Circuit-QED-based scalable architectures for quantum information processing with superconducting qubits

We discuss different ways of generating entanglement in the original picture of circuit QED (XcQED) and several restrictions that arise in the context of a large-scale quantum architecture. To alleviate some of the issues posed by the presence of the nonlinearities inherent to these systems, we introduce a layout for circuit QED, wherein an artificial atom is coupled to a quantized radiation field via its longitudinal degree of freedom (ZcQED). This system is akin to ion traps used in atomic physics, but it relies on fixed coupling between the atom and the resonator. We describe a scalable architecture for processing quantum information with superconducting qubits, which is free from any type of residual interaction between the atomic and photonic degrees of freedom. Tunable interactions can be realized based on sideband transitions, and the system can be operated out of the Lamb-Dicke regime, allowing it to benefit from the possibility of achieving large coupling strengths between atoms and resonators. We also discuss a readout scheme that does not require any extra circuits and allows a qubit-specific measurement of the state of the quantum register inspired by the electron shelving technique. This scheme is quantum nondemolition (QND)-like, and allows for single-shot determination of the qubit states.

Figure 1

(a) Schematic representation of a cavity QED experiment: a real atom interacts via its electric dipole moment with the electric field of the quantized radiation field of a 3D cavity. (b) Possible implementation of a XcQED experiment with a flux qubit: the artificial atom is transversely coupled to the resonator by inductive coupling to the main loop of the qubit. In this configuration, the qubit can be driven longitudinally (i.e., along ${\sigma }_z)$ by modulating the flux through the $\alpha$-loop (split junction) with a microwave drive.

Figure 2

Matrix elements for sideband transitions (red sideband [(a) and (c)] and blue sideband [(b) and (d)]) for a single qubit transversely coupled to a resonator (XcQED - Rabi Hamiltonian) vs the coupling $g$ for different values of $\Delta ({\omega }_r/h=10\mathrm{GHz}$ and $\Delta /h$ ranges from 1 to $8\mathrm{GHz}$ in steps of $1\mathrm{GHz})$ in (a) and (b), and vs the gap $\Delta$ for different values of $g({\omega }_r/h=10\mathrm{GHz}$ and $g/h$ ranges from 100 to $500\mathrm{MHz}$ in steps of $50\mathrm{MHz})$ in (c) and (d): results obtained by numerical diagonalization of the Rabi Hamiltonian (dots) are compared with perturbative expansion (lines). For numerical diagonalizations, the resonator Hilbert space is truncated to $n=80$ Fock states.

Figure 3

Matrix elements of the parametrically induced iSWAP gate between two qubits coupled to a common resonator (XcQED - Rabi Hamiltonian) vs the asymmetry $\rho$ between their gap $[{\Delta }_1=\Delta (1+\rho ),{\Delta }_2=\Delta (1-\rho )]$, for different values of $\Delta ({\omega }_r/h=20\mathrm{GHz},g_1/h=g_2/h=100\mathrm{MHz}$, and $\Delta /h$ ranges from 1 to $10\mathrm{GHz}$ in steps of $1\mathrm{GHz})$: results obtained from numerical diagonalization of the Rabi Hamiltonian (dots) are compared with perturbative expansion (lines). For numerical diagonalizations, the resonator Hilbert space is truncated to $n=50$ Fock states.

Figure 4

Residual ${\sigma }_z{\sigma }_z$ interaction between two qubits coupled to a common resonator (XcQED) for the Jaynes-Cummings Hamiltonian [(a) and (b)] and the Rabi Hamiltonian [(c) and (d)] vs the asymmetry $\rho$ between their gap $[{\Delta }_1=\Delta (1+\rho ),{\Delta }_2=\Delta (1-\rho )]$, for different values of $\Delta ({\omega }_r/h=20\mathrm{GHz},g_1/h=g_2/h=100\mathrm{MHz})$: results obtained from numerical diagonalization (dots) are compared with perturbative expansion (lines). In (a) and (c), $\Delta /h$ ranges from 1 to $10\mathrm{GHz}$ in steps of $1\mathrm{GHz}$, whereas for (b) and (d), $\Delta /h$ ranges from 11 to $19\mathrm{GHz}$ in steps of $1\mathrm{GHz}$. For numerical diagonalizations, the resonator Hilbert space is truncated to $n=40$ Fock states.

Figure 5

(a) Schematic representation of a single ion confined in a linear quadrupole RF trap: the oscillator corresponds to the quantized center-of-mass motion of the ion in the trap (not represented). (b) Possible implementation of a ZcQED experiment with a flux qubit: the coupling to the longitudinal degree of freedom of the artificial atom can be achieved by inductively coupling the resonator to the $\alpha$-loop of the qubit (see Appendix pp1-s1).

Figure 6

Schematic representation of the effect of the coupling of a single qubit to the resonator in phase space within the framework of ZcQED. The amplitude of the effective displacement $\left|\alpha \right|$ due to the coupling does not depend upon the state of the qubit, which explains why there are no dispersive or Lamb shifts.

Figure 7

Schematic representation of the effect of the coupling to two qubits on the resonator in phase space [ZppZ configuration]: the amplitude of the effective displacement $\left|\alpha \right|$ due to the coupling depends on the state of the qubits $({\alpha }_{+}=(g_1+g_2)$ and ${\alpha }_{-}=(g_1-g_2))$, which explains the presence of the residual ${\sigma }_z{\sigma }_z$ interaction.

Figure 8

(a) Schematic representation of an array of qubits on a 2D square lattice where coupling is mediated by two resonators fixedly coupled [Zp(qq)pZ configuration]. (b) Schematic representation of an enlarged unit cell among the 2D array: each physical qubit (say $p_0$ at the center) is surrounded by eight resonators, thereby complicating the justification of the rotating wave approximation for single-qubit operations and sideband transitions.

Figure 9

(a) Matrix elements for single-photon sideband transitions between qubit 1 and its nearest-neighbor (NN) resonator vs the coupling strength $g_1$, for different values of the coupling $g_c$ between both resonators: their values for red sideband (i.e., $|\langle {\downarrow }_1{1}_{\alpha }{0}_{\beta }{\psi }_2|{\sigma }_1^x|{\uparrow }_1{0}_{\alpha }{0}_{\beta }{\psi }_2\rangle |)$ and blue sideband (i.e., $|\langle {\uparrow }_1{1}_{\alpha }{0}_{\beta }{\psi }_2|{\sigma }_1^x|{\downarrow }_1{0}_{\alpha }{0}_{\beta }{\psi }_2\rangle |)$ are identical. (b) The same is shown for sideband transitions between qubit 1 and its next-nearest-neighbor (NNN) resonator (either red or blue sideband, i.e., $|\langle {\downarrow }_1{0}_{\alpha }{1}_{\beta }{\psi }_2|{\sigma }_1^x|{\uparrow }_1{0}_{\alpha }{0}_{\beta }{\psi }_2\rangle |$ and $|\langle {\uparrow }_1{0}_{\alpha }{1}_{\beta }{\psi }_2|{\sigma }_1^x|{\downarrow }_1{0}_{\alpha }{0}_{\beta }{\psi }_2\rangle |$, respectively). In both cases, results obtained from numerical diagonalization (dots) are compared with the exact analytic formula (lines) $({\omega }_1/h=4\mathrm{GHz},{\omega }_2/h=6\mathrm{GHz}$, and $g_c/h$ ranges from $0.2$ to $1\mathrm{GHz}$ in steps of $0.2\mathrm{GHz})$. The values of ${\Delta }_1,{\Delta }_2$, and $g_2$ are irrelevant, and $\left|{\psi }_2\right.〉$ is an arbitrary state of qubit 2. For numerical diagonalizations, the resonator Hilbert space is truncated to $n=20$ Fock states.

Figure 10

Matrix element for the sideband transition $({\sigma }_1^{+}{a}^{†}b+{\sigma }_1^{-}a{b}^{†})$ used in the conditional-phase accumulation mechanism $({\omega }_1/h=4\mathrm{GHz},{\omega }_2/h=6\mathrm{GHz}$, and $g_c/h$ ranges from $0.2$ to $1\mathrm{GHz}$ in steps of $0.2\mathrm{GHz}$, the values of ${\Delta }_1,{\Delta }_2$, and $g_2$ are irrelevant). Results obtained from numerical diagonalization (dots) are compared with the exact analytic formula (lines). For numerical diagonalizations, the resonator Hilbert space is truncated to $n=20$ Fock states.

Figure 11

Energy level diagram representing the effect of combined red and blue sideband transitions associated with the transition from the first excited state to the second excited state in the case of a three-level atom, for an arbitrary initial state of the qubit.

Figure 12

Energy level diagram representing the effect of combined single-qubit and two-resonator sideband transitions in the case of a two-level atom, for an arbitrary initial state of the qubit.

Figure 13

The dependence of the matrix elements associated with the sideband transitions involved in the readout process vs the Fock state of (a) the nearest-neighbor resonator, and (c) the next-nearest-neighbor resonator. The intracavity mean photon number vs the length of the bichromatic microwave pulse applied for readout in the case where we fill (b) the nearest-neighbor resonator, and (d) the next-nearest-neighbor resonator. The considered parameters of the device are ${\omega }_1/h=4\mathrm{GHz},{\omega }_2/h=6\mathrm{GHz},g_1/h=1\mathrm{GHz}$, and $g_c/h=1\mathrm{GHz}$, while the value of $g_2$ is irrelevant.

Figure 14

Schematic representation of a tunable gap flux qubit inductively coupled to a quantum harmonic oscillator (lumped element resonator in this case). The interaction with the longitudinal degree of freedom of the artificial atom is achieved by coupling the flux through the split $\alpha$-junction and the resonator via mutual inductance.

Figure 15

(a) Energy levels of a gradiometric flux qubit as a function of the magnetic frustration $f_{\varepsilon }(\alpha =0.5,\chi =0.05$, and $f_{\alpha }=0.2)$. (b) Dependence of the gap $\Delta$ on the magnetic flux threading the $\alpha$-loop $f_{\alpha }$ for different values of $\chi (\alpha =0.5)$. Dependence of (c) the gap $\Delta$ and (d) the longitudinal coupling $g_{\parallel }$ on $f_{\alpha }$ for different values of $\alpha (\chi =0.05)$. Other parameters are as follows: $C=8\mathrm{fF},E_J/h=121\mathrm{GHz},C_r=250\mathrm{fF},L_r=1\mathrm{nH}$, and $M_{\alpha }=50\mathrm{pH}$. We assume that there is one fluxon trapped in the flux trapping loop $(N=n=1)$, and that the ratio of the total qubit loop area to the $\alpha$-loop area ${\mathcal{A}}_{\Sigma }/{\mathcal{A}}_{\alpha }$ is taken as being equal to 50. These results are obtained by numerical diagonalization of the circuit Hamiltonian with $n_{\mathrm{ch}}=15$ charge states.

Figure 16

Schematic representation of a tunable gap flux qubit capacitively coupled to a quantum harmonic oscillator: the interaction to the longitudinal degree of freedom of the artificial atom is achieved by coupling the electric potentials of the island defined by the split $\alpha$-junction and the resonator via a capacitance.

Figure 17

Tunability of the gap vs the reduced gate charge $n_g$ of the island defined by the split $\alpha$-junction (a) for different values of $\alpha (C_c=0\mathrm{fF})$ and (b) for different values of the coupling capacitance $C_c(\alpha =0.4)$. (c) Dependence of the longitudinal coupling $g_{\parallel }$ on $n_g$ for different values of the coupling capacitance $C_c(\alpha =0.4)$. (d) Longitudinal coupling $g_{\parallel }$ as a function of the coupling capacitance $C_c(\alpha =0.4$ and $n_g=0.25)$. Other parameters are as follows: $C=8\mathrm{fF},E_J/h=121\mathrm{GHz},C_r=100\mathrm{fF}$, and $L_r=10\mathrm{nH}$. These results are obtained by numerical diagonalization of the circuit Hamiltonian with $n_{\mathrm{ch}}=7$ charge states.