Qubit state measurements are an essential part of any quantum computer, constituting the readout. Accurate measurements are also an integral component of one-way quantum computation and of error correction, which is needed for fault-tolerant quantum computation 1 . Here, we present a state measurement for neutral-atom qubits based on coherent spatial splitting of the atoms' wavefunctions. It is reminiscent of the Stern-Gerlach experiment 2 , but carried out in light traps. For around 160 qubits in a three-dimensional array, we achieve a measurement fidelity of 0.9994, which is roughly 20 times lower error than in previous measurements of neutral-atom arrays 3,4 . It also greatly exceeds the measurement fidelity of other arrays with more than four qubits, including those with ion and superconducting qubits 5,6 . Our measurement fidelity is essentially independent of the number of qubits measured, and since the measurement causes no loss, we can reuse the atoms. We also demonstrate that we can replace atoms lost to background gas collisions during the experiment 7 .

Neutral atoms are promising qubit candidates, because they are identical, have long coherence times, and can be readily scaled to large arrays with spacings of several micrometres so that they can be individually addressed 8 . A simple way to measure the states of atom qubits is to resonantly push (clear) atoms in one qubit state out of the trap with light and detect those that remain. The detection fidelity for the remaining atoms can exceed 0.9997 (ref. 9 ), but the scheme suffers from two considerable drawbacks. First, qubit loss during the computation is indistinguishable from the atom being in the cleared state, making the loss rate the de facto limit on the fidelity of the state measurement. Second, about half the atoms are cleared during measurement, necessitating reloading of atoms, thus making it hard to adapt this method to error correction. A few alternative approaches have been used to achieve lossless state detection. Atoms in each state can in turn be made to fluoresce on a cycling transition 3,4,10,11 . However, since the atoms cannot be cooled during the measurement, this method is a balance between detecting enough photons to identify the atom's state and those photons heating the atom out of the trap. The best results on small, 1D alkali-atom ensembles have had 0.987 fidelity and 2% heating loss 3 or 0.98 fidelity and 1% heating loss 4 . One-dimensional arrays of alkaline earth atoms have achieved 0.98 fidelity, limited by off-resonant scattering of trap light, with 0.5% heating loss 12 . Photon collection efficiency improves when a single atom is trapped in a high-finesse cavity 13,14 , where 0.9992 fidelity has been achieved 14 , but this enhancement is hard to scale to more atoms. In a quantum gas microscope, a large magnetic field gradient was applied to coherently separate atoms in two different internal states, reminiscent of the Stern-Gerlach experiment, after which the two states were trapped and detected at adjacent lattice sites 15 . A fidelity of 0.98 was reached, limited by lattice phase fluctuations. Our technique is also a Stern-Gerlachtype approach, but without magnetic field gradients.

Our state detection scheme is conceptually illustrated in Fig. 1a. We start with atoms in a 3D lattice with a large spacing in all directions (lattice XYZ) in an unknown superposition of two internal states. To detect the state of each atom, we adiabatically transform the state-independent X lattice into two state-dependent potentials that move in opposite directions 7,16,17 . The two state components of each atom follow their respective potentials, spatially splitting the wavefunction in two. Next, we replace X with X S , a lattice with an order of magnitude shorter lattice spacing. The two parts of the wavefunction are each localized to within a couple of X S lattice sites. We then image the atoms with cooling light 9 , which projects the wavefunction of each atom onto a single site in X S YZ, and measures its location. Mapping the internal state to spatial position in this way avoids the detection-heating trade-off that has limited previous measurements.

The X, Y and Z lattices in our experiment are created by pairs of 839 nm laser beams, crossing at 10° angles as shown in Fig. 1b, linearly polarized perpendicular to their plane of propagation. The lattice beams are slightly mutually shifted in frequency (by tens of megahertz), and together form an approximately cubic 3D lattice with 4.8 μm spacing and 190 μK depth. The vibrational frequency of an atom trapped near the bottom of a lattice site is 15 kHz (ref. 9 ). Caesium atoms are loaded into the lattice from a magneto-optic trap. We can either use the approximately 40% random occupancy we start with, or sort atoms to fully fill a sublattice 7 . We detect atoms in five planes by imaging phase-scrambled (see Supplementary Information) polarization gradient cooling light one Z plane at a time, which takes 830 ms in total (see Supplementary Information). Projection sideband cooling 7,18 leaves 89% of the atoms in their 3D vibrational ground states and more than 99.9% of the atoms in the |F = 4,m F = -4〉 hyperfine ground state, where F and m F are the hyperfine and magnetic quantum numbers, respectively. We can transfer atoms into any other magnetic sublevel with a series of adiabatic fast passage (AFP) microwave pulses 19 , and we can create superpositions of sublevels using Blackman pulses 19 .

The 3D lattice initially traps all hyperfine sublevels nearly identically (see Fig. 1a (i)). We make the X lattice sublevel-dependent by rotating the polarization of one of its beams, which we accomplish with two Pockels cells and a λ/4 plate 7 . Atoms in different hyperfine levels with the same sign of m F , such as |F = 4,m F = -4〉 and |F = 3,m F = -3〉, move in opposite directions. A nearly π/2 polarization rotation separates the two states by almost half the X lattice spacing (see Fig. 1a (ii)). Ramping the voltage on the Pockels cell within 300 μs keeps the motion adiabatic. At the end of the ramp we suddenly turn on a retro-reflected approximately 150 μm waist laser beam that forms a lattice, X S , with 0.42 μm spacing (see Fig. 1b and Fig. 1a (iii)). To avoid mutual interference, the wavelength of the X S

light is slightly different (by tens of megahertz) from the other lattice beams. The spatial phase of X S relative to X is not controlled. After the initial turn on, the X S power is increased within 78 μs so that the vibrational frequency in the ground state is raised from 43 kHz to 98 kHz, a sequence empirically adjusted to avoid site hopping in X S . We then turn off X and turn on the polarization gradient cooling light to cool and image the atoms. The wavefunction of the atom is thus projected into a single site of the X S YZ lattice (see Fig. 1a (iv)).

In Fig. 2a we show the results of measurements starting with all atoms in |F = 4, m F = -4〉 after optical pumping. For Fig. 2b, the atoms start in |F = 3,m F = -3〉 after an AFP pulse and an F = 4 stateclearing laser pulse to ensure clean state preparation. The single plane images are typical and the grid vertices demark the initial atom locations. The detected atoms are clearly shifted to the left for |F = 4,m F = -4〉 and to the right for |F = 3,m F = -3〉. The locations of the atoms are fitted to a floating x centre (see Supplementary Information), and the distributions of these positions are shown in the histograms, which quantify the qualitative shifts visible in the images. For Fig. 2c, the atoms start in a superposition of the two internal states. We set the line that separates atoms in the two states at -0.2 μm, equidistant from the two peaks, and colour the histograms and the occupancy maps accordingly.

The Gaussian root-mean-square widths of the displacement distributions in Fig. 2 are 203 nm, which is about half an X S lattice spacing (210 nm). This width mostly results from the random phase relationship between X and X S . We infer that the asymmetry of the displacement centres results from slight imperfections in the X polarization (see Supplementary Information). Across the entire 9 × 9 × 5 array, ×

x (µm)

x (µm)

x (µm)

x (µm)

x (µm)

x (µm) which is consistent with this being the dominant source of error.

If the X polarization were to be improved, we could move the atoms faster and thus reduce the scattering and the associated state detection error. We could also employ a 'throw and catch' method. In this scheme, we would suddenly change an X beam polarization, let the atoms be accelerated by the shifted lattices, and then shut off X to avoid spontaneous emission. When the atoms have travelled sufficiently far, X S would be turned on with cooling light to catch the atoms. We estimate that this could reduce our error from spontaneous emission by a factor of five.

After the state measurement, we adiabatically turn X back on with its standard polarization and then turn off X S with essentially the reverse sequence that we used to turn it on (see Fig. 1a (v) and Supplementary Information). One quarter oscillation cycle later, when the atoms are centred in X (see Fig. 1a (vi)), we recool them and take another set of pictures to determine the final occupancy. The final occupancy map facilitates the occupancy determination in X S YZ, bringing its error below 10 -4 (see Supplementary Information). Between the initial occupancy determination and the state detection, 2% of the atoms are lost, which is consistent with independent measurements of the loss rate due to collisions with the background gas. We infer that the state detection measurement itself causes no loss. We also see no evidence of loss or site hopping due to the process of transferring the atoms back to X.

With essentially lossless detection, we can re-initialize our qubits after state measurement, a procedure we demonstrate in Fig. 3. We first sort to get a perfectly filled 5 × 5 × 2 pattern (see Fig. 3a) 7 . We then prepare an equal superposition of the two stretched states, execute state detection in X S YZ (see Fig. 3b), and reload into XYZ (see Fig. 3c). Atom loss to a background gas collision can occur in either stage; in this implementation, one atom was lost between state detection and reloading. We re-sort, filling the vacancy with an atom from the reservoir region (within 5 × 5 × 5 but outside 5 × 5 × 2) (see Fig. 3d). The ability to fix qubit loss errors will ultimately be an important element of quantum error correction.

Because of their insensitivity to noise, it is preferable to quantum compute in the clock states, |F = 4,m F = 0〉 and |F = 3,m F = -0〉 (ref. 20 ). However, these states do not move in the state-dependent lattice. To generalize our detection method, we use two AFP pulses to transfer atoms from superpositions of clock states to superpositions of |F = 4,m F = -2〉 and |F = 3,m F = -2〉, and detect the state from there (see Supplementary Information). We generate a range of clock state superpositions by transferring all the atoms to |F = 3,m F = 0〉 and executing a π/2-π-π-π/2 sequence on the clock transition, scanning the phase of the final pulse. We do this in a one-quarterdepth XYZ lattice, which is preferred for minimizing decoherence while maintaining gate fidelity. Before state detection, we adiabatically raise the lattice to full power. The result is shown in Fig. 4a, with 480 μs between the π/2 pulses. The fraction of atoms found in the wrong state at phase π is

, which is to say we found no errors in 2,200 atom detections, consistent with as good a fidelity as we measured starting from the stretched states. The error at phase 0 is larger, × . We attribute the poorer performance when measuring from the |F = 4,m F = 0〉 state to Rabi frequency differences between the various AFP microwave transitions (see Supplementary Information). Adding the ability to dynamically change the microwave polarization should avoid this issue completely.

To further make use of our state detection method, we measure our qubit coherence time by adding additional intermediate π pulses at 20 ms intervals (see Supplementary Information) and performing the spin-echo measurement described above with longer evolution times. From the observed fringe contrast as a function of time, shown in Fig. 4b, we find that the coherence times ( ′ T 2 and T 1 ) for atoms in a 5 × 5 × 5 volume is 12.6 s. The improvement over our previously reported coherence time of 7.4 s (refs. 719 ) results from At each point, the populations of both states are determined in the same state measurement. The data points at phase 2π are the same as those at phase 0. The solid curves are fits using a sine function, and the fit amplitude gives the fringe contrast. b, Magnified regions of a. At phases π and 0, perfect state preparation and detection would lead to all atoms being in the same state. At π phase, there were no errors among the 2,200 atoms we measured. We attribute the 2 × 10 -3 error at 0 phase to the lower quality of some AFP pulses (see Supplementary Information).

Error bars in a and b represent 1 s.d. and are due to counting statistics. The error bars for the data points at phase π are smaller than the size of the symbols. c, Semi-logarithmic plot of fringe contrasts for spin-echo sequences, for a range of evolution times. The fitted exponential time constant, which gives the coherence time of the qubits, is 12.6 s. The one standard deviation error bars from the fringe contrast fits are smaller than the size of the symbols.

NAture PHySIcS better cooling and a farther-detuned optical lattice, both of which minimize spontaneous emission from the lattice. The concept of our state measurement method, where internal states are mapped to atom positions, can be generalized to atoms in reconfigurable dipole-trap arrays [21][22][23][24] . Each dipole trap can be formed by overlapping two traps with opposite circular polarization. For state detection, the two traps can be spatially separated. If needed, dipole heating could be minimized by making the two traps linearly polarized before the atoms are imaged.

While it can now be used for final state readout, we ultimately plan to adapt this measurement technique to quantum error correction, where state detection of only a subset of the atoms is required [25][26][27] . The atoms to be detected would be selectively transferred to larger m F states. During the motion, the clock state qubits will remain weakly trapped at their original position. Transferring them to X S will probably require matching the nodes of X to nodes of X S . Speed concerns will require fluorescent detection to be replaced with phase-contrast imaging 28 , which would act as a form of holography and allow all planes to be imaged at once. Preventing the undetected qubits from interacting with detection light will require the qubits to be shifted out of resonance or transferred to dark states, or the use of a second atomic species for measurement 29 . Our state detection error approaches the commonly accepted threshold of 10 -4 for fault-tolerant quantum computation 30 , and it already comfortably surpasses the thresholds of about 10 -3 used for some surface codes 31 .