Fully microwave-tunable universal gates in superconducting qubits with linear couplings and fixed transition frequencies

A register of quantum bits with fixed transition frequencies and weakly coupled to one another through simple linear circuit elements is an experimentally minimal architecture for a small-scale superconducting quantum information processor. Presently, the known schemes for implementing two-qubit gates in this system require microwave signals having amplitudes and frequencies precisely tuned to meet a resonance condition, leaving only the signal phases as free experimentally adjustable parameters. Here, we report a minimal and robust microwave scheme to generate fast, tunable universal two-qubit gates: simply irradiate one qubit (the “control”) at the transition frequency of another (the “target”). The effective coupling between them is then switched on by tuning only the frequency of this single drive tone; the drive amplitude adjusts the effective coupling strength; and the drive phase selects the particular two-qubit gate implemented. This cross-resonance effect turns on linearly with the ratio of the drive amplitude $\Omega$ to the qubit-qubit detuning $\Delta$, as compared with earlier proposals that turn on as ${\left(\Omega /\Delta \right)}^4$.

Figure 1

(Color online) Left: FLICFORQ-style qubits (circles) have fixed transition frequencies ${\omega }_j$ and fixed linear off-diagonal coupling ${\omega }_{xx}$ (magenta bar). One- and two-qubit gates are performed by applying exclusively microwave control signals. Circuit parameters are selected to satisfy the weak-coupling condition ${\omega }_{xx}⪡\left|\Delta \right|\equiv \left|{\omega }_1-{\omega }_2\right|$, allowing the interaction to be neglected in the absence of irradiation up to an error oscillation of frequency ${\left({\omega }_{xx}^2+{\Delta }^2\right)}^{1/2}$ and amplitude ${\omega }_{xx}/\Delta$. Cross resonance: a universal two-qubit gate can be implemented in this system by simply applying a microwave drive (green arrow) to the “control” qubit (Q1, red) at the transition frequency of the “target” (Q2, purple). This procedure generates the effective Hamiltonian ${\mathcal{H}}^{\text{eff}}=\frac{{\Omega }_1{\omega }_{xx}}{4\Delta }\left({\sigma }_1^z{\sigma }_2^x\text{cos}{\varphi }_1+{\sigma }_1^z{\sigma }_2^y\text{sin}{\varphi }_1\right)$ and with ${\varphi }_1=0$ the universal gate ${\left[ZX\right]}^{1/2}$ is realized in time $t=\pi \Delta /\left({\Omega }_1{\omega }_{xx}\right)$. Right: cross resonance in the dressed state picture. When Q1 is irradiated at ${\omega }_1^{\text{rf}}={\omega }_2$ the $\text{Q}1+\mu$-wave photons system may undergo spontaneous transitions (Ref. 20) at ${\omega }_2$. The interaction is switched on by tuning only the frequency of a single tone applied to one of the qubits and the resulting effective interaction strength turns on linearly with ${\Omega }_1/\Delta$.

Figure 2

(Color online) Effective interaction strength generated by cross-resonance driving as a function of drive strength-to-detuning ratio ${\Omega }_1/\Delta$ (solid); generalized off-resonant FLICFORQ scheme of Refs. 15, 17 (dashed); and double-resonant driving of original FLICFORQ (Ref. 14) (red point). Equal drive amplitudes $\left({\Omega }_1={\Omega }_2\right)$ on each qubit are assumed for the latter two.

Figure 3

(Color online) Cross-resonance two-qubit gate time (time required to generate fully entangled state from product state and vice versa) for three practical parameter sets as a function of absolute drive amplitude $\Omega$. Green: $\left\{{\omega }_1=12.8,{\omega }_2=16.1,{\omega }_{xx}=0.13\right\}/2\pi$; red: $\left\{{\omega }_1=9.8,{\omega }_2=16.1,{\omega }_{xx}=0.4\right\}/2\pi$; and blue: $\left\{{\omega }_1=15.8,{\omega }_2=16.1,{\omega }_{xx}=0.05\right\}/2\pi$; all in gigahertz. Solid lines are from analytics; points are extracted from simulation of full Hamiltonian. Inset: entanglement vs time for sample point indicated by arrow at $\Omega =0.5\text{GHz}$.

Figure 4

(Color online) Cross resonance in presence of microwave cross-talk. Left: cross-talk leads to a fraction $\alpha$ of the signal applied to Q1 to be felt also by Q2. A signal at ${\omega }_1^{\text{rf}}$ then need not be precisely resonant with ${\omega }_2$ in order to switch on the coupling. Right: dressed state manifolds for a single cross-coupled signal applied to both Q1 and Q2. As both the signal frequency and amplitude are varied, a resonance is maintained between the central transitions of each $\text{qubit}+\text{photons}$ system.

Figure 5

(Color online) Effective cross-resonance coupling strength ${\omega }_{xx}^{\text{eff}}$ in presence of microwave cross-talk as a function of drive frequency ${\tilde{\omega }}_1^{\text{rf}}$ and drive amplitude ${\tilde{\Omega }}_1$ [curly bar indicating normalization to mean qubit transition frequency $\overline{\omega }=\left({\omega }_1+{\omega }_2\right)/2$]. The microwave signal is applied intentionally to Q1 and a fraction $\alpha$ is felt also by Q2. Qubit transition frequencies are indicated by red $\left({\tilde{\omega }}_1=0.98\right)$ and purple $\left({\tilde{\omega }}_2=1.02\right)$ arrows. When $\alpha =0$ (not shown), the cross-resonance effect is broadened beyond finite qubit lifetime effects only by $\sqrt{N}$ fluctuations in the drive signal photon number. At top is case where $\alpha =0.2$; at bottom $\alpha =0.8$, corresponding (roughly) to the experiment of Ref. 2. In the latter case the range of drive frequencies effective at switching on the coupling is dramatically broadened. When $\alpha =1$ (not shown), ${\omega }_{xx}^{\text{eff}}$ is symmetric and centered at $\overline{\omega }$.