Quantum emulation of a spin system with topologically protected ground states using superconducting quantum circuits
Using superconducting quantum circuit elements, we propose an approach to experimentally construct a Kitaev lattice, which is an anisotropic spin model on a honeycomb lattice with three types of nearest-neighbor interactions and having topologically protected ground states. We study two particular parameter regimes to demonstrate both vortex and bond-state excitations. Our proposal outlines an experimentally realizable artificial many-body system that exhibits exotic topological properties.
pacs:75.10.Jm, 85.25.-j, 05.30.Pr
Interesting phenomena, such as the Aharonov-Bohm effect and Berry phases, can occur in physical systems with nontrivial topology in real or parameter space. Topological quantum systems are now attracting considerable interest because of their fundamental importance in diverse areas ranging from quantum field theory to semiconductor physics,RMP () with the most recent example being the exploration of topological insulators.Kane_PRL05 (); Bernevig () These topological physical systems may also have potential applications because they are robust against local perturbations. Specifically, a topologically protected quantum state degeneracy cannot be lifted by any local interactions.RMP (); Wen () It is therefore natural to consider using topological phases for applications requiring a high degree of quantum coherence.RMP () For example, it has recently been pointed out that non-Abelian anyonsLeinaas (); Wilczek (); Wilczek_book () in a fractional quantum Hall system can lead to topological quantum computing.Sarma05 () Anyons are neither bosons nor fermions, but obey anyonic braiding statistics.Leinaas (); Wilczek (); Wilczek_book () Unfortunately, they have not yet been convincingly observed experimentally in any physical system.
Instead of only looking for naturally existing topological phases, one could also design artificial lattice structures that possess desired topological phases. One example is the Kitaev honeycomb model,Kitaev () which requires that the spin (natural or artificial) at each node of a honeycomb lattice interacts with its three nearest neighbors through three different interactions: , , and . Depending on the bond parameters, this anisotropic spin model supports both Abelian and non-Abelian anyons. Kitaev () Its realization could potentially lead to experimental demonstration of anyons and implementation of topological quantum computing. However, the requirement for anisotropic interactions is tremendously demanding and generally cannot be satisfied by natural spin lattices.
Various artificial lattices may possess interesting topological phases. For instance, it has been proposed that a triangular Josephson junction array may have a two-fold degenerate ground state that is topologically protected.Ioffe_Nat02 (); Albuquerque_PRB08 () A recent proposal suggests the use of capacitively coupled Josephson junction arrays to simulate a two-component fermion model that has topological excitations.Xue_PRA09 () There is also a suggestion that a Josephson junction array with properly designed interactions and topology can be local-noise resistant.Gladchenko () With respect to the physical realization of the Kitaev model, there are proposals using neutral atoms in optical lattices.Duan (); Zoller (); Sarma07 () One similarity among all of these proposals, whether based on Josephson junction arrays or on optical lattices, is that they all require extremely low temperatures and precise single-atom manipulations. The reason is that topologically interesting properties are not generally contained in the symmetry of the system Hamiltonian. Instead they are only emergent properties at very low temperatures.
Here we propose a quantum emulation of the Kitaev lattice using superconducting quantum circuits (Ref. YSN_preprint, gives a brief summary of this work). As for the topic of quantum analog simulations, see Ref. Buluta, for a brief review. In our superconducting network, a Josephson charge qubit is placed at each node of a honeycomb lattice. These charge qubits behave like artificial spins and are tunable via external fields.YN05 (); Mak (); Wendin () Each charge qubit interacts with its three nearest neighbors through three different types of circuit elements. One advantage of our proposal is that some circuit elements involved and their functionalities at low energies have already been demonstrated experimentally—for example, the and couplings between charge qubits have been studied in experiments.NEC (); Yamamoto () Here we show theoretically that they can indeed provide the needed anisotropic interactions when included in a honeycomb lattice. We then identify the ground states of this network in two different parameter regimes and show that it can have both vortex and bond-state excitations. We also describe how they can be generated using spin-pair operations.
Ii Kitaev lattice based on superconducting quantum circuits
At low energies, superconducting (SC) qubits can behave as artificial spins. Among the varieties of SC qubits (charge, flux, phase,YN05 (); Mak (); Wendin () and other hybrids YTN06 (); YHAN07 (); transmon ()), only charge qubits are known to interact with each other in all the individual forms of , , and (via a mutual inductance, an oscillator, and a capacitance, respectively).YTN (); Schon (); NEC (); Bruder () Therefore, to emulate a Kitaev honeycomb lattice, we propose to build a two-dimensional SC circuit network based on SC charge qubits. More specifically, on a honeycomb lattice a charge qubit is placed on each node [Fig. 1(b)], and one of the three circuit elements is inserted along each bond of the lattice (denoted as the -, -, or -type bond). A building block of this lattice is shown in Fig. 1(a), which consists of four charge qubits that are connected via an -, a -, and a -type bond. Each charge qubit is a Cooper-pair box connected to a superconducting ring by two identical Josephson junctions to give it tunability: Each qubit is controlled by both the magnetic flux piercing the SQUID loop and the voltage applied via the gate capacitance .
Naively, a circuit element should maintain its basic characteristics when inserted in a larger network, at least in the linear regime. However, as it has been shown in previous studies of hybrid qubits,YTN06 (); YHAN07 (); transmon () a superconducting qubit based on one particular variable (for example, charge) can acquire characters of another (for example, flux) when additional circuit elements are added to it. Therefore, here we first clarify whether the different circuit elements in our honeycomb network maintain their basic individual characteristics (particularly the forms and strengths of the interactions) at low energies when lumped together.
We first write down the Lagrangian of the quantum circuits, choosing the average phase drop across the two Josephson junctions of each charge qubit as the canonical coordinates. After identifying the corresponding canonical momenta, we then derive (this derivation is shown in the appendix) the total Hamiltonian of the quantum circuits as
Here the free Hamiltonian of the th charge qubit is
where is the charging energy of the Cooper pair box, with the total capacitance , and ; the number operator of the Cooper pairs in the th box (which is conjugate to ); the reduced offset charge induced by the gate voltage ; and the effective Josephson coupling energy of the th charge qubit, with the flux quantum.
The three nearest-neighbor couplings, shown as the , , and bonds in Fig. 1(a), are given by
Here is the critical current through the Josephson junctions of the charge qubits (we assume identical junctions for simplicity), while is the circulating supercurrent in the SQUID loop of the th charge qubit. Note that the coupling strength between nodes 1 and 3 (along a -link), , is affected by the mutual capacitance that connects qubit 1 (3) with its nearest-neighbor along a -link. Compared to the case of two qubits coupled by an oscillator,Schon () where , the capacitive inter-node coupling along the -link in the present circuit greatly increases the inter-node coupling along the -link because usually . This is an important and positive consequence when multiple circuit elements are introduced to create different inter-node interactions.
At low temperatures, only the lowest-energy states of a superconducting circuit element are involved in the system dynamics, which is quantum mechanical. For the particular case of a charge qubit, where , the lowest-energy eigenstates are mixtures of having zero and one Cooper pair in the box, when the gate voltage is near the optimal point (i.e., ). Defining and as the two charge states having zero and one extra Cooper pair in the box, we now have a two-level system as a quantum bit, or qubit. In the spin- representation based on these charge states and ( is the index of the nodes), the system variables can be expressed as
Here we consider the simple case with (i.e., all gate voltages on the different nodes are identical: ) and for all qubits. The low-energy Hamiltonian of the system is then reduced to
with . The reduced Hamiltonian (6) is the Kitaev model with an effective magnetic field with - and -components. Here and play the role of a “magnetic” field. Since , to maintain finite inter-qubit couplings, cannot vanish. Therefore our Hamiltonian represents a Kitaev model in an always-finite magnetic field, although the field direction can be adjusted. This Hamiltonian has an extremely complex quantum phase diagram because of all the (experimentally) adjustable parameters. Here we are particularly interested in whether it has topologically-interesting phases and when such topological properties might emerge.
Iii Vortex and bond-state excitations
Below we focus on two particular parameter regimes of the finite-field Kitaev model of (6), under the general condition that the -bond interaction dominates over the other interactions. In particular, when , we identify a vortex state excitation. This case is described in Section A below. When but is of the same order as and , we identify a new excitation that we call the bond state. We describe this case in Sec. III.B. The vortex state is a known topological excitation in the zero-field Kitaev model, while the bond state is specific to the finite-field Kitaev model.
iii.1 Kitaev lattice with dominant -bonds in a weak “magnetic” field
We first consider the case when
and treat as the perturbation. Using perturbation theory in the Green function formalism, Kitaev () one can construct an effective Hamiltonian for the lattice:
where is the excitation energy of the state , i.e., the energy difference between states and . Here is the ground state of the unperturbed Hamiltonian, i.e., Hamiltonian (6) with the perturbation term excluded. Note that the effective Hamiltonian only contains contributions from the second-order terms because both the first- and third-order terms vanish. Including the zeroth-order term (unperturbed Hamiltonian), the total Hamiltonian of the system can be written as
where the effective - and -couplings are
Below we focus on the Abelian excitations. When , the dominant part of the Hamiltonian is that along the vertical links,
Under , the two spins along each -link tend to be aligned opposite to each other ( or ) in order to lower their energies. Indeed, the highly degenerate ground state of is an arbitrary vector in the Hilbert subspace spanned by , where denotes the total number of -links and . Within the ground-state subspace of and up to fourth order,Kitaev () the effective Hamiltonian of the Kitaev lattice takes the form
Here is the plaquette operator for a given plaquette [see Fig. 1(b)]. The operator for any plaquette commutes with the unperturbed Hamiltonian :
so that as well, and the ground states of form a subset of the degenerate ground states of . It is straightforward to show that , or . Since , to minimize the energy of , we need
In other words, the eigenvalues of the operators in the ground state are for all plaquettes .
When some plaquettes undergo transformations that lead to , the system gets into an excited state. The lowest-energy excitation corresponds to the generation of a pair of vortices when for two neighboring plaquettes. In this excitation process, each of the two neighboring plaquettes acquires a phase , which is equivalent to the addition of a flux quantum through each plaquette. As shown in Fig. 2(a), such an excitation can be generated by applying either of the following two spin-pair operators on the ground state :
Here the two operators () and () act on the ground state at the bottom and top sites of the th -link, respectively. This pair of vortices, generated by either or , are topological states with an excitation energy of
For example, if GHz and both and are one tenth of , this gap would be about 1 MHz, corresponding to a temperature of 0.1 mK. This small gap requires an extremely low experimental temperature for suppressing the thermal activation of the ground state to the vortex states. Note that a different perturbative approach Vidal () shows that in the parameter region , the spin-pair operators and generally create both vortex and fermionic excitations. However, in the limit of , the dominant excitations are vortex states, Vidal () which is consistent with the conclusion drawn above.
iii.2 Kitaev lattice with dominant -bonds in a uniform “magnetic” field along the -direction
If we stay in the regime where the -bond couplings are dominant (), but place each charge qubit at the optimal point where , so that , a different quantum phase arises when is comparable to . In other words, we now consider the regime
Here the zeroth-order Hamiltonian is again the coupling along the -bonds: (notice that here the coupling strength is , not ), with the same highly degenerate ground state as discussed in the previous subsection. To clarify the low-energy excitation spectrum in this regime, we again use perturbation theory in the Green’s function formalism to remove the linear terms and derive an effective Hamiltonian in the ground state sub-Hilbert space of . Up to second-order, the effective Hamiltonian takes the form
The spin pair operator at a -bond (again the two Pauli operators act on the bottom and top nodes of the particular -bond) commutes with the zeroth-order Hamiltonian (although it anti-commutes with the four plaquette operators connected to this -bond). Similar to , the pair operator also has two eigenvalues . Thus the ground state of should satisfy for all the -bonds in the system. In other words,
for all -bonds. Since no two -bonds share a node in the honeycomb lattice, and the lattice is completely covered by all the -bonds, we can solve the eigenstates of of each -bond and obtain the ground state of as
This is a nondegenerate ground state, which forms a simple subset of the highly degenerate ground states of . It is maximally entangled within each -bond, but not entangled at all between different -bonds. In other words, the two-spin correlation function decays to identically zero beyond a -bond. The lattice is now an ensemble of maximally entangled “spin” pairs that are completely independent from each other. This ground state is reminiscent of (and simpler than) the dimerized valence bond solid state discussed in the context of spin Hamiltonians.Affleck_87 (); Majumdar_69 () There valence bond states refer to a singlet for the electron spins, which is dictated by the Coulomb interaction and Pauli principle between electrons.
When the pair operators and are separately applied to the ground state at the th -bond [see Fig. 2(b)], the excited states
are called a bond state—while the pair operators are different, the states they generate are only different by an overall phase because is a factored state for all -bonds. A bond state at the th -bond corresponds to the change of at that particular bond. It is above the ground state in energy. Notice that a bond state is an excitation that is completely localized to a particular -bond. Furthermore, bond states are generated by the same pair operators that generate the vortex excitations, although the ground states of the system are different in these two cases. In contrast to , the ground state is nondegenerate, and the bond state excitations are very different from the vortex states. This transition from vortex excitations to bond states occurs when we vary the parameters of the system (i.e., tuning to and reduce from close to so that increases to the same magnitude as and/or ), during which the topological property of the system changes.
Iv The braiding of excitations
A vortex looping around another vortex can produce either a sign change or no sign change to the wave function. The first case is denoted as an -type vortex looping around an -type vortex, and the second case corresponds to an -type vortex looping around an -type vortex (see, e.g., Ref. Sarma07, for a more detailed discussion). This indicates anyonic statistics between the and vortex states. Therefore, braiding, which refers to moving one quasi-particle around another, is an important tool to determine the statistics of the quasi-particles (in the present case the vortices). Here we show an alternative procedure for braiding an excitation with another, which can be applied to both vortex and bond states.
Let us consider two particular evolutions for the system. The first evolution contains spin-pair operations applied to the ground state at three successive -bonds along the vertical path , as shown in Fig. 3(a). Here for the vortex case and for the bond-state case. The second evolution contains spin-pair operations applied at four successive -bonds along the horizontal path as shown in Fig. 3(a). After these two operations in series, the state of the system is , where
Now we turn the evolutions backward by applying and to the system successively, so as to fuse RMP (); Kitaev () the excitations to the vacuum (i.e., the ground state) [see Fig. 3(b)]. The final state of the system is now
When the paths and intersect at a -bond, such as in the example given in Fig. 3(a), where and anti-commute, and anti-commute as well: . The final state thus becomes
In other words, a phase flip resulted from the evolutions. For vortex excitations, this is equivalent to the case of an -type vortex looping around an -type vortex, as shown in Ref. Sarma07, . In contrast, when similar operations are applied but the paths and do not intersect at a -bond [see Fig. 3(c), for example], , so that
yielding no phase flip in the final state as compared to the initial state. For vortex excitations, this is equivalent to the case of an -type vortex looping around another -type vortex.
The braiding of excitations, i.e. whether there is or there is no phase flip, can be revealed by means of Ramsey-type interference. Sarma07 (); Pachos () To achieve this, one can keep the same as above, but use
i.e., each is replaced by half of the rotation. In the braiding case shown in Fig. 3(a),
at the crossing point of paths and . Thus,
Therefore, the braiding of excitations can be distinguished by verifying if an excited state occurs at the crossing point of paths and .
While the vortex state described by Eq. (13) and the bond state described by Eq. (22) are very different excitations, they have similar braiding properties. In the braiding procedure shown above, the system is initially in the vacuum (either or ); after the braiding operations in Eq. (28), the system is fused to the vacuum again, but with a sign change to the ground-state wave function no matter which ground state the system starts with. In order to distinguish the difference between the vortex and bond-state excitations, one needs to focus on the intermediate steps of the braiding operations. Take in Eq. (27) as an example. When it is applied to , the spin-pair operator in it first creates a pair of vortices, and then the other spin-pair operations and successively move one vortex downward along the vertical path . The final state is also a pair of vortices, but the two vortices are separated by three -bonds in the vertical direction [see Fig. 3(a)]. Importantly, this pair of vortices is degenerate with the pair of vortices . However, in sharp contrast to the vortex case, when in Eq. (27) is applied to , each of the spin-pair operations , creates a bond state and the final state is nondegenerate with the bond state .
V Implementation of quantum rotations
As indicated in previous sections, single-qubit rotations are needed to create vortex and bond-state excitations, and to perform braiding operations. Below we show that these quantum rotations of individual qubits in the honeycomb lattice can be achieved via electrical and magnetic controls. The key is to reduce the coupling between a specific qubit and its neighboring qubits to such a degree that its single-qubit dynamics dominates for a period of time.
To generate a rotation at a particular charge qubit, we consider the following approach by controlling both the magnetic flux through SQUID loops and the local electric field. Specifically, when the magnetic flux in the SQUID loop of each charge qubit is set to , and , so that the honeycomb lattice is now decoupled into a series of one-dimensional chains. For a charge qubit , thus . We further assume that , so that as well. One can now shift the gate voltage for a period of time at the th lattice point far away from the usual working point of the Kitaev lattice, so that the corresponding single-qubit energy (instead of ) is much larger than both and . Such a parameter regime should be reasonably easy to achieve for a charge qubit. This operation of shifting should yield a local -type rotation on the th qubit:
When (where ), , so the operation on the th qubit is given by
while half of the rotation is
The corresponding inverse rotations can be achieved by shifting the gate voltage to .
A rotation of a particular charge qubit can be generated by a similar approach. Specifically, when and , one has , , and . Again the honeycomb lattice is separated into a series of one-dimensional chains. Here we assume that , achievable in this charge-qubit system, which allows us to perform a single-qubit rotation driven by . When , for a time we switch off the flux in the SQUID loop of the th qubit (the working point of this Kitaev lattice is usually at ), producing a local -type rotation on the th qubit:
The rotation on the th qubit is
where . Note that when the flux in the SQUID loop of the th qubit is switched off to produce a local -rotation, the flux in the SQUID loop of the nearest-neighbor qubit that is connected to the th qubit via an oscillator should be simultaneously shifted to a value around , so as to keep between these two qubits much smaller than .
With both and rotations available for the th qubit, the rotation is given by
Therefore, one can construct all the wanted operations and for generating both vortex and bond-state excitations by using the single-qubit rotations and .
In order to obtain accurate - and -type single-qubit rotations, we assume that and are much larger than the inter-qubit coupling. Actually this somewhat stringent condition can be loosened for realistic systems. As shown in Ref. Wei, , accurate effective single-qubit rotations can still be achieved using techniques from nuclear magnetic resonance when the inter-qubit coupling is small compared to single-qubit parameters (instead of much smaller than and ).
Vi Discussion and conclusion
In this paper, our main objective is to construct an experimentally feasible proposal to emulate the Kitaev model on a network made of superconducting nanocircuits. To focus on the topological properties of the system, we choose the limit of identical qubits and identical coupling strength. Furthermore, we fix the mutual inductances and the capacitances of the various circuit elements involved. There are basically two tunable parameters: the gate voltage on the Cooper pair boxes () for each charge qubit, and the magnetic flux through the SQUID loops connected to the Cooper pair boxes. Within the regime where -bonds dominate in interaction energy scale ( much larger than all other couplings, including , , , and ), we have explored two limiting cases: one with weak effective magnetic fields (, , ), the other with the effective field only along the -direction. We have identified some properties of the relevant ground states, and the low-energy excitations, the vortex and bond states. However, much more study is needed to completely clarify the energy spectrum, the phase diagram, and the dynamics of this superconducting network.
One observation we have made is that the vortex excitations and bond-state excitations can be generated using the same spin-pair operations, starting from different ground states ( and ) that depend on the system parameters. We have also shown that while is highly entangled, is only entangled locally but not globally. This quantum phase transition requires more extensive studies to identify the critical point and related critical phenomena, such as how system entanglement changes near the transition point, and most importantly how its topological properties change. It would also be worthwhile to investigate the system spectrum (from vortex excitation to bond state excitation) and dynamics during this transition, similar to our study of quantum phase transitions between Abelian and non-Abelian phases of the Kitaev model.Shi_PRB09 () While such studies are generally numerically intensive, it would help reveal the exotic topological properties of this many-body model.
With the elementary building blocks given in Fig. 1(a), one can construct Kitaev spin models on other types of lattices as well (see, e.g., Refs. Yao, and Sun, ). In particular, it has been shown that in the absence of a magnetic field, the Kitaev model on a decorated honeycomb lattice Yao () can support gapped non-Abelian anyons. The quantum analog simulation of Kitaev models on different lattices using superconducting circuits should shed light on the novel properties of these topological systems.
There are two important open issues in the study of building a superconducting qubit network to emulate a spin lattice. One is the role played by the decoherence of individual qubits, and the other is the measurement of correlated states on a qubit network. It is well known that charge qubits suffer from fast decoherence. However, it is not clear how decoherence would affect the topological excitations. Indeed, the faster decoherence of charge qubits may allow the system to reach its ground state faster. Furthermore, topological excitations are supposed to be robust against local fluctuations, so that decoherence in individual nodes may not easily destroy excitations such as the vortex state. Quantum measurement is another open issue in the study of collective states, whether ground states or low-energy excitations, of a qubit lattice. While single-qubit measurement of superconducting qubits can now be done with quite high fidelity,Holfheinz_Nature (); DiCarlo_Nature () and two-qubit correlation measurements have been done,Bell_UCSB () measuring multi-qubit correlations requires further theoretical and experimental studies. We hope that our proposal acts as another incentive for researchers in the field of superconducting qubits to look for ways to perform measurements that can reveal quantum correlations.
In conclusion, we have proposed an approach to emulate the Kitaev model on a honeycomb lattice using superconducting quantum circuits, and shown that the low-energy dynamics of the superconducting network should follow a finite-field Kitaev model Hamiltonian. We analytically study two particular limits for system parameters, explore their ground state characteristics, and identify their low-energy excitations as vortex states and bond states. We further show that both vortex- and bond-state excitations can be generated using the same spin-pair operations, starting from different ground states. Our proposal points to an experimentally realizable many-body system for the quantum emulation of the Kitaev honeycomb spin model.
Acknowledgements.We thank J. Vidal and Yong-Shi Wu for useful discussions. J.Q.Y. and X.F.S. were supported by the National Basic Research Program of China Grant Nos. 2009CB929300 and 2006CB921205, and the National Natural Science Foundation of China Grant Nos. 10625416 and 10534060. X.H. and F.N. acknowledge support by the National Security Agency and the Laboratory for Physical Sciences through the US Army Research Office, X.H. acknowledges support and hospitality by the Kavli Institute of Theoretical Physics at the University of California at Santa Barbara, and F.N. thanks support by the National Science Foundation Grant No. 0726909.
Appendix A Derivation of the Hamiltonian
Below we derive the Hamiltonian of the honeycomb lattice constructed with superconducting quantum circuits as described in Fig. 1. For simplicity all charge qubits have the same parameters. Furthermore, the mutual inductances , the oscillators, and the mutual capacitances for the -, -, and -couplings are also identical, respectively. Since the self inductance of the SQUID loop in each charge qubit is small, the voltage drop across this loop inductance can be ignored as compared with the voltage drops across the Josephson junctions in the loop. Also, we assume that the capacitance of the oscillator is much larger than the gate capacitance and the mutual capacitance, i.e., . The total electrical energy of the qubit lattice can be written as (the 1-2 ad 1-3 couplings are magnetic and will be discussed later)
where the summation is over all the -links. the term contains the charging energies of the nodes on either end of a -link in the building block, together with the coupling across the link. It is given by
and . Here is the average voltage drop across the two Josephson junctions of the th charge qubit, and () is the voltage drop across the oscillator connected to qubit 1 (4).
The Langrangian of the qubit lattice is
where is the total potential energy of the system, including Josephson coupling energy and magnetic energy in all the inductors in the network. To derive the system Hamiltonian, we choose and as the canonical coordinates. The corresponding canonical momenta are thus
where the subscript 5 denotes the qubit which is connected to qubit 2 via the mutual capacitance . In the limit of , . Thus one has
with , and
The Hamiltonian of the honeycomb lattice is thus
We now perform two gauge transformations, so that and become:
After these gauge transformations, and become