Universal Quantum Computing Using Electronuclear Wavefunctions of Rare-Earth Ions

We propose a scheme for universal quantum computing based on Kramers rare-earth ions. Their nuclear spins in the presence of a Zeeman-split electronic crystal field ground state act as “passive” qubits that store quantum information. The “active” qubits are switched on optically by fast coherent transitions to excited crystal field states with a magnetic moment, and the magnetic dipole interaction between these states is used to implement controlled not (cnot) gates. We compare our proposal with others, noting particularly the much improved cnot gate time as compared with a $\mathrm{Si}$:$\mathrm{P}$ proposal, also relying on magnetic dipole interactions between active qubits, and rare-earth schemes depending on the dipole blockade for qubits spaced by more than of the order of $1\mathrm{nm}$.

Popular Summary

Nuclear spins and electronic crystal-field excitations of rare-earth ions in insulating crystals have a long history of exploitation for quantum science and technology. Here we show that they are ideal qubit hosts for a programmable quantum processor, and we propose a scheme to implement a universal set of quantum gates between them.

The scheme builds on the extremely long lifetime of nuclear spins in the presence of rare-earth electronic spins to realize “passive” memory qubits. A magnetic field polarizes the electronic spins, which efficiently prevents entanglement spreading between the qubits, even though the electronic interactions are not shielded. At first sight, this looks like a bug, whereas in fact, it is a feature, allowing qubits to be effectively “shielded” (on demand) locally without the need for complicated operations. We use optical pulses to efficiently map the quantum information from nuclear spins to excited, long-lived crystal field states with electronic magnetism. This implements single-qubit gates up to three orders of magnitude faster than direct nuclear spin manipulations. Two such “activated” qubits can interact via strong magnetic dipolar interactions, enabling the implementation of fast two-qubit gates, up to two orders of magnitude faster than previous proposals exploiting optical activation of atoms in crystals. The long-range nature of the dipolar interaction does not limit gates to nearest-neighbor qubits. Instead, it allows for a large number of qubits to be connected, even while crosstalk with nontargeted qubits in their neighborhood is robustly suppressed.

The next step is to implement our concepts in experiments. We suggest that rare-earth ions implanted in isotopically enriched silicon could meet the requirements for a manufacturable quantum processor. Here, as for other platforms, hybrid architectures including photonic cavity resonators will need to be explored for efficient coupling of photons to the individual qubits.

Figure 1

Basic principle of the quantum computing scheme. The passive (memory) qubit consists of nuclear degrees of freedom in the presence of polarized electronic RE spins. Even though the electronic spins create substantial internal fields, they are much smaller than the external field, such that the electronic system remains in its gapped, fully polarized ground state and spin flips are suppressed. The nuclear qubits can be selectively activated by optical excitation to a (locally tunable) electronic CF doublet with a large magnetic moment. Two activated qubits communicate via the electronic (magnetic) dipolar interaction, allowing for fast two-qubit gates over distances of several 10 nm, as long as the dipolar interaction is sufficiently strong as compared with the inverse lifetime of the active qubits. This enables gates between non-nearest neighbors.

Figure 2

The optimal level scheme of a Kramers RE ion in a CF field, showing the energy scales governing Hamiltonian (1). We use the ground state and one excited spin-orbit manifold, requiring the lowest-lying states of both manifolds to be CF doublets with different anisotropy directions, which we then Zeeman split with an external field. From the ground state doublet we only need the lower Zeeman state. The coupling to a nuclear spin with $I=1/2$ splits the electronic states into pairs of hyperfine states, where the nuclear spins are either aligned or antialigned with the electronic spin. The spatial density of the RE ions is adjusted such that the dipolar interactions $J_{\mathrm{dip}}$ have by far the smallest energy scale (apart from the nuclear Zeeman energy). An analogous bound applies to the operation temperature $T$.

Figure 3

Two possible choices for the mapping between passive and active qubits: (a) electronuclear qubit and (b) purely electronic qubit. Activation of a qubit is achieved by two $\pi$ pulses as discussed in the main text. Each pulse leaves one hyperfine state of the passive qubit untouched and takes the other state into one of two target hyperfine states within the excited electronic doublet. These two states constitute the active qubit. The two target states have opposite (electronic) magnetic moments, and thus the active qubits can interact over a long distance.

Figure 4

Implementation of a fast $X$ gate via an excited state. The blue arrows denote $\pi$ pulses that switch the population of the states they connect. The pulses are applied in the order of the indicated numbers. The differing $g$-factor anisotropies in the ground and excited doublets ensure sizable matrix elements for all three transitions.

Figure 5

Two possible active qubit choices that are found by minimizing the number of resonant channels and maximizing the lifetime of the active qubits: (a) the electronuclear qubit and (b) the electronic qubit. We use here the same level structure as for the excited doublet in Fig. 3, with the indices $1$ and $2$ labeling the two ions on which the cnot gate acts. The red arrows denote the electronuclear and electronic “spin flip-flop” matrix elements for (a) and (b), respectively. Such a simultaneous swap of both RE qubits is exactly resonant. The blue and green arrows in (b) denote the nuclear and electronuclear flip-flops, respectively, which are nearly resonant up to the nuclear Zeeman energy. These processes limit the fidelity of the cnot gate in the most favorable case of active qubits with $g_{\perp }=0$, for which the electronic qubit (b) minimizes the gate error. In most host materials $g_{\perp }$ is nonzero and the electronuclear qubit (a) should be used.

Figure 6

Pulse sequence for the dipole-blockade scheme. If the control qubit is in state $|0⟩$, it is excited, which shifts the levels of the target qubit out of resonance with the swapping pulses. A swap is thus only carried out if the control qubit is in state $|1⟩$. The indices “ctrl” and “tar” denote the control and target qubits, respectively.

Figure 7

The speedup factor $t_{\mathrm{gate}}^{\mathrm{db}}/t_{\mathrm{gate}}$ by which the gate time of our cnot implementation is reduced as compared with that of a dipole-blockade scheme carried out with Gaussian pulses (assuming that the dipolar interaction is smaller than the hyperfine splitting and the largest achievable Rabi frequencies). The speedup increases logarithmically with the maximally admissible gate error $1-\mathcal{F}$ that we assume to be given by the probability of a nonresonant excitation (rather than by the decoherence time of the qubits). The oscillatory behavior traces back to Rabi oscillations in the unwanted transition probability.

Figure 8

The maximal error of an optimized $\pi$ pulse with truncated Gaussian envelope as a function of the pulse time $T_{\pi }=2T_{\mathrm{cut}}$ is plotted for a fixed dipolar coupling $\mathrm{\Delta }\omega$. The error decreases exponentially with the pulse time on top of which there are Rabi-like oscillations with period $4\pi /\mathrm{\Delta }\omega$. The fits to the numerically optimized pulse times are of the form (A7).

Figure 9

Sketch of our RE qubit setup with a qubit spacing of order 10 nm. The qubits are activated with laser pulses. A grid of electrical gates, which can be switched on and off, induces a Stark shift on selected qubits. The difference $\mathrm{\Delta }{\omega }_{\mathrm{St}}$ in the Stark shift between neighboring RE ions, which is proportional to the electric field gradient, allows one to address single qubits. Note that the sketch is not true to scale, in particular the thickness of the wires and their distance to the qubits are scaled down to make the figure readable.

Figure 10

Sketch of our choice of passive and active qubits in ${}^{167}{\mathrm{Er}}^{3+}:{\mathrm{Y}}_2{\mathrm{SiO}}_5$. The nuclear spin states denote the projections in the electronic polarization directions, i.e., they are not the same for the ground state and excited doublets. Because of the similar $g$-factor anisotropies in the ground and excited doublets, the transitions indicated with blue arrows are a factor of 3.5 faster than the transitions indicated with green arrows.