Resonator–zero-qubit architecture for superconducting qubits

We analyze the performance of the resonator–zero-qubit (RezQu) architecture in which the qubits are complemented by memory resonators and coupled via a resonator bus. Separating the stored information from the rest of the processing circuit by at least two coupling steps and the zero qubit state results in a significant increase in the on-to-off ratio and a reduction in the idling error. Assuming no decoherence, we calculate such idling error, as well as the errors for the move operation and tunneling measurement, and show that the RezQu architecture can provide the high-fidelity performance required for medium-scale quantum information processing.

Figure 1

Schematic diagram of the RezQu architecture: $m$, memory resonators; $q$, qubits; $b$, bus. We assume frequencies of $\sim$7 GHz for the memories and $\sim$6 GHz for the bus, and qubit frequencies are varied between these values.

Figure 2

Frequency ${\Omega }_{ZZ}$ for a truncated memory-qubit-bus system [the idling error is ${\left({\Omega }_{ZZ}t\right)}^2$] for two values of the coupling: $g_m/2\pi =g_b/2\pi =25$ MHz (blue lines) and $g_m/2\pi =g_b/2\pi =50$ MHz (red lines). Solid lines show the results of exact diagonalization of the RWA Hamiltonian, (3), with and without $g_d$. The effect of $g_d$ is not visible (smaller than the line thickness). Blue and red dashed lines show the analytical result, (6).

Figure 3

Illustration of a piecewise-linear tune/detune pulse shape (qubit frequency as a function of time) for the move operation $\mathrm{qubit}\to \mathrm{memory}$ in a three-component $mqb$ system. The front ramp consists of two straight segments. The solid (blue) line shows the result of a four-parameter numerical optimization in which the slope of the first straight segment, the qubit frequency at the end of the first straight segment, the duration $t_2-t_1$ of the flat part, and its overshoot $D$ have been optimized. This gives $\mathcal{E}=0$ up to machine accuracy. The dashed (red) line shows the analytical design based on Eqs. (17) and (22); in this case, $\mathcal{E}=5\times {10}^{-4}$. System parameters: ${\omega }_m/2\pi =7$ GHz, ${\omega }_b/2\pi =6$ GHz, ${\omega }_q\left(0\right)/2\pi =6.7$ GHz, ${\omega }_q\left(t_{\mathrm{f}}\right)/2\pi =6.5$ GHz, and $g_m/2\pi =g_b/2\pi =25$ MHz. The slopes of the second front segment and of the final ramp have been fixed at 500 MHz/ns.

Figure 4

Implementation of the $\mathrm{qubit}\to \mathrm{memory}$ move operation in a three-component $mqb$ system using a pulse with error-function-shaped ramps (sum of two time-shifted error functions for the front ramp). Four parameters of the pulse shape (upper panel) are optimized: the time shift and amplitude ratio for the front-ramp error functions, the duration of the middle part of the pulse, and the overshoot magnitude. Error functions are produced by integrating Gaussians with standard deviation $\sigma =1$ ns; the beginning and end of the pulse are at 3$\sigma$ from the nearest error-function centers (shown by vertical lines in the upper panel). The middle and lower panels show the time dependence of the level populations in the bare-state basis and comoving eigenbasis. The move error, (23), is zero up to machine accuracy.

Figure 5

Time dependence of the squared amplitudes ${\left|\alpha \right|}^2$ and ${\left|\beta \right|}^2$ of the $mq$ state $\left|\psi \right(t)\rangle =\alpha (t\left)\right|10\rangle +\beta \left(t\right)|01\rangle$, decaying in the process of tunneling measurement. Blue curves correspond to the system initially prepared in the bare state $\left|\psi \right(0)\rangle =|01\rangle$, while for red curves the initial state is the eigenstate $|\psi \left(0\right)\rangle =|\overline{01}\rangle$. For $t\gg {\Gamma }^{-1}$ the measurement error, (27), is mainly the residual occupation ${\left|\alpha \left(t\right)\right|}^2$ of the memory resonator (solid curves). For the depicted system parameters, $|{\alpha }_{\mathrm{eigen}}{\left(t\right)|}^2/{\left|{\alpha }_{\mathrm{bare}}\left(t\right)\right|}^2\approx 0.025$ at $t\gtrsim 10$ ns; i.e., the error of the eigenstate measurement is 40 times smaller than the error of the bare-state measurement.