Programmable quantum processor implemented with superconducting circuit
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Nian-Quan Jiang,∗, Xi Liang, Ming-Feng WangCollege of Mathematics and Physics, Wenzhou University, Wenzhou 325035, China
First author contact:∗Author to whom any correspondence should be addressed. Received:2020-10-20Revised:2021-02-10Accepted:2021-02-10Online:2021-03-15
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
PDF (296KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Nian-Quan Jiang, Xi Liang, Ming-Feng Wang. Programmable quantum processor implemented with superconducting circuit. Communications in Theoretical Physics, 2021, 73(5): 055102- doi:10.1088/1572-9494/abe4cb
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 Schrödinger 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 ${{\rm{10}}}^{{\rm{8}}}$ is 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 witheach 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 symmetricallydistribution of the qubits enable the same delay of the propagation ofelectromagnetic 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 we demonstrate 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.
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 capacitor ${C}_{mi},$ the inductance and resistance of the metal body is very small and can be neglected, and then arbitrary two qubits ${Q}_{i}$ and ${Q}_{j}$ are coupled through the capacitance ${C}_{ij}={C}_{mi}{C}_{mj}/\left({C}_{mi}+{C}_{mj}\right),$ thus the $N$ qubits are directly connected with each other by capacitors. Each qubit ${Q}_{i}$ can be controlled by gate voltage ${V}_{gi},$ flux ${{\rm{\Phi }}}_{ei}$ and microwave XY pulses [21] and can be measured with a readout resonator R [21, 24]. The normalized charges induced on the qubit ${Q}_{i}$ can be tuned by changing gate voltage Vgi. The effective Josephson energy ${E}_{Ji}$ of qubit ${Q}_{i}$ is controlled by the external magnetic flux ${{\rm{\Phi }}}_{ei}$ through 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 qubit ${Q}_{i}$ with a coupling capacitor ${C}_{Xi}.$ The gate voltage given by the XY control line is ${V}_{Xi}.$ Each qubit can be measured by using a coplanar waveguide resonator R (blue), the capacitance between its input port and the island is ${C}_{Ri}$ and the gate voltage given by the resonator is ${V}_{Ri}.$ A shunted capacitor ${C}_{Bi}$ is used to ensure ${C}_{mi}+{C}_{Bi}\gg {C}_{i}$ to get long coherence time for the qubit, where ${C}_{i}$ is the Josephson capacitance of qubit ${Q}_{i}.$ The coupling between arbitrary two qubits ${Q}_{i}$ and ${Q}_{j}$ can be turned on or off by changing the fluxes so that ${\omega }_{i}={\omega }_{j},$${\omega }_{i}+{\omega }_{j}\gg {J}_{ij}$ or $\left|{\omega }_{i}-{\omega }_{j}\right|\gg {J}_{ij},$ respectively. The circuit dynamics of the system is governed by the Hamiltonian (which is detailed in supplementary material II, stacks.iop.org/CTP/73/055102/mmedia [25]) (the constant terms are omitted):$\begin{eqnarray}\begin{array}{l}{H}=\displaystyle \sum _{i=1}^{N}[{E}_{ci}{({\hat{n}}_{i}-{n}_{gi})}^{2}-{E}_{Ji}\,\cos \,{\varphi }_{i}]\\ \,\,+\,\displaystyle \sum _{i\lt j;i,j=1}^{N}{E}_{ij}({\hat{n}}_{i}-{n}_{gi})({\hat{n}}_{j}-{n}_{gj}),\end{array}\end{eqnarray}$where ${E}_{ci}=2{e}^{2}(1+{C}_{mi}^{2}/\beta {D}_{i})/{D}_{i}$ and ${E}_{ij}=4{e}^{2}{C}_{mi}{C}_{mj}/\beta {D}_{i}{D}_{j}$ ($i\ne j,$$i,j=1,2,\cdots ,N$) are the effective Cooper-pair charging energy of qubit $i$ and the coupling energy between qubits $i$ and $j,$ respectively. $e$ is the charge on the electron. ${D}_{i}={C}_{i}+{C}_{gi}+{C}_{mi}+{C}_{Bi}+{C}_{Ri}+{C}_{Xi}$ is the sum of all capacitances connected to the island of qubit $i$ including Josephson capacitance ${C}_{i}$ and gate capacitance ${C}_{gi}.$$\beta $ is related to all the capacitances with $\beta =\displaystyle {\sum }_{i=1}^{N}{C}_{mi}({D}_{i}-{C}_{mi})/{D}_{i}.$${\hat{n}}_{i}$ is the number of excess Cooper pairs in the island. ${n}_{gi}\,=-({C}_{gi}{V}_{gi}+{C}_{Ri}{V}_{Ri}+{C}_{Xi}{V}_{Xi})/2e$ is the normalized charges induced on the qubit by gate voltages. ${\varphi }_{i}$ is the phase difference across the junctions of the qubit (assume the two junctions of the qubit are identical). The effective Josephson coupling energy ${E}_{Ji}$ is tunable by the external flux ${{\rm{\Phi }}}_{ei}$ between $2{E}_{Ji}^{0}$ and zero:$\begin{eqnarray}{E}_{Ji}=2{E}_{Ji}^{0}\,\cos (\pi {{\rm{\Phi }}}_{ei}/{{\rm{\Phi }}}_{0}),\end{eqnarray}$where ${E}_{Ji}^{0}$ is the Josephson coupling energy of single junction in the qubit Qi, ${{\rm{\Phi }}}_{0}=h/2e$ is the flux quantum.
Figure 1.
New window|Download| PPT slide 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.
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 non-nearest 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 of ${H}_{0}$ (${H}_{0}=\displaystyle {\sum }_{i=1}^{N}[{E}_{ci}{({\hat{n}}_{i}-{n}_{gi})}^{2}-{E}_{Ji}\,\cos \,{\varphi }_{i}]$), i.e. $\left|{0}_{i}\right\rangle $ and $\left|{1}_{i}\right\rangle $ ($i=1,2,\cdots ,N.$), then the Hamiltonian (1) reduces to [25]$\begin{eqnarray}\begin{array}{l}H=-\displaystyle \sum _{i=1}^{N}\displaystyle \frac{1}{2}\hslash {\omega }_{i}{\sigma }_{i}^{z}+\displaystyle \sum _{i\lt j;i,j=1}^{N}\displaystyle \frac{{E}_{ij}}{4}\left(\sin \,{\theta }_{i}{\sigma }_{i}^{z}-\,\cos \,{\theta }_{i}{\sigma }_{i}^{x}-{\alpha }_{i}\right)\\ \,\,\times \,\left(\sin \,{\theta }_{j}{\sigma }_{j}^{z}-\,\cos \,{\theta }_{j}{\sigma }_{j}^{x}-{\alpha }_{j}\right),\end{array}\end{eqnarray}$where ${\omega }_{{\rm{i}}}=\sqrt{{\alpha }_{i}^{2}{E}_{{\rm{c}}i}^{2}+{E}_{Ji}^{2}}/\hslash ,$$\sin \,{\theta }_{i}={\alpha }_{i}{E}_{ci}/\sqrt{{\alpha }_{i}^{2}{E}_{{\rm{c}}i}^{2}+{E}_{Ji}^{2}},$$\cos \,{\theta }_{i}={E}_{Ji}/\sqrt{{\alpha }_{i}^{2}{E}_{{\rm{c}}i}^{2}+{E}_{Ji}^{2}},$ ${\sigma }_{i}^{z}=\left|{0}_{i}\right\rangle \left\langle {0}_{i}\right|-\left|{1}_{i}\right\rangle \left\langle {1}_{i}\right|,$${\sigma }_{i}^{x}\,=\left|{1}_{i}\right\rangle \left\langle {0}_{i}\right|+\left|{0}_{i}\right\rangle \left\langle {1}_{i}\right|$ and ${\alpha }_{{\rm{i}}}=1-2{n}_{gi}.$ The Hamiltonian (3) shows that there exist complex interactions including zz-type, zx-type and xx-type couplings, and there are lots of high-frequency terms ${\sigma }_{i}^{x}$ (with oscillation frequency ${\omega }_{{i}}$ in the interaction picture with respect to ${H}_{0}=-\displaystyle {\sum }_{{i}=1}^{N}\tfrac{1}{2}\hslash {\omega }_{{i}}{\sigma }_{{i}}^{z}$). In particular, the unwanted zz-type couplings can never be removed via any approximation methods due to the ${\sigma }_{i}^{z}$ in ${H}_{0}.$ Because these couplings and high-frequency terms are induced by the capacitor between qubits, they also exist in lots of current schemes in which there exist capacitive connections between superconducting qubits [15, 18, 20]. We need some of these couplings to implement quantum operations, but the unwanted couplings (crosstalk) and high-frequency terms will lead to decoherence and gate errors, which is exactly the reasons why it is difficult to achieve high-fidelity quantum computations in various schemes [15, 18]. Here, to suppress the unwanted couplings (crosstalk) and high-frequency terms, we apply gate power supplies to control each of the qubits. We tune the gate voltages Vgi$({i}=1,2,\cdots ,N.)$ so that ${n}_{gi}\approx \tfrac{1}{2},$ and then ${\alpha }_{{i}}\approx 0,$$\sin \,{\theta }_{{i}}\approx 0$ and $\cos \,{\theta }_{{i}}\approx 1,$ thus the unwanted terms containing ${\sigma }_{i}^{z}{\sigma }_{j}^{z},$${\sigma }_{i}^{z}{\sigma }_{j}^{x}$ and ${\sigma }_{i}^{x}$$(i\ne j;\,i,j=1,2,\cdots ,N)$ are very small and can be neglected. Especially when ${n}_{gi}=\tfrac{1}{2},$ the Hamiltonian (3) reduces to$\begin{eqnarray}H=-\displaystyle \sum _{i=1}^{N}\displaystyle \frac{1}{2}\hslash {\omega }_{i}{\sigma }_{i}^{z}+\displaystyle \sum _{i\lt j;i,j=1}^{N}\hslash {J}_{ij}{\sigma }_{i}^{x}{\sigma }_{j}^{x},\end{eqnarray}$with ${J}_{ij}={E}_{ij}/4\hslash .$ The Hamiltonian (4) demonstrates an ideal coupling between arbitrary qubits where all the unwanted crosstalk and harmful high-frequency terms are disappeared, and then it has the potential to achieve high-fidelity coupling operations. In the following, using the Hamiltonian (4) we show a full set of tools for performing universal quantum computations. Assume $\left|{\omega }_{i}-{\omega }_{j}\right|\gg {J}_{ij},$ in the rotating-wave approximation (RWA) the interaction term ${\sigma }_{i}^{x}{\sigma }_{j}^{x}$ in equation (4) can be neglected, and then the couplings between qubits i and j are turned off. On the other hand, when ${\omega }_{i}={\omega }_{j}$ and ${\omega }_{i}+{\omega }_{j}\gg {J}_{ij},$ in the RWA the term ${\sigma }_{i}^{x}{\sigma }_{j}^{x}$ will be equal to ${\sigma }_{i}^{+}{\sigma }_{j}^{-}+{\sigma }_{i}^{-}{\sigma }_{j}^{+}$ with ${\sigma }_{j}^{\pm }=\left({\sigma }_{j}^{x}\pm {\rm{i}}{\sigma }_{j}^{y}\right)/2,$ and then the couplings between qubits i and j are turned on and some quantum gates such as $\sqrt{i{\rm{S}}{\rm{W}}{\rm{A}}{\rm{P}}}$ can be performed after certain interaction time [25]. Because the couplings between arbitrary qubits are included in (4), the Hamiltonian allows us to perform arbitrary operations between or among arbitrary qubits, we will show that in the remainder of the article.
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 ($i=1,2,\cdots ,N$), and tune the fluxes Φei at low temperature to get $\hslash {\omega }_{i}\gg {k}_{B}T$ and $\left|{\omega }_{i}-{\omega }_{j}\right|\gg {J}_{ij}$ ($i\ne j;$$i,j=1,2,\cdots ,N$). When the evolving time $t\gg \pi /\left|{\omega }_{i}-{\omega }_{j}\right|,$ in the RWA the Hamiltonian (4) reads $H=-\displaystyle {\sum }_{i=1}^{N}\tfrac{1}{2}\hslash {\omega }_{i}{\sigma }_{i}^{z}.$ After sufficient time, the residual interaction relaxes all the qubits to the ground states, i.e. the system will be prepared in the initial state $\left|{1}_{1}{1}_{2}\cdots {1}_{{N}}\right\rangle .$
Then the single-qubit gate operations have to be performed. Using pulses on the microwave (XY) line in figure 1, rotations around the X and Y axes in the Bloch sphere representation can be performed [21, 24]. To operate single qubit i we set the gate voltages of all the qubits to keep ${n}_{gi}=1/2,$ while tune the fluxes to satisfy $\left|{\omega }_{i}-{\omega }_{j}\right|\gg {J}_{ij}$ and $\left|{\omega }_{j}-{\omega }_{k}\right|\gg {J}_{jk},$ where $j,k\ne i;$$j\ne k;$$i,j,k=1,2,\cdots ,N.$ Then single-qubit gates on qubit $i$ can be performed using microwave pluses in the similar way in [21].
Two-qubit gate operation on arbitrary two qubits $i$ and $j$ can be achieved by controlling the gate voltages and magnetic fluxes. Tune the fluxes of all qubits to get ${\omega }_{i}={\omega }_{j}$ and $\left|{\omega }_{i}+{\omega }_{j}\right|\gg {J}_{ij},$ while, for arbitrary pair of qubits beyond $i$ and $j,$ their frequency difference is much larger than their coupling strength. Meanwhile, the gate voltages of all the qubits in the system are remained at degeneracy points. In the RWA, the Hamiltonian of the system reduces to$\begin{eqnarray}H=-\displaystyle \sum _{i=1}^{N}\displaystyle \frac{1}{2}\hslash {\omega }_{i}{\sigma }_{i}^{z}+\hslash {J}_{ij}({\sigma }_{i}^{+}{\sigma }_{j}^{-}+{\sigma }_{i}^{-}{\sigma }_{j}^{+}).\end{eqnarray}$After a period of evolving time$\tau =\pi /(2{J}_{ij}),$ the Hamiltonian (5) will produce a swapping operation between $\left|{1}_{i}{0}_{j}\right\rangle $ and $\left|{0}_{i}{1}_{j}\right\rangle .$ When the evolving time $\tau =\pi /(4{J}_{ij}),$ the Hamiltonian (5) should correspond to a $\sqrt{i{\rm{SWAP}}},$ which is a universal two qubit gate for the states $\left|{0}_{i}{0}_{j}\right\rangle ,$$\left|{0}_{i}{1}_{j}\right\rangle ,$$\left|{1}_{i}{0}_{j}\right\rangle ,$$\left|{1}_{i}{1}_{j}\right\rangle .$ After that, ${{\rm{\Phi }}}_{ei}$ and ${{\rm{\Phi }}}_{ej}$ are set back to satisfy $\left|{\omega }_{i}-{\omega }_{j}\right|\gg {J}_{ij}$ and then the coupling between qubits $i$ and $j$ is turned off.
Operations on multiple pairs of qubits in parallel can be achieved as follows. If we want to simultaneously perform two-qubit universal gates on $k$ pairs of qubits: ${i}_{1}$ and ${j}_{1},$${i}_{2}$ and ${j}_{2},$ …, ${i}_{k}$ and ${j}_{k},$ respectively. We tune the magnetic fluxes of all qubits so that ${\omega }_{{i}_{m}}={\omega }_{{j}_{m}},$${\omega }_{{i}_{m}}+{\omega }_{{j}_{m}}\gg {J}_{{i}_{m}{j}_{m}},$$m=1,2,\cdots ,\,k,$ while, for any other pair of two qubits beyond the above k pairs, their frequency difference is much larger than their coupling strength. The gate voltages of all the qubits are tuned to satisfy ${n}_{gi}=1/2.$ In the RWA, the Hamiltonian of the system is described by$\begin{eqnarray}H=-\displaystyle \sum _{k=1}^{N}\displaystyle \frac{1}{2}\hslash {\omega }_{k}{\sigma }_{k}^{z}+\displaystyle \sum _{m=1}^{k}\hslash {J}_{{i}_{m}{j}_{m}}({\sigma }_{{i}_{m}}^{+}{\sigma }_{{j}_{m}}^{-}+{\sigma }_{{i}_{m}}^{-}{\sigma }_{{j}_{m}}^{+}).\end{eqnarray}$After a period of evolving time $\tau =\pi /(2{J}_{{i}_{m}{j}_{m}}),$ (for convenience, assume ${J}_{{i}_{m}{j}_{m}}=J,$$m=1,2,\cdots ,k$), the Hamiltonian (6) will produce $k$ pairs of swapping operations between $\left|{1}_{{i}_{m}}{0}_{{j}_{m}}\right\rangle $ and $\left|{0}_{{i}_{m}}{1}_{{j}_{m}}\right\rangle .$ When evolving time $\tau =\pi /(4{J}_{{i}_{m}{j}_{m}}),$ the Hamiltonian (6) should correspond to $k$ universal two qubit $\sqrt{i{\rm{SWAP}}}$ gates, and then $k$ pairs 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 satisfy $\left|{\omega }_{{i}_{m}}-{\omega }_{{j}_{m}}\right|\gg {J}_{{i}_{m}{j}_{m}}$ and the interactions are turned off.
To perform coupling operations on a group of qubits (more than two qubits, say qubits 1 through $k,$$k\gt 2$), we properly tune the fluxes of the qubits to get ${\omega }_{i}={\omega }_{j}$ and $2{\omega }_{i}\gg {J}_{ij}$ ($i\ne 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 keep ${n}_{gi}=1/2.$ In the RWA, the Hamiltonian of the system is governed by$\begin{eqnarray}H=-\displaystyle \sum _{i=1}^{N}\displaystyle \frac{\hslash {\omega }_{i}}{2}{\sigma }_{i}^{z}+\displaystyle \sum _{i\lt j;\,i,j=1}^{k}\hslash {J}_{ij}\left({\sigma }_{i}^{+}{\sigma }_{j}^{-}+{\sigma }_{i}^{-}{\sigma }_{j}^{+}\right).\end{eqnarray}$After a period of evolving time $\tau =\pi /(4{J}_{ij}),$ (for convenience, assume Jij≡J12$i\ne j,i,j=1,2,\cdots ,k$), the Hamiltonian (7) should correspond to a series of $\sqrt{i{\rm{SWAP}}}$ gates, which operate in parallel on arbitrary pair of qubits $i$ and $j$ in 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 $\left|{\omega }_{i}-{\omega }_{j}\right|\gg {J}_{ij}$ and 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 conditions ${\omega }_{i}={\omega }_{j}$ and $2{\omega }_{i}\gg {J}_{ij},$ 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 keep ${n}_{gi}=1/2.$ Assume $l$ groups of qubits are selected to be operated, and there are ${k}_{i}$ qubits in group $i$ ($i=1,2,\cdots ,l$), i.e. qubits ${i}_{1},{i}_{2},\cdots ,{i}_{{k}_{i}}.$ In the RWA, the Hamiltonian of the system is governed by$\begin{eqnarray}H=-\displaystyle \sum _{i=1}^{N}\displaystyle \frac{\hslash {\omega }_{i}}{2}{\sigma }_{i}^{z}+\displaystyle \sum _{i=1}^{l}\displaystyle \sum _{m\lt m^{\prime} ;m,m^{\prime} =1}^{{k}_{i}}\hslash {J}_{{i}_{m}{i}_{m^{\prime} }}\left({\sigma }_{{i}_{m}}^{+}{\sigma }_{{i}_{m^{\prime} }}^{-}+{\sigma }_{{i}_{m}}^{-}{\sigma }_{{i}_{m^{\prime} }}^{+}\right).\end{eqnarray}$After the evolving time $\tau =\pi /(4{J}_{{i}_{m},{i}_{m^{\prime} }}),$ any two qubits ${i}_{m}$ and ${i}_{m^{\prime} }$ in group $i$ are coupled by a $\sqrt{i{\rm{SWAP}}}.$ In the case of ${J}_{{i}_{m}{i}_{m^{\prime} }}=J$ ($m\ne m^{\prime} ,$$m,m^{\prime} =1,2,\cdots ,{k}_{i},$$i=1,2,\cdots ,l$), arbitrary two qubits in the same group are coupled by a $\sqrt{i{\rm{SWAP}}}$ operation after the evolving time $\tau =\pi /(4J),$ and $l$ groups are evolved in parallel. In the meantime, the coupling for any two qubits coming from different groups is turned off. After that, the system is set back to the idle state where the interactions are turned off.
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 conditions ${\omega }_{i}={\omega }_{j}$ and $2{\omega }_{i}\gg {J}_{ij}$ for any pair of qubits $i$ and $j$ which 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 that $k$ pairs of qubits and $k^{\prime} $ groups of qubits are selected to be operated, i.e. qubits ${i}_{m}$ and ${j}_{m},$ ($m=1,2,\cdots ,k$), and qubits ${l}_{1},\,{l}_{2},\cdots ,\,{l}_{{n}_{l}},$ ($l=1,2,\cdots ,k^{\prime} $) with ${i}_{m},\,{j}_{m},\,{l}_{r}\in \{1,2,\cdots ,N\},$ and there are ${n}_{l}$ qubits in group $l.$ The gate voltages of all the qubits in the system are set to keep ${n}_{gi}=1/2.$ In the RWA, the Hamiltonian of the system is$\begin{eqnarray}\begin{array}{l}H=-\displaystyle \sum _{i=1}^{N}\displaystyle \frac{\hslash {\omega }_{i}}{2}{\sigma }_{i}^{z}+\displaystyle \sum _{m=1}^{k}\hslash {J}_{{i}_{m}{j}_{m}}\left({\sigma }_{{i}_{m}}^{+}{\sigma }_{{j}_{m}}^{-}+{\sigma }_{{i}_{m}}^{-}{\sigma }_{{j}_{m}}^{+}\right)\\ \,\,+\,\displaystyle \sum _{l=1}^{k^{\prime} }\displaystyle \sum _{r\lt r^{\prime} ;r,r^{\prime} =1}^{{n}_{l}}\hslash {J}_{{l}_{r}{l}_{r^{\prime} }}\left({\sigma }_{{l}_{r}}^{+}{\sigma }_{{l}_{r^{\prime} }}^{-}+{\sigma }_{{l}_{r}}^{-}{\sigma }_{{l}_{r^{\prime} }}^{+}\right).\end{array}\end{eqnarray}$After the evolving time $\tau =\pi /(4J)$ (for convenience, we assume ${J}_{{i}_{m}{j}_{m}}={J}_{{l}_{r}{l}_{r^{\prime} }}=J$), any two qubits both from the same pair or the same group are coupled by a $\sqrt{i{\rm{SWAP}}},$ and 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 capacitors ${C}_{Bi}$ satisfy the conditions ${C}_{mi}+{C}_{Bi}\gg {C}_{i}$ ($i=1,2,\cdots ,N$), in this case the qubits are similar to the Xmon qubits [21, 26, 27], meanwhile, we choose proper values of ${{\rm{\Phi }}}_{ei}$ so that the qubits are operated at a large ${E}_{Ji}/{E}_{ci}$ ratio, e.g. ${E}_{ci}/h\sim 50\,{\rm{MHz}},$${E}_{Ji}/h\geqslant 5\,{\rm{GHz}},$ and ${E}_{Ji}/{E}_{ci}\geqslant 100,$ then the relaxation times of the order of ${10}^{2}\,\mu {\rm{s}}$ will 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 $\left|{\omega }_{i}-{\omega }_{j}\right|\gg {J}_{ij},$$i\ne j,$ then the static coupling will be effectively reduced. With current technology the Josephson energy $2{E}_{Ji}^{0}/h\approx 50\,{\rm{GHz}}$ is experimentally accessible [28], then the frequency ${\omega }_{i}/2\pi $ of the qubits are tunable in the range of 0–50 GHz We choose ${\omega }_{i}/2\pi $ be in the range of 5–50 GHz and $\left|{\omega }_{i}-{\omega }_{j}\right|\sim {10}^{3}{J}_{ij}.$ 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 $\left|{\omega }_{i}-{\omega }_{j}\right|\sim \,\max \{{10}^{3}{J}_{ij}/k,50{J}_{ij}\}$ for the qubits that will be operated after a period of time $t=k\tau ,$ where $k$ is an integer and $\tau =\pi /(2{J}_{ij})$ is the evolving time for one coupling operation. Choose the interaction strength ${J}_{ij}=10\pi \,{\rm{MHz}},$ the timescale of a coupling operation should be $\tau =50\,{\rm{ns}}.$ 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 time ${\tau }_{{\rm{coh}}}\sim {10}^{2}\,\mu {\rm{s}}$ [26, 27] we can perform operations nearly ${10}^{3}$ times. 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 ${{\rm{\Phi }}}_{ei}$ when we increase and decrease the value of the frequency ${\omega }_{i}/2\pi .$ 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.
,∗, Xi Liang, Ming-Feng WangCollege of Mathematics and Physics, 
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