Creating Quantum Cluster States embedded in ancillary baths
aa r X i v : . [ qu a n t - ph ] D ec Creating Quantum Cluster States embedded in ancillary baths
Jun Jing , , , Mark S. Byrd , and Lian-Ao Wu ∗ Department of Physics, Zhejiang University,Hangzhou 310027, Zhejiang, China Department of Theoretical Physics and History of Science,The Basque Country University (EHU/UPV),PO Box 644, 48080 Bilbao, and Ikerbasque,Basque Foundation for Science, 48011 Bilbao, Spain Institute of Atomic and Molecular Physics,Jilin University, Changchun 130012, Jilin, China Physics Department, Southern Illinois University, Carbondale, IL, 62901-4401, USA (Dated: October 2, 2018)
Abstract
We propose a systematic and explicit method for the inverse engineering of the dynamics of anopen quantum systems with no auxiliary Hamiltonian nor the prerequisite of adiabatic passage. Inparticular, we exploit the Lindblad dissipators in order to create a resource state or subspace ofinterest in the presence of decoherence. In a conceptual shift, the Lindblad dissipators, includingmultiple interactions that are central to determine the steady state in the long-time limit for an openquantum system, can be guided to produce a useful practical resource to achieve an arbitrary targetstate or subspace. More importantly, with the help of gate and circuit-based quantum control, weprovide an explicit, programmable, and polynomially efficient control sequence to create a clusterstate or graph state useful for one-way quantum computing. ∗ Corresponding author: [email protected]
ResultsCreating an open quantum system state.
In the field of open quantum system dy-namics, the most general Markovian master equation can be written in a Lindblad form.Maintaining the translational invariance and positivity, the Lindblad master equation is animportant and reliable tool for the treatment of irreversible and non-unitary processes ofopen-system states, covering the dissipation and pure-decoherence processes in the quantummeasurement process. Even beyond the Markovian regime, it is practically used to find thesteady state or subspace in the long-time limit for systems under stationary non-Markoviannoises. Consider a system coupled to a multi-mode bosonic bath or field via the systemoperator L . Suppose the environment is at zero temperature , in the rotating frame we canarrive at a general master equation of the Lindblad form (see the derivation in Method ), ∂ t ρ S = L ρ S = γ (cid:20) Lρ S L † − (cid:8) L † L, ρ S (cid:9)(cid:21) . (1)The Stark effect is ignored, which is not relevant to the non-unitary evolution of the system.Here γ = 2 π R ∞ dωJ ( ω ) is the decay rate, where J ( ω ) is the spectral function of the bath.The spectral function is obtained by the Fourier transformation of the two-point correlationfunction f ( t, s ) = P j | g j | e − iω j ( t − s ) , where g j ( g ∗ j ) is the coupling strength between system3nd the j -th mode with eigenfrequency ω j . For a structured environment, γ could be time-dependent, but would become asymptotically time-independent for times longer than thetimescale of correlation function. Equation (1) shows that the dark states of the system-environment interaction, i.e., those that satisfy L | Φ i = 0, will survive after the decoherenceprocess determined by the dissipator L . On the other hand, the dissipator serves as a filterto remove those states that are not in the steady-state subspace.Extended to a more general situation with multiple environmental interactions, the mi-croscopic Hamiltonian (see Method ) becomes H I ( t ) = P j [ L j B † j ( t ) + h.c. ]. The right handside of Lindblad equation (1) is thus generalized into a summation over dissipators with dif-ferent L j ’s. Consider an N -dimensional system ρ S = P mn ρ mn | m ih n | , h m | n i = δ mn , whichis under the irreversible dynamics determined by L j>k = | ϕ j ih j | with | ϕ j i = P kp =1 a jp | p i (itis not necessarily a normalized vector, so a jp ’s can be arbitrary). Evidently, we have L j ρ S = L j ρ S L † j − (cid:16) L † j L j ρ S + ρ S L † j L j (cid:17) = ρ jj | ϕ j ih ϕ j | − || ϕ j i| N X n =1 ( ρ jn | j ih n | + ρ nj | n ih j | ) . (2)Therefore the requirement of the long-time limit L ρ S = P Nj = k +1 L j ρ S = 0 will give rise tothe vanishing of both population and coherence terms outside of the chosen k × k subspaceof interest. When k = 1, the system is eventually engineered to a target pure state, whichis immune to the external disturbance and insensitive to the initial state.Notably the effect of dissipator is invariant under simultaneous unitary transformation ofthe operators and states: ˜ L ρ S = U L ρ S U † with ˜ L = U LU † and ˜ ρ S = U ρ S U † . This propertyaids in the experimental implementation. For instance, let us consider an open quantumtwo-level system with the target state an eigenstate | ϕ i = | + i ≡ ( | i + | i ) / √ X ( X |±i = ±|±i ), where X, Y, Z are Pauli matrices along the directions x, y, z ,respectively. From the protocol in Eq. (2), one can see that the Hamiltonian and Lindbladoperator could be given by H = ωX and L = Z − iY , respectively. After a sufficientlylong time, this dissipator creates the superposed state in a single two-level system (one-dimensional DFS), | + i . In fact, if this is observed in the rotating frame with respect to theunitary transformation U = r −
11 1 , then this describes the well-known dissipative process for the single qubit system with H ∝ Z L ∝ σ − = ( X − iY ) /
2. Under the condition that the environmental degrees of freedomhave a much faster relaxation timescale, the steady state of the system is governed byEq. (1) and the preparation time would be sped up by strengthening the coupling betweenthe system and environment, as allowed by the quantum speed limits [40, 41].
Creating cluster state via dissipators.
Cluster or graph states are special class ofstates that are useful in quantum error-correcting codes, entanglement measurement and pu-rification, and for characterization of computational resources in measurement based quan-tum computing models. The one-dimensional graph state consisting of n qubits can beexpressed as | ϕ n i = 12 n/ n O q =1 ( | i q Z q +1 + | i q ) , (3)with the convention Z n +1 ≡
1. For the two-qubit case, this may be written, up to localunitary transformations, as | ϕ i = 1 √ | i | i + | i | i ) . (4)In quantum computing, a graph state can be represented by a qubit-network of m nodes.Each node of this network (graph) is associated with a qubit prepared in the state of | + i andeach edge between two qubits Q and Q is acted on by a controlled phase gate, i.e., | G n i ≡ U | + i ⊗ n , where U = Q Q ,Q U with Q , Q ∈ { , , , · · · , n } and U = diag([1 , , , − | G i = | ϕ i up to a local unitary transformation. Similarly, one mayobtain for n = 3 , | ϕ i = ( | i + | i ) / √ | ϕ i = ( | i + | i + | i−| i ) / | ϕ i as in Eq. (4), we can employ threeLindblad operators L j = | ϕ ih φ j | , provided h ϕ | φ j i = 0 and h φ j | φ k i = δ jk , with j, k = 1 , , L = i ( X Y + Y X ) − ( Z + Z ) ,L = i ( Z Y + Y Z ) + ( X + X ) ,L = ( Z X − X Z ) − i ( Y − Y ) , (5)respectively, where the subscripts of X, Y, Z imply the indexes of qubits. At the same line,it is intuitively clear that for creating a general cluster state with n qubits, the operators5 j , j = 1 , , · · · , n −
1, can be constructed as L j ≡ X α W ( jα ) = X α W ( jα )1 W ( jα )2 · · · W ( jα ) n , (6)where W k ∈ { X, Y, Z, I } , I is the identity operator, and α is the index of the n -bodyinteraction operators of order O ( n ).To be polynomially efficient , this method can be dramatically improved by utilizing aspecial Lindblad operator L ′ = | φ i ( P β> a β h φ β | ), where | φ β i ’s constitute a complete setof eigenstates for the n -qubit system and a β ’s are non-vanishing constants. It is alwayspossible to find a proper unitary transformation to get the cluster state | ϕ n i from a purestate connected to | φ i . In another words, the protocol/algorithm based on Eq. (6) withmany dissipators can be reformulated as one with a single dissipator. This special L can beobtained as follows. The steady state determined by the vanishing dissipator in Eq. (1) canbe always transformed into a diagonal form as D = U ρ S U † , where U is a unitary matrix.Accordingly, L is written as L ′ = U L U † . If D = P β ≥ P β | φ β ih φ β | , then the Lindbladoperator can be written as L ′ = | φ i ( P β> a β h φ β | ). Thus L ′ D = ( P β> P β | a β | ) | φ ih φ | + P β ′ ,β> P β ′ a ∗ β a β ′ | φ β ih φ β ′ | = 0, which yields P β> = 0 for arbitrarily chosen coefficients a β so that the steady state is a pure state | φ i . One should keep in mind that any pure statesatisfying L | Φ i = 0 is a dark state of the dissipator of Eq. (1), and is therefore decoupledfrom the collective dissipation of the n -qubit system. However, the above specified L ′ isnot the unique solution of this type. Now returning to the previous frame, the Lindbladdissipator L = U † L ′ U can be used to drive the system into a pure state U † | φ i since theunitary transformation does not influence the purity of the steady state. Thus the conditionsto realize Eq. (6) are relaxed to L = X α W ( α )1 W ( α )2 · · · W ( α ) n , (7) without requiring an exponentially-increasing numbers of operations.Physically, our protocol is available for a bosonic zero-temperature environment. Onecan use the above composite Lindblad dissipator to prepare the system in a pure state (notnecessarily the ground state and could be the graph state | ϕ n i ) according to | ϕ n i = U † | φ i .For the two-qubit cluster state in Eq. (4), the corresponding Lindblad operator is L = P j =1 a j L j , where L j ’s have been described in Eq. (5).6 IG. 1: (color online) Quantum circuit of realizing a special 3-qubit interaction with the environ-ment B given by Eq. (8). T SB = e iθY B . U A = e i π A , where A stands for Pauli operators X j , Y j ,and Z j or their tensor products. Creating many-body interactions.
The Lindblad master equation (1) is obtainedfrom the partial trace over the total unitary transformation
U ρ S ρ B U † . The full time-evolution operator U = T ← exp[ − i R t dsH I ( s )] can be implemented using the Trotter for-mula by U ( δ t ) ≈ exp[ − i ( LB † + h.c. ) δ t ] = Q α exp[ − i ( W ( α ) B † + h.c. ) δ t ]. From Eq. (1), it isclear that the dissipator-creation control protocol must be based on the system operators in P α W ( α ) B † that connect the system to the bath. The constituent W B † (here for simplicity,the subscript of W is omitted and W is assumed to be unitary) involves a many-body in-teraction [42, 43] which is not naturally occurring. However, it can be constructed from theavailable interactions by a control protocol as follows. Let us start from the Hamiltoniandescribing the interaction of the first qubit and the environment W B † , which is alwaysattainable. Then, supposing that the interaction between neighboring qubits p, q contains acontrollable term W p W q , such as X p X q (in reality such control can be obtained as discussedbelow). For any set of operators satisfying SU (2) commutation relations, such as X, Y, Z ,we have the following useful formula: T Z ◦ e iθY ≡ e i ( π/ Z e iθY e − i ( π/ Z = e iθX , where the operators could also be permuted cyclically, e.g., Z → Y → X → Z . Using7his property, we can efficiently establish the many-body interactions in an open system byalternately switching relevant interactions, e.g., T Z ◦ { T X X ◦ [ T Y ◦ ( T Z X ◦ e iθY B )] } , (8)which can, in turn, be used to recursively generate e iθX X X B . The process provided byEq. (8) is demonstrated by a circuit in Fig. 1 for the coupling term of 3-qubits to theenvironment simultaneously, in which the dashed frame distinguishes the circuit generating e iθX X B for 2-qubits coupled to the environment. A similar process can be used even formore long-range interactions of a many-qubit system coupled to a bosonic environment.Notably as shown in Fig. 1, we need merely up to 2-qubit gates.The two-body and even many-body interaction terms (quantum gates) have alreadybeen simulated in the ion-trap systems. For example, the time-evolution operator U XX =exp( iθX p X q ) can be implemented by two Mølmer − Sørensen gates [44, 45] applied to the twosystem qubits (denoted as p and q ) and an ancilla qubit (denoted as 0) initially prepared in | i along with a single-qubit rotation on the ancilla qubit, U MS ( − π/ ,
0) exp( − iθZ ) U MS ( π/ , U MS ( µ, ν ) = exp[ − iµ (cos νS x + sin νS y ) / S x = X + X p + X q and S y = Y + Y p + Y q .Using properties of the group SU (2), arbitrary two-body interactions can be simulated withthe help of local unitary transformations by quantum gates [46]. Discussion
Dissipators in the Markovian master equation or non-Markovian master equation presentedin the Lindblad form are usually regarded as decoherence or non-unitary evolution of openquantum systems which are detrimental. In this work, we have shown how to efficientlyguide an open quantum system into one of the zero eigenvalue eigenstates (dark states) byusing certain dissipators acting as generators of the desired state or subspace.In particular, we have presented an explicit control protocol to create cluster states, orgraph states, suitable for one-way quantum computing on an n -qubit system. Using circuitdiagrams with up to the two-qubit quantum gates, the relevant Lindblad operators as well asthe full evolution operator can be decomposed into O ( n ) elements of many-body interactionsbetween qubits and the bath. Our protocol can be realized in a polynomially efficient wayand could also be implemented using available quantum gates in ion-trap systems. Possibleextensions of this work include a protocol to generate subsystem codes and to optimize theoperations to improve the inverse engineering efficiency of the desired states.8 ethod The total Hamiltonian describing both system and multi-mode environment in the rotatingframe can be generally written as H I ( t ) = LB † ( t ) + L † B ( t ) , (9)where L and B ( t ) = P j g j a j e − iω j t are the Lindblad operator and the bath operator, respec-tively. a j is the annihilation operator for the j -th environmental mode with eigenfrequency ω j and g j is its coupling strength with the system. The corresponding Born-Markov masterequation reads ∂ t ρ S ( t ) = − Z ∞ ds Tr B { [ H I ( t ) , [ H I ( s ) , ρ S ( t ) ρ B ]] } , (10)which is based on a popular assumption of no initial correlations between the system andenvironment. Tracing over the environmental degrees (Tr B ) plays the role of an averagemeasurement over the environment. A straightforward derivation from Eq. (10) yields ∂ t ρ S = − [ F ( t )( L † Lρ S − Lρ S L † ) + G ( t )( ρ S LL † − L † ρ S L ) + h.c. ] , where F ( t ) = R ∞ ds Tr B [ B ( t ) B † ( s ) ρ B ] and G ( t ) = R ∞ ds Tr B [ B † ( t ) B ( s ) ρ B ]. Suppose the en-vironment is at zero temperature , then F ( t ) = P j | g j | R t dse − iω j ( t − s ) and G ( t ) = 0. Ignoringthe Stark effect, which is not relevant to the non-unitary evolution of the system, we arriveat a master equation of the Lindblad form, ∂ t ρ S = L ρ S = γ (cid:20) Lρ S L † − (cid:8) L † L, ρ S (cid:9)(cid:21) , (11)where γ = 2 π R ∞ dωJ ( ω ) is the decay rate. Here J ( ω ) is the spectral function of thebath coupled to the system via the system operator L , which is obtained by the Fouriertransformation of the two-point correlation function f ( t, s ) = P j | g j | e − iω j ( t − s ) .The universality of Eq. (11) or Eq. (1) can alternatively be shown using the quantum-state-diffusion (QSD) equation [38, 39] which is a dynamical equation for the stochasticwavefunction of the system subject to the influence from the environment. Starting fromEq. (9) and assuming a zero-temperature environment, the QSD equation reads, ∂ t ψ t ( z ∗ ) = (cid:20) Lz ∗ t − L † Z t dsf ( t, s ) O ( t, s, z ∗ ) (cid:21) ψ t ( z ∗ ) . (12)The stochastic process z ∗ t ≡ − i P j g ∗ j z ∗ j e iω j t is the result of the aforementioned partial traceover the environment, where z ∗ j is a Gaussian random number indicating a random coherent9tate of the j -th environmental mode. The system operator O ( t, s, z ∗ ) embodies the system-environment interaction. The density matrix of the system is obtained by ensemble average ρ S = M [ | ψ t ih ψ t | ]. Using a Markov approximation, O ( t → s, s, z ∗ ) → L , which correspondsto the environmental correlation function with f ( t, s ) = γδ ( t − s ). In this limit, the Novikovtheorem will ensure that Eq. (12) can be used to represent the same open dynamics asindicated by Eq. (1). [1] Breuer, H. P., Petruccione, F. The Theory of Open Quantum Systems , (Oxford UniversityPress Oxford, 2002).[2] Albash, T., and Lidar, D. A. Decoherence in adiabatic quantum computation, Phys. Rev. A , 062320 (2015).[3] Viola, L., Knill, E., Lloyd, S. Dynamical decoupling of open quantum systems, Phys. Rev.Lett. , 2417 (1999).[4] Uhrig, G. S. Keeping a quantum bit alive by optimized π -pulse sequences, Phys. Rev. Lett. , 100504 (2007).[5] Gaitan, F. Quantum Error Correction and Fault Toleratn Quantum Computing , (CRC press,Boca Raton, 2008).[6] Lidar D. A., and Brun, T. A.
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We acknowledge grant support from the Basque Government (grant IT472-10), the Span-ish MICINN (No. FIS2012-36673-C03-03), the National Science Foundation of China No.11575071, and UMass Boston (P20150000029279).
Author contributions
J.J. performed analyzed results and prepared figures. L.-A.W. contributed to the conceptionand development of the research problem. All authors (J.J., M.S.B. and L.-A.W.) discussedthe results and physical implications, and wrote the manuscript.