Universal distributed quantum computing on superconducting qutrits with dark photons
Ming Hua, Ming-Jie Tao, Ahmed Alsaedi, Tasawar Hayat, Fu-Guo Deng
aa r X i v : . [ qu a n t - ph ] S e p Universal distributed quantum computing on superconducting qutrits with darkphotons ∗ Ming Hua , , , Ming-Jie Tao , Ahmed Alsaedi , Tasawar Hayat , , and Fu-Guo Deng , , † Department of Physics, Applied Optics Beijing Area Major Laboratory,Beijing Normal University, Beijing 100875, China NAAM-Research Group, Department of Mathematics, Faculty of Science,King Abdulaziz University, Jeddah 21589, Saudi Arabia Department of Applied Physics, School of Science,Tianjin Polytechnic University, Tianjin 300387, China Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan
We present a one-step scheme to construct the controlled-phase gate deterministically on remotetransmon qutrits coupled to different resonators connected by a superconducting transmission linefor an universal distributed quantum computing. Different from previous works on remote super-conducting qubits, the present gate is implemented with coherent evolutions of the entire system inthe all-resonance regime assisted by the dark photons to robust against the transmission line loss,which allows the possibility of the complex designation of a long-length transmission line to link lotsof circuit QEDs. The length of the transmission line can reach the scale of several meters, whichmakes our scheme is suitable for the large-scale distributed quantum computing. This gate is a fastquantum entangling operation with a high fidelity of about 99%. Compare with previous works inother quantum systems for a distributed quantum computing, under the all-resonance regime, thepresent proposal does not require classical pulses and ancillary qubits, which relaxes the difficultyof its implementation largely.
I. INTRODUCTION
Quantum computation (QC) [1, 2], as an interdisciplinary research of computer science and quantum mechanics,has attracted much attention in recent years. It can implement the famous Shor’s algorithm [3] for the factorizationof an n-bit integer exponentially faster than the classical algorithms and the Grover’s algorithm [4] or the optimalLong’s algorithm [5] for unsorted database search. Various quantum systems have been used to implement QC, suchas photons [6–12], nuclear magnetic resonance [13–15], diamond nitrogen-vacancy center [16–21], and cavity quantumelectrodynamics (QED) [22]. Among the quantum systems, circuit QED [23, 24], composed of a superconductingqubit (SQ) coupled to a superconducting resonator (SR), provides a good platform for the implementation of QCbecause of its good ability of the large-scale integration and the accurate manipulation on the SQ [25, 26].Circuit QED has been studied a lot for achieving the basic tasks of QC on SQs or SRs, such as the construction ofthe single-qubit and the universal quantum gates [27–33], entangled state generation [34–42], and the measurementand the non-demolition detection on SQs or SRs [43–46]. The types for integrating the SQs and the SRs mainlycontain some SQs coupled to a SR bus [47] or some SRs coupled to a SR bus [48] or a SQ [49–51]. At present, it ishard to integrate lots of SQs or SRs in a quatum-bus-based processor to achieve the complex universal QC. Furtherscaling up the number of SQs or SRs requires linking the distant circuit QED systems to form a quantum network[52–64] introduced by the distributed quantum computing [65], in which a quantum computer can be seen as a quantumnetwork of distant local processors with only a few qubits and are connected by quantum transmission lines (TL).As the key problem in the realization of the distributed quantum computing, quantum entanglement and universalquantum gate on remote qubits have been discussed in some other systems [66–71]. For example, Cirac et al. [72]proposed a scheme to achieve the ideal quantum transmission between atoms trapped at spatially separated nodesin 1997. In 2004, Xiao et al [73] realized the controlled phase (c-phase) gate between two rare-earth ions embeddedin the respective microsphere cavities assisted by a single-photon pulse in sequence. In 2011, L¨u et al [74] proposedtwo schemes to complete the entanglement generation and quantum-state transfer between two spatially separatedsemiconductor quantum dot molecules.To achieve the universal quantum gate on distant qubits coupled to different cavities connected by the TLs[75–78],realistic flying-photon qubit or adiabatic processes and the local operations are required. On one hand, there are someworks which studied the quantum network by using the dark photon in the TL in other quantum systems. In 2007,Yin et al [75] presented some schemes to achieve the state transfer and quantum entangling gates deterministically ∗ Published in Ann. Phys. (Berlin) , 1700402 (2018). † Corresponding author:[email protected] between the remote multiple two-level atoms trapped in different cavities connected by an optical fiber, in which thec-phase gate should be completed by using the “dipole blockade” effect among atoms in a cavity and it needs not topopulate the realistic photons in the fiber. In 2014, Clader [76] presented an adiabatic scheme to transfer a microwavequantum state from one cavity to another, assisted by an optical fiber which is robust against both mechanical andfiber loss. On the other hand, one should transfer the microwave photon to the optical photons to link the remote SQs.In 2015, Yin et al. [77] proposed a scheme to achieve the quantum networking of SQs based on the optomechanicalinterface.To implement the distributed quantum computing on remote SQs coupled to different SRs connected by a supercon-ducting TL, one should overcome the decay of the TL as the more the complicated designation and a longer length forthe TL is, the bigger the decay of the photon in it becomes. In this paper, we propose a scheme for the constructionof the c-phase gate on two remote transmon qutrits coupled to different SRs connected by a superconducting TL forthe distributed quantum computing on SQs. Our scheme works in the all-resonance regime by letting the frequenciesof the qutrits and the resonators equal to each other. The scheme can be achieved with just one step assisted bythe dark photons in the TL, without requiring classical pulses and ancillary qubits, which relaxes the difficulty of itsimplementation in experiment largely. Far different from the c-phase gate on two remote superconducting resonatorsconstructed in Ref.[31] which is completed with three resonance steps between resonators and a qubit and can beextended to achieve the gate on two remote superconducting qubits by coupling them to the two remote resonators,respectively, we use a superconducting TL instead of the superconducting qubit as a quantum bus. Here, using thedark photons in TL to reduce the requirement of the quality factor of the TL allows the complex designation of aTL to link lots of remote circuit QEDs and the length of the TL (the distance between two remote superconductingqubits) can reach the scale of several meters. The fidelity of the present c-phase gate is beyond 99% by using thenumerical simulation with the feasible parameters.
II. BASIC THEORIES
Let us consider a distributed quantum computing composed of two remote superconducting qubits q and q coupled to two single-mode high-quality superconducting resonators r a and r b , respectively, which are connected bya superconducting TL r f , shown in Fig. 1. The Hamiltonian of this device is (in the interaction picture with ¯ h = 1) H = H a + H b + H a ( b ) f = g a ( a + σ − e − iδ a t + aσ +1 e iδ a t ) + g b ( b + σ − e − iδ b t + bσ +2 e iδ b t )+ ∞ X j =1 g If,j (cid:2) f j ( a + + ( − j e iφ b + ) + H.c. (cid:3) , (1)where H a , H b , and H a ( b ) f are the interaction Hamiltonians of the subsystems composed of q and r a , q and r b , and r f and r a ( r b ), respectively. H a ( b ) f applies to the high-finesse resonators and resonant operations over the time scalemuch longer than the TL’s round-trip time [79]. δ IJ = ω I − ω J ( I = a , b and J = 1,2, f ). ω a , ω b , and ω f are thetransition frequencies of resonators r a , r b , and the TL r f , respectively. ω and ω are the transition frequencies ofthe qubits q and q , respectively. a + , b + , and f + are the creation operators of the resonators r a , r b , and the TL r f ,respectively. σ +1 and σ +2 are the creation operators of the transitions | g i ↔ | e i and | g i ↔ | e i of the qubits q and q , respectively. | g i and | e i are the ground and the first excited states of the qubit q , respectively. g a and g b are the coupling strength between q and r a and that between q and r b , respectively. g If,j is the coupling strengthbetween r a ( b ) and the mode j of the TL r f . φ is the phase induced by the propagating field through the TL r f oflength l with the relation φ = 2 πωl/c in which c is the speed of light.In the short TL limit (2 Lκ a ( b ) f ) / (2 πc ) ≤
1, only one resonant mode f of the TL r f interacts with the resonators’modes ( L is the length of r f and κ a ( b ) f is the decay rate of the resonator r a ( b ) into a continuum of TL modes) [75].The Hamiltonian H can be reduced to H int = g a ( a + σ − e − iδ a t + aσ +1 e iδ a t ) + g b ( b + σ − e − iδ b t + aσ +2 e iδ b t ) + g af ( f + a + f a + ) + g bf ( f + b + f b + ) . (2)In the Schr¨odinger picture, this Hamiltonian can be rewritten as H ′ = ω a a + a + ω b b + b + ω f f + f + ω σ +1 σ − + ω σ +2 σ − + g a ( a + σ − + aσ +1 ) + g b ( b + σ − + bσ +2 ) + g af ( f + a + f a + ) + g bf ( f + b + f b + ) . (3) (cid:1869) (cid:2869) (cid:1869) (cid:2870) (cid:1870) (cid:3028) (cid:1870) (cid:3029) (cid:1870) (cid:3033) (a) (b) (cid:513)(cid:883)(cid:1767) (cid:3030) (cid:3126) (cid:2033) (cid:3028) (cid:513)(cid:883)(cid:1767) (cid:3028) (cid:513)(cid:883)(cid:1767) (cid:3033) (cid:513)(cid:883)(cid:1767) (cid:3029) (cid:513)(cid:882)(cid:1767) (cid:3028) (cid:513)(cid:882)(cid:1767) (cid:3033) (cid:513)(cid:882)(cid:1767) (cid:3029) (cid:513)(cid:883)(cid:1767) (cid:3030) (cid:513)(cid:883)(cid:1767) (cid:3030) (cid:3127) (cid:513)(cid:882)(cid:1767) (cid:3030) (cid:513)(cid:882)(cid:1767) (cid:3030) (cid:3126) (cid:513)(cid:882)(cid:1767) (cid:3030) (cid:3127) (cid:2033) (cid:3029) (cid:2033) (cid:3033) (cid:2033) (cid:3030) (cid:3126) (cid:2033) (cid:3030) (cid:2033) (cid:3030) (cid:3127) FIG. 1: (a) Setup for a distributed quantum computing composed of two remote qubits q and q coupled to different resonators r a and r b connected by a transmission line r f . (b) Illustrations of the energy splitting of the subsystem composed of r a , r b ,and r f . To construct the c-phase gate on the remote transmon qutrits below, we consider the all-resonance condition with ω a = ω b = ω f = ω = ω = ω by letting the frequencies of the qubits and the resonators and the TL equal to eachother and g af = g bf = g by letting the coupling strength between r a and r f equal to the one of r b and r f . If onetakes the canonical transformations C ± = ( a + b ± √ f ) and C = √ ( a − b ) [80, 81], the Hamiltonian H ′ can berepresented as H ′′ = ωσ +1 σ − + ωσ +2 σ − + ωC + C + (cid:16) ω + √ g (cid:17) C + C ++ + (cid:16) ω − √ g (cid:17) C − C + − + 12 h g a (cid:16) C + + C − + √ c (cid:17) σ +1 + g a (cid:16) C ++ + C + − + √ C + (cid:17) σ − + g b (cid:16) C + + C − − √ C (cid:17) σ +2 + g b (cid:16) C ++ + C + − − √ C + (cid:17) σ − i . (4)Here the modes C and C ± are three bosonic modes and they are not coupled to each other. From Eq. (4), the energylevel of the subsystem composed of r a , r b , and r f are split into three different parts with frequencies ω c + , ω c − , and ω c signed by the modes C + , C − , and C , respectively, as shown in Fig. 1 (b). Because of the contributions of the fieldsof r a and r b , the three modes C and C ± interact with the two qubits q and q . When g ≫ { g a , g b } , the excitationsof modes C ± are highly suppressed as ω ± √ g detune with the resonance modes ( C , q , and q with frequency of ω ) largely, which means the modes C ± are the dark ones to the frequency ω f of r f , and the Hamiltonian H ′′ can bereduced to H ′′′ = ωσ +1 σ − + ωσ +2 σ − + ωC + C + 1 √ (cid:2) g a (cid:0) Cσ +1 + C + σ − (cid:1) − g b (cid:0) Cσ +2 + C + σ − (cid:1)(cid:3) . (5)It can be written as H eff = 1 √ (cid:2) g a ( Cσ +1 + C + σ − ) − g b ( Cσ +2 + C + σ − ) (cid:3) (6)in the interaction picture. Here, only the mode C = √ ( a − b ) is left, which means that the TL can not be populatedin the all-resonance regime in our system. The interaction between two remote two-energy-level qubits expressed by H eff will be used to construct the all-resonance c-phase gate on the two remote qutrits below. (cid:513)(cid:882)(cid:1767) (cid:3033) (cid:513)(cid:1859)(cid:1767) (cid:2870) (cid:1859) (cid:2869)(cid:482)(cid:3034)(cid:3032)(cid:3028) (cid:1869) (cid:2869) (cid:513)(cid:1859)(cid:1767) (cid:2869) (cid:1859) (cid:2869)(cid:482)(cid:3032)(cid:3046)(cid:3028) (cid:1859) (cid:3033)(cid:3028) (cid:1859) (cid:3033)(cid:3029) (cid:1859) (cid:2870)(cid:482)(cid:3034)(cid:3032)(cid:3029) (cid:1859) (cid:2870)(cid:482)(cid:3032)(cid:3046)(cid:3029) (cid:1869) (cid:2870) (cid:1870) (cid:3033) (cid:513)(cid:1857)(cid:1767) (cid:2869) (cid:513)(cid:1871)(cid:1767) (cid:2869) (cid:513)(cid:883)(cid:1767) (cid:3033) (cid:513)(cid:1857)(cid:1767) (cid:2870) (cid:513)(cid:1871)(cid:1767) (cid:2870) FIG. 2: Illustrations of interactions between q and r a , r f and r a ( r b ), and q and r b , respectively, for the construction of thec-phase gate on two remote transmon qutrits q and q . III. C-PHASE GATE ON THE TWO REMOTE QUTRITS q AND q To construct the c-phase gate on the two remote transmon qutrits q and q in the device shown in Fig. 1, oneshould consider the evolutions of states | g i | g i , | g i | e i , | e i | g i , and | e i | e i simultaneously, and generate a minusphase on one of them only. According to Eq. (6) and considering the two lowest transitions of the qubits onlydiscussed in the section II, the phase of the state | g i | g i doesn’t evolve with time. The phase of the state | e i | e i is the product of the phases of states | g i | e i and | e i | g i which are both evolve with time. There are three statesevolve with time. Based on these relationships among the phases of the four states, one cannot construct the one-stepall-resonance c-phase gate.Here, we consider the second excited energy level | s i of q and take ω ge / (2 π ) = ω es / (2 π ) = ω a / (2 π ) = ω b / (2 π ) = ω f / (2 π ). The illustrations of the interactions between q and r a , r f and r a ( r b ), and q and r b for constructing thec-phase gate on q and q are shown in Fig. 2. The Hamiltonian of the whole system can be expressed as H q = g a ge (cid:16) a + σ − ge e − iδ a ge t + aσ +1; ge e iδ a ge t (cid:17) + g a es (cid:16) a + σ − es e − iδ a es t + aσ +1; es e iδ a es t (cid:17) + g b ,ge (cid:16) b + σ − ge e − iδ b ge t + bσ +2; ge e iδ b ge t (cid:17) + g b es (cid:16) b + σ − es e − iδ b es t + bσ +2; es e iδ b es t (cid:17) + g af (cid:0) f + a + f a + (cid:1) + g bf (cid:0) f + b + f b + (cid:1) , (7)in which σ +1(2); ge and σ +1(2); es are the creation operators of the transitions | g i ↔ | e i and | e i ↔ | s i ofthe qutrit q , respectively. g a ( b )1(2); ge and g a ( b )1(2); es (cid:16) g a ( b )1(2); es = √ g a ( b )1(2); ge (cid:17) are the coupling strengths between the twotransitions of q and r a ( b ) , respectively. δ a ( b )1(2); ge = ω ge − ω a ( b ) and δ a ( b )1(2); es = ω es − ω a ( b ) . ω ge ( ω ef ) isthe frequency of the transition | g i ↔ | e i (cid:0) | e i ↔ | s i (cid:1) of the qutrit q . | s i is the second excited stateof q . In order to get two states of the qubits evolve with time only, one should take ω ge − ω es ≫ { g a ge , g b ge } to ignore the dispersive coupling between the transition | e i ↔ | s i of the qutrit q and r a and the one between thetransition | g i ↔ | e i of the qutrit q and r b . The Hamiltonian H q can be reduced to H ′ q = g a ge ( a + σ − ge + aσ +1; ge ) + g b es ( b + σ − es + bσ +2; es ) + g af ( f + a + f a + ) + g bf ( f + b + f b + ) . (8)To achieve the one-step all-resonance c-phase gate by using the dark photon in the superconducting TL, we takethe same canonical transformations as the ones in Sec. II and g af = g bf ≫ { g a ge , g b es } , the Hamiltonian H ′ q becomes H ′ eff = 1 √ (cid:2) g a ge ( Cσ +1; ge + C + σ − ge ) − g b es ( Cσ +2; es + C + σ − es ) (cid:3) . (9)Suppose that | ψ i = | g i | g i | i c , | ψ i = | g i | e i | i c , | ψ i = | e i | g i | i c , and | ψ i = | e i | e i | i c ( | i c ≡ | i a | i b | i f )are the initial states of the system undergoes the Hamiltonian H ′ eff , respectively, one can get their evolutions as | Ψ ( t ) i = e iH ′ eff t | g i | g i | i c = | g i | g i | i c , (10) | Ψ ( t ) i = e − iH ′ eff t | g i | e i | i c = | g i | e i | i c , (11) | Ψ ( t ) i = e − iH ′ eff t | e i | g i | i c = cos (cid:18) g a ge √ t (cid:19) | e i | g i | i c + sin (cid:18) g a ge √ t (cid:19) | g i | g i | i c , (12) | Ψ ( t ) i = e iH ′ eff t | e i | e i | i c = 1 G ′ " ( g b es ) + ( g a ge ) cos r G ′ t ! | e i | e i | i c − g a ge g b es G ′ " cos r G ′ t ! − | g i | s i | i c − ig a ge √ G ′ sin r G ′ t ! | g i | e i | i c , (13)where G ′ = ( g a ge ) + ( g b es ) . From the evolutions of the four states, one can construct the c-phase gate on the tworemote qutrits q and q . In detail, we suppose the initial state of the system described by H ′ eff is | Ψ cp i = (cos θ | g i + sin θ | e i ) ⊗ (cos θ | g i + sin θ | e i ) ⊗ | i c . (14)According to Eqs. (10) and (11), one can keep the states | g i | g i | i c and | g i | e i | i c unchanged. By taking the proper g a ge and g b ge to satisfy g a ge √ t = (2 k − π and q G ′ t = 2 mπ ( k, m = 1 , , , · · · ) simultaneously, one can achieve thecondition that when the state | e i | g i | i c undergoes an odd number of periods and generates a minus phase (from Eq.(12)), the state | e i | e i | i c undergoes an even number of periods and keeps unchanged (from Eq. (13)) meanwhile.That is, the state of the system evolves from | Ψ cp i to the final state | Ψ cpf i = ( α | g i | g i + α | g i | e i − α | e i | g i + α | e i | e i ) ⊗ | i c , (15)which is just the target state after our c-phase gate operation on q and q with the initial state | Ψ cp i .Here α = cos θ cos θ , α = cos θ sin θ , α = sin θ cos θ , and α = sin θ sin θ . In the basis {| g i | g i , | g i | e i , | e i | g i , | e i | e i } , the matrix of the c-phase gate is U cp = − . (16)To get the c-phase gate within a short time, we take k = m = 1, that is, g es / (2 π ) = √ g ge / (2 π ) = √ g ge / (2 π ).Supposing that the initial state of the system is | Ψ max i = ( | ψ i + | ψ i + | ψ i + | ψ i ), one can get the state | Ψ cpmax i = ( | ψ i + | ψ i−| ψ i + | ψ i ) after our c-phase gate operation on the two remote qutrits q and q with a maximal fidelityof 99 .
8% (∆ = 25), 99 .
6% (∆ = 10), and 98 .
8% (∆ = 5) within gt = 0 .
705 ( ω ge − ω es = ω ge − ω es = 90 g a ge ),by using the definition F cpmax = |h Ψ cpmax | e − iH q t | Ψ max i| , (17)as shown in Fig. 3. Here ∆ ≡ g a ( b ) f /g a ge . One can tune the frequencies of transmon qutrits by using individual fluxbias lines [82], which let the frequencies of qutrits resonate or detune with resonators to turn on or off the operationof our gate. IV. POSSIBLE EXPERIMENTAL IMPLEMENTATION AND FIDELITY
In experiment, a high quality factor Q ∼ × of a 1D SR has been demonstrated [83]. By considering therelation between the decay rate κ , Q , and the frequency of resonator ω r with κ = ω r /Q [23], the best life time of a g t F i de li t y ∆ =25 ∆ =10 ∆ =5 FIG. 3: The fidelity of our c-phase gate varies with gt and different ∆ = g a ( b ) f /g a ge . photon in a superconducting resonator can reach ∼ µ s. The coherence time of a transmon qubit [82, 84–86] canalso reach 50 µ s by using titanium nitride [87]. The tunable range of the transition frequency of a transmon qubit canreach 2 . ω ge / (2 π ) − ω es / (2 π ) = ω ge / (2 π ) − ω es / (2 π ) = 0 .
72 GHz [89], which lets us ignore the detune interaction between each qutrit and theircorresponding resonator, compared with the small coupling strength between them. As for the SRs and the TL r f , onecan couple them by using the SQUID, which can reach a coupling strength of g a ( b ) f / (2 π ) ∼
200 MHz theoretically withreasonable experimental parameters [90]. Moreover, one can also use the capacitance coupling between resonatorsand the superconducting TL. With the reasonable parameters ω a / (2 π ) = ω b / (2 π ) = ω f / (2 π ) = 6 GHz, the couplingcapacitance C = 13 . C r = 2pF [23, 91], the capacitance coupling strength can reach g a ( b ) f / (2 π ) = 40 MHz (which will be discussed below for theconstruction of the c-phase gate with ∆ = 5). It is worth noticing that a coupling strength between a SR and asuperconducting TL has been realized with about 32 MHz [54].To show the feasibility of our scheme for the construction of the c-phase gate on two remote qutrits, we numericallysimulate the fidelity of the scheme based on the parameters realized in experiments or predicted theoretically withreasonable experimental parameters.The dynamics of the quantum system undergoes the Hamiltonian H q is determined by the master equation dρdt = − i [ H q , ρ ] + κ a D [ a ] ρ + κ b D [ b ] ρ + κ f D [ f ] ρ + X l =1 , { γ l ; ge D [ σ − l ; ge ] ρ + γ l ; es D [ σ − l ; es ] ρ + γ φl ; e ( σ l ; ee ρσ l ; ee − σ l ; ee ρ/ − ρσ l ; ee /
2) + γ φl ; s ( σ l ; ss ρσ l ; ss − σ l ; ss ρ/ − ρσ l ; ss / } . (18)Here, κ a,b,f is the decay rate of the resonator r a,b,f . γ l ; ge ( γ l ; es ) and γ φl ; e ( γ φl ; s ) are the energy relaxation and thedephase rates of the transition | e i l ↔ | g i l ( | s i l ↔ | e i l ) of the transmon qutrits q l ( l = 1 , γ − l ; ge =( γ φl ; ge ) − = ( γ φl ; es ) − = 2 γ − l ; es [92], σ l ; ee = | e i l h e | , and σ l ; ss = | s i l h s | . D [ L ] ρ = (2 LρL + − L + Lρ − ρL + L ) /
2. Because ofthe competition relation between the coupling strength g a ge and the decay rates and that between the decoherencetime of resonators and qutrits, for different γ l ; ge and κ a,b,f , one should choose different g a ge to reach the maximalfidelity of our scheme for constructing the c-phase gate on remote qutrits q and q . The coupling strengths g a ge / (2 π )chosen below are the optimal ones which correspond to the highest fidelities of the gate when we fix g a ( b ) f / (2 π ) = 200MHz and γ − l ; ge = κ a,b,f = 50 µ s by considering the set of discretized g a ge values, varying from 1 to 100MHz in stepsof 1MHz.To show the feasibility of our c-phase gate on remote qutrits q and q with decoherence time and the decay timeof the qutrits and the resonators, we numerically simulate the fidelity of | Ψ cpmax i after our c-phase gate operations onthe whole system (the initial state of the system is | Ψ max i ) by using the definition F cp = h Ψ cpmax | ρ ( t ) | Ψ cpmax i , (19)in which the effects from the unresonant parts H = g e,fa, ( aσ +1; e,f e iδ e,fa, t + a + σ − e,f e − iδ e,fa, t ) (20)and H = g g,eb, ( bσ +2; g,e e iδ g,eb, t + b + σ − g,e e − iδ g,eb, t ) (21)are considered. Parameters chosen here are ω ge / (2 π ) = ω es / (2 π ) = ω a / (2 π ) = ω b / (2 π ) = ω f / (2 π ) = 6 GHz, q g b ge / (2 π ) = g a ge / (2 π ) = 8 MHz. γ − l ; ge = κ − a,b,f = 50 µ s. As shown in Fig. 4 (a), the fidelity of the state | Ψ cpmax i can reach 99 .
28% within 88 . | Ψ cpmax i from the coupling strength, the anharmonicity, the decoherence time,and the frequency of the qutrits and the decay time of resonators as shown in Fig. 4 (b)-(f). In each figure in Fig. 4,parameters are kept unchanged except for the one signed in the axis of abscissas. The influences from g a ge is shownin Fig. 4 (b) and the small change of g a ge influences the fidelity little. Fig. 4 (c) indicates that the anharmonicityof q should be chosen to let the transition frequency ω ge detune with ω f + √ g a ( b ) f largely, which is required whenwe reduce the Hamiltonian from H q to H ′′ eff . Because of the four-step coupling between the two qutrits, accurateresonance between the two remote qutrits is required as shown in Fig. 4 (d). In the large-scale integration of oursystem, the interaction between qutrits can be turned off conveniently by tuning the frequencies of the qutrits. InFig. 4 (e), we give the influences on the fidelity of the state from the decay time of r f . It can be seen that when κ − f >
10 ns, κ − f influences the fidelity of the state little. To show the possible influence from the decay rates ofthe resonators and the decoherence time of the qutrits, we calculate the average gate fidelity of the c-phase gate withdifferent Γ − = γ − l ; ge = κ − a,b,f , shown in Fig. 4 (f) by using the average-gate-fidelity definition F = ( 12 π ) Z π Z π h Ψ cpf | ρ ( t ) | Ψ cpf i dθ dθ . (22)Here, ρ ( t ) is the realistic density operator after our c-phase gate operation on the initial state Ψ cp with the Hamiltonian H . It is worth noticing that the decay time 20 µ s corresponds to the typically quality factor Q ∼ × of a 1Dsuperconducting resonator [93] and the coherence time 20 µ s of a transmon qutrit can also be readily realized inexperiment [86]. Although the coupling strength g a ( b ) f / (2 π ) taken here is 200 MHz (predicted theoretically in [90])which satisfies ∆ = 25 and it has not been realized in experiments, if we take ∆ = 5 ( g a ( b ) f / (2 π ) = 40 MHz) and themore of the actual situation of the life time of SRs and qutrits with Γ − = 20 µ s, the fidelity of our c-phase gate canalso reach a high value of 98% (compared with the fidelities between ∆ = 25 and ∆ = 5 as shown in Fig. 3) whichshould be enhanced further by taking corresponding optimal parameters. Corresponding to the operation time of thec-phase gate construction, the length of the superconducting TL can, in principle, reach the scale of several meters. V. DISCUSSION AND SUMMARY
On one hand, in order to use the dark photons in the TL to achieve the c-phase gate on qutrits q and q , oneshould take g a ( b )1(2); ge ≪ g a ( b ) f . On the other hand, the small coupling strength of g a ( b )1(2); ge used here compared withanharmonicites of transmon qutrits (hundreds of megahertz) does not require the large anharmonicities of the qutrits.Moreover, to achieve our c-phase scheme, the Ξ-type energy level of the qutrits is required. Besides the transmonqutrit, the superconducting charge qutrit [94, 95] or phase qutrit [96] can also been applied to our scheme. By using thetransmon qutrit or the phase qutrit with ω ge / (2 π ) > ω ef / (2 π ), one should take the proper anharmonicity of q to letthe transition frequency ω ge detune with ω f + √ g a ( b ) f largely. By using the charge qutrit with ω ge / (2 π ) < ω ef / (2 π ),one should take ω ge detune with ω f − √ g a ( b ) f largely. That is, when the frequency ω ge ∼ ω f + √ g a ( b ) f , the effectiveHamiltonians H eff and H ′ eff can not be obtained as the mode C ± can not be suppressed effectively.
70 80 90 100 11080859095100 t (ns) F c p ( % ) (a) g a ge (MHz) F c p ( % ) (b) ω ge − ω ef (GHz) F c p ( % ) (c) ω ef (GHz) F c p ( % ) (d) κ − f (ns) × −1 −2 F c p ( % ) (e)
10 20 30 40 5096979899100 γ − ( µ s) F c p ( % ) (f) FIG. 4: (a) The fidelity of the c-phase gate on q and q varies with the operation time t . (b)-(f) The relations between thefidelity of the gate and g a ge , ω ge − ω ef , ω ef , κ − f , and γ − , respectively. In summary, we have proposed a one-step scheme to achieve the c-phase gate on two remote transmon qutritscoupled to different resonators connected by a superconducting TL. The scheme works in the all-resonance regimewith just one step, which leads to a fast operation and can be demonstrated in experiment easily. Moreover, ourscheme is robust against the TL loss by using the dark photon. That is, the superconducting TL needs not to bepopulated, which is convenient to be extended to a large-scale integration condition by the complex designation of along-length TL to link lots of remote circuit QEDs.
Acknowledgments
F.G. Deng was supported by the National Natural Science Foundation of China under Grants No. 11474026,No. 11647042, and No. 11674033, the Fundamental Research Funds for the Central Universities under Grant No.2015KJJCA01. M. Hua was supported by the National Natural Science Foundation of China under Grants No.11647042 and No. 11704281. [1] M. A. Nilsen and I. L. Chuang,
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