A 10-qubit solid-state spin register with quantum memory up to one minute

TL;DR

This paper demonstrates a 10-qubit solid-state spin register with long coherence times up to 63 seconds, enabling advanced quantum information processing and networks with high-fidelity control and entanglement.

Contribution

The authors develop decoherence-protected gates and realize a large, fully connected 10-qubit register with long coherence times, surpassing previous multi-qubit control capabilities.

Findings

Achieved a 10-qubit quantum register in diamond with full connectivity.

Demonstrated genuine multipartite entanglement among up to 7 qubits.

Stored two-qubit entangled states for over 10 seconds.

Abstract

Spins associated to 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 multi-qubit registers is challenging due to the need for quantum gates that avoid crosstalk 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 10-qubit quantum register consisting of the electron spin of a nitrogen-vacancy center and 9 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 7 qubits. Finally, we investigate the register as a multi-qubit memory. We show coherence times up to 63(2) seconds - the longest reported for a single solid-state qubit - and demonstrate that two-qubit entangled states can be stored for over 10 seconds. 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.

Equations

$$H={\omega }_L{I}_z+A_{\parallel }{S}_z{I}_z+A_{\perp }{S}_z{I}_x,$$

$$\begin{array}{rl}H=& |0⟩⟨0|\otimes ({\omega }_L-{\omega }_1)I_z\\ & +|1⟩⟨1|\otimes \Omega (\cos (\varphi )I_x+\sin (\varphi )I_y),\end{array}$$

$$\phi_{k}^{\prime}=\left\{\begin{array}[]{ll}(k-1)\phi_{\tau}+\pi&\quad k\text{ odd}\\ (k-1)\phi_{\tau}&\quad k\text{ even},\end{array}\right.$$

$$\begin{array}{rl}{V}_{\text{CROT}}=& |0⟩⟨0|\otimes R_{\phi }(N\Omega \tau )\\ & +|1⟩⟨1|\otimes R_{\phi }(-N\Omega \tau ),\end{array}$$

$$|{\text{GHZ}}_N⟩=\frac{1}{\sqrt{2}}\left(|0⟩\otimes {|+⟩}^{\otimes (N-1)}+|1⟩\otimes {|-⟩}^{\otimes (N-1)}\right).$$

Full text

I INTRODUCTION

Electron and nuclear spins associated with single defects in solids provide a promising platform for quantum networks and quantum computations Awschalom et al. (2018); Zwanenburg et al. (2013). In these hybrid registers, different types of spins fulfill different roles. Electron spins offer fast control De Lange et al. (2010); Fuchs et al. (2009); Christle et al. (2015); Seo et al. (2016); Sukachev et al. (2017); Iwasaki et al. (2015); Siyushev et al. (2017); Becker et al. (2018); Pingault et al. (2017); Trusheim et al. (2018); Rugar et al. (2018) and high fidelity readout Robledo et al. (2011); Pla et al. (2013); Sukachev et al. (2017), and can be used to control and connect nuclear spins Cramer et al. (2016); Waldherr et al. (2014); Pla et al. (2013); Wolfowicz et al. (2016); Tosi et al. (2017); Metsch et al. (2019); Hensen et al. (2019). Furthermore, electron-electron couplings enable on-chip connectivity between defects Dolde et al. (2013); Yamamoto et al. (2013); Tosi et al. (2017), whilst coupling to photons Yang et al. (2016); Togan et al. (2010); Christle et al. (2017); Sipahigil et al. (2016); Trusheim et al. (2018); Evans et al. (2018) allows for the realization of long-range entanglement links Bernien et al. (2013); Hensen et al. (2015); Humphreys et al. (2018). Nuclear spins provide additional qubits with long coherence times that can be used to store and process quantum states Maurer et al. (2012); Muhonen et al. (2014); Waldherr et al. (2014); Cramer et al. (2016); Dehollain et al. (2016); Yang et al. (2016); Hensen et al. (2019).

Recent experiments have demonstrated various schemes for high-fidelity two-qubit gates Dehollain et al. (2016); Van der Sar et al. (2012); Taminiau et al. (2014); Rong et al. (2015); Zaiser et al. (2016); Unden et al. (2018); Huang et al. (2019), as well as basic quantum algorithms Van der Sar et al. (2012); Kalb et al. (2017) and error correction codes Waldherr et al. (2014); Cramer et al. (2016). However, to date, these systems have been restricted to few-qubit registers: the largest reported entangled state contains 3 qubits Waldherr et al. (2014); Cramer et al. (2016); van Dam et al. (2019). Larger quantum registers are desired for investigating advanced algorithms and quantum networks Terhal (2015); Nickerson et al. (2013, 2014). Such multi-qubit registers are challenging to realize due to the required gates that selectively control the qubits and at the same time decouple unwanted interactions in order to protect coherence in the complete register.

In this work, we develop a novel gate scheme based upon selective phase-controlled driving of nuclear spins interleaved with decoupling sequences on an electron spin. These gates enable high-fidelity control of hitherto inaccessible nuclear spin qubits. We combine these gates with previously developed control techniques Robledo et al. (2011); Cramer et al. (2016); Abobeih et al. (2018) to realize a 10-qubit register composed of a diamond nitrogen-vacancy (NV) center, its 14N nuclear spin and 8 13C spins (Fig. 1). We show that the register is fully connected by preparing entangled states for all possible pairs of qubits. Furthermore, by also decoupling nuclear-nuclear interactions through echo sequences, we generate $N$-qubit Greenberger-Horne-Zeilinger (GHZ) states, and witness genuine multipartite entanglement for up to 7 spins. Finally, we investigate the coherence properties of the register. We show that single qubit states can be stored for up to 63(2) seconds and two-qubit entangled states can be stored for over 10 seconds.

II TWO-QUBIT GATES: THEORY

We consider an NV center in diamond and surrounding 13C nuclear spins. To realize a multi-qubit register, we design single-qubit gates and electron-nuclear two-qubit gates to control the NV 14N spin and several individual 13C spins. Key challenges in these hybrid systems of multiple coupled spins are to maintain coherence on the electron spin qubit and to avoid unwanted crosstalk. In particular, the electron spin continuously couples to all 13C spins through the hyperfine interaction, and the dynamics of the electron spin and nuclear spins typically occur on very different timescales Van der Sar et al. (2012). To address these issues, a variety of decoherence-protected gates, in which decoupling sequences on the electron spin are combined with nuclear spin control, have been investigated Van der Sar et al. (2012); Kolkowitz et al. (2012); Taminiau et al. (2012); Zhao et al. (2012); Taminiau et al. (2014); Zaiser et al. (2016); Wang et al. (2017); Pfender et al. (2017); Unden et al. (2018). Here we develop and demonstrate a novel electron-nuclear two-qubit gate based upon phase-controlled radio-frequency (RF) driving of nuclear spins, interleaved with dynamical decoupling (DD) of the electron spin. We will refer to this scheme as a DDRF gate. Our scheme enables the control of additional 13C spins while offering improved flexibility in dynamical decoupling to optimize the electron spin coherence and avoid unwanted crosstalk.

To design a selective two-qubit gate, we utilize the hyperfine interaction which couples each nuclear spin to the electron spin. As this interaction depends on the relative position of the spin to the NV, different nuclear spins can be distinguished by their precession frequencies Kolkowitz et al. (2012); Taminiau et al. (2012); Zhao et al. (2012). In the interaction picture with respect to the electron energy splitting, and neglecting non-secular terms, the Hamiltonian describing the electron and a single 13C nuclear spin is given by Kolkowitz et al. (2012); Taminiau et al. (2012); Zhao et al. (2012)

[TABLE]

where $\omega_{L}=\gamma B_{z}$ is the nuclear Larmor frequency set by the external magnetic field $B_{z}$ along the NV axis, $\gamma$ is the 13C gyromagnetic ratio, $S_{\alpha}$ and $I_{\alpha}$ are the spin-1 and spin-1/2 operators of the electron and nuclear spins respectively, and $A_{\parallel}$ and $A_{\bot}$ are the parallel and perpendicular hyperfine components.

To control the nuclear spin, we apply RF pulses of Rabi frequency $\Omega$, phase $\phi$ and frequency $\omega$. To target a specific nuclear spin, we set $\omega$ = $\omega_{1}$, where $\omega_{1}=\sqrt{(\omega_{L}-A_{\parallel})^{2}+A_{\perp}^{2}}$ is the nuclear spin precession frequency when the electron is in the $m_{s}=-1$ spin projection. In the following we assume $(\omega_{L}-\omega_{1})\gg\Omega$, such that driving of the nuclear spin is negligible while the electron is in the $m_{s}=0$ spin projection, and set $A_{\perp}=0$ for simplicity (see the Supplemental Material sup for the general case). Considering only the $m_{s}=\{0,-1\}$ subspace, with the addition of RF driving and in a rotating frame at the RF frequency, the Hamiltonian of Eq. 1 becomes Van der Sar et al. (2012); sup

[TABLE]

where $\ket{0}$ ($\ket{1}$) indicates the electron $m_{s}=0$ ($m_{s}=-1$) spin projection. In this picture, for the electron in state $\ket{0}$, the nuclear spin undergoes precession around the $\hat{z}$-axis at frequency $(\omega_{L}-\omega_{1})=A_{\parallel}$. Conversely, while the electron is in the state $\ket{1}$, the nuclear spin is driven around a rotation axis in the $\hat{x}$-$\hat{y}$ plane defined by the phase of the RF field $\phi$.

To simultaneously decouple the electron spin from the environment, we interleave the RF pulses in a sequence of the form ($\tau$ - $\pi$ - $2\tau$ - $\pi$ - $\tau$)N/2, where $\pi$ is a $\pi$-pulse on the electron spin, $2\tau$ is the interpulse delay, and $N$ is the total number of electron decoupling pulses (Fig. 2(a)) Kolkowitz et al. (2012); Taminiau et al. (2012); Zhao et al. (2012). We consider the evolution of the nuclear spin during this sequence separately for the two initial electron eigenstates: $\ket{0}$ and $\ket{1}$ Kolkowitz et al. (2012); Taminiau et al. (2012); Zhao et al. (2012). We label each successive RF pulse by integer $k=1,...,K$, where $K=N+1$ is the total number of RF pulses. If the inital electron spin state is $\ket{0}$, only the even $k$ RF pulses will be resonant and drive the nuclear spin (Fig. 2(b)). Conversely, for initial state $\ket{1}$, the odd $k$ pulses are resonant. The desired nuclear spin evolution can now be created by setting the phases $\phi_{k}$ of the RF pulses.

We construct both an unconditional rotation (single-qubit gate) and a conditional rotation (two-qubit gate). To ensure that the sequential RF rotations build up constructively, the phases of each RF pulse should be set to account for the periods of nuclear spin precession between them, which build up in integer multiples of $\phi_{\tau}=(\omega_{L}-\omega_{1})\tau$. For the case where the electron starts in the state $\ket{0}$ (even $k$), the required sequence of phases is $\phi_{\tau},3\phi_{\tau},5\phi_{\tau},\dots$, while for the case where the electron starts in the state $\ket{1}$ (odd $k$) we require the sequence $0,2\phi_{\tau},4\phi_{\tau},\dots$. The required phases are therefore given by sup

[TABLE]

where the (optional) $\pi$ phase shift for the odd $k$ sequence converts the unconditional rotation into a conditional rotation. By adding a further phase $\varphi$ to all pulses, we can also set the rotation axis of the gate. The RF pulse phases are thus summarized by $\phi_{k}=\varphi+\phi_{k}^{\prime}$.

With this choice of phases, the total evolution of the two-qubit system is given by $V=V_{z}\cdot V_{\text{CROT}}$. Here, $V_{z}$ is an unconditional rotation of the nuclear spin around $z$ sup and $V_{\text{CROT}}$ is a conditional rotation of the nuclear spin depending on the electron state, given by

[TABLE]

where $R_{\varphi}(\theta)=e^{-i\theta(\cos(\varphi)I_{x}+\sin(\varphi)I_{y})/\hbar}$. $V_{\text{CROT}}$ describes a controlled rotation of the nuclear spin with tuneable rotation angle (set by $N$, $\Omega$ and $\tau$) and rotation axis (set by $\varphi$). Setting $N\Omega\tau=\pi/2$, a maximally entangling two-qubit operation is achieved, equivalent to a controlled-not (CNOT) gate up to local rotations. Example dynamics for a nuclear spin evolving under such a sequence are shown in Figs. 2(c) and (d).

Our design has several advantages. First, the gate allows nuclear spins with small or negligible $A_{\perp}$ to be controlled, thereby increasing the number of accessible nuclear spin qubits. Second, because the targeted dynamics are achieved by setting the RF phases and amplitudes, the interpulse delay $\tau$ of the decoupling sequence can be freely optimized to protect the electron coherence. This is in contrast to the gates described in van der Sar et al. Van der Sar et al. (2012), for which $\tau$ is restricted to a specific resonance condition for each spin, making multi-qubit control challenging. Third, because our method does not rely on an average frequency shift over the two electron spin states Taminiau et al. (2014), our gates can also be used for selective control of nuclear spins coupled to spin-1/2 defects (such as the negatively-charged group-IV color centers Metsch et al. (2019); Müller et al. (2014); Sukachev et al. (2017); Evans et al. (2018); Siyushev et al. (2017); Rugar et al. (2018); Thiering and Gali (2018)), and via a contact hyperfine coupling, such as for donor spins in silicon Dehollain et al. (2016) and SiMOS quantum dots Hensen et al. (2019). Finally, because control is achieved through the RF field, a multitude of avenues for future investigation are opened up, such as parallelizing gates by frequency multiplexing and using shaped and composite pulses to mitigate dephasing and crosstalk Vandersypen and Chuang (2005); Dolde et al. (2014); Rong et al. (2015).

III TWO-QUBIT GATES: EXPERIMENT

Our experiments are performed at $3.7\,$K using a single NV center in diamond with natural abundance of carbon isotopes ($1.1\%$ 13C). Further details of the sample and experimental setup can be found in the Supplemental Material sup . As a starting point, we use the DDRF gate to identify and characterize 13C nuclear spin qubits surrounding the NV center. If the electron spin is prepared in a superposition state and the RF frequency is resonant with a nuclear spin in the environment, the entangling interaction (Eq. 4) decoheres the electron spin. Therefore, varying the RF frequency ($\omega$) performs spectroscopy of the nuclear spin environment. Fig. 3 shows that multiple dips in the electron coherence can be observed, indicating selective interactions with several individual nuclear spins. Importantly, like other RF-based approaches Zaiser et al. (2016); Pfender et al. (2017), the DDRF sequence is sensitive to nuclear spins with small or negligible $A_{\perp}$. Besides extending the number of qubits that can be controlled with a single NV center, this also enables the detection of additional spins when using the NV as a quantum sensor, which we exploit in parallel work to realize 3D imaging of large spin clusters Abobeih et al. (2019).

To verify the control offered by the DDRF two-qubit gate, we first demonstrate high fidelity ancilla-based initialization and readout by preparation and tomography of a maximally entangled electron-nuclear state. To test the gate, we select a 13C spin (spin C1, Fig. 3) with a strong parallel hyperfine component of $A_{\parallel}/2\pi=213.154(1)\,$kHz, but a weak perpendicular hyperfine component $A_{\perp}/2\pi=3.0(4)\,$kHz sup . We exploit the freedom in choosing the interpulse delay by setting $\tau$ to an integer multiple of the 13C Larmor period, $\tau_{L}=2\pi/\omega_{L}$, so that unwanted interactions between the electron spin and other 13C spins in the environment are effectively decoupled Childress et al. (2006); Abobeih et al. (2018).

The sequence to perform the state preparation and tomography experiment is shown in Fig. 4(a) Taminiau et al. (2014); Cramer et al. (2016). We first initialize the electron spin in the state $\ket{0}$ by resonant optical excitation Robledo et al. (2011). We then swap the state of the electron spin onto the 13C spin and reset the electron spin. Next, we prepare the electron in a superposition state before performing the DDRF controlled-rotation gate, ideally preparing the electron-nuclear Bell state $\ket{\psi_{\text{Bell}}}=(\ket{0+}+\ket{1-})/\sqrt{2}$, where $\ket{\pm}=(\ket{\downarrow}\pm\ket{\uparrow})/\sqrt{2}$.

To perform quantum state tomography on the two-qubit state, we first measure the electron spin along a chosen axis by appropriate basis rotations followed by $Z$-basis optical readout Robledo et al. (2011). To mitigate potential dephasing of the nuclear spin induced by the electron spin measurement, we make the electron spin measurement non-destructive by using a short, weak laser pulse and conditioning progression of the sequence on the outcome $\ket{0}$ (detection of a photon) Cramer et al. (2016); Kalb et al. (2017). Following appropriate basis rotations, we then use the electron spin to measure the nuclear spin in the $X$-basis Cramer et al. (2016). In this measurement the electron is read out in a single-shot with average fidelity $0.945(2)$ Robledo et al. (2011). In order to reconstruct the full electron-nuclear state, we perform the sequence with and without an additional electron $\pi$-pulse before the first readout sup .

The reconstructed density matrix from quantum state tomography is shown in Fig. 4(b). The prepared state $\rho$ exhibits a fidelity with the target Bell state of $\mathcal{F}_{\text{Bell}}=\bra{\psi_{\text{Bell}}}\rho\ket{\psi_{\text{Bell}}}=0.972(8)$. Based upon a simple depolarizing noise model, we estimate the two-qubit gate fidelity to be $\mathcal{F}_{\text{gate}}=0.991(9)$ sup . Additional characterization measurements in combination with numerical simulations indicate that the remaining infidelity can be mostly attributed to electron spin dephasing due to noise from the electronic hardware sup .

IV A 10-QUBIT SOLID-STATE SPIN REGISTER

We now show how the combination of our DDRF gate with previously developed gates and control techniques Taminiau et al. (2014); Cramer et al. (2016) enables high-fidelity control of a 10-qubit hybrid spin register associated to a single NV-center. Our register is composed of the electron and 14N spins of the NV-center, along with 8 13C nuclear spins (Fig. 1). Our quantum register is connected via the central electron spin. To demonstrate this, we first show that all nuclear spins can be entangled with the electron spin by following the protocol shown in Fig. 4(a). For the case of the nitrogen spin, initialization is performed by a measurement-based scheme which heralds the preparation in a particular eigenstate. Compared to previous work Kalb et al. (2016), we realize an improved initialization fidelity ($\mathcal{F}_{\text{init}}=0.997(11)$) by pre-preparing the electron in the $m_{s}=-1$ state instead of a mixed state of $m_{s}=-1$ and $+1$, and by repeating the measurement-based initialization sequence twice sup . After initialization, we work in the $m_{I}=\{0,-1\}$ subspace, and perform operations analogous to those for the 13C nuclear spins, including the two-qubit gates using the DDRF scheme. Genuine entanglement is probed by measuring the non-zero matrix elements of the target state, and confirmed by negativity of the entanglement witness $\mathcal{W}_{\text{Bell}}=\mathds{1}-2\ket{\psi_{\text{Bell}}}\bra{\psi_{\text{Bell}}}$ Gühne and Tóth (2009).

Next, we show that the register is fully connected by preparing entangled states for all possible pairs of spins. To prepare nuclear-nuclear entanglement, we implement a probabilistic measurement-based scheme Pfaff et al. (2013), as shown in Fig. 5(a). We first prepare a three-qubit GHZ state comprising the electron and two nuclear spins, $\ket{\text{GHZ}_{3}}=(\ket{0++}+\ket{1--})/\sqrt{2}$, before performing a non-destructive $X$-basis measurement on the electron spin. The measurement ideally prepares the Bell state $\ket{\Phi^{+}}=(\ket{++}+\ket{--})/\sqrt{2}$ on the targeted pair of nuclear spins. Finally, we measure the necessary expectation values in order to reconstruct the non-zero matrix elements of this state and confirm bipartite entanglement (Fig. 5(b)).

The measured Bell state fidelities, ranging from 0.63(3) to 0.97(1), are shown in Fig. 6. We attribute the variations in the measured values to differences in the two-qubit gate fidelities for each spin. In particular, the lower values measured for 13C spins C7 and C8 are due to short coherence times in combination with long two-qubit gate durations, necessitated by close spectral proximity to other spins sup . All data is measured using a single set of gate parameters, and using a single hardware configuration, rather than separately optimizing for each pair of qubits.

V GENERATION OF N-QUBIT GHZ STATES

Quantum information processing tasks such as computations and error correction will require the execution of complex algorithms comprising a large number of qubits. An important requirement for a quantum processor is thus the ability to perform operations on many of its constituents within a single algorithm. We test this capability by generating $N$-qubit GHZ type states, defined as

[TABLE]

To generate such states, we follow the sequence shown in Fig. 7(a). First, $N-1$ nuclear spins are initialized in the state $\ket{\uparrow}$. Next, we prepare the electron spin in a superposition state, and perform sequential controlled rotation gates between the electron and nuclear spins.

Characterizing the full quantum state for a system of this size is an expensive task due to the dimensionality of the associated Hilbert space. However, we can determine if the state exhibits genuine multipartite entanglement of all $N$ qubits using an entanglement witness with a reduced subset of measurement bases Gühne and Tóth (2009). For a GHZ state with system size $N$, there exist 2N operators from which the non-zero elements of the density matrix can be reconstructed by linear inversion, and from which a fidelity with the target state can be calculated. Negativity of the entanglement witness $\mathcal{W}_{\text{GHZ}}=\mathds{1}/2-\ket{\text{GHZ}_{N}}\bra{\text{GHZ}_{N}}$ heralds genuine multipartite entanglement Gühne and Tóth (2009). We measure the required expectation values of products of Pauli operators on the register via multi-qubit parity measurements. In these experiments, the readout sequence is modified slightly. Prior to the readout of the electron spin state, we rotate the nitrogen such that the desired measurement basis is mapped to the $Z$-basis. This ensures that the population in the measurement basis is protected from dephasing caused by optically reading out the electron spin Blok et al. (2014); sup .

As the number of qubits is increased, a new challenge arises: the total sequence time becomes comparable to, or even exceeds the natural dephasing times ($T_{2}^{*}$) of the nuclear spins. In order to preserve the nuclear spin coherence, we insert spin (Hahn) echo pulses (RF $\pi$-pulses) into the sequence to refocus each spin at the point of the next operation performed upon it. In the Supplemental Material sup , we derive a general solution that can be used to algorithmically construct echo sequences that avoid any overlap in gates and that minimize idle time with the electron spin in a superposition.

In Figs. 7(b,c), we show measurements for $N=5$ and $N=7$ qubits. In Fig. 7(d), we present the measured fidelities with the target GHZ states for 2 to 8 qubits, along with theoretical values as predicted by a depolarizing-noise model based on the individual two-qubit gate fidelities sup . The growing discrepancy between the measured and predicted values for larger $N$ suggests residual crosstalk between the qubits, which is not taken into account in the model. For registers comprising up to 7 spins we observe negativity of the witness $\mathcal{W}_{\text{GHZ}}$, revealing genuine $N$-qubit entanglement of up to 7 qubits with high statistical significance.

VI A LONG LIVED QUANTUM MEMORY

The nuclear spin qubits surrounding the NV center are promising candidates for quantum memories with long coherence times Maurer et al. (2012); Kalb et al. (2018). Here we investigate the coherence properties of single-qubit states and two-qubit entangled states under dynamical decoupling. We show that single qubit states can be stored for up to 63(2) seconds and two-qubit entangled states can be stored for over 10 seconds, thereby demonstrating a long-lived quantum memory within our 10-qubit register.

We first investigate the coherence of individual nuclear spin qubits under dynamical decoupling. After initializing the nuclear spin in the state $\ket{+}$, we prepare the electron in the state $\ket{1}$ (electron $T_{1}=3.6(3)\ \times\ 10^{3}$ s Abobeih et al. (2018)). This has two effects. Firstly, it allows us to perform selective RF $\pi$-pulses on the target nuclear spin. Secondly, the magnetic field gradient imposed by the electron-nuclear hyperfine interaction induces a frozen core, which suppresses flip-flop interactions between nuclear spins Khutsishvili (1962); Guichard et al. (2015) and thereby reduces the noise the spins are exposed to.

The observed spin-echo coherence times $T_{2}^{\alpha=1}$, with $\alpha$ the number of RF pulses, vary between $0.26(3)$ s to $0.77(4)$ s for the 8 13C spins. For the 14N spin we find $2.3(2)$ s, consistent with the smaller gyromagnetic ratio by factor $3.4$. The range of coherence times observed for the 13C spins is likely caused by differences in the microscopic environment of each spin. In particular, 13C spins close to the NV center are in the heart of the frozen core, and, generally tend to couple predominantly to the part of the spin environment for which the dynamics are also suppressed most strongly. Spins farther from the NV tend to couple more strongly to the spin environment outside the frozen core. This explanation is consistent with the observation that the spin with the longest $T_{2}^{\alpha=1}$ of $0.77(4)$ s is located closest (C1, $r=0.53(5)$ nm Abobeih et al. (2019)) to the vacancy lattice site, while the shortest $T_{2}^{\alpha=1}$ of $0.26(3)$ s is found for a spin at a larger distance (C8, $r=1.04(4)$ nm Abobeih et al. (2019)).

As expected, increasing the number of decoupling pulses leads to an increase in the measured coherence times. Examples of the measured decay curves are shown in Fig. 8(a) (C5) and (b) (14N), where $\alpha$ is varied from 1 to 256. For $\alpha=256$ pulses, the decay time of C5 reaches $T_{2}^{\alpha=256}=12.9(4)\,$s, while for the 14N spin, we measure $T_{2}^{\alpha=256}=63(2)\,$s. For the other 13C spins for which we measure $T_{2}^{\alpha=256}$, we find a range of values from $4(1)$ to $25(4)$ seconds sup . These coherence times are the longest reported for individual qubits in the solid state and exceed values for isolated nuclear spin qubits in isotopically purified materials Maurer et al. (2012); Muhonen et al. (2014); Yang et al. (2016). More importantly, however, in our register we realize these long coherence times while maintaining access to 10 coupled spin qubits.

We exploit the multi-qubit nature of the register to investigate the storage of entangled states of two 13C spin qubits. After preparing the Bell state $\ket{\Phi^{+}}=(\ket{++}+\ket{--})/\sqrt{2}$ following the sequence shown in Fig. 5(a), we again prepare the electron in the state $\ket{1}$. We then measure the Bell state fidelity as a function of total evolution time for $\alpha=1$ to $\alpha=256$ pulses. Note that since $\ket{\Phi^{+}}$ is an eigenstate of $ZZ$, its evolution is not affected by the coupling between the two qubits ($1.32(4)$ Hz Abobeih et al. (2019)). The measured fidelities are plotted in Fig. 8(c). For $\alpha=256$ decoupling pulses, we confirm the preservation of entanglement for $>10\,$s, as quantified by a fidelity exceeding $0.5$ with the desired Bell state.

With the capability to store multi-qubit quantum states, it becomes important to consider additional effects that may affect their coherence, such as the presence of correlated noise. As a first experimental step towards understanding such effects, we use entangled states of nuclear spins to explore spatial correlations within the noise environment. We perform experiments on two pairs of 13C spins. We prepare two Bell states for each pair, one exhibiting even $ZZ$ parity, which, written in the $Z$-basis, is given by $\ket{\Phi^{+}}=(\ket{\downarrow\downarrow}+\ket{\uparrow\uparrow})/\sqrt{2}$, and another exhibiting odd $ZZ$ parity, $\ket{\Psi^{-}}=(\ket{\downarrow\uparrow}-\ket{\uparrow\downarrow})/\sqrt{2}$. The difference in the coherence times of these two states gives an indication of the amount of correlation in the noise experienced by the two spins Kwiatkowski and Cywiński (2018). In the case of perfectly correlated noise, one would expect the state $\ket{\Phi^{+}}$ to decay at four times the single qubit decay rate (superdecoherence), while the state $\ket{\Psi^{-}}$ would form a decoherence-free subspace Lidar et al. (1998); Reiserer et al. (2016). In contrast, for completely uncorrelated noise, the coherence times for the two states would be identical.

We measure the coherence times for the two Bell states, varying the total evolution time for the case of a single spin-echo pulse ($\alpha=1$) with the electron spin prepared in the state $\ket{1}$. In Fig. 8(d), we plot the normalized coherence signal for both Bell states and for both pairs of qubits. A statistically significant difference between the decay curves for the two Bell states is found for both pairs, where the odd-parity state $\ket{\Psi^{-}}$ decays more slowly than the even-parity state $\ket{\Phi^{+}}$, indicating partly correlated noise in the system. We can relate the size of the effect to the distance between the spins in the pairs, which has been characterized in separate work Abobeih et al. (2019). This reveals that the pair with a smaller separation (C1 and C6, distance $0.96(3)\,$nm) shows more correlation than the pair with a larger separation (C5 and C2, $1.38(7)\,$nm). This observation is consistent with the idea that proximity generally promotes correlated noise, although large deviations from this rule are expected to be possible for specific cases Kwiatkowski and Cywiński (2018). Characterizing such correlated noise provides new opportunities to investigate the physics of decoherence in spin baths Kwiatkowski and Cywiński (2018), and to develop and test quantum error correction schemes that are tailored for specific correlated noise in such systems Monz et al. (2011); Layden et al. (2019).

VII CONCLUSION

In conclusion, we have developed a novel electron-nuclear two-qubit gate and applied these gates to realize a 10-qubit solid-state spin register that can store quantum states for up to one minute. The techniques developed in this work can be readily implemented for multi-qubit control in a variety of other donor and defect platforms, including spin-1/2 Metsch et al. (2019); Müller et al. (2014); Sukachev et al. (2017); Evans et al. (2018); Siyushev et al. (2017); Rugar et al. (2018); Thiering and Gali (2018) and contact hyperfine Dehollain et al. (2016); Hensen et al. (2019) systems, for which many previous gate designs are challenging to apply Kolkowitz et al. (2012); Taminiau et al. (2012); Zhao et al. (2012); Taminiau et al. (2014). Further improvements in selectivity and fidelity of the gates are anticipated to be possible by (optimal) shaping of the RF pulses Vandersypen and Chuang (2005); Dolde et al. (2014); Rong et al. (2015) and by reducing electronic noise. Additionally, the use of direct RF driving opens the possibility to perform gates in parallel on multiple qubits. Combined with already demonstrated long-range optical entanglement Bernien et al. (2013); Hensen et al. (2015); Humphreys et al. (2018), our multi-qubit register paves the way for the realization of rudimentary few-node quantum networks comprising tens of qubits. This will enable the investigation of basic error correction codes and algorithms over quantum networks Terhal (2015); Nickerson et al. (2013, 2014). Finally, looking beyond quantum information, the gate sequences developed here also enable new quantum sensing methods Abobeih et al. (2019).

ACKNOWLEDGEMENTS

We thank T. van der Sar for valuable discussions and preliminary experiments. We thank V. V. Dobrovitski and R. Hanson for valuable discussions, R. F. L. Vermeulen and R. N. Schouten for assistance with the RF electronics, and M. Eschen for assistance with the experimental setup. We acknowledge support from the Netherlands Organization for Scientific Research (NWO) through a Vidi grant.