Scalable quantum memory in the ultrastrong coupling regime
SScalable quantum memory in the ultrastrong coupling regime
T. H. Kyaw a , S. Felicetti, G. Romero, E. Solano,
2, 3 and L.-C. Kwek
1, 4, 51
Centre for Quantum Technologies, National University of Singapore,3 Science Drive 2, Singapore 117543, Singapore Department of Physical Chemistry, University of the Basque Country UPV/EHU,Apartado 644, E-48080 Bilbao, Spain IKERBASQUE, Basque Foundation for Science,Maria Diaz de Haro 3, 48013 Bilbao, Spain Institute of Advanced Studies, Nanyang Technological University,60 Nanyang View, Singapore 639673, Singapore National Institute of Education, Nanyang Technological University,1 Nanyang Walk, Singapore 637616, Singapore (Dated: September 5, 2018)
Abstract
Circuit quantum electrodynamics, consisting of superconducting artificial atoms coupled to on-chip res-onators, represents a prime candidate to implement the scalable quantum computing architecture because ofthe presence of good tunability and controllability. Furthermore, recent advances have pushed the technol-ogy towards the ultrastrong coupling regime of light-matter interaction, where the qubit-resonator couplingstrength reaches a considerable fraction of the resonator frequency. Here, we propose a qubit-resonator sys-tem operating in that regime, as a quantum memory device and study the storage and retrieval of quantuminformation in and from the Z parity-protected quantum memory, within experimentally feasible schemes.We are also convinced that our proposal might pave a way to realize a scalable quantum random-accessmemory due to its fast storage and readout performances. a Corresponding author: [email protected] a r X i v : . [ qu a n t - ph ] M a r nalogous to the classical computer processors, a quantum processor inevitably requires mem-ory cell elements to store arbitrary quantum states in efficient and faithful manner. In particu-lar, these memory devices might be needed and useful for storing and retrieving qubits in a fasttimescale between the quantum processor and the memory elements, similar to a classical random-access memory . That would require ability to store information for a short time with fast storageand readout responses. Here, we propose a quantum memory implemented on circuit quantumelectrodynamics (cQED) with fast storage and retrieval responses. The memory cell operatesin the ultrastrong coupling (USC) regime of light-matter interaction , where the qubit-resonatorcoupling strength approaches a significant fraction of the resonator frequency. In addition, a keyingredient towards realizing our scheme is the existense of the Z parity symmetry , whichallows us to encode quantum information in partiy-protected states that are robust against certainenvironmental noises .In this report, we propose a scalable quantum memory cell element based on the cQED architec-ture, comprising of a superconducting flux qubit galvanically coupled to a microwave resonator.In particular, we study the storage and retrieval of single- and two-qubit states, while the inputstates are in the form of flying microwave photons . These processes can be carried out withgood fidelity even with the presence of noise. We are also convinced that our memory can bescaled up to store large number of qubits since the cQED architecture provides very high levelof controllability and scalability . In this way, we believe our proposal might pave a waytowards scalable quantum random-access memory (QRAM) and distributed quantum inter-connects , which in turn might steer towards novel applications ranging from entangled-statecryptography , teleportation , purification , fault-tolerant quantum computation to quan-tum simulations.The qubit-resonator system operating in the USC regime, as shown in Fig. 1, exhibits a Z parity symmetry and its dynamics is governed by the quantum Rabi Hamiltonian H Rabi = (cid:126) ω eg σ z + (cid:126) ω cav a † a + (cid:126) Ω σ x ( a + a † ) , (1)where ω eg , ω cav , and Ω stand for the qubit frequency, cavity frequency, and qubit-resonator cou-pling strength, respectively. In addition, a ( a † ) is the bosonic annihilation(creation) operator, and σ x,z are the Pauli matrices of the qubit. A compelling feature of Hamiltonian (1) is that for ratios2 /ω cav (cid:38) . , the ground and first excited states can be approximated as | ψ G (cid:105) (cid:39) √ | − α (cid:105)| + (cid:105) − | α (cid:105)|−(cid:105) ) , | ψ E (cid:105) (cid:39) √ | − α (cid:105)| + (cid:105) + | α (cid:105)|−(cid:105) ) , (2)where | α (cid:105) is a coherent state for the resonator field with amplitude | α | = Ω /ω cav , and |±(cid:105) =( | e (cid:105) ± | g (cid:105) ) / √ are the eigenstates of σ x . The states | ψ G/E (cid:105) form a robust parity-protected qubit whose coherence time can be up to τ coh (cid:38) /ω eg . In the following, we outline a protocol thatallows storage and retrieval of quantum information to and from this qubit. It is achieved by adi-abatically tuning the qubit-resonator coupling strength, from the Jaynes-Cummings (JC) to USCregime. In particular, we propose a USC memory cell element (see Fig. 1a) that can be designedby the flux-qubit architecture presented in Ref. 33, which provides a tunable qubit-resonatorcoupling (see Supplementary information). The latter can be implemented by using a supercon-ducting quantum interference device (SQUID) as proposed for qubit-qubit coupling in Refs. 34,35. Results
Generating and catching flying qubits . Photons propagating through linear devices are well-suited as information carriers because they possess long coherence length and can be encoded withuseful information. In our case, a flying microwave photon is generated from cQED platforms ,and a qubit is encoded in a linear superposition of zero ( | F (cid:105) ) and one photon ( | F (cid:105) ) Fock states.Recently, it has been pointed out in Ref. 19 that if a photonic wave packet emitted from a sourcehas a temporally symmetric profile, it overcomes the impedance mismatch problem when a flyingqubit impinges onto a resonator. With all these latest advancement in cQED technologies, weenvision our memory cell be located on the pathway of a single microwave photon to accomplishquantum information storage (see Fig. 1).
Storage and retrieval processes . The storage of quantum information into our USC memory cellis realized within three steps. At first, we cool down the system to reach its ground state in the USCregime. Secondly, the qubit frequency is tuned to be off-resonant with the resonator frequency, i.e., ω cav > ω eg , while the qubit-resonator coupling stregth Ω is adiabatically tuned towards the strongcoupling regime where Ω /ω cav (cid:28) , where the coupling is much larger than any decoherencerate in the system. In this regime, the ground and first excited states of the qubit-resonator systemare | ψ (cid:105) = | g (cid:105) ⊗ | (cid:105) and | ψ (cid:105) = | e (cid:105) ⊗ | (cid:105) , respectively. Here, the states | g (cid:105) and | e (cid:105) stand for3he ground and excited states of the qubit, while | (cid:105) stands for the vacuum state of the resonator.Since we have adiabatically tuned the coupling from the USC to the strong coupling regime, ourinitial USC ground state is then mapped to the JC ground state, i.e., | ψ (cid:105) = | g (cid:105) ⊗ | (cid:105) . At thisstage, our memory cell is ready for information storage. When a flying qubit with an unknownquantum state | Ψ F (cid:105) = α F | F (cid:105) + β F | F (cid:105) comes in contact with the cell as shown in Fig. 1a, theencoded information from the flying qubit is transferred to the flux qubit due to the JC dynamics.Therefore, the state of our system becomes | ψ s (cid:105) = ( α F | g (cid:105) + β F | e (cid:105) ) ⊗ | (cid:105) . At last, we turn onthe qubit-resonator coupling adiabatically towards the USC regime. For simplicity, we considera linear adiabatic switching scheme such that Ω( t ) = (cos( f ) − ∆ f sin( f ) t/T )Ω , with T totalevolution time and f = φ ext /φ . Here, φ ext is an external magnetic flux and φ = h/ e is the fluxquantum (see Methods and Supplementary information for detailed definition). In Fig. 2a,b, weshow the storage and retrieval processes for a quantum state | ψ s (cid:105) = α F | ψ (cid:105) + β F | ψ (cid:105) , and Fig. 2c,dshow the ground state | ψ (cid:105) and the first excited state | ψ (cid:105) adiabatically follow the instantaneouseigenstates such that | ψ (cid:105) → | ψ G (cid:105) and | ψ (cid:105) → | ψ E (cid:105) . In this manner, we can encode importantinformation onto the parity-protected qubit basis. Retrieval (decoding) process is reverse of thestorage process and is achieved by adiabatically switching off the qubit-resonator coupling strengthfrom the USC to SC regime.We note that the time for storage and retrieval of quantum information is several order ofmagnitude faster than the coherence time of the parity-protected qubit, which is about T coh ∼ µ sfor a coupling strength Ω /ω eg ∼ . . For instance, if we consider a flux qubit with energy ω eg / π ∼ , and a cavity of frequency ω cav / π ∼ , our system reaches the USCregime with Ω /ω cav = 0 . . For the linear adiabatic switching scheme with the above parameters,we estimate total time for storage/retrieval of a qubit is about ˜ T ≈ − ns.At the end of an adiabatic evolution, the state | ˜ ψ (cid:105) = α F | ψ G (cid:105) + β F | ψ E (cid:105) is desired. However, thestate after the evolution might become | ˜ ψ ( T ) (cid:105) = α F | ψ G ( T ) (cid:105) + β F e iθ ( T ) | ψ E ( T ) (cid:105) , with a relativephase θ ( T ) resulting from the dynamical and geometrical effects . Hence, we need to keep trackof a relative phase during the storage and retrieval processes.In order to find out which phase θ ( t ) optimizes the processes, in Fig. 3a,b, we plot the fidelity F (Ω , θ ) = |(cid:104) ˜ ψ | ψ ( t ) (cid:105)| between the state | ˜ ψ (cid:105) = α F | ψ G (cid:105) + β F e iθ | ψ E (cid:105) and the state | ψ ( t ) (cid:105) , whichhas adiabatically evolved from the initial state | ψ s (cid:105) = α F | ψ (cid:105) + β F | ψ (cid:105) . In these simulations, wefind the fidelity over the landscape of θ ∈ [0 , π ] versus the qubit-resonator coupling strength Ω( t ) ,for two different total evolution time T = 105 /ω cav (Fig. 3a), and T = 120 /ω cav (Fig. 3b). White4ines show the phase θ opt , which optimizes the fidelity F for both cases. Notice that the maximumfidelity and the optimal phase θ depend strongly on the system parameters and the total evolutiontime T . Thus, we require, for each USC memory cell, to find out the parameter T that maximizesthe fidelity only once. When T is known, the cell can always be operated at that specific parameterfor storing and retrieving unknown quantum states. Therefore, the time T might be a benchmarkto characterize our potential USC quantum memory devices, in the same way as hard disk drivesof the classical computer are being characterized by their seek time and latency.Additionally, storage and retrieval of entangled states in two separate USC cells is feasible.To demonstrate such a process, we let two bosonic fields to interact via the SQUID, simulat-ing a Hong-Ou-Mandel setup as shown in Fig. 1b. Let us suppose that we have an initialstate | ψ (cid:105) = | (cid:105) ˆ a (cid:48) | (cid:105) ˆ b (cid:48) . After experiencing a beam splitter interaction, we have two-photonentangled state | ψ (cid:48) (cid:105) = √ ( | (cid:105) ˆ a | (cid:105) ˆ b + | (cid:105) ˆ a | (cid:105) ˆ b ) , which enters two cavities c and c , each con-taining a flux qubit prepared in its ground state. This process allows the cavities to be pre-pared in the state | Ψ (cid:105) = √ ( | (cid:105) c | (cid:105) c + | (cid:105) c | (cid:105) c ) ⊗ | gg (cid:105) . Following the same procedure,we tune the qubits towards resonance with its respective cavity such that we arrive at the state | Ψ (cid:105) = √ ( | ge (cid:105) + | eg (cid:105) ) ⊗ | (cid:105) c c . With our protocol, the state is eventually mapped to a parity-protected state | ˜Ψ (cid:105) = √ ( | ψ G (cid:105)| ψ E (cid:105) + | ψ E (cid:105)| ψ G (cid:105) ) . In Fig. 3c,d, we show the numerical simulationsfor the storage and retrieval processes of the entangled state | Ψ (cid:105) . Discussion
We have presented the basic tools for building a quantum memory based on a cQED architec-ture that operates in the USC regime of light-matter interaction. The storage/retrieval processfor unknown quantum states, be single-qubit or two-qubit entangled states, can be accomplishedby adiabatically switching on/off the qubit-resonator coupling strength. As a scope, we proposethe large-scale quantum memory network shown in Fig. 4(a), where each edge of the memorynetwork is constituted with our memory cell element. This architecture can pave the way forthe implementation of a scalable QRAM, which might benefit from the fast storage and readoutperformances of superconducting circuits. In addition, each node in the network is connected toa SQUID that allows to selectively switch on/off interaction between neighbouring microwavecavities , in order to implement quantum state transfer processes within the same layer (seeSupplementary information). Ultimately, we would like to achieve a multilayer circuit architec-ture, where a quantum processor layer interfaces with the proposed memory layer as shown in5ig.4. Methods
Switchable quit-resonator coupling strength . In the cQED architecture composed of a fluxqubit galvanically coupled to an inhomogeneous resonator, the Hamiltonian that describes thedynamics reads H = (cid:126) ω eg σ z + (cid:126) ω cav a † a + H int , (3)with an effective tunable interaction Hamiltonian H int = − E J β cos (cid:16) π φ ext φ (cid:17) (cid:88) n =1 , (∆ ψ ) n (cid:88) µ = x,y,z c ( n ) µ σ µ , (4)where E J is the Josephson energy, β is a parameter that depends on the Josephson junctions size, φ = h/ e is the flux quantum, and φ ext is an external flux through a superconducting loop. Thelatter in turn allows to switch on/off the qubit-resonator coupling strength. ∆ ψ stands for thephase slip shared by the resonator and the flux qubit. And, the coefficients c ( n ) µ ’s can be tuned at will via additional external fluxes (see Supplementary information). Adiabatic evolution . We obtain the effective system Hamiltonian H = (cid:126) ω eg σ jz + (cid:126) ω cav a † a +(cos( f ) − ∆ f sin( f ) t/T )Ω σ jx ( a † + a ) from Eqs. (3) and (4), if we consider an external flux thatvaries linearly with time according to φ ext = ¯ φ + (∆ φ ) t/T , where ¯ φ is an offset flux and ∆ φ isa small flux amplitude.We remark that all our simulations presented so far assume no loss in both the qubit and res-onator. Nonetheless, the open system analysis of a USC system can be carried out by studyingdynamics of the microscopic master equation (see Supplementary information). In Fig. 5, weshow numerical results for the storage and retrieval processes of an arbitrary superposed state | ψ s (cid:105) in presence of external noises. With our scheme and a simple decoherence model, we estimatefidelity of F s = 0 . at the end of the retrieval process. Pritchett, E. J. & Geller, M. R. Quantum memory for superconducting qubits.
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Acknowledgments
The authors acknowledge support from the National Research Foundation & Ministry of Educa-tion, Singapore; Spanish MINECO FIS2012-36673-C03-02; UPV/EHU UFI 11/55; Basque Gov-ernment IT472-10; and CCQED, PROMISCE, SCALEQIT EU projects. We also thank LorenzoMaccone and Seth Lloyd for their helpful comments.
Author contributions
All authors T.H.K., S.F., G.R., E.S. and L.-C.K. contributed equally to the results.
Additional informationSupplementary information accompanies this paper at
Competing financial interests:
The authors declare no competing financial interests.9 igure 1. Schematic of circuit-QED design for storage and retrieval of an unknown single-and two-qubit states. (a) A USC memory cell element, composed of a qubit-resonator systemoperating at the USC regime. (b) Two flying microwave photons, with modes ˆ a (cid:48) and ˆ b (cid:48) , come inand pass through a beam splitter (BS) implemented by a superconducting quantum interferencedevice (SQUID) to form a two-qubit entangled state, which is then stored in two USC qubitslocated at a distance apart. Figure 2. Fidelity plots. (a) Storage and (b) retrieval processes for a quantum state | ψ s (cid:105) = α F | ψ (cid:105) + β F | ψ (cid:105) . In both cases, we plot the fidelity between the initial | ψ (cid:105) s and the instantaneousstate | ψ ( t ) (cid:105) , i.e., F s = |(cid:104) ψ s | ψ ( t ) (cid:105)| . Any arbitrary state | ψ (cid:105) = u | ψ (cid:105) + v | ψ (cid:105) can be stored andretrieved with unit fidelity. (c) Fidelity between the approximated ground state in Eq. (2) andthe instantaneous ground state F G = |(cid:104) ψ G | ψ G ( t ) (cid:105)| . (d) Fidelity between the approximated firstexcited state in Eq. (2) and the instantaneous first excited state F E = |(cid:104) ψ E | ψ E ( t ) (cid:105)| . For all thesimulations, we choose the system parameters as ω cav = 1 , ω eg = 0 . ω cav , Ω = ω cav , and thetotal evolution T = 105 /ω cav . Figure 3. Contour plots of the fidelity F = |(cid:104) ˜ ψ | ψ ( t ) (cid:105)| between the state | ˜ ψ (cid:105) = α F | ψ G (cid:105) + β F e iθ | ψ E (cid:105) and the state | ψ ( t ) (cid:105) , which has adiabatically evolved from the initial state | ψ s (cid:105) = α F | ψ (cid:105) + β F | ψ (cid:105) . (a) The total evolution time is set to T = 105 /ω cav . (b) The evolution timeis T = 120 /ω cav . For the above cases, the black lines stand for the phase which maximized thefidelity F . In these simulations, the parameters are ω cav = 1 , ω eg = 0 . ω cav , and Ω = ω cav . (c)Storage process for an entangled state | Ψ (cid:105) = √ ( | ge (cid:105) + | eg (cid:105) ) ⊗ | (cid:105) c c . (d) Retrieval process.In both cases, we plot the fidelity between the initial state | Ψ (cid:105) , and the instantaneous state | ψ ( t ) (cid:105) , ¯ F = |(cid:104) Ψ | ψ ( t ) (cid:105)| . In the simulations (c) and (d), we have chosen ω cav = 1 , ω eg = 0 . ω cav , Ω = ω cav , and the evolution time T = 105 /ω cav .10 igure 4. A scalable quantum network. (a) The light-matter interface operating at the ultra-strong coupling regime may be envisioned as a set of microwave cavities connected, at the nodes,by SQUID devices that allow to switch on/off the cavity-cavity interaction. Notice that each cavityis represented with different colors (red, black, blue and green) that stand for different lengths toassure the manipulation of specific pairwise interactions (see Supplementary information). Inaddition, on each edge of the memory array, there is a memory cell made of a USC entity (bluesquare) to store an arbitrary quantum information in a specific location. (b) Integrated quantumprocessor . A 2D cavity grid with a qubit distribution (rectangular boxes) represented in variouscolors is shown here. It was previously shown in Ref. 39 that such a cavity grid may provide ascalable fault-tolerant quantum computing architecture with minimal swapping overhead. Datatransfer between the two layers may be done via cavity buses (vertical black colored lines con-necting layer a and b).