Electromechanical quantum simulators

Digital quantum simulators are among the most appealing applications of a quantum computer. Here we propose a universal, scalable, and integrated quantum computing platform based on tunable nonlinear electromechanical nano-oscillators. It is shown that very high operational fidelities for single- and two-qubits gates can be achieved in a minimal architecture, where qubits are encoded in the anharmonic vibrational modes of mechanical nanoresonators, whose effective coupling is mediated by virtual fluctuations of an intermediate superconducting artificial atom. An effective scheme to induce large single-phonon nonlinearities in nanoelectromechanical devices is explicitly discussed, thus opening the route to experimental investigation in this direction. Finally, we explicitly show the very high fidelities that can be reached for the digital quantum simulation of model Hamiltonians, by using realistic experimental parameters in state-of-the-art devices, and considering the transverse field Ising model as a paradigmatic example.

Figure 1

(a) Schematic representation of the circuit building block for digital quantum simulation proposed in this work: two electromechanical oscillators (NR) mutually coupled through a transmon within a superconducting circuit; each element is tunable through either external voltage bias (the NR) or external magnetic fields (the transmon), and ground state cooling of the NRs can be achieved by coupling to a transmission line resonator, or a lumped LC circuit element; the intrinsic scalability of this setup is sketched below. (b) Analog circuit model of the elementary building block shown in (a).

Figure 2

Nonlinearity induced on a nanoresonator by a superconducting circuit. (a) Nonlinear shift between the ground-to-first and the first-to-second excited states transitions $\delta ={\omega }_{21}-{\omega }_{10}$. (b) Components of the corresponding wave function. In both panels, ${\omega }_{\text{NR}}=100$ MHz, ${\mathrm{\Omega }}_{\text{SC}}=500$ MHz.

Figure 3

(a) Fidelity of a single-qubit $x$ rotation by a $\pi /2$ angle, and (b) fidelity of the two-qubit $\sqrt{\text{iSWAP}}$ gate, as functions of the pure dephasing rates of the electromechanical resonators and the transmon, respectively. For the two-qubits gate, we assume ${\gamma }_{1,d}={\gamma }_{2,d}={\gamma }_{\text{NR},d}$; all the other simulation parameters are reported in the text.

Figure 4

(a) Exact time evolution of the total magnetization of a $S=1$ system (full line) undergoing oscillations between opposite polarized states (describing, e.g., quantum tunneling across an anisotropy barrier), compared to the digital quantum simulation of the target Hamiltonian (8) for infinite and finite $T_2$ time of the electromechanical qubits, respectively (points). (b) Exact Trotter evolution ($N=10$) of the total $x$ polarization in the model Hamiltonian (9) (full line), compared to the corresponding quantum simulation for different values of the qubits $T_2$ time (points).

Figure 5

Comparison between two possible models for the single-phonon nonlinearity of the nanomechanical resonators. Panel on the right focuses on the region of interest for our setup.

Figure 6

Fidelity of single quibit $R_x\left(\pi \right)$ gate as a function of the nonlinear shift between the first and second energy gaps. Gaussian tuning pulses were used (see Appendix pp3) to activate and control the gates.

Figure 7

Dissipation and coherence times for the coupled nanomechanical resonator and superconducting circuit system, obtained from Bloch-Redfield simulations. The values of overall $T_1$ and $T_2$ time scales are expressed in units of the corresponding typical time scales for the superconducting circuit assumed in the simulation, as taken from the literature (i.e., $T_{1,\text{SC}}=1$ ms and $T_{2,\text{SC}}\simeq 10\mu \mathrm{s}$).

Figure 8

Numerical simulations of single- and two-qubit quantum gates in an electromechanical circuit. Here the system undergoes unitary evolution, with ${\omega }_1=85$ MHz and ${\omega }_2=75$ MHz. With $p_{ab}$ we denote the component of the two-qubit wave function on the corresponding Fock state, $p_{ab}={\left|\langle ab|\psi \rangle \right|}^2$. (Bottom left) $R_x\left(\pi \right)$ rotation performed on qubit 1 while qubit 2 is kept isolated and the transmon frequency, $\mathrm{\Omega }=10$ GHz, is left unchanged. (Top left) The Gaussian oscillating pulse acting on qubit 1 needed for the $R_x\left(\pi \right)$ gate. The peak amplitude is 0.3 MHz. (Bottom right) A typical time evolution displaying a $\sqrt{\text{iSWAP}}$ operation, with a short idle phase before and after the gate. (Top right) Frequency shifts operated on qubit 2 and on the transmon during the time evolution, including the rephasing stage on the qubit. Notice that the renormalization shift ${\lambda }_2$ on the qubit is not included.