Lyapunov Methods

The stationary states of Eq. (4.1) have been studied extensively for the purpose of state stabilization [160, 171, 170, 154, 6]. For a multipartite quantum system, the method of using dissipative dynamics to engineer quantum states has been generalized to the notion ofdissipatively quasi-locally stabilizable(DQLS) states [172, 173]. The theory of DQLS states proposes a systematic approach to determine whether a given multipartite state is asymptotically stabilizable if local dissipation controls can be engineered. Furthermore, if the quantum state satisfies the DQLS condition, the required multipartite system Hamiltonian and the system-environment coupling operators can be constructively derived.

The aforementioned results are based on (4.1). Alternatively, the desired states can be stabilized by studying the evolution of certain operators. Since the expectation of an operatorV _∈B(H )at the stateρ is calculated by⟨V⟩_ρ =trace(Vρ), the evolution of

the operatorV(t)can be defined via the relation_⟨V(t)_⟩ρ0 =_⟨V_⟩ρ

t. Note thatV =V(0).

The generator of this Markov process is given by [83] G(V(t)) = −i[V(t),H(t)] +L(V(t)) = −i[V(t),H(t)] + J

∑

j=1 L†_j(t)V(t)L_j(t) −1 2L † j(t)Lj(t)V(t)− 1 2V(t)L † j(t)Lj(t). (4.12) A large class of quantum states, including graph states and cluster states (which are DQLS states as well), can be encoded as the ground states of a multipartite operator taking the form V =∑K_i=1Vi [143, 185, 172]. As a result, state stabilization can be achieved by engineering the dissipation such that the system converges to the ground states asymptotically. The merit of this formulation is that the ground-state stability of

V can be established by Lyapunov-type operator inequalities. This scenario has been considered before in [139], where a scalability condition is used to prove the ground-state stability ofV when eachV_iis stabilized individually. However the scalability condition

78 Chapter 4. Quantum Reservoir Engineering

proposed in [139] does not hold for certain applications, especially when_{V_i}consist of two-body interactions. An illustrative example can be found in Section 4.3.

Note that ground-state stability does not necessarily guarantee that a particular ground state is stable against errors, because the erroneous state may return to a different ground state under the dissipative dynamics. However, we can prove in Section 4.3 that if certain types of errors occur to one of the ground states, the dissipation control can steer the erroneous state back to the initial ground state exactly, without any measurement or active feedback. In this regard, this result can be considered as an addition to the existing physical literature onautomatic quantum error correction(AQEC) [14, 90, 26, 88, 82]. The results of Section 4.3 are intended to deal with a specialized case, where we apply algebraic methods to achieve the stabilization of the states by imposing scalable Lyapunov-type conditions on the operators. If these conditions are satisfied, then the ground states of the system are DQLS andV is frustration-free[185, 173, 139].

The multipartite quantum system is defined onH =NM_m=1H_m which is a tensor product of Hilbert spaces _{H_m} (eachH_m is associated with a subsystem). Unless otherwise noted, we will the following assumption throughout this chapter.

Assumption 4.2.1 V can be decomposed asV =∑K_i=1Vi, andViis defined on a subset of

{Hm}. {Vi}are orthogonal projections, i.e. Vi2=Vi and[Vi,Vj] =0,i̸= j. EachViis associated with a set of dissipation controls_{L_j}. EachL_j allows the decomposition

L_j=∑iUi,jViwithUi,jbeing a unitary operator.

Remark 4.2.1 {V_i} can be regarded as quasi-local operators [172] since they are de- fined on a subset of_{H_m}. Therefore, the stabilizing dynamics in this chapter can be considered as specific realizations for the stabilization of DQLS states.

V_i2 =V_{i ≥}0 is a natural assumption that holds for many applications, e.g. the dissipation control of stabilizer states [143, 185, 139]. _{V_i}being commutative is an intuitive assumption which enables _{V_i} to share common ground states. Physical examples of the decomposition includes the dissipation control of graph states [185].

We also make the following assumption.

Assumption 4.2.2 The system Hamiltonian can be written asH=∑K_i=1Hi, where each

H_isatisfiesH_i=V_i−g_iI. Here₋g_iis the smallest eigenvalue of the Hermitian operator

4.2 Background 79

Remark 4.2.2 It is experimentally possible to engineer Hamiltonian on a multipartite quantum system, e.g. [93].

As shown in the next section, the two assumptions allow a concise and scalable stability analysis based on the generator (4.17). In addition, we have two definitions as follows.

Definition 4.2.2 V_i is said to be a two-body operator if it can be decomposed asV_i=

X_m

1⊗Xm2, withXm1,Xm2 defined on two Hilbert spacesHm1,Hm2, respectively.

Definition 4.2.3 The generator of the evolution ofV_ithat is induced by a single dissipation controlL_iis defined by G(V_i)_Li=−i[V_i,H] +L†_iV_iL_i−1 2L † iLiVi− 1 2V L † iLi. (4.13)

Remark 4.2.3 Eq. (4.13) is the generator ofV_icontrolled by a single coupling operator

L_i. IfV_iis also affected by other coupling operators_{L_j,j_̸=i_}, then we haveG(V_i) =

G(V_i)_Li+∑j_̸=iL † jViLj−12L † jLjVi−12ViL † jLj. We also recall one theorem from [83]:

Theorem 4.2.1 If an operatorX _≥0 satisfies the following inequality

G(X)_{≤ −}cX, c>0, (4.14)

then the system will asymptotically converge toZ_X.

Remark 4.2.4 The algebraic condition (4.21) uses X =X(0), H =H(0) and _{L_j =

L_j(0)_}. The satisfaction of this condition implies that lim_t→∞⟨X(t)_⟩=0. The other algebraic conditions of this chapter also use the operators at the initial time. For the details of the Heisenberg-picture stability theory, please refer to [136].

The following theorem can be derived using Theorem 4.2.1, Assumption 4.2.1 and 4.2.2.

Theorem 4.2.2 [139] If the following condition

J

∑

j=1

80 Chapter 4. Quantum Reservoir Engineering

• • •

Fig. 4.1 EachV_ihas 0 as its lowest energy. The system may be stabilized to the ground state ofV₁under a local dissipation control. The local dissipation control is scalable if it maintains or decreases the energy of_{V_i,i_̸=1_}, i.e. the system is steered towards the ground states of_{V_i,i_̸=1_}under the local dissipation control ofV₁. However for instance, if the local dissipation control ofV₁increases the energy ofV₂, then_{L_i,i_̸=1_} must be able to compensate this increase in order to stabilizeV₂.

holds, thenG(V_i)≤ −c_iV_iandV_iis asymptotically ground-state stable under the dissipa- tion control of_{L_j}. In particular, we say_{U_i,j}stabilizeV_iif Eq. (4.30) holds.

Now we recall the scalability condition derived in [139].

Theorem 4.2.3 Suppose for eachV_i, there exists_{L_i,j}such thatG(V_i)≤ −c_iV_i,c_i>0 holds. The ground-state stability ofV can be implied if the intuitive scalability condition

∑

i′_̸=i

∑

j G(V_i)L i′,j ≤ 0, (4.16)

can be established for alli.

Remark 4.2.5 V is asymptotically ground-state stable if the local dissipation controls of _{V

i′,i

′

̸

=i_} do not increase the expectation ofV_i (Fig. 4.1). However, Eq. (4.23) is easily violated if_{V_i}involve two-body interactions. IfV_iis a two-body operator, the local controlL_i,j that stabilizesV_i may not act onV_i′,i′ ̸=i trivially. To be more

specific, ifV_i=X_m

1⊗Xm2, then there exists at least oneVi′,i

′

̸

=iwhich is defined on Hm1, otherwise the resulting ground state cannot be entangled inHm1. As a result, the

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