Blueprint for a Molecular-Spin Quantum Processor

The implementation of a universal quantum processor still poses fundamental issues related to error mitigation and correction, which demand investigation of also platforms and computing schemes alternative to the main stream. A possibility is offered by employing multilevel logical units (qudits), naturally provided by molecular spins. Here we present the blueprint of a molecular spin quantum processor consisting of single molecular nanomagnets, acting as qudits, placed within superconducting resonators adapted to the size and interactions of these molecules to achieve a strong single spin-to-photon coupling. We show how to implement a universal set of gates in such a platform and to readout the final qudit state. Single-qudit unitaries (potentially embedding multiple qubits) are implemented by fast classical drives, while an alternative scheme is introduced to obtain two-qubit gates via resonant photon exchange. The latter is compared to the dispersive approach, finding in general a significant improvement. The performance of the platform is assessed by realistic numerical simulations of gate sequences, such as Deutsch-Josza and quantum simulation algorithms. The very good results demonstrate the feasibility of the molecular route towards a universal quantum processor.

Figure 1

Lowest energy levels for the two qudits used in the reported simulations, with $D_1/2\pi =7.1$ GHz, $D_2/2\pi =7.7$ GHz, $g_i=2$. Blue double arrows indicate allowed EPR transitions for qudit 2 between $|m_2⟩\to |m_2\pm 1⟩$ levels, induced by resonant (low-power) microwave classical pulses in the range $5$–$20$ GHz. An analogous situation is found for qudit 1 (not shown for clarity). The thick arrow on the right indicates the bare resonator frequency $\hslash {\omega }_0$.

Figure 2

Scheme of the two-qudit resonant gate: dark (bright) boxes indicate empty (single photon) resonators. Two photon states (included in the simulations) are never populated. Energy levels of the two qudits are indicated by red lines. In this example, we use two levels to encode the logical state of $q_1$ (left) and four to encode $q_2$ (right), while the second excited state of $q_1$ (dotted line, $|e⟩$) is an auxiliary one exploited during the gate. The dashed green line indicates the resonator frequency, which is varied at each step. We illustrate the steps of the gate for a system prepared in $|01⟩$ (crosses). (a) Idle state, with no photons (dark) and the resonator out of resonance with respect to all qudit transitions. (b) The resonator is brought into resonance with $E_{01}^{(q_2)}$ and a photon is emitted if $q_2$ is prepared in $|1⟩$. (c) A photon is now in the resonator (bright) and can be absorbed emitted if $q_1$ is prepared in $|0⟩$, once the resonator is brought into resonance with $E_{0e}^{(q_1)}$. (d) The resonator is brought again into resonance with $E_{01}^{(q_2)}$ and the photon is absorbed, ending up again with $|01⟩$ and an additional phase.

Figure 3

Scheme of the resonant readout of a qubit in state $|0⟩$ (left) or $|1⟩$ (right). (a) Starting with no photons (dark cavity) the resonator is brought into resonance with $E_{01}$. If the qubit is in state $|1⟩$ (left) a photon is emitted and the cavity becomes bright (b), bringing the qubit to the ground state. At this step, a single-photon counter detects the possible presence of the photon (and annihilates it). In the case of positive outcome, a classical drive sends back the qubit state to $|1⟩$ (c), thus implementing a projective measurement on the qubit.

Figure 4

Schematic layout (a) and electric circuit (b) of the MSQP elementary unit, which consists of an on-chip $LC$ superconducting resonator with a few magnetic molecules deposited on top. Nanosized constrictions concentrate the magnetic field at specific regions, where the coupling to single molecules, with spins ${\mathbf{S}}_1$ and ${\mathbf{S}}_2$ is enhanced. Transmission lines introduce electromagnetic pulses to coherently control each spin qudit and to tune the flux though a dc SQUID in series with the inductor. The latter serves to tune the resonator characteristic frequency. An additional transmission line reads out the state of the resonator, which provides information on the spin-qudit states.

Figure 5

(a) Layout of a $7.5$-GHz $LC$ superconducting resonator with a 22-$\text{μ}\mathrm{m}$-long and 4-$\text{μ}\mathrm{m}$-wide inductor and two parallel plate capacitors. (b) Two-dimensional plot of the distribution of currents at resonance. (c) Spatial dependence of the microwave magnetic field generated by the resonator for different inductor widths $w$.

Figure 6

Coupling to photons of individual spin qudits, located 1 nm above the inductor line of the $LC$ resonator shown in Fig. 5. Results calculated for spins with easy axis [$D<0$ in Eq. (2)] and easy plane ($D>0$) are shown in, respectively, the bottom and top panels. They correspond to a different spin ground state and, therefore, to different resonant spin transitions. The solid line gives the coupling obtained for a simple $S=1/2$ spin qubit.

Figure 7

Decomposition of the circuit for the Deutsch-Jozsa (two-qubit) algorithm into a sequence of pulses on a four-level qudit. (a) Schematic representation of the two-qubit circuit and of the sequence of qudit operations needed to implement each unitary. Each arrow color represents one rotation direction [parameter $\varphi$ in Eq. (5)]: black for $Y$, blue for $X$, and red for $Z$. (b) Error dependence with the driving magnetic field, $B_1$, and with the decoherence time $T_2$. The error is computed as $\mathcal{E}=Tr\left(\rho P\right)$ and $P$ is the projector onto the subspace in which the final wave function must not be contained according to the applied function, $U_{f_i}$.

Figure 8

Error $1-\mathcal{F}$ (log scale) in the implementation of the resonant controlled-$Z$ gate as a function of the resonator quality factor $Q$ and of the qudits’ coherence time $T_2$. Random components on $|p⟩|q⟩$ initial state are given by $|0⟩|q⟩=(0.31,0.46,0.48,0.37)$ and $|1⟩|q⟩=(0.37,0.25,0.25,0.24)$ with $q=0,1,2,3$.

Figure 9

Fidelity in the implementation of the controlled-$Z$ (CZ) gate in the resonant regime (solid lines) and of the $i$swap gate in the dispersive regime (dotted) as a function of the qudits’ coherence time $T_2$, for different values (colors) of the resonator quality factor $Q$.

Figure 10

Comparison between resonant (left) and dispersive (right) approaches in the digital quantum simulation of the Heisenberg model, and associated fidelity (bottom panels) as a function of the simulated time. The system is initialized in $(|00⟩+|01⟩)/\sqrt{2}$. In the top panels we report the time evolution of the expectation values of $s_{z1}$ (red) and $s_{z2}$ (black). The resonator quality factor is fixed to $Q={10}^6$.

Figure 11

Quantum simulation of the time evolution of the system magnetisation $⟨S_z⟩\equiv ⟨s_{z,1}+s_{z,2}⟩$ for the transverse-field Ising model on two qubits. Dashed (continuous) line represents the exact evolution (after Trotter decomposition, $n=6$). Colored points are the results of full numerical simulation, including the effect of photon loss, parametrized by the resonator quality factor $Q$, and of spin pure dephasing, parametrized by the coherence time $T_2$.