=\sum_j L_j\rho L_j^\dagger -\frac{1}{2}\left\{L_j^\dagger L_j,\rho\right\} \end{eqnarray*}$$

with Lindblad operators L_j acting locally on a constant number of qubits. We ask whether the completely positive trace-preserving map (CPTPM) $\mathrm {e}^{t\mathcal {L}}$ generated by $\mathcal {L}$ is an (approximate) encoder for sufficiently large times t, i.e. whether it can realize the map (1). Our result is the following.

Theorem 1. Let $\mathcal {Q}\subset (\mathbb {C}^2)^{\otimes 2L^2}$ be the toric code consisting of 2L² qubits on the edges of a periodic L × L lattice. Consider the partition of the qubits into disjoint sets $\mathcal {A}\cup \mathcal {B}\cup \mathcal {B}'\cup \mathcal {C}\cup \mathcal {C}'\cup \mathcal {D}$ shown in figure 1. That is

• $\mathcal {A}=\{A_1,A_2\}$ are two neighboring qubits having a common adjacent vertex v_* and plaquette p_*;
• $\mathcal {B}=\{B_1,\ldots ,B_{L-1}\}$ and $\mathcal {C}=\{C_1,\ldots ,C_{L-1}\}$ are located along a vertical, respectively horizontal, line passing through A₁;
• $\mathcal {B}'=\{B'_1,\ldots ,B'_{L-1}\}$ and $\mathcal {C}'=\{C'_1,\ldots ,C'_{L-1}\}$ are located along a horizontal, respectively vertical, line passing through A₂ and
• $\mathcal {D}$ are the remaining 2(L − 1)² qubits.

There is a geometrically local Liouvillian $\mathcal {L}$ (with four-qubit Lindblad operators) such that the following holds: for any state $\rho _\mathcal {D}$ on $\mathcal {D}\cong (\mathbb {C}^{2})^{\otimes 2(L-1)^2}$ and $|\Psi \rangle \in (\mathbb {C}^2)^{\otimes 2}$, and for any  > 0, we have

$$\begin{eqnarray*} &&\Big\|\mathrm{e}^{t\mathcal{L}}\left(|\Psi\rangle\langle\Psi|_{\mathcal{A}}\otimes|+\rangle\langle+|^{\otimes 2(L-1)}_{\mathcal{B}\mathcal{B}'}\otimes|0\rangle\langle0|^{\otimes 2(L-1)}_{\mathcal{CC}'}\otimes\rho_{\mathcal{D}}\right)-|\overline{\Psi}\rangle\langle\overline{\Psi}|\Big\|_1\leqslant\epsilon \end{eqnarray*}$$

whenever

$$\begin{equation} t\geqslant (4\ln(2)) L+2\ln(16\epsilon^{-2}). \end{equation} \tag{ 2 }$$

In this expression, we use the trace norm $\|A\|_1=\mathop {\mathrm {tr}}\nolimits \sqrt {A^\dagger A}$ for Hermitian operators.

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We will give a detailed description of the relevant Liouvillian in section 2. The evolution $\mathrm {e}^{t\mathcal {L}}$ implements a continuous-time version of a local error correction process somewhat analogous to Toom's rule: excitations move toward a single plaquette/vertex where they annihilate. An analogous ground-state preparation scheme for more general frustration-free Hamiltonians was discussed in [18]. However, guarantees about encoded information appear to be harder to obtain in their generic scheme. Furthermore because their construction requires the injectivity [14] property of the associated projected entangled pair state (PEPS) description (and hence blocking of sites), the resulting locality of the Lindblad operators will be slightly worse. In contrast, our scheme directly exploits the stabilizer structure of the underlying code, resulting in a comparatively simple Liouvillian. Indeed, our construction is optimal in terms of locality, i.e. the number of particles involved in each Lindblad term [12]. Related locality constraints for pure steady states were derived⁴ in [15, 16]. In our setup, the initial product state determines in a transparent way which code state is prepared. Our work goes beyond the mere characterization of steady states by adding two key ingredients: the consideration of logical observables (which are preserved) and an analysis of the convergence toward the ground space of the toric code.

The bound O(L) on the convergence time established by theorem 1 improves on the O(L²) upper bound predicted for the analogous construction in [18], without a guarantee on the logical information. In fact, it is tight: there are initial states ρ which thermalize slowly, i.e. $\mathrm {e}^{t\mathcal {L}}(\rho )$ is far away from the code space $\mathcal {Q}$ for any time t ≪ L. In a separate work [10], we provide a general no-go theorem in this direction: dissipative state preparation of topologically ordered states requires at least a linear amount of time in L if the Liouvillian is local. Combined with theorem 1, this implies that the construction presented here is optimal in terms of preparation time among the entire class of local Liouvillians.

In summary, our work shows that dissipative processes can be used to implement an encoder for the toric code. Intriguingly, this encoder is more time-efficient than the best known unitary circuit. We stress, however, that both types of encoders need to be supplemented with additional mechanisms in the presence of noise, especially if the encoded information is further processed. As discussed in [13], local Liouvillians such as the one considered here are not suitable for the preservation of encoded information.

2.1. A Liouvillian for a general stabilizer code

We first describe a generic Liouvillian associated with a stabilizer code $\mathcal{Q}$ with stabilizer generators $\{S_j\}_{j\in \mathcal{S}}$. We will subsequently specialize this to the toric code. Let ${\bf P}_{j}^{\pm }=\frac {1}{2} (I\pm S_{j})$ be the projections onto the ±1 eigenspaces of the stabilizer S_j. The code space $\mathcal {Q}$ is the ground space of the Hamiltonian

$$\begin{eqnarray*} H&=\frac{1}{2}\left(|\mathcal{S}|\cdot I-\sum_j S_j\right)=\sum_{j}{\bf P}_j^- \end{eqnarray*}$$

(the global energy shift is introduced for convenience). Our goal is to implement a local error-correction strategy by dissipative evolution. Concretely, we associate a unitary Pauli correction operator C_j with each stabilizer generator S_j such that C_j and S_j anticommute, i.e. {C_j,S_j} = 0. We define the CPTPMs

$$\begin{eqnarray*} \mathcal{T}_j (\rho) &={\bf P}_j^+\rho{\bf P}_j^+ + C_j{\bf P}_j^-\rho{\bf P}_j^- C_j^\dagger. \end{eqnarray*}$$

Note that by definition, $\mathcal {T}_j$ lowers the energy of the term P⁻_j in the Hamiltonian, i.e.

$$\begin{equation} \mathop{\mathrm{tr}}\nolimits(\mathcal{T}_j(\rho){\bf P}^-_j)\leqslant \mathop{\mathrm{tr}}\nolimits(\rho {\bf P}^-_j)\quad{\rm for \ any \ state} \ \rho. \end{equation} \tag{ 3 }$$

While after application of $\mathcal {T}_j$, the stabilizer constraint defined by S_j is satisfied, its application may create a non-trivial syndrome (excitation) for a neighboring stabilizer S_k, k ≠ j. By design (i.e. the choice of correction operators for the toric code discussed below), repeated application of all $\{\mathcal {T}_j\}_{j\in \mathcal{S}}$ eventually removes all excitations, resulting in a state supported on the code space $\mathcal {Q}$. It will be convenient to introduce the averaged CPTPM

$$\begin{equation} \mathcal{T}_{\mathsf{av}}(\rho)=\frac{1}{|\mathcal{S}|} \sum_j \mathcal{T}_j \end{equation} \tag{ 4 }$$

which randomly chooses a syndrome and applies the associated correction map.

By the correspondence discussed in [20], each local CPTPM $\mathcal {T}_j$ defines a local Liouvillian $\mathcal {L}_j=\mathcal {T}_j-\mathsf{id}$. We are interested in the evolution under $\mathcal {L}=\sum _j \mathcal{L}_j$, which, by definition, is a sum of constant-strength local terms. Observe that $\mathcal {L}=|\mathcal {S}| (\mathcal {T}_{\mathsf{av}}-\mathsf {id})$.

2.2. Construction for the toric code

We now consider the toric code with qubits on the edges of an L × L (periodic) square lattice. We separate the Hamiltonian into

$$\begin{eqnarray*}&&\fl H=H^{(p)}+H^{(v)} \quad {\rm where} \ H^{(p)}=\frac{1}{2} \left(L^2 I-\sum_{p\in\mathcal{S}^{(p)}} S_p\right)\quad {\rm and }\quad H^{(v)}=\frac{1}{2}\left(L^2 I-\sum_{v\in\mathcal{S}^{(v)}} S_v\right), \end{eqnarray*}$$

where the former includes all plaquette and the latter includes all vertex terms. Here, we have taken ${{S}}_{p}$ and ${{S}}_{v}$ to correspond to plaquette and vertex stabilizers, respectively,

In other words, ${{S}}_{p}$ is the product of Z-type Pauli operators acting on the edges bounding plaquette $p\in {\mathcal{S}}^{(p)}$ and ${{S}}_{v}$ is the product of Pauli operators acting on the qubits incident to vertex $v \in {\mathcal{S}}^{(v)}$. To define the associated Pauli correction operators $\{C_v\}_{v\in \mathcal{S}^{(v)}}$ and $\{C_p\}_{p\in \mathcal{S}^{(p)}}$, let us partition the set of vertices and plaquettes into

$$\begin{eqnarray*} &&\mathcal{S}^{(p)}=\{p_*\}\cup \mathcal{S}^{(p)}_\rightarrow\cup \mathcal{S}^{(p)}_\uparrow\quad{\rm and } \quad \mathcal{S}^{(v)}=\{v_*\}\cup \mathcal{S}^{(v)}_\leftarrow\cup \mathcal{S}^{(v)}_\downarrow. \end{eqnarray*}$$

Here p_* is a single plaquette, $\mathcal {S}^{(p)}_{\rightarrow }$ consists of the L − 1 plaquettes lying on a fundamental cycle of the torus (which we refer to as the 'equator', running 'horizontally' or 'east–west' along the torus) on which p_* is located, whereas $\mathcal {S}^{(p)}_{\uparrow }$ are the remaining L² − L plaquettes. The vertex v_* as well as the sets $\mathcal {S}^{(v)}_\leftarrow$ and $\mathcal {S}^{(v)}_\downarrow$ are defined similarly on the dual lattice, see figure 2.

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The subscript associated with these sets indicates the direction of movement of an excitation under application of the local correction map. For example, a magnetic excitation (caused by an X-error) on a plaquette $p\in \mathcal{S}^{(p)}_{\uparrow }$ will move to the neighboring plaquette to the north of p under application of the correction map C_p. That is, we define

We will set

$$\begin{eqnarray*} &&C_{v_*}=C_{p_*}=I, \end{eqnarray*}$$

corresponding to a trivial correction operation (this is simply done for convenience). The Liouvillian $\mathcal {L}$ is then defined as in section 2.1.

The proof of theorem 1 relies on two basic statements. The first one concerns the logical information encoded in the state: for suitable initial states, this information is preserved during the evolution. Recall that the toric code has two encoded qubits. Let

$$\begin{eqnarray*} \bar{X}_1&=\left(\bigotimes\limits_{b\in\mathcal{B}} X_b\right)\otimes X_{A_1},\quad \bar{Z}_1=\left(\bigotimes\limits_{c\in\mathcal{C}} Z_c\right)\otimes Z_{A_1},\quad \bar{Y}_1={\rm i}\bar{X}_1\bar{Z}_1\\ \bar{X}_2&=\left(\bigotimes\limits_{b\in\mathcal{B}'} X_b\right)\otimes X_{A_2},\quad \bar{Z}_2=\left(\bigotimes\limits_{c\in\mathcal{C}'} Z_c\right)\otimes X_{A_2},\quad \bar{Y}_2={\rm i}\bar{X}_2\bar{Z}_2 \end{eqnarray*}$$

be the two-qubit logical Pauli operators defined by figure 1, and let us set

$$\begin{eqnarray*} &&\bar{\mathcal{P}}_2=\left\{ \bar{P}_1\bar{P}_2\ |\ \bar{P}_1\in \{I,\bar{X}_1,\bar{Y}_1,\bar{Z}_1\},\, \bar{P_2}\in \{I, \bar{X}_2,\bar{Y}_2,\bar{Z}_2\}\right\}, \end{eqnarray*}$$

where I is the identity operator. These operators play a crucial role in the following statement.

Lemma 1 For any two-qubit Pauli operator P = P₁⊗P₂, P_j∈{I,X_j,Y_j,Z_j}, let $\overline {P}=\overline {P}_1\overline {P}_2$, $\overline {P}_j\in \bar {\mathcal {P}}_2$ be its corresponding logical counterpart. Consider an initial state of the form

$$\begin{equation} \rho_0=|\Psi\rangle\langle\Psi|_{\mathcal{A}}\otimes|+\rangle\langle+|^{\otimes 2(L-1)}_{\mathcal{B}\mathcal{B}'}\otimes|0\rangle\langle0|^{\otimes 2(L-1)}_{\mathcal{CC}'}\otimes\rho_{\mathcal{D}}, \end{equation} \tag{ 5 }$$

where the state $\rho _\mathcal {D}$ is arbitrary on $(\mathbb {C}^2)^{\otimes 2(L-1)^2}$. Then

$$\begin{equation} \mathop{\mathrm{tr}}\nolimits(\overline{P}\mathrm{e}^{t\mathcal{L}}(\rho_0))=\langle\Psi|P|\Psi\rangle\quad{\rm for \ all} \ t \ {\rm and \ all \ }P\in\mathcal{P}_2. \end{equation} \tag{ 6 }$$

Proof. Let P = P₁⊗P₂ be arbitrary. Observe first that the rhs of (6) is equal to

$$\begin{equation} \langle\Psi|P|\Psi\rangle=\mathop{\mathrm{tr}}\nolimits(\overline{P}\rho_0)\ \end{equation} \tag{ 7 }$$

because of the definition of $\overline {P}$ (cf figure 2) and the form of ρ₀, i.e. the fact that the qubits in $\mathcal {B}\mathcal {B}'$ and $\mathcal {CC}'$ are +1-eigenstates of the single-qubit X and Z operators, respectively. In other words, it suffices to show the expectation value $\mathop {\mathrm {tr}}\nolimits (\overline {P}\mathrm {e}^{t\mathcal {L}}(\rho _0))$ is time independent for initial states ρ₀ of the form (5). We claim that an even stronger statement holds: we have

$$\begin{equation} (\mathrm{e}^{t\mathcal{L}})^\dagger (\overline{P})=\overline{P}\quad{\rm for \ any} \ \overline{P}\in\overline{\mathcal{P}}_2, \end{equation} \tag{ 8 }$$

i.e. any observables of the form $\overline {P}$ do not evolve in the Heisenberg picture.

To prove (8), observe that the one-parameter family of unital maps $\{(\mathrm {e}^{t\mathcal {L}})^\dagger \}_{t\geqslant 0}$ is generated by the adjoint $\mathcal {L}^\dagger$ of the Liouvillian, hence (8) is equivalent to

$$\begin{eqnarray*} &&\mathcal{L}^\dagger(\overline{P})=0. \end{eqnarray*}$$

Since $\mathcal {L}=\sum _{j}\mathcal {L}_j$ is a sum of Liouvillians $\mathcal {L}_j=\mathcal {T}_j-\mathsf {id}$ associated with stabilizers j, it suffices to verify that for every j, we have $\mathcal {T}_j^\dagger (\overline {P})=\overline {P}$ or

$$\begin{equation} \textbf{P}_j^+\overline{P}\textbf{P}_j^+ +\textbf{P}_j^-C_j^\dagger \overline{P} C_j\textbf{P}_j^-=\overline{P}. \end{equation} \tag{ 9 }$$

Because $\overline {P}$ is a logical operator, it commutes with each stabilizer, and hence also the projections P^±_j. Equality (9) is in fact implied by $[C_j^\dagger ,\overline {P}]=0$, which we can verify by a case-by-case analysis for single-qubit (logical) operators $\overline {P}\in \{\bar {X}_1,\bar {Z}_1,\bar {X}_2,\bar {Z}_2\}$ (the general case then follows since a product $\overline {P}_1\overline {P}_2$ commutes with C^†_j if each factor $\overline {P}_j$ does). Consider for example the case where $\overline {P}=\bar {Z}_1$. We then have to consider two cases:

• (i)  
  The correction j = p is associated with a plaquette. In this case C_p is a Pauli-X (or the identity). However, inspection of the support of $\bar {Z}_1$ (see figure 1) and the location of the correction operators (see figure 2(a)) reveals that no correction operation C_p acts on the support of $\bar {Z}_1$, hence $[C_p^\dagger ,\bar {Z}_1]=0$ as desired.
• (ii)  
  The correction j = v is associated with a vertex. In this case C_v is a Pauli-Z (or the identity), hence $[C_v^\dagger ,\bar {Z}_1]=0$ holds trivially.

□

The second fundamental statement is about convergence to the ground space.

Lemma 2 Let Q be the projection onto the code space of the toric code, and let Q^⊥ = I − Q be the projection onto the orthogonal complement. Let ρ be an arbitrary initial state. Then we have

$$\begin{equation} \mathop{\mathrm{tr}}\nolimits(\mathsf{Q}^\bot \mathrm{e}^{t\mathcal{L}}(\rho))\leqslant \epsilon\quad{\rm for \ all} \ t\geqslant (4\ln(2)) L + 2\ln(1/\epsilon). \end{equation} \tag{ 10 }$$

We have not optimized the constants in this bound as we are interested in the overall (linear) scaling in L. The proof strategy is different from the arguments in [18] and may be of independent interest.

Proof. Consider the function $f{:}\ \mathcal {S}\backslash \{v_*,p_*\}\rightarrow \mathbb {N}\cup \{0\}$ defined as follows:

$$\begin{eqnarray*} {\rm for \ a \ plaquette} \ p{:} & f(p) +1 ={\rm length \ of \ a \ path \ moving \ north}\mbox{-}{\rm east \ from} \ p \ {\rm to } \ \ p^* \\ {\rm for \ a \ vertex} \ v{:} & f(v) +1 ={\rm length \ of \ a \ path \ moving \ south}\mbox{-}{\rm west \ from} \ v \ {\rm to} \ v^*. \end{eqnarray*}$$

In other words, the function expresses the axial distance to v* and p*, respectively. For a vertex v (plaquette p), the quantity f(v) (f(p)) is the number of vertices (plaquettes) traversed by an electric (magnetic) excitation on the primary (dual) lattice before reaching v_* (p_*) along a path (v = v₁,v₂,...,v_f(v),v_*) (or (p = p₁,p₂,...,p_f(p),p_*)). For the special vertex v* and the plaquette p*, we set f(v*) = f(p*) = −1 for convenience (alternatively, we could omit the discussion of the corresponding trivial correction operations altogether as the stabilizer generators are linearly dependent).

The key property of the function f is the fact that it is compatible with the way excitations are propagated under the correction operations. More precisely, for each stabilizer S_j, let $\mathsf{Pred}(j)$ be the set of correction operations that anticommute with it, i.e.

$$\begin{eqnarray*} &&\mathsf{Pred}(j):=\{k\in\mathcal{S} \ |\ k\neq j \ {\rm and } \ \{C_k,S_j\}_+ =0\}. \end{eqnarray*}$$

Then f has the property that

$$\begin{equation} k\in\mathsf{Pred}(j) \ {\rm implies} \ f(k) \geqslant f(j)+1. \end{equation} \tag{ 11 }$$

Namely, whenever an excitation is created by C_k at S_j, one can be certain that a higher valued excitation has been removed at S_k. We will argue that for any α ⩾ 1, we have

$$\begin{equation} \mathop{\mathrm{tr}}\nolimits(\mathsf{Q}^\bot \mathrm{e}^{t\mathcal{L}}(\rho))\leqslant \mathrm{e}^{-(1-\alpha^{-1}m)t}\sum_{j\in\mathcal{S}}\alpha^{f(j)}, \end{equation} \tag{ 12 }$$

where $m:=\max _{k\in \mathcal{S}}|\{ j | k \in \mathsf {Pred}(j) \}|$ is the maximal number of stabilizers a single correction operator can excite. In the case of the toric code, m = 1, hence we obtain $\mathop {\mathrm {tr}}\nolimits (\mathsf {Q}^\bot \mathrm {e}^{t\mathcal {L}}(\rho ))\leqslant \mathrm {e}^{-t/2} 2^{2L}$ (implying the claim) by choosing α = 2 and observing that

$$\begin{eqnarray*} &&\sum_{j\in\mathcal{S}}\alpha^{f(j)} = 2\alpha^{-1}\sum_{r,s=0}^{L-1}\alpha^{r+s}=2\alpha^{-1}\left(\frac{\alpha^L-1}{\alpha-1}\right)^2 < \frac{2\alpha^{2L-1}}{(\alpha-1)^2}\Big|_{\alpha=2} = 2^{2L}. \end{eqnarray*}$$

To prove (12), consider the observable

$$\begin{eqnarray*} &&D=\sum_{j\in\mathcal{S}\backslash\{v_*,p_*\}}\alpha^{f(j)} \textbf{P}_j^-. \end{eqnarray*}$$

Clearly D ⩾ Q^⊥ for any α ⩾ 1, hence it suffices to show that the expectation value $\mathop {\mathrm {tr}}\nolimits \left ( D\mathrm { e}^{t\mathcal {L}}(\rho )\right )$ is upper bounded by the rhs of (12), i.e.

$$\begin{equation} \mathop{\mathrm{tr}}\nolimits(D\mathrm{e}^{t\mathcal{L}}(\rho))\leqslant \mathrm{e}^{-(1-\alpha^{-1}m)t}\sum_{j\in\mathcal{S}}\alpha^{f(j)}. \end{equation} \tag{ 13 }$$

Consider the Heisenberg evolution of the projection operators { P_j}_j. Since Pauli operators either commute or anticommute, a straightforward calculation gives

$$\begin{eqnarray*} \mathcal{L}^\dagger_k({\textbf{P}}_j^-)=\left\{\begin{array}{@{}ll@{}}0& {\rm if}~[C_k,S_j]=0,\\ (I-{\textbf{P}}_j^-){\textbf{P}}_k^--{\textbf{P}}_j^-{\textbf{P}}_k^-& {\rm if}~ \{C_k,S_j\}=0.\end{array}\right. \end{eqnarray*}$$

In particular, the expectation values behave classically under the designed Liouvillian, i.e. (writing $\langle X\rangle _t=\mathop {\mathrm {tr}}\nolimits (X\mathrm {e}^{t\mathcal {L}}(\rho ))$ for brevity)

$$\begin{eqnarray} \frac{{\rm d}\langle {\textbf{P}}_j^- \rangle_t }{{\rm d}t}&=& -\langle {\textbf{P}}_j^- \rangle_t +\sum_{k \in {\mathsf{Pred}}(j)} \langle(I-{\textbf{P}}_j^-) {\textbf{P}}_{k}^-\rangle_t-\langle {\textbf{P}}_k^- {\textbf{P}}_j^- \rangle_t\nonumber\\ &\leqslant & -\langle {\textbf{P}}_j^- \rangle_t + \sum_{k \in {\mathsf{Pred}}(j)}\langle {\textbf{P}}_k^- \rangle_t. \end{eqnarray} \tag{ 14 }$$

According to (14), we have

$$\begin{eqnarray*} \frac{{\rm d}\langle D \rangle_t}{{\rm d}t}&\leqslant& - \langle D\rangle_t+\sum_j \alpha^{f(j)}\sum_{k\in{\mathsf{Pred}}(j)} \langle{\textbf{P}}_k^-\rangle_t\\ &\leqslant& - \langle D\rangle_t+\alpha^{-1}\sum_k \alpha^{f(k)} \langle{\textbf{P}}_k^-\rangle_t \sum_{j : k\in{\mathsf{Pred}}(j)} 1, \end{eqnarray*}$$

where we used property (11) on the function f. According to the definition of m and D, this implies

$$\begin{eqnarray*} \frac{{\rm d}\langle D\rangle_t}{{\rm d}t}&\leqslant -(1-\alpha^{-1}m)\langle D\rangle_t, \end{eqnarray*}$$

i.e. the expectation value decays exponentially. The claim (13) then follows from the fact that P⁻_j ⩽ I for all $j\in \mathcal{S}$ since these are projections, hence $\mathop {\mathrm {tr}}\nolimits (D\rho )=\langle D\rangle _0\leqslant \sum _{j\in \mathcal{S}}\alpha ^{f(j)}$.□

With lemmas 1 and 2, we are ready to prove our main result.

Proof of theorem 1. Consider an initial state ρ₀ of the form (5) and assume that t ⩾ (4 ln(2))L + 2 ln(⁻¹). By lemma 2, we have $\mathop {\mathrm {tr}}\nolimits (\mathsf {Q}^\bot \mathrm {e}^{t\mathcal {L}}(\rho _0))\leqslant \epsilon$. With the gentle measurement lemma (see e.g. [19, lemma 9.4.1]), this implies

$$\begin{equation} \|\mathrm{e}^{t\mathcal{L}}(\rho_0)-\bar{\rho}'_t\|_1\leqslant 2\sqrt{\epsilon}, \end{equation} \tag{ 15 }$$

where $\bar{\rho}'_t$ is defined as $\bar {\rho }'_t:= \frac {{{\mathsf{Q}}} \mathrm {e}^{t\mathcal {L}}(\rho _0){\mathsf Q}}{\mathop {\mathrm {tr}}\nolimits ({\mathsf Q} \mathrm {e}^{t\mathcal {L}}(\rho _0))}$. Observe that the state $\bar {\rho }'_t$ is supported entirely on the code space $\mathcal {Q}$.

Note that for any two states ρ,σ we have $\|\rho -\sigma \|_1=\max _{\| P \| \leqslant 1}\mathop {\mathrm {tr}}\nolimits (P(\rho -\sigma ))$ and therefore $|\mathop {\mathrm {tr}}\nolimits (P(\rho -\sigma ))|\leqslant \|\rho -\sigma \|_1$ for any normalized operator P. Hence (15) implies that for any normalized logical operator $\overline {P}$, we have

$$\begin{equation} \big|\mathop{\mathrm{tr}}\nolimits\left(\overline{P}(\mathrm{e}^{t\mathcal{L}}(\rho_0)-\bar{\rho}_t')\right)\big|\leqslant 2\sqrt{\epsilon}. \end{equation} \tag{ 16 }$$

Let $|\overline {\Psi }\rangle \in \mathcal{Q}$ be the target encoded state, i.e. the state satisfying (cf (7) and (8))

$$\begin{eqnarray*} &&\langle\overline{\Psi}|\overline{P}|\overline{\Psi}\rangle=\langle\Psi|P|\Psi\rangle=\mathop{\mathrm{tr}}\nolimits(\overline{P}\rho_0)=\mathop{\mathrm{tr}}\nolimits(\overline{P}\mathrm{e}^{t\mathcal{L}}(\rho_0))\quad{\rm for \ any \ logical \ operator} \ \overline{P}. \end{eqnarray*}$$

Combining this with (16) shows that $|\overline {\Psi }\rangle \langle \overline {\Psi }|$ and $\bar {\rho }'_t$ have approximately the same expectation values for logical operators. Since both states are supported on the ground space $\mathcal {Q}$ for which we have access to a full algebra of logical observables (linear combinations of logical Pauli observables), one may choose⁵ a normalized logical operator $\overline {P}$ such that $\overline {P}\mathsf {Q} =\mathop {\mathrm {sgn}}\nolimits (|\overline {\Psi }\rangle \langle \overline {\Psi }|-\bar {\rho }'_t)$ which achieves the maximization defining one-norm distance

$$\begin{equation}\fl \big\||\overline{\Psi}\rangle\langle\overline{\Psi}|-\bar{\rho}'_t\big\|_1 = \mathop{\mathrm{tr}}\nolimits[\mathop{\mathrm{sgn}}\nolimits(|\overline{\Psi}\rangle\langle\overline{\Psi}|-\bar{\rho}'_t)(|\overline{\Psi}\rangle\langle\overline{\Psi}|-\bar{\rho}'_t)]= \big|\mathop{\mathrm{tr}}\nolimits(\overline{P}(|\overline{\Psi}\rangle\langle\overline{\Psi}|-\bar{\rho}'_t))\big| \leqslant 2\sqrt{\epsilon}. \end{equation} \tag{ 17 }$$

Combining (17) with (15) and using the triangle inequality, we conclude that

$$\begin{eqnarray*} &&\|\mathrm{e}^{t\mathcal{L}}(\rho_0)-|\overline{\Psi}\rangle\langle\overline{\Psi}|\|_1\leqslant 4\sqrt{\epsilon}\quad {\rm if } \ t\geqslant (4\ln(2)) L+2\ln (\epsilon^{-1}). \end{eqnarray*}$$

The claim of theorem 1 follows immediately from this statement.

We have presented a quasi-local time-independent Liouvillian which generates a dissipative encoder: it encodes two physical qubits into the ground space of the toric code. Its key features are translation-invariance (up to boundary conditions), optimal locality of the Lindblad operators, as well as optimal, i.e. linear convergence (encoding) time of the resulting dissipative encoder as a function of the lattice size. This illustrates the power of engineered dissipation for state preparation in a non-trivial setting with degeneracy.

Since the Liouvillian is constituted of quasi-local CPTPMs associated with each stabilizer, one may interpret⁶ the resulting evolution $\mathrm {e}^{t\mathcal {L}}$ as the repeated application of the 'average' correction map (cf (4)). This viewpoint immediately results in a preparation algorithm for surface code states. The latter inherits the properties of the dissipative evolution: it is translation-invariant up to boundary conditions, composed of feedback-free quasi-local maps and requires only a minimal number of iterations to converge. These features make it an attractive candidate for potential experimental realizations.

4.1. Towards fault tolerance

JD and RK gratefully acknowledge funding provided by NSERC. RK would like to thank John Preskill for discussions, and the Institute for Quantum Information and Matter for their hospitality. FP would like to thank Iman Marvian, Ignacio Cirac and John Preskill for helpful discussions and comments. FP acknowledges funding provided by the Institute for Quantum Information and Matter, a NSF Physics Frontiers Center with support of the Gordon and Betty Moore Foundation (grants numbers PHY-0803371 and PHY-1125565).