Abstract
We propose a quantum processor for the scalable quantum computation on microwave photons in distant onedimensional superconducting resonators. It is composed of a common resonator R acting as a quantum bus and some distant resonators r_{j} coupled to the bus in different positions assisted by superconducting quantum interferometer devices (SQUID), different from previous processors. R is coupled to one transmon qutrit and the coupling strengths between r_{j} and R can be fully tuned by the external flux through the SQUID. To show the processor can be used to achieve universal quantum computation effectively, we present a scheme to complete the highfidelity quantum state transfer between two distant microwavephoton resonators and another one for the highfidelity controlledphase gate on them. By using the technique for catching and releasing the microwave photons from resonators, our processor may play an important role in quantum communication as well.
Introduction
Quantum computation^{1}, which can implement the famous Shor’s algorithm^{2} for integer factorization and Grover/Long algorithm^{3,4} for unsorted database search, has attracted much attention in recent years. There are some interesting systems which have been used to realize quantum computation, such as photons^{5,6}, quantum dots^{7,8}, nuclear magnetic resonance^{9,10,11}, diamond nitrogenvacancy center^{12,13} and cavity quantum electrodynamics (QED)^{1}. Achieving quantum computation, quantum state transfer^{14,15} and universal quantum gates have been studied a lot, especially the twoqubit controlledphase (cphase) gate or its equivalent (controllednot gate) which can be used to construct a universal quantum computation assisted by singlequbit operations^{1}. To construct the highefficiency and highfidelity quantum state transfer and the cphase gate on fields or atoms, cavity QED, composed of a twoenergylevel atom coupled to a singlemode filed, has been studied a lot.
Simulating cavity QED, circuit QED^{16,17,18,19,20,21,22,23,24,25,26,27}, composed of a superconducting qubit coupled to a superconducting resonator, plays an important role in quantum computation because of its good ability for the largescale integration^{28,29,30,31,32,33}. By far, some important tasks of quantum computation based on the superconducting qubits have been realized in experiments. For example, DiCarlo et al. demonstrated a cphase gate on two transmon qubits^{34} in 2009 and they prepared and measured the entanglement on three qubits in a superconducting circuit^{35} in 2010. In 2012, Lucero et al.^{36} computed the prime factors with a Josephson phase qubit quantum processor and Reed et al.^{37} constructed a controlledcontrolled phase gate to realize a threequbit quantum error correction with superconducting circuits. In 2014, Barends et al.^{38} realized the singlequbit gate and the cphase gate on adjacent Xmon qubits with high fidelities of 99.94% and 99.4%, respectively.
Interestingly, the quality factor of a onedimensional (1D) superconducting resonator^{39} has been enhanced to 10^{6}, which makes the resonator as a good carrier for quantum information processing^{40,41,42,43,44,45,46,47,48,49,50,51,52}. For instance, Houck et al.^{53} generated single microwave photons in a circuit in 2007. In 2008, Hofheinz et al.^{54} generated the Fock states in a superconducting quantum circuit. In 2010, Johnson et al.^{55} realized the quantum nondemolition detection of single microwave photons in a resonator. In 2011, Wang et al.^{56} deterministically generated the entanglement of photons in two superconducting microwave resonators and Strauch et al.^{57} proposed a scheme to prepare the NOON state on two resonators. In 2013, Yang et al.^{58} presented two schemes for generating the entanglement between microwave photons and qubits. Recently, Hua et al.^{59} proposed some schemes to construct the universal cphase and ccphase gates on resonators.
There have been some theoretic studies on constructing the multiresonator quantum entanglement and the universal quantum gate on local microwavephoton resonators in a processor composed of some resonators coupled to a superconducting qubit^{57,58,59,60,61}. In this paper, we propose a quantum processor for quantum computation on distant resonators with the tunable coupling engineering^{62,63} between the superconducting resonator and the quantum bus. There is just one superconducting transmon qutrit q in our processor, which is coupled to a common resonator R (acts as a quantum bus). Different from the processors in previous works^{57,58,59,60,61}, the resonators r_{j} (j = 1, 2) (act as the information carriers) in our processor are coupled to the quantum bus R, not the qutrit, which makes it have the capability of integrating some distant resonators^{35} by coupling them to the bus in different positions. In contrast with the resonatorzeroqubit architecture by Galiautdinov et al.^{31}, the resonators in our processor are used for quantum information processing, not the memory elements. It does not require more superconducting qubits. With our processor, we present an effective scheme for the quantum state transfer between r_{1} and r_{2} with the Fock states and and another for the cphase gate on two resonators by using the resonance operations between R and r_{j} and that between R and q. The fidelities of our quantum state transfer and cphase gate are 99.97% and 99.66%, respectively. By catching and releasing the microwave photons from resonators^{64}, our processor maybe play an important role in quantum communication.
Results
Quantum processor composed of resonators and a quantum bus
Our quantum processor is composed of some distant highquality 1D superconducting resonators r_{j} and a highquality 1D superconducting resonator R, shown in Fig. 1. The common resonator R acts as a quantum bus for quantum information processing and it is capacitively coupled to a Ξ type threeenergylevel superconducting transmon qutrit q whose frequency can be tuned by an external magnetic field. The qutrit is placed at the maximum of the voltage standing wave of R (not be drawn here). The simple superconducting quantum interferometer device (SQUID) with two Josephson junctions inserted between r_{j} and R serves as the tunablecoupling function between them. The SQUID variables are not independent and introduce no new modes^{63}. Here, the SQUIDs are not sensitive to the charge noise and can achieve a full tunability. Besides, the plasma frequencies of SQUIDs should be larger than the frequencies of the resonators. r_{j} are laid far enough to each other to avoid their direct interaction generated by mutual capacitances and mutual inductive coupling. In the interaction picture, the Hamiltonian of the processor is (ħ = 1, under the rotatingwave approximation)
Here, and Δ_{j} = ω_{j} − ω_{R}. ω_{R} and ω_{j} are the the first mode frequencies of R and r_{j}, respectively. ω_{g,e(e,f)} is the frequency of the transmon qutrit q with the transition in which , and are the ground, the first excited and the second excited states of the qutrit, respectively. and are the creation operators of R and r_{j}, respectively. and are the creation operators of the two transitions of q, respectively. g_{g,e} and g_{e,f} are the coupling strengths between R and the two transitions of q, respectively. g_{j} is the coupling strength between r_{j} and R, which is contributed by their capacitive and inductive and can be tuned by the external flux through the SQUID^{63}. By controlling the time dependence of the coupling, the crosstalk between resonators can be switched on and off.
The evolution of our processor can be described by the master equation^{65}
Here, the operator D[L]ρ = (2LρL^{+} − L^{+}Lρ − ρL^{+}L)/2 (L = a, b, , ). and . κ_{1}, κ_{2} and κ_{R} are the decay rates of the resonators r_{1}, r_{2} and R, respectively. γ_{g,e} (γ_{e,f}) is the energy relaxation rate of the qutrit with the transition . γ_{ϕ,e} (γ_{ϕ,f}) is the dephasing rate of the level of the qutrit. To achieve the resonance operations between R and r_{j}, the transition frequencies of all the resonators are taken equal to each other.
Quantum state transfer between r_{1} and r_{2}
Our quantumstatetransfer protocol between r_{1} and r_{2} can be completed with two resonance operations between the quantum bus R and the resonator r_{j}. The interaction between R and r_{j} can be described as
In our scheme, the states , and are required only. Here, the state keeps unchanged with the evolution . and are the Fock states of R and r_{j}, respectively. and . For the resonance condition between R and r_{j} (Δ_{j} = 0) and if we take the initial state of the subsystem composed of R and r_{j} to be , the state of the system composed of R and r_{j} can be expressed as (further details can be found in the method)
Our scheme for the quantum state transfer between the two resonators r_{1} and r_{2} can be accomplished with twostep resonance operations described in detail as follows.
Initially, we assume the initial state of the processor is
which means r_{1} is in the state , R and r_{2} are all in the vacuum state and q is in the ground state. First, tuning the transition frequency of q to detune with R largely and turning off (on) the coupling strength between R and r_{2} (r_{1}) by using the external flux through their SQUIDs, the state of the processor can evolve into
after a time of g_{1}t = π/2.
Second, keeping the frequency of q detune with R largely, turning off g_{1} and turning on g_{2}, the state of the processor can evolve from Eq. (6) to
within a time of g_{2}t = π/2. Here, we complete the quantum state transfer as
If the operation time of the second step is taken as g_{2}t = 3π/2, the final state after the information transfer is
This is just the result of the quantum state transfer between the two resonators r_{1} and r_{2} from the initial state .
Controlledphase gate on r_{1} and r_{2}
Cphase gate is an important universal twoqubit gate. In the basis of two resonators and , a matrix of the gate can be expressed as
which means a minus phase should be generated if and only if the two qubits are in the state . In our processor, the cphase gate on the resonators r_{1} and r_{2} can be completed with five steps by combining the resonance operations between the quantum bus R and the resonator r_{j} and those between R and q with the two transitions and
By taking the coupling strength between q and R much smaller than the anharmonicity of q (g_{g,e} ≪ ω_{g,e} − ω_{e,f}), the interactions between R and q with the two transitions of and can be reduced into those of two individual twoenergylevel qubits with R, whose Hamiltonians are
and
respectively. In the condition of resonance interactions between R and q with the transitions (Δ_{g,e} = 0) and (Δ_{e,f} = 0), the timeevolution operation of the system undergoing the Hamiltonians and are^{66}
and
respectively.
Supposing the initial state of the processor is
Here, the amplitudes α_{1} = cosθ_{1} cosθ_{2}, α_{2} = cosθ_{1} sinθ_{2}, α_{3} = sinθ_{1} cosθ_{2} and α_{4} = sinθ_{1} sinθ_{2}. The five steps for the construction of our cphase gate on r_{1} and r_{2} can be described in detail as follows.
First, turning on the coupling strength between R and r_{1} with g_{1} = g_{g,e} and turning off the interaction between R and r_{2}, the state of the processor can evolve from to
with an operation time of ^{67}.
Second, tuning the frequency of q to detune with R largely and turning off the coupling between R and r_{1}, one can get the state of the processor as
after the time of g_{2}t = π/2 when the coupling between R and r_{2} is turned on.
Third, resonating R and q with the transition of with a time of g_{e,f}t = π and keeping R uncoupled to r_{1} and r_{2}, the state of the the processor becomes
Fourth, repeating the second step, one can get the state of the processor as
Fifth, repeating the first step, we can get the state
This is just the result of our cphase gate on r_{1} and r_{2} with the initial state .
Possible experimental implementation
Resonance operation between a superconducting qubit and a 1D superconducting resonator has been used to achieve some basic tasks in quantum information processing, such as generating Fock states in a superconducting quantum circuit^{54}, realizing the NOON state entanglement on two superconducting microwave resonators^{56}, constructing the resonant quantum gates on charge qubits in circuit QED^{68} or on resonators^{59} and completing a fast scheme to generate NOON state entanglement on two resonators^{69}. To get a highfidelity resonant operation between the qubit and the resonator, the magnetic flux with fast tunability is required.
To show the performance of our schemes for quantum state transfer and the cphase gate, we simulate their fidelities by using the whole Hamiltonian in each step. In our simulations, the parameters are chosen as: g_{1}/(2π) and g_{2}/(2π) can be tuned from 0 MHz to 50 MHz, ω_{R}/(2π) = 6.65 GHz^{63}, δ = ω_{g,e}/(2π) − ω_{e,f}/(2π) = 0.72 GHz^{70}, g_{g,e}/(2π) = MHz, μs and μs. The transition frequency of a transmon qutrit can be tune with a range of about 2.5 GHz^{71}, which is enough for us to make it detune with R largely. The maximal values of g_{1}/(2π) and g_{2}/(2π) taken by us are 50 MHz as the rotationwave approximation can work well when the coupling strength is much smaller than the frequency of R and a theoretic predict of the coupling strength between two superconducting resonators can reach 1.2 GHz^{63}.
The process for the generation of the initial states of and are not included in our simulations. To prepare the initial states, one should perform a proper singlequbit operation on q and send the information from q to r_{j} by using the resonance operation, the same as the one in the first step for the construction of our cphase gate. Here, the interactions which do not attend the resonance operation should be turned off. The singlequbit operation on a superconducting qubit has been realized in experiment with a quantum error smaller than 0.0006^{38}, which has little influence on our schemes. By taking the energy relaxation rate of the qutrit, the decay rates of resonators and g_{g,e} and g_{j} into account, the generation of the initial states just increases a little error value of the fidelities of the quantum state transfer and the cphase gate.
Fidelity for our quantum state transfer
We numerically simulate the populations (vary with the operation time) of a microwave photon in r_{1}, R and r_{2}, shown in Fig. 2. The definition of the population is
Here m = 1, 2, 3. , and . ρ(t) is the realistic density operator of the processor for the quantum state transfer from the initial state . The parameters taken in the first step in our scheme are: ω_{g,e}/(2π) = 5 GHz, g_{1}/(2π) = 50 MHz, g_{2}/(2π) = 0 MHz. In the second step, the parameters are: g_{1}/(2π) = 0 MHz, g_{2}/(2π) = 50 MHz and the other parameters are the same as the ones in the first step.
From the numerical simulation, the quantum state transfer between r_{1} and r_{2} with θ = π/4 can reach a fidelity of 99.97% within 10 ns by using the definition of the fidelity as with the initial state . In the inset in Fig. 2, we give the three conditions of the populations with different decay rates of r_{1}, r_{2} and R.
Fidelity for our cphase gate
We calculate the fidelity of our cphase gate by using the averagegatefidelity definition^{72,73}
Here, is the final state of the processor by using the ideal cphase gate operation on the initial state . ρ(t) is the realistic density operator after our cphase gate operation on the initial state with the Hamiltonian H. The fidelity of our cphase gate reaches 99.66% within 91.5 ns by using the parameters taken in each step as shown in Table 1. Here, if we take θ_{1} = θ_{2} = π/4 in Eq. (15) as an example, the density operators of and the real final state are shown in Fig. 3(a,b), respectively.
Actually, the fidelity of our cphase gate is influenced by the decay rates κ of the resonators, the energy relaxation rate Γ of q and the anharmonicity δ of q, shown in Fig. 4. In Fig. 4(a), we show the fidelity of the gate varying with the decay rates and the energy relaxation rate of the resonators and q (κ = Γ). The fidelity of the gate is numerically simulated by using different optimal parameters corresponding to different Γ (keeping δ = 0.72 GHz unchanged) as the competition between the operation time (leads to the error from the coherence time of the qutrit) and the coupling strength between the qutrit and the bus R (leads to the error from the anharmonicity of the qutrit). Here, in order to choose Γ^{−1} = 10, 20, 30, 40 and 50 μs, we take , 19, 13, 13 and 13 MHz, respectively. The corresponding operation times are t = 58.1, 65.8, 91.5, 91.5 and 91.5 ns, respectively. By using κ = ω_{r}/Q (ω_{r} is the frequency of the resonator)^{16}, κ^{−1} = 50 μs corresponds to a quality factor Q ~ 2.08 × 10^{6} of the resonators. In Fig. 4(b), the anharmonicity of the qutrit influences the fidelity with a small value as the coupling strength g_{g,e} is much smaller than δ, which means that the transmon qutrit in our processor does not require a large anharmonicity.
Conclusion
To show our processor can be used for an effective quantum computation based on resonators, we have given the scheme to achieve the quantum state transfer between two resonators and the one for the cphase gate on them. These two schemes are just based on the Fock states and of the resonators r_{j}. The fidelities of our quantum state transfer and cphase gate reach 99.97% and 99.66% within 10 ns and 91.5 ns, respectively. In our processor, a singlequbit operation on the resonator r_{j} can be achieved with the following steps: 1), one should transfer the information from r_{j} to the qutrit with the resonance operation between them. 2), one can take the singlequbit gate on the qutrit. 3), one should transfer the information from the qutrit to r_{j}. It is worth noticing that there are two steps with resonance operations in our scheme for the singlequbit operation on a microwavephoton resonator. Each resonance operation can generate a −phase for the state of the resonator r_{j} or the state of the qutrit. The two steps with resonance operations can just eliminate this unwanted phase generated by each resonance operation as (−1)^{2} = 1. So, the singlequbit operation on the qutrit is convenient without considering the additional phase generated by the resonance operations. To readout the information of the photon states in r_{j}, one can also transfer the information of the photon from r_{j} (based on the Fock states and ) to the qutrit (based on the states and and then readout the state of the qutrit. To achieve the quantum nondemolition detection on the resonator r_{j}, one can use a lowquality resonator coupled to the qutrit q to detect the information in the quantum bus R^{55} which comes from r_{j}. By using the resonators which can catch and release the microwave photons^{64}, our processor maybe play an important role in quantum communication.
In summary, we have proposed a quantum processor composed of some 1D superconducting resonators r_{j} (quantum information carriers) which are coupled to a common 1D superconducting resonator R (the quantum bus), not the superconducting transmon qutrit, which makes it have the capability of integrating some distant resonators for quantum information processing on microwave photons assisted by circuit QED. With this processor, we have presented a scheme for the highfidelity state transfer between two resonators. Also, we have given a scheme for the cphase gate on two resonators with the resonance operations. With feasible parameters in experiment, the fidelities of our two schemes are 99.97% and 99.66%, respectively. Maybe this processor can play an important role in quantum communication in future.
Methods
Interaction between a resonator and a qubit
In the interaction picture, the Hamiltonian of a system composed of a twoenergylevel qubit coupled to a resonator (QR system) can be written as (under the rotatingwave approximation):
Here, g is the coupling strength between the qubit and the resonator. and a^{+} are the create operators of the qubit and the resonator, respectively. Δ = ω_{q} − ω_{r}. ω_{q} (ω_{r}) is the transition frequency of the qubit (resonator).
The state of the QR system can be solved with the equation of motion
in which is a linear combination of the states and , that is,
Here, c_{e,n}(t) and c_{g,n}(t) are the slowly varying probability amplitudes. is the Fock state of the resonator. Because the only transitions between and can be caused by the Hamiltonian H_{I}, we just need to consider the evolutions of c_{e,n}(t) and c_{g,n+1}(t).
By combining Eqs. (24) and (25), one can get
A general solution for these amplitudes is
Here Ω^{2} = 4g^{2}(n + 1) + Δ^{2}.
Additional Information
How to cite this article: Hua, M. et al. Quantum state transfer and controlledphase gate on onedimensional superconducting resonators assisted by a quantum bus. Sci. Rep. 6, 22037; doi: 10.1038/srep22037 (2016).
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Acknowledgements
The simulations were coded in PYTHON using the QUTIP library. This work is supported by the China Postdoctoral Science Foundation under Grant No. 2015M581061, the National Natural Science Foundation of China under Grants No. 11474026 and the Fundamental Research Funds for the Central Universities under Grant No. 2015KJJCA01.
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M.H. and M.J. completed the calculation and prepared the figures. M.H. and F.G. wrote the main manuscript text. F.G. supervised the whole project. All authors reviewed the manuscript.
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Hua, M., Tao, MJ. & Deng, FG. Quantum state transfer and controlledphase gate on onedimensional superconducting resonators assisted by a quantum bus. Sci Rep 6, 22037 (2016). https://doi.org/10.1038/srep22037
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DOI: https://doi.org/10.1038/srep22037
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