A Ten-Qubit Solid-State Spin Register with Quantum Memory up to One Minute

Spins associated with single defects in solids provide promising qubits for quantum-information processing and quantum networks. Recent experiments have demonstrated long coherence times, high-fidelity operations, and long-range entanglement. However, control has so far been limited to a few qubits, with entangled states of three spins demonstrated. Realizing larger multiqubit registers is challenging due to the need for quantum gates that avoid cross talk and protect the coherence of the complete register. In this paper, we present novel decoherence-protected gates that combine dynamical decoupling of an electron spin with selective phase-controlled driving of nuclear spins. We use these gates to realize a ten-qubit quantum register consisting of the electron spin of a nitrogen-vacancy center and nine nuclear spins in diamond. We show that the register is fully connected by generating entanglement between all 45 possible qubit pairs and realize genuine multipartite entangled states with up to seven qubits. Finally, we investigate the register as a multiqubit memory. We demonstrate the protection of an arbitrary single-qubit state for over 75 s—the longest reported for a single solid-state qubit—and show that two-qubit entanglement can be preserved for over 10 s. Our results enable the control of large quantum registers with long coherence times and therefore open the door to advanced quantum algorithms and quantum networks with solid-state spin qubits.

Popular Summary

The computational power of quantum networks is expected to far exceed that of classical technologies. Among the most promising quantum bits (qubits) for such networks are the spins of individual particles in solids, such as electrons and nuclei. Elementary control of such qubits and links within quantum networks have been demonstrated, but the largest entangled quantum states reported to date have contained just three spins. Larger quantum registers are essential to realizing advanced computational power. However, controlling individual spins within complex and strongly interacting spin systems is a significant challenge. In this study, we demonstrate a fully controllable ten-qubit register.

Working at 3.7 K, we study the electron spin and the nuclear spin of a single nitrogen-vacancy center in diamond, plus eight nearby carbon-13 nuclear spins. We show that the system is fully connected by generating entangled states between all 45 possible spin pairs, and we prepare genuine seven-spin entanglement. The spins that we study are excellent quantum memories that can store quantum states for up to one minute, the longest coherence time reported for solid-state spin qubits.

We expect that our methods can also be applied to other spin platforms in diamond, silicon, and silicon carbide. Our findings pave the way for advanced quantum algorithms and large multiqubit quantum networks based on tens of solid-state spin qubits.

Figure 1

Illustration of the ten-qubit register developed in this work. The electron spin of a single NV center in diamond acts as a central qubit and is connected by two-qubit gates to the intrinsic ${}^{14}\mathrm{N}$ nuclear spin and a further eight ${}^{13}\mathrm{C}$ nuclear spins surrounding the NV center.

Figure 2

(a) Illustration of the pulse sequence employed to realize a DDrf gate. Dynamical decoupling pulses on the electron spin (purple) are interleaved with rf pulses (yellow), which selectively drive a single nuclear spin. (b) Illustration showing that the initial state of the electron spin determines which rf pulses are resonant with the nuclear spin. If the electron spin starts in $|1⟩$ ($m_s=-1$), the odd rf pulses (red) are resonant. For initial electron state $|0⟩$ ($m_s=0$), the even (blue) rf pulses are resonant. The phase of each rf pulse is adapted to create the desired nuclear spin evolution, accounting for periods of free precession according to Eq. (3). (c) Nuclear spin trajectory on the Bloch sphere for a conditional rotation with $N=8$ electron decoupling pulses. Starting from the initial nuclear state $|\uparrow ⟩$ (yellow), the red (blue) path shows the nuclear spin evolution for the case where the electron starts in the state $|1⟩$ ($|0⟩$). The final state vectors are antiparallel along the equator; therefore, the gate is a maximally entangling two-qubit gate. (d) Top-down view of (c).

Figure 3

(a) Nuclear spin spectroscopy. After preparing the electron in a superposition state, the DDrf gate [controlled $\pm \pi /2$ rotation; see Eq. (4)] is applied for different rf frequencies $\omega$. The electron spin is then measured along a basis in the equatorial plane defined by angle $\phi$ (see inset). Each data point in (a) corresponds to the fitted amplitude $A$ of the function $f(\phi )=a+A\cos (\phi +{\phi }_0)$, where $\phi$ is swept from 0 to 360 deg, and ${\phi }_0$ accounts for deterministic phase shifts induced on the electron by the rf field. By fitting the amplitude, we distinguish such deterministic phase shifts from loss of coherence due to entangling interactions. The signals due to interaction with the eight ${}^{13}\mathrm{C}$ spins used in this work are labeled. The dashed gray line indicates the ${}^{13}\mathrm{C}$ Larmor frequency ${\omega }_L$. A detailed analysis of the spectrum is given in the Supplemental Material [53]. (b),(c) Example phase sweeps for two data points highlighted in red (b) and orange (c) in (a). Solid lines are fits to $f(\phi )$. The DDrf gate parameters are $N=48$ and $\tau =8{\tau }_L$, where ${\tau }_L=2\pi /{\omega }_L$ ($\approx 2.3\mu \mathrm{s}$).

Figure 4

(a) Experimental sequence to prepare an electron-nuclear Bell state and determine the expectation value of the two-qubit operator $ZX$. A series of single- and two-qubit gates are used to initialize the nuclear spin [16, 37]. A subsequent $\pi /2$ rotation and two-qubit gate generate the Bell state $|{\psi }_{\text{Bell}}⟩=(|0+⟩+|1-⟩)/\sqrt{2}$. A measurement of the electron spin in the $Z$ basis is followed by an $X$-basis measurement of the nuclear spin through the electron spin. These measurements are separated by a nuclear spin echo, which is implemented to mitigate dephasing of the nuclear spin. The entire sequence is applied with and without an additional electron $\pi$ pulse (dashed box) before the first electron readout in order to reconstruct the electron state while ensuring that the measurement does not disturb the nuclear spin state [16, 42]. (b) Density matrix of the electron-nuclear state after applying the sequence shown in (a) to qubit C1, reconstructed with state tomography. We correct for infidelities in the readout sequence characterized in separate measurements [53]. The DDrf gate parameters are $N=8$, $\tau =17{\tau }_L\approx 39.4\mu \mathrm{s}$, $\mathrm{\Omega }/2\pi =1.09(3)\mathrm{kHz}$, and the total gate duration is $629\mu \mathrm{s}$ compared with the nuclear spin $T_2^{*}=12.0(6)\mathrm{ms}$. We use error function pulse envelopes with a $7.5\text{−}\mu \mathrm{s}$ rise or fall time for each rf pulse to mitigate pulse distortions induced by the rf electronics [53]. The fidelity with the target Bell state is measured to be ${\mathcal{F}}_{\text{Bell}}=0.972(8)$. Lighter blue shading indicates the density matrix for the ideal state $|{\psi }_{\text{Bell}}⟩$.

Figure 5

(a) Experimental sequence for the preparation of a nuclear-nuclear Bell state and measurement of the two-qubit operator $ZZ$. After preparation of the electron-nuclear-nuclear GHZ state $|{\mathrm{GHZ}}_3⟩=(|0++⟩+|1--⟩)/\sqrt{2}$, an $X$-basis measurement on the ancilla (electron spin) projects the nuclear spins into the Bell state $|{\mathrm{\Phi }}^{+}⟩=(|++⟩+|--⟩)/\sqrt{2}$. Measurement of the two-qubit correlations between the nuclear spins is then performed through the electron spin. Spin echoes (dashed boxes) built into the measurement sequence protect the nuclear spins from dephasing errors. (b) Measured expectation values (nonzero terms of the ideal state only) for the electron-nuclear-nuclear state $|{\mathrm{GHZ}}_3⟩$ and for the nuclear-nuclear state $|{\mathrm{\Phi }}^{+}⟩$. Blue (purple) bars show the experimental (ideal) expectation values for each operator. The nuclear-nuclear correlations are well preserved after a nondestructive measurement of the electron spin in the $X$ basis.

Figure 6

Measured Bell state fidelities for all pairs of qubits in the ten-qubit register. Genuine entanglement is confirmed in all cases, as witnessed by a fidelity exceeding 0.5 with the target state. Qubits C1, C7, C8, and ${}^{14}\mathrm{N}$ are controlled using DDrf gates (Sec. 2). Qubits C2, C3, C4, C5, and C6 are controlled using the methods described in Taminiau et al. [37], as their hyperfine interaction parameters enable high-fidelity control using previously optimized gates.

Figure 7

(a) Experimental sequence to prepare a seven-qubit GHZ state $|{\mathrm{GHZ}}_7⟩$ (purple) and determine the expectation value of the seven-qubit operator $XYYYYZZ$ (orange). The measurement sequence is broken down into basis rotations (BR 1,2), an electron readout (RO), nuclear spin echoes (echo 1,2), and a multiqubit readout of the nuclear spins. All operations are applied sequentially (in the same way as shown in Fig. 5), but some are shown in parallel for clarity. (b),(c) Bar plots showing the measured expectation values (nonzero terms of the ideal state only) after preparing the five-spin (b) and seven-spin (c) GHZ states. The colors indicate the number of qubits involved, i.e., the number of (nonidentity) operators in the expectation value (inset). Gray bars show the ideal expectation values. See the Supplemental Material [53] for the operator corresponding to each bar. The fidelity with the target state is 0.804(6) (b) and 0.589(5) (c), confirming genuine multipartite entanglement in both cases. (d) Plot of GHZ state fidelity against the number of constituent qubits. A value above 0.5 confirms genuine $N$-qubit entanglement. The blue points are the measured data, while the green points are theoretical predictions assuming a simple depolarizing noise model whose parameters are extracted from single- and two-qubit experiments. Numerical values are given in the Supplemental Material [53].

Figure 8

(a) Dynamical decoupling for spin C5. With $\alpha =256$ pulses, the average state fidelity of the six cardinal states is measured to be 0.73(2) after 16.8 s, above the limit of $\frac{2}{3}$ for a classical memory with a confidence of 99.7% (upper-tailed $Z$ test). Solid lines are fit to the function $f(t)=A+B{e}^{-(t/T{)}^n}$. The offset $A$ is fixed using the average fidelity of the input states $|\uparrow ⟩$ and $|\downarrow ⟩$, which show no decay on these timescales. $B$, $T$, and $n$ are fit parameters which account for the decay of the fidelity due to interactions with the nuclear spin bath, external noise, and pulse errors. (b) Dynamical decoupling of the ${}^{14}\mathrm{N}$ spin. For $\alpha =256$ pulses, the average state fidelity at 75.3 s is 0.73(3), which is above the bound for a classical memory with 99.4% confidence. (c) Dynamical decoupling of a pair of ${}^{13}\mathrm{C}$ spins prepared in the Bell state $|{\mathrm{\Phi }}^{+}⟩$. Solid lines are fits to $f(t)$, but with $A$ as a free parameter to account for the observed decrease in the $ZZ$ correlations at large pulse numbers, likely due to pulse errors. With 256 decoupling pulses, genuine two-qubit entanglement is witnessed at times up to 10.2 s, where we observe a fidelity of 0.57(2) with the target Bell state (99.9% confidence of entanglement). In addition, interpolation of the fit yields 11.3(8) s as the point where the fidelity crosses 0.5 [53]. (d) Normalized coherence $(⟨XX⟩\pm ⟨YY⟩)/2\mathcal{N}$, where $\mathcal{N}$ is a normalization factor, for two pairs of ${}^{13}\mathrm{C}$ spins prepared in both the even- and odd-parity Bell states $|{\mathrm{\Phi }}^{+}⟩=(|\downarrow \downarrow ⟩+|\uparrow \uparrow ⟩)/\sqrt{2}$ and $|{\mathrm{\Psi }}^{-}⟩=(|\downarrow \uparrow ⟩-|\downarrow \uparrow ⟩)/\sqrt{2}$. Solid lines are fits to $f(t)$ with $A=0$ and $B=1$. For pair 1, the fitted decay times $T$ are 0.45(2) and 0.54(1) s for the states $|{\mathrm{\Phi }}^{+}⟩$ and $|{\mathrm{\Psi }}^{-}⟩$, respectively. For pair 2, the equivalent values are 0.46(2) and 0.70(3) s.