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A new reference 31 has been added to the main-text reference list and all subsequent references have been renumbered. Shor, P. in Proc. Foundations Comput. Cory, D. et al. Experimental quantum error correction.

Article ADS CAS Google Scholar. Knill, E. Benchmarking quantum computers: the five-qubit error correcting code. Boulant, N. Experimental implementation of a concatenated quantum error-correcting code. Article ADS Google Scholar. Moussa, O. Demonstration of sufficient control for two rounds of quantum error correction in a solid state ensemble quantum information processor.

Schindler, P. Experimental repetitive quantum error correction. Science— Reed, M. Realization of three-qubit quantum error correction with superconducting circuits.

Nature— Kane, B. A silicon-based nuclear spin quantum computer. Morton, J. Solid-state quantum memory using the 31 P nuclear spin. Neumann, P. Single-shot readout of a single nuclear spin.

Koehl, W. Room temperature coherent control of defect spin qubits in silicon carbide. Nature84—87 Pla, J. A single-atom electron spin qubit in silicon. Taminiau, T. Detection and control of individual nuclear spins using a weakly coupled electron spin.

Pfaff, W. Demonstration of entanglement-by-measurement of solid-state qubits. Nature Phys. Dolde, F. Room-temperature entanglement between single defect spins in diamond. Chekhovich, E. Nuclear spin effects in semiconductor quantum dots. Nature Mater. Bernien, H. Heralded entanglement between solid-state qubits separated by three metres.

Nature86—90 Maurer, P. Room-temperature quantum bit memory exceeding one second. High-fidelity readout and control of a nuclear spin qubit in silicon. Yin, C. Optical addressing of an individual erbium ion in silicon.

Nature91—94 Kolesov, R. Optical detection of a single rare-earth ion in a crystal. Nature Commun. Le Gall, C. Optical Stark effect and dressed exciton states in a Mn-doped CdTe quantum dot.

Dutt, M. Quantum register based on individual electronic and nuclear spin qubits in diamond. Article Google Scholar.

Multipartite entanglement among single spins in diamond. Robledo, L. High-fidelity projective read-out of a solid-state spin quantum register. Universal control and error correction in multi-qubit spin registers in diamond. Dréau, A. Single-shot readout of multiple nuclear spin qubits in diamond under ambient conditions.

Machnes, S. Comparing, optimizing, and benchmarking quantum-control algorithms in a unifying programming framework. A 84 Dür, W.

Three qubits can be entangled in two inequivalent ways. A 62 Article ADS MathSciNet Google Scholar. Mermin, N. Extreme quantum entanglement in a superposition of macroscopically distinct states. Article ADS MathSciNet CAS Google Scholar.

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Waldherr, G.

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Our findings pave the way for advanced quantum algorithms and large multiqubit quantum networks based on tens of solid-state spin qubits.

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 N 14 nuclear spin and a further eight C 13 nuclear spins surrounding the NV center.

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. The phase of each rf pulse is adapted to create the desired nuclear spin evolution, accounting for periods of free precession according to Eq.

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. a Nuclear spin spectroscopy. The electron spin is then measured along a basis in the equatorial plane defined by angle φ see inset.

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 C 13 spins used in this work are labeled. The dashed gray line indicates the C 13 Larmor frequency ω 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 φ. a Experimental sequence to prepare an electron-nuclear Bell state and determine the expectation value of the two-qubit operator Z X. A series of single- and two-qubit gates are used to initialize the nuclear spin [ 16, 37 ].

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 π 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 ].

We use error function pulse envelopes with a 7. a Experimental sequence for the preparation of a nuclear-nuclear Bell state and measurement of the two-qubit operator Z Z.

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.

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. 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. Qubits C1, C7, C8, and N 14 are controlled using DDrf gates Sec.

Qubits C2, C3, C4, C5, and C6 are controlled using the methods described in Taminiau et al. 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. 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. 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. d Plot of GHZ state fidelity against the number of constituent qubits.

A value above 0. 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 ].

a Dynamical decoupling for spin C5. 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 N 14 spin. Solid lines are fits to f t , but with A as a free parameter to account for the observed decrease in the Z Z correlations at large pulse numbers, likely due to pulse errors.

With decoupling pulses, genuine two-qubit entanglement is witnessed at times up to In addition, interpolation of the fit yields For pair 1, the fitted decay times T are 0. For pair 2, the equivalent values are 0.

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author s and the published article's title, journal citation, and DOI are maintained.

Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures. Physical Review X Highlights Recent Subjects Accepted Collections Authors Referees Search Press About Editorial Team.

Featured in Physics Open Access. A Ten-Qubit Solid-State Spin Register with Quantum Memory up to One Minute C. Bradley, J.

Randall, M. Abobeih, R. Berrevoets, M. Degen, M. Bakker, M. Markham, D. Twitchen, and T. d Rabi flops on a nuclear spin qubit, taken by setting the qubit frequency which defines the exact drive frequency of EOM1 to 2. The intensity profile of each addressing beam closely resembles that of an Airy disk, with the location of the second intensity minimum approximately coinciding with the qubit spacing.

The beams are detuned from the 3 P 1 manifold of states in order to drive two-photon Raman transitions between the nuclear-spin ground states. As shown in Fig. We achieve full amplitude and phase control over the two-photon transition at each site in the array by adjusting the radio-frequency tones driving the AODs that correspond to addressing each qubit.

Importantly, the AODs are oriented such that the detuning between the beams is both finite and constant across the array of sites—this offers a separate degree of freedom for actuating the two-photon coupling, while also ensuring that atoms can be driven in parallel.

On Phoenix, we can apply operations in parallel only on atoms in a single column or row and serially apply drives to qubits in separate columns or rows. Specifically for all data presented here, the register of 21 qubits is addressed by column: this means that up to seven qubits are driven simultaneously and all 21 can be addressed with three groups of pulses.

This approach ensures that we can have full control over the operation applied to each qubit, independent of the drive applied to any other qubits.

Turning on the two-photon drive is accomplished by driving a pair of electro-optic phase modulators EOMs , one in each of the two addressing beams see Fig.

The first-order sidebands of the two EOMs drive the primary Raman process. The EOMs enable fast pulse shaping and rapid adjustments to the intermediate state detuning. Adjusting the drive frequency of either EOM effectively tunes the relative frequency of the two beams, which we can use to find the qubit transition, as seen in Fig.

By varying the length of the EOM drive, we then observe coherent Rabi oscillations between the two qubit states see Fig. We note that our choice of Rabi frequency of 1. With the ability to site-selectively drive individual qubits, we attempt to bound the spin relaxation timescale T 1 within the qubit manifold in a single experiment by performing standard state preparation on the full register of qubits and an additional π rotation on 11 of the qubits using a checkerboard pattern.

As can be seen in Fig. Solid lines are sinusoidal curves with phase offset and artificial detuning fixed to their programmed values. Each point comprises about counts.

All data points have error bars representing the standard error of the mean, but in most cases these are smaller than the point markers. We now turn our attention to experiments that are sensitive to the phase coherence of the qubit manifold by encoding a superposition state and reading out that superposition after some delay Dephasing in the qubit manifold would generally reduce the contrast of these oscillations.

In a first Ramsey experiment, we emphasize our ability to individually drive qubits within the computational array in parallel. In Fig. As a result, every individual qubit should have a different Ramsey oscillation.

Note again that all seven qubits in a single column were driven simultaneously in this experiment. The solid lines are sinusoidal fits with frequency and phase offset fixed to their programmed values, with only amplitude and vertical offset left as free parameters.

Their agreement with the data demonstrates our ability to fully control phase and frequency for every qubit. The Ramsey oscillations are expected to decay on an exponential timescale as the qubits dephase.

To measure the decay, we take similar snapshots of the Ramsey oscillations same artificial detuning, time span, and point spacing , but with an exponentially increasing time offset. To more clearly display these data, Fig.

In contrast to Fig. The solid curve is a simultaneous fit to all the measurements, highlighting the phase stability of the Ramsey oscillations. This indicates not only that changes to the relative phase for the drive beams are small on the seconds timescale, but also that the qubit frequency is not drifting significantly on the timescale of the entire experiment taken in pieces over the course of 2 days.

The ability to see coherent oscillations when averaging the signal across the entire array highlights the uniformity of the qubit frequency across the array, but the slight reduction in contrast likely indicates slight miscalibrations of the pulse area used to encode and read out the phase of each qubit.

Here we opt to explicitly adjust the phase logarithmically in t R to give familiar sinusoidal oscillations when plotted on a semilog plot. While the array-averaged data have been fit to estimate the overall coherence time of each qubit, the fits are consistent with the fitted values from individual site data.

Future work is underway to optimize the operational parameters of this system, as well as to explore methods to mitigate undesired scattering that currently limits the performance of our system.

Specifically, we anticipate that the use of larger magnetic fields and different state detunings will improve undesired scattering rates from the stark shifting beam during qubit manipulations.

Additionally, we can take advantage of pulse shaping and composite pulses to improve the uniformity of rotational area for each atom in the array, which currently limits the contrast of the array-averaged oscillations presented.

In conclusion, we have demonstrated the encoding of a qubit in the nuclear spin degree of freedom of individually trapped neutral atoms.

Furthermore, we have introduced a platform that can assemble an individually-addressable register of nuclear spin qubits and is compatible with increased computational array sizes, as well as reduced gate operation times.

Future work in both strontium and other elements with similar level structures will tackle increasing the qubit coherence time via a combination of lower noise local oscillators as well as magnetic shielding while increasing the driven Rabi rates by multiple orders of magnitude with the goal of reaching system coherence times that are 10 8 times the length of the individual gates.

The ability to individually encode, manipulate, and read out these qubits is an important first step in demonstrating this platform as a leading contender for the realization of a universal quantum computer. The process of initializing the qubit starts with producing a strontium atomic beam in an ultra-high vacuum UHV system.

The atomic beam is slowed by optical forces from a Zeeman slower and 2D magneto-optical trap MOT. The atoms are then further cooled by a second 3D MOT, overlapped with the nm MOT.

The modulated light efficiently captures hotter atoms from the blue MOT 16 , When this frequency modulation stops, the narrow linewidth of the nm transition is fully utilized to cool the atoms into co-located optical tweezer traps.

For each trap loading cycle, we subsequently run many sequences of atom rearrangement, state-preparation, gates, and measurements as described below; also see the sequence diagram in Fig.

Our choice of tweezer array size is mainly limited by the maximum trap depth which can be achieved with a finite amount of laser power. Operating at lower tweezer depth results in two deleterious effects.

First, the maximum scattering rate which can be achieved is necessarily lower in shallower traps, since larger scattering rates increase the probability that an atom will escape from a weakly confining potential during imaging. We have found that the maximum achievable scattering rate supporting a fixed survival probability between consecutive images scales approximately linearly with tweezer trap depth, meaning that shallower tweezers require longer camera exposure times to achieve acceptable SNR.

Second, loading into optical tweezers is less efficient at lower trap depth. In order for an atom to fall into the potential defined by an optical tweezer with high probability, the trap depth must exceed the equilibrium temperature of atoms during their final stage of MOT cooling.

The atoms are then optically cooled to lower motional states of the trap using the Sisyphus cooling mechanism Efficient cooling with a global beam is best achieved when operating with uniform-intensity traps, since variations in trap-induced light shifts across the array remain small compared to the linewidth of the cooling light.

We then apply a sequence of gates to the qubits, as described in the main text, before performing a projective measurement. To read out the individual nuclear spin states, we shelve one of them into a metastable clock state in the 3 P 0 manifold prior to applying a first imaging pulse.

In order to monitor and correct for atom loss, we post-select by repumping the shelved atoms to the ground states and then applying a second imaging pulse. The broad linewidth of the imaging transition not only allows for rapid photon scattering for detection but also causes detrimental heating of the trapped atoms that can lead to atom loss.

To avoid dislodging atoms from their respective tweezers, we counteract this heating by applying Sisyphus cooling simultaneously 19 , in addition to carefully setting the intensity and frequency of the imaging light.

The tweezer traps are produced at the focal plane of a custom high-NA 0. A phase mask is imprinted on the trap light by a spatial light modulator SLM and optically relayed onto the back focal plane of this objective, generating nearly arbitrary and reconfigurable two-dimensional arrays of optical tweezers.

The spatial phase imparted by the SLM is optically Fourier transformed by the microscope objective to create a grid of focused spots. We use the weighted Gerchberg—Saxton algorithm to calculate the appropriate phase mask for the SLM Rearrangement is performed using a single focused beam derived from the same Ti:Sapphire laser source used to create the static computing traps.

This beam is steered using a pair of crossed AODs, which are driven by RF waveforms generated by custom FPGA hardware. In order to rearrange atoms into a desired target pattern, an image is first taken which establishes the locations of initially occupied sites. This image is used to calculate a set of moves to fill target sites from the initially occupied sites according to the compression algorithm Moves are then performed in three steps: 1 ramp up the intensity of the rearrangement beam, 2 translate the rearrangement beam from the initial site to the target site using linear frequency chirps on the AODs, and 3 ramp down the intensity of the rearrangement beam.

Population in this manifold experiences Raman scattering due to nm trapping light, and ultimately decays into the ground state by way of the 5s5p 3 P 1 manifold In this approach, the POVM M ideal for an ideal single-qubit projective measurement in the computational basis is.

From this, one finds the intuitive result that the non-ideal POVM is given as. The measurement correction matrix is then. such that the measured vector of probabilities P exp i.

Additionally, error bars for a particular measurement are corrected by using this equation to propagate the uncertainties in m exp and q. Typically, one would run dedicated experiments to measure the probabilities p and q and construct the POVM.

In fact, our measurement of q is updated from one point to the next during an experimental parameter scan, since the undriven atoms are always read out. To calculate q for a set of experimental shots, we start by summing the bright counts from all the undriven atoms, with each count conditioned on the corresponding atom remaining trapped throughout the measurement.

This sum is then normalized by the total counts from all the undriven atoms, conditioned in the same way as before. In summary, our procedure amounts to using atoms outside the computational array to find an average value of q for each condition of a parameter sweep and then applying that value to atoms within the computational array in post-processing.

Since these corrections have been applied to the measurements shown in the main text, in Fig. Uncorrected measurements of a the nuclear-spin qubit resonance originally shown in Fig.

Solid lines represent fits that use the same underlying models as in the main text to fit these uncorrected data points. Information regarding error bars and counts-per-point are provided in the main text.

Raman scattering from 3 P 0 can be reduced by minimizing the amount of time atoms spend in the clock state before imaging. One strategy for achieving this, recently implemented in 88 Sr 40 , is to remove the ground state population via a pulse of resonant nm light immediately after the clock pulse is performed.

This strategy was used to create the data presented in Fig. Future measurements can push non-POVM-corrected fidelity higher and atom loss lower by collecting photons more efficiently, e.

Another approach involves using the narrow 3 P 1 intercombination line to sequentially image atoms in each qubit level eliminating the need for shelving in the clock state altogether , although the smaller scattering rate in this case would lower the readout speed Finally, this error is only a limitation when single-shot measurements are required e.

For all fitted data presented, we perform a weighted least-squares optimization with the indicated functional form.

Any fixed parameters for the fits are indicated in the text and are set with independent measurements e. All fits reported in the main text are performed after applying the POVM correction to the data based on the shelving probability of the atoms outside the computational array including the propagation of errors through the POVM correction , as discussed above.

The reported error bars on fit parameters are the standard error from the weighted least squares optimization fitting routine. As we are driving each qubit individually, it is important to consider the cross-talk of the drive lasers from one site driving rotations of its neighboring atoms.

While further characterization of this cross-talk with dedicated experiments will be required for demonstrating high-fidelity operations, we can use the data presented in Fig. Specifically, we compare the population of the qubits in the computational array that are undriven and thus neighboring a driven qubit, depicted as red boxes in Fig.

The color of each box represents indicates qubits that are driven blue or undriven red within the computational array for the data presented in Fig.

Additionally, the shading of the sites around the perimeter of the computational array indicates whether that site neighbored a driven qubit or an undriven qubit. All qubits more than one site away from the computational array are categorized as background atoms and are assumed to be undriven by the gate beams.

We do not include signal from trap sites immediately neighboring the array in this analysis because the statistics are quite low those sites will be rearranged into the computational array first.

This 1. If the beam were a perfect gaussian, this would be consistent with a larger-than-expected waist of ~2. However, we know the beams are apertured into the objective, which leads to the actual profile including rings around the central spot. Combining this effect with imperfections in the beam shape due to, e.

Future work will aim to fully characterize the cross-talk between neighboring qubits, as well as to modify the beam shaping optics to minimize the cross-talk between neighboring qubits, a requirement for realizing high-fidelity local qubit manipulations.

While the rearrangement process should not couple the nuclear spin states, the process can lead to heating of the atoms, which would degrade the shelving pulse fidelity, resulting in the increased signal Further studies will aim to determine how much of this population imbalance is due to cross-talk of the manipulation beams compared to different state preparation and shelving performance in the computational array.

The processed data used to create all figures in this article are included in Supplementary Data files. The raw data i. No custom code was used that is central to the conclusions of the manuscript. The data analysis routines can be made available upon reasonable request to the corresponding authors.

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Synthetic three-dimensional atomic structures assembled atom by atom.

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