Programmable quantum processor implemented with superconducting circuit

Nian-Quan Jiang,Xi Liang and Ming-Feng Wang

College of Mathematics and Physics,Wenzhou University,Wenzhou 325035,China

Abstract A quantum processor might execute certain computational tasks exponentially faster than a classical processor.Here,using superconducting quantum circuits we design a powerful universal quantum processor with the structure of symmetric all-to-all capacitive connection.We present the Hamiltonian and use it to demonstrate a full set of qubit operations needed in the programmable universal quantum computations.With the device the unwanted crosstalk and ZZ-type couplings between qubits can be effectively suppressed by tuning gate voltages,and the design allows efficient and high-quality couplings of qubits.Within available technology,the scheme may enable a practical programmable universal quantum computer.

Keywords: programmable quantum processor,universal quantum gate,superconducting qubit

1.Introduction

Quantum computers execute certain computational tasks exponentially faster than classical computers[1,2],so much attention has been attracted to explore quantum processors in recent years.Some schemes of implementing the processors have been presented with different physical platforms,such as ions [3-6],photons [7-11],superconducting qubits [12-17].In particular,with superconducting qubits,several significant progresses have recently been made by Song et al[15]and by the groups such as the Google AI Quantum team (GAQT) [18].The GAQT has achieved a 53 quantum processor which reached the regime of quantum supremacy [18],and Song et al have created an 18-qubit GHZ state and a 20-qubit Schr?dinger cat state[15].The GAQT’s device consists of an array of qubits where each qubit is coupled to four nearest neighbors,and it works with surface code quantum computing.With this design,the programmable quantum computations are achieved,but it is troublesome to couple non-nearest neighbor qubits,and it requires a very large number of physical qubits to perform practical quantum computations (of order 108is probably the smallest number needed for a practical factoring computer,for example)[19].The Song’s processor is an all-to-all connected circuit architecture where arbitrary two qubits can be coupled by resonator-mediated interactions,thus it has higher efficiency and controllability in coupling qubits.But,in the scheme the unwanted ZZ-type couplings between qubits are inevitable (besides Z-crosstalk caused by control lines) [20],and there exists two types of couplings which interfere with each other and then cause harmful crosstalk(see the supplementary material which is available here stacks.iop.org/CTP/73/055102/mmedia) [20].So,although quite significant progresses have been made in recent years,realizing practical universal quantum computer still requires further theoretical and technical leaps to achieve more effective couplings and to engineer higher fidelity qubit gates [21-23].

In this article,we introduce a powerful superconducting quantum processor with symmetric all-to-all capacitive connectivity,in which unwanted ZZ-type couplings and harmful crosstalk are suppressed and efficient and parallel operations of qubits are available.The symmetrically distribution of the qubits enable the same delay of the propagation of electromagnetic field between qubits,and then the decoherence caused by different delay is reduced.We present the Hamiltonian of the system and from which we achieve a full set of operations needed in performing universal quantum computations.The article is arranged as follows: in section 2 wedemonstrate the architecture and the Hamiltonian of the device.In section 3 we show a full set of tools to perform all operations needed in universal quantum computations.We make the discussions and conclusions in section 4.

Figure 1.Design of a programmable universal quantum processor.All the qubits are connected with each other via capacitors,the frequency of each qubit is controlled by external magnetic flux,each qubit can be measured by using a coplanar waveguide resonator R,the normalized charges induced on the qubit can be tuned by changing gate voltage.Single-qubit operations can be performed using pulses on the microwave XY control line,and the coupling between arbitrary two qubits can be turned on or off by changing the fluxes.

2.Architecture and Hamiltonian

The quantum processor is illustrated with the symmetric architecture in figure 1,which consists of N superconducting qubits.Each of the qubits is connected to a solid metal body (orange circular area denoted by ‘o’) by a capacitorCmi,the inductance and resistance of the metal body is very small and can be neglected,and then arbitrary two qubitsQiandQjare coupled through the capacitanceCij=Cm i Cmj/(Cmi+Cmj),thus theNqubits are directly connected with each other by capacitors.Each qubitQican be controlled by gate voltageVgi,fulx Φeiand microwave XY pulses[21]and can be measured with a readout resonator R [21,24].The normalized charges induced on the qubitQican be tuned by changing gate voltage Vgi.The effective Josephson energyEJiof qubitQiis controlled by the external magnetic flux Φeithrough the superconducting loop of the qubit.Single-qubit operations can be performed using pulses on the microwave XY control line (blue),which is connected to the island of qubitQiwith a coupling capacitorCXi.The gate voltage given by the XY control line isVXi.Each qubit can be measured by using a coplanar waveguide resonator R (blue),the capacitance between its input port and the island isCRiand the gate voltage given by the resonator isVRi.A shunted capacitorCBiis used to ensureCmi+CBi?Cito get long coherence time for the qubit,whereCiis the Josephson capacitance of qubitQi.The coupling between arbitrary two qubitsQiandQjcan be turned on or off by changing the fulxes so thatωi=ωj,ωi+ωj?Jijor∣ωi-ωj∣ ?Jij,respectively.The circuit dynamics of the system is governed by the Hamiltonian (which is detailed in supplementary material II,stacks.iop.org/CTP/73/055102/mmedia[25]) (the constant terms are omitted):

whereis the Josephson coupling energy of single junction in the qubitis the fulx quantum.

Equation (1) shows that there is only one type of interaction between qubits in the system,and all of the qubits are coupled with each other.Thus,the design has the potential to avoid the interference crosstalk caused by complex interactions more than two types as in some current schemes [15,20],meanwhile,the all-to-all structure where any two of the qubits are nearest neighbors with capacitive connections allows the most efficient coupling operations for arbitrary qubits,which could overcome the shortages of inefficiency in coupling nonnearest neighbor qubits in some schemes based on the surface code [18,19].When each of the qubits is encoded as the two lowest quantum eigenstates of the resonant circuit,we truncate the Hamiltonian(1)to the subspace spanned by the eigen states ofand∣ 1i〉(i= 1,2,… ,N.),then the Hamiltonian (1) reduces to [25]

3.A set of tools to perform all operations

Based on the theories mentioned above we now demonstrate various operations needed for performing programmable universal quantum computers (PUQCs).Firstly,the system has to be prepared in an initial state.For this we tune the gate voltages of all the qubits to keep ngi= 1/2 (i= 1,2,… ,N),and tune the fluxes Φeiat low temperature to get?ωi?kB Tand∣ωi-ωj∣ ?Jij(i≠j;i,j= 1,2,… ,N).When the evolving timet?π/∣ωi-ωj∣ ,in the RWA the Hamiltonian (4)readsAfter sufficient time,the residual interaction relaxes all the qubits to the ground states,i.e.the system will be prepared in the initial state∣ 1112… 1N〉.

Then the single-qubit gate operations have to be performed.Using pulses on the microwave(XY)line in figure 1,rotations around the X and Y axes in the Bloch sphere representation can be performed [21,24].To operate single qubit i we set the gate voltages of all the qubits to keepngi=1 /2,while tune the fluxes to satisfy∣ωi-ωj∣ ?Jijand∣ωj-ωk∣?Jjk,wherej,k≠i;j≠k;i,j,k= 1,2,… ,N.Then single-qubit gates on qubitican be performed using microwave pluses in the similar way in [21].

Two-qubit gate operation on arbitrary two qubitsiandjcan be achieved by controlling the gate voltages and magnetic fluxes.Tune the fluxes of all qubits to getωi=ωjand∣ωi+ωj∣ ?Jij,while,for arbitrary pair of qubits beyondiandj,their frequency difference is much larger than their coupling strength.Meanwhile,the gate voltages of all the qubits in the system are remained at degeneracy points.In the RWA,the Hamiltonian of the system reduces to

After a period of evolving timeτ=π/(2Jij),the Hamiltonian(5) will produce a swapping operation between∣ 1i0j〉and∣0i1j〉.When the evolving timeτ=π/(4Jij),the Hamiltonian(5)should correspond to awhich is a universal two qubit gate for the states∣ 0i0j〉,∣0i1j〉,∣ 1i0j〉,∣1i1j〉.After that,Φeiand Φejare set back to satisfy∣ωi-ωj∣ ?Jijand then the coupling between qubitsiandjis turned off.

Operations on multiple pairs of qubits in parallel can be achieved as follows.If we want to simultaneously perform twoqubit universal gates onkpairs of qubits:i1andj1,i2andj2,…,ikandjk,respectively.We tune the magnetic fulxes of all qubits so thatωim=ωjm,ωim+ωjm?Jim jm,m= 1,2,… ,k,while,for any other pair of two qubits beyond the above k pairs,their frequency difference is much larger than their coupling strength.The gate voltages of all the qubits are tuned to satisfyngi=1 /2.In the RWA,the Hamiltonian of the system is described by

After a period of evolving timeτ=π/ (2Jim jm),(for convenience,assumeJim jm=J,m= 1,2,… ,k),the Hamiltonian(6)will producekpairs of swapping operations between∣1im0jm〉andWhen evolving timethe Hamiltonian(6)should correspond tokuniversal two qubitgates,and thenkpairs of qubits are coupled respectively in parallel,but any two qubits come from different pairs are not coupled.After that,the fluxes are set back to satisfyand the interactions are turned off.

To perform coupling operations on a group of qubits(more than two qubits,say qubits 1 throughk,k＞ 2),we properly tune the fluxes of the qubits to getωi=ωjand 2ω i?Jij(i≠j) for arbitrary pair of qubits in the selected group,while,for any other two qubits which are not both from the group,their frequency difference is much larger than their coupling strength.The gate voltages of all qubits are set to keepngi= 1 /2.In the RWA,the Hamiltonian of the system is governed by

After a period of evolving timeτ=π/(4Jij),(for convenience,assume Jij≡ J12i≠j,i,j=1,2,… ,k),the Hamiltonian (7) should correspond to a series ofgates,which operate in parallel on arbitrary pair of qubitsiandjin the group,thus any qubit in the group is coupled with the others.Meanwhile,for any pair of qubits which are not both from the group,the coupling between them is turned off.After that,the fluxes are set back to satisfy∣ωi-ωj∣ ?Jijand then the interactions are turned off.

To perform couplings on multiple groups of qubits in parallel,we tune the magnetic fluxes of all qubits to make the frequencies of any pair of qubits in the same group satisfying the conditionsωi=ωjand2ωi?Jij,while,for any other pair of qubits in the system,their frequency difference is much larger than their coupling strength.At the same time,the gate voltages of all qubits are set to keepngi=1 /2.Assumelgroups of qubits are selected to be operated,and there arekiqubits in groupi(i= 1,2,… ,l),i.e.qubitsi1,i2,… ,iki.In the RWA,the Hamiltonian of the system is governed by

To perform couplings on multiple pairs of qubits and on multiple groups of qubits in Parallel,we tune the magnetic fluxes of all qubits to make the frequencies satisfying the conditionsωi=ωjand2ωi?Jijfor any pair of qubitsiandjwhich both come from the same selected pair or the same selected group,while,for any other pair of qubits in the system,their frequency difference is much larger than their coupling strength.Assume thatkpairs of qubits andk′ groups of qubits are selected to be operated,i.e.qubitsimandjm,(m= 1,2,… ,k),and qubitsl1,l2,… ,lnl,(l= 1,2,… ,k′)withi m,jm,lr∈{1,2,… ,N},and there arenlqubits in groupl.The gate voltages of all the qubits in the system are set to keepngi=1 /2.In the RWA,the Hamiltonian of the system is

After the evolving timeτ=π/ (4J)(for convenience,we assumeJim jm=Jlr lr′=J),any two qubits both from the same pair or the same group are coupled by aand all these couplings are performed in parallel.Meanwhile,the coupling between any two qubits not both from the same pair or the same group are turned off.After that,the system is set back to the idle state where the interactions are turned off.

4.Discussions and conclusions

Several conditions have been assumed in the design in order to obtain a controlled manipulation of qubits.Here we discuss the appropriate range of parameters.To get long coherence time of the qubits,we let the shunted capacitorsCBisatisfy the conditionsCmi+CBi?Ci(i= 1,2,… ,N),in this case the qubits are similar to the Xmon qubits[21,26,27],meanwhile,we choose proper values of Φeiso that the qubits are operated at a largeEJ i/Eciratio,e.g.Ec i/h～50 MHz,EJ i/h≥5 GHz,andEJ i/Eci≥100,then the relaxation times of the order of10μ2swill be possible[26,27].In the scheme,all the qubits are coupled and then the static coupling between the qubits will be inevitable.To decrease the affect of this problem,we can tune the frequency of all the qubits,which are not implemented couplings between them,to meet the RWA conditions∣ωi-ωj∣ ?Jij,i≠j,then the static coupling will be effectively reduced.With current technology the Josephson energyGHz is experimentally accessible [28],then the frequencyω i/2πof the qubits are tunable in the range of 0-50 GHz We chooseωi/2πbe in the range of 5-50 GHz and∣ωi-ωj∣ ～103Jij.But,in a system with large number of qubits,we should meet the RWA conditions and overcome the difficulty of frequency crowding simultaneously,in this case we can choose∣ωi-ωj∣ ～ max {1 03Ji j/k,50Jij} for the qubits that will be operated after a period of timet=kτ,wherekis an integer andτ=π/ (2J ij)is the evolving time for one coupling operation.Choose the interaction strengthJij= 10πMHz,the timescale of a coupling operation should beτ= 50 ns.These parameters ensure that the RWA conditions can be satisfied and the frequency crowding is effectively decreased simultaneously.With the above parameters,the number of qubits in a computer can be chosen in the range of 50-100,and within the available coherence timeτcoh～102μs [26,27] we can perform operations nearly 103times.In the scheme we also should consider the fact that the gate operations are realized in the way of switching on the coupling between qubits by bringing them to resonance,these operations may influence other qubits during the process and then cause crosstalk.To solve this problem,we can employ rapid changes of flux Φeiwhen we increase and decrease the value of the frequencyωi/2π.With the parameters mentioned above,our scheme will enable a practical PUQC.

In summary,we have demonstrated a powerful universal quantum processor with symmetric all-to-all structure where any two of the qubits are directly connected by capacitors.The unwanted couplings or crosstalk and harmful high-frequency terms can be effectively suppressed by tuning the gate voltages.The design enables couplings between or among arbitrary qubits and allows high-fidelity quantum operations.The incorporation of local flux control,gate voltage regulation and microwave pulse enables the preparation of the initial state of the system,single-qubit gate operations,two-qubit gate operations between arbitrary two qubits,multiple two-qubit gate operations on multiple pairs of qubits in parallel,coupling operations on every pair of qubits in a selected group in parallel,coupling operations on every pair of qubits in any group of selected multiple groups in parallel,coupling operations on every pair of qubits in any group of selected multiple groups or in any of selected pairs in parallel.Within the current technology,the present scheme may allow a practical PUQC.

Acknowledgments

We thank Heng Fan,Shi-Ping Zhao,Yu-xi Liu,and yi-rong Jing for helpful discussions.