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Quantum state teleportation is commonly used in designs for large-scale fault-tolerant quantum computers. Using Quantinuum's H2 trapped-ion quantum processor, we implement the first demonstration of a fault-tolerant state teleportation circuit for a quantum error correction code - in particular, the planar topological [[7,1,3]] color code, or Steane code. The circuits use up to 30 trapped ions at the physical layer qubits and employ real-time quantum error correction - decoding mid-circuit measurement of syndromes and implementing corrections during the protocol. We conduct experiments on several variations of logical teleportation circuits using both transversal gates and lattice surgery protocols. Among the many measurements we report on, we measure the logical process fidelity of the transversal teleportation circuit to be 0.975(2) and the logical process fidelity of the lattice surgery teleportation circuit to be 0.851(9). Additionally, we run a teleportation circuit that is equivalent to Knill-style quantum error correction and measure the process fidelity to be 0.989(2).

The promise of quantum computers hinges on the ability to scale to large system sizes, e.g., to run quantum computations consisting of more than 100 million operations fault-tolerantly. This in turn requires suppressing errors to levels inversely proportional to the size of the computation. As a step towards this ambitious goal, we present experiments on a trapped-ion QCCD processor where, through the use of fault-tolerant encoding and error correction, we are able to suppress logical error rates to levels below the physical error rates. In particular, we entangled logical qubits encoded in the [[7,1,3]] code with error rates 9.8 times to 500 times lower than at the physical level, and entangled logical qubits encoded in a [[12,2,4]] code with error rates 4.7 times to 800 times lower than at the physical level, depending on the judicious use of post-selection. Moreover, we demonstrate repeated error correction with the [[12,2,4]] code, with logical error rates below physical circuit baselines corresponding to repeated CNOTs, and show evidence that the error rate per error correction cycle, which consists of over 100 physical CNOTs, approaches the error rate of two physical CNOTs. These results signify an important transition from noisy intermediate scale quantum computing to reliable quantum computing, and demonstrate advanced capabilities toward large-scale fault-tolerant quantum computing., Comment: 13 pages, 8 figures

We describe and benchmark a new quantum charge-coupled device (QCCD) trapped-ion quantum computer based on a linear trap with periodic boundary conditions, which resembles a race track. The new system successfully incorporates several technologies crucial to future scalability, including electrode broadcasting, multi-layer RF routing, and magneto-optical trap (MOT) loading, while maintaining, and in some cases exceeding, the gate fidelities of previous QCCD systems. The system is initially operated with 32 qubits, but future upgrades will allow for more. We benchmark the performance of primitive operations, including an average state preparation and measurement error of 1.6(1)$\times 10^{-3}$, an average single-qubit gate infidelity of $2.5(3)\times 10^{-5}$, and an average two-qubit gate infidelity of $1.84(5)\times 10^{-3}$. The system-level performance of the quantum processor is assessed with mirror benchmarking, linear cross-entropy benchmarking, a quantum volume measurement of $\mathrm{QV}=2^{16}$, and the creation of 32-qubit entanglement in a GHZ state. We also tested application benchmarks including Hamiltonian simulation, QAOA, error correction on a repetition code, and dynamics simulations using qubit reuse. We also discuss future upgrades to the new system aimed at adding more qubits and capabilities., Comment: 24 pages, 24 figures. Made some minor edits and added several more authors

We compare two different implementations of fault-tolerant entangling gates on logical qubits. In one instance, a twelve-qubit trapped-ion quantum computer is used to implement a non-transversal logical CNOT gate between two five qubit codes. The operation is evaluated with varying degrees of fault tolerance, which are provided by including quantum error correction circuit primitives known as flagging and pieceable fault tolerance. In the second instance, a twenty-qubit trapped-ion quantum computer is used to implement a transversal logical CNOT gate on two [[7,1,3]] color codes. The two codes were implemented on different but similar devices, and in both instances, all of the quantum error correction primitives, including the determination of corrections via decoding, are implemented during runtime using a classical compute environment that is tightly integrated with the quantum processor. For different combinations of the primitives, logical state fidelity measurements are made after applying the gate to different input states, providing bounds on the process fidelity. We find the highest fidelity operations with the color code, with the fault-tolerant SPAM operation achieving fidelities of 0.99939(15) and 0.99959(13) when preparing eigenstates of the logical X and Z operators, which is higher than the average physical qubit SPAM fidelities of 0.9968(2) and 0.9970(1) for the physical X and Z bases, respectively. When combined with a logical transversal CNOT gate, we find the color code to perform the sequence--state preparation, CNOT, measure out--with an average fidelity bounded by [0.9957,0.9963]. The logical fidelity bounds are higher than the analogous physical-level fidelity bounds, which we find to be [0.9850,0.9903], reflecting multiple physical noise sources such as SPAM errors for two qubits, several single-qubit gates, a two-qubit gate and some amount of memory error.

Mid-circuit measurement and reset are crucial primitives in quantum computation, but such operations require strong interactions with selected qubits while maintaining isolation of neighboring qubits, which is a significant challenge in many systems. For trapped ion systems, measurement is performed with laser-induced fluorescence. Stray light from the detection beam and fluorescence from the measured ions can be significant sources of decoherence for unmeasured qubits. We present a technique using ion micromotion to reduce these sources of decoherence by over an order of magnitude. We benchmark the performance with a new method, based on randomized benchmarking, to estimate the magnitude of crosstalk errors on nearby qubits. Using the Honeywell System Model H0, we demonstrate measurement and reset on select qubits with low crosstalk errors on neighboring qubits., Comment: 13 pages, 7 figures

Correcting errors in real time is essential for reliable large-scale quantum computations. Realizing this high-level function requires a system capable of several low-level primitives, including single-qubit and two-qubit operations, mid-circuit measurements of subsets of qubits, real-time processing of measurement outcomes, and the ability to condition subsequent gate operations on those measurements. In this work, we use a ten qubit QCCD trapped-ion quantum computer to encode a single logical qubit using the $[[7,1,3]]$ color code, first proposed by Steane~\cite{steane1996error}. The logical qubit is initialized into the eigenstates of three mutually unbiased bases using an encoding circuit, and we measure an average logical SPAM error of $1.7(6) \times 10^{-3}$, compared to the average physical SPAM error $2.4(8) \times 10^{-3}$ of our qubits. We then perform multiple syndrome measurements on the encoded qubit, using a real-time decoder to determine any necessary corrections that are done either as software updates to the Pauli frame or as physically applied gates. Moreover, these procedures are done repeatedly while maintaining coherence, demonstrating a dynamically protected logical qubit memory. Additionally, we demonstrate non-Clifford qubit operations by encoding a logical magic state with an error rate below the threshold required for magic state distillation. Finally, we present system-level simulations that allow us to identify key hardware upgrades that may enable the system to reach the pseudo-threshold.

The trapped-ion QCCD (quantum charge-coupled device) architecture proposal lays out a blueprint for a universal quantum computer. The design begins with electrodes patterned on a two-dimensional surface configured to trap multiple arrays of ions (or ion crystals). Communication within the ion crystal network allows for the machine to be scaled while keeping the number of ions in each crystal to a small number, thereby preserving the low error rates demonstrated in trapped-ion experiments. By proposing to communicate quantum information by moving the ions through space to interact with other distant ions, the architecture creates a quantum computer endowed with full-connectivity. However, engineering this fully-connected computer introduces a host of difficulties that have precluded the architecture from being fully realized in the twenty years since its proposal. Using a Honeywell cryogenic surface trap, we report on the integration of all necessary ingredients of the QCCD architecture into a programmable trapped-ion quantum computer. Using four and six qubit circuits, the system level performance of the processor is quantified by the fidelity of a teleported CNOT gate utilizing mid-circuit measurement and a quantum volume measurement of $2^6=64$. By demonstrating that the low error rates achievable in small ion crystals can be successfully integrated with a scalable trap design, parallel optical delivery, and fast ion transport, the QCCD architecture is shown to be a viable path toward large quantum computers. Atomic ions provide perfectly identical, high-fidelity qubits. Our work shows that the QCCD architecture built around these qubits will provide high performance quantum computers, likely enabling important near-term demonstrations such as quantum error correction and quantum advantage.

To date, the highest fidelity quantum logic gates between two qubits have been achieved with variations on the geometric-phase gate in trapped ions, with the two leading variants being the Molmer-Sorensen gate and the light-shift (LS) gate. Both of these approaches have their respective advantages and challenges. For example, the latter is technically simpler and is natively insensitive to optical phases, but it has not been made to work directly on a clock-state qubit. We present a new technique for implementing the LS gate that combines the best features of these two approaches: By using a small ($\sim {\rm MHz}$) detuning from a narrow (dipole-forbidden) optical transition, we are able to operate an LS gate directly on hyperfine clock states, achieving gate fidelities of $99.74(4)\%$ using modest laser power at visible wavelengths. Current gate infidelities appear to be dominated by technical noise, and theoretical modeling suggests a path towards gate fidelity above $99.99\%$., Comment: 4+ pages, 3 figures, and supplemental material

We present a new and simplified two-qubit randomized benchmarking procedure that operates only in the symmetric subspace of a pair of qubits and is well suited for benchmarking trapped-ion systems. By performing benchmarking only in the symmetric subspace, we drastically reduce the experimental complexity, number of gates required, and run time. The protocol is demonstrated on trapped ions using collective single-qubit rotations and the Molmer-Sorenson (MS) interaction to estimate an entangling gate error of $2(1) \times 10^{-3}$. We analyze the expected errors in the MS gate and find that population remains mostly in the symmetric subspace. The errors that mix symmetric and anti-symmetric subspaces appear as leakage and we characterize them by combining our protocol with recently proposed leakage benchmarking. Generalizations and limitations of the protocol are also discussed., Comment: 16 pages, 4 figures

We report high-fidelity laser-beam-induced quantum logic gates on magnetic-field-insensitive qubits comprised of hyperfine states in $^{9}$Be$^+$ ions with a memory coherence time of more than 1 s. We demonstrate single-qubit gates with error per gate of $3.8(1)\times 10^{-5}$. By creating a Bell state with a deterministic two-qubit gate, we deduce a gate error of $8(4)\times10^{-4}$. We characterize the errors in our implementation and discuss methods to further reduce imperfections towards values that are compatible with fault-tolerant processing at realistic overhead., Comment: 12 pages, 7 figures

Entangled states are a crucial resource for quantum-based technologies such as quantum computers and quantum communication systems (1,2). Exploring new methods for entanglement generation is important for diversifying and eventually improving current approaches. Here, we create entanglement in atomic ions by applying laser fields to constrain the evolution to a restricted number of states, in an approach that has become known as "quantum Zeno dynamics" (3-5). With two trapped $^9\rm{Be}^+$ ions, we obtain Bell state fidelities up to $0.990^{+2}_{-5}$, with three ions, a W-state (6) fidelity of $0.910^{+4}_{-7}$ is obtained. Compared to other methods of producing entanglement in trapped ions, this procedure is relatively insensitive to certain imperfections such as fluctuations in laser intensity, laser frequency, and ion-motion frequencies., Comment: 44 pages, 10 figures

Precision control over hybrid physical systems at the quantum level is important for the realization of many quantum-based technologies. In the field of quantum information processing (QIP) and quantum networking, various proposals discuss the possibility of hybrid architectures where specific tasks are delegated to the most suitable subsystem. For example, in quantum networks, it may be advantageous to transfer information from a subsystem that has good memory properties to another subsystem that is more efficient at transporting information between nodes in the network. For trapped-ions, a hybrid system formed of different species introduces extra degrees of freedom that can be exploited to expand and refine the control of the system. Ions of different elements have previously been used in QIP experiments for sympathetic cooling, creation of entanglement through dissipation, and quantum non-demolition (QND) measurement of one species with another. Here, we demonstrate an entangling quantum gate between ions of different elements which can serve as an important building block of QIP, quantum networking, precision spectroscopy, metrology, and quantum simulation. A geometric phase gate between a $^9$Be$^+$ ion and a $^{25}$Mg$^+$ ion is realized through an effective spin-spin interaction generated by state-dependent forces induced with laser beams. Combined with single-qubit gates and same-species entangling gates, this mixed-element entangling gate provides a complete set of gates over such a hybrid system for universal QIP. Using a sequence of such gates, we demonstrate a Controlled-NOT (CNOT) gate and a SWAP gate. We further demonstrate the robustness of these gates against thermal excitation and show improved detection in quantum logic spectroscopy (QLS). We also observe a strong violation of a CHSH-type Bell inequality on entangled states composed of different ion species., Comment: 7 pages, 4 figures

We investigate the dynamics of mixed-species ion crystals during transport between spatially distinct locations in a linear Paul trap in the diabatic regime. In a general mixed-species crystal, all degrees of freedom along the direction of transport are excited by an accelerating well, so unlike the case of same-species ions, where only the center-of-mass-mode is excited, several degrees of freedom have to be simultaneously controlled by the transport protocol. We design protocols that lead to low final excitations in the diabatic regime using invariant-based inverse-engineering for two different-species ions and also show how to extend this approach to longer mixed-species ion strings. Fast transport of mixed-species ion strings can significantly reduce the time overhead in certain architectures for scalable quantum information processing with trapped ions., Comment: 6 pages, 4 figures

Entangled states are a key resource in fundamental quantum physics, quantum cryp-tography, and quantum computation [1].To date, controlled unitary interactions applied to a quantum system, so-called "quantum gates", have been the most widely used method to deterministically create entanglement [2]. These processes require high-fidelity state preparation as well as minimizing the decoherence that inevitably arises from coupling between the system and the environment and imperfect control of the system parameters. Here, on the contrary, we combine unitary processes with engineered dissipation to deterministically produce and stabilize an approximate Bell state of two trapped-ion qubits independent of their initial state. While previous works along this line involved the application of sequences of multiple time-dependent gates [3] or generated entanglement of atomic ensembles dissipatively but relied on a measurement record for steady-state entanglement [4], we implement the process in a continuous time-independent fashion, analogous to optical pumping of atomic states. By continuously driving the system towards steady-state, the entanglement is stabilized even in the presence of experimental noise and decoherence. Our demonstration of an entangled steady state of two qubits represents a step towards dissipative state engineering, dissipative quantum computation, and dissipative phase transitions [5-7]. Following this approach, engineered coupling to the environment may be applied to a broad range of experimental systems to achieve desired quantum dynamics or steady states. Indeed, concurrently with this work, an entangled steady state of two superconducting qubits was demonstrated using dissipation [8]., Comment: 25 pages, 5 figures

We demonstrate a trapped-ion entangling-gate scheme proposed by Bermudez et al. [Phys. Rev. A 85, 040302 (2012)]. Simultaneous excitation of a strong carrier and a single-sideband transition enables deterministic creation of entangled states. The method works for magnetic field-insensitive states, is robust against thermal excitations, includes dynamical decoupling from qubit dephasing errors, and provides simplifications in experimental implementation compared to some other entangling gates with trapped ions. We achieve a Bell state fidelity of 0.974(4) and identify the main sources of error., Comment: 5 pages, 4 figures, 1 table

We use electromagnetically induced transparency (EIT) laser cooling to cool motional modes of a linear ion chain. As a demonstration, we apply EIT cooling on $^{24}Mg^+$ ions to cool the axial modes of a $^9Be^+$ - $^{24}Mg^+$ ion pair and a $^9Be^+$ - $^{24}Mg^+$ - $^{24}Mg^+$ - $^9Be^+$ ion chain, thereby sympathetically cooling the $^{9}$Be$^{+}$ ions. Compared to previous implementations of conventional Raman sideband cooling, we achieve approximately an order-of-magnitude reduction in the duration required to cool the modes to near the ground state and significant reduction in required laser intensity., Comment: 5 pages, 4 figures

We investigate the motional dynamics of single and multiple ions during transport between and separation into spatially distinct locations in a multi-zone linear Paul trap. A single 9Be+ ion in a 2 MHz harmonic well located in one zone was laser-cooled to near its ground state of motion and transported 370 micrometers by moving the well to another zone. This was accomplished in 8 microseconds, corresponding to 16 periods of oscillation. Starting from a state with n=0.1 quanta, during transport the ion was excited to a displaced coherent state with n=1.6 quanta but on completion was returned close to its motional ground state with n=0.2. Similar results were achieved for the transport of two ions. We also separated chains of up to 9 ions from one potential well to two distinct potential wells. With two ions this was accomplished in 55 microseconds, with final excitations of about 2 quanta for each ion. Fast coherent transport and separation can significantly reduce the time overhead in certain architectures for scalable quantum information processing with trapped ions., Comment: 5 pages, 5 figures

The ideal (i.e. noninteracting), homogeneous Fermi gas, with its characteristic sharp Fermi surface in the momentum distribution, is a fundamental concept relevant to the behavior of many systems. With trapped Fermi gases of ultracold atoms, one can realize and probe a nearly ideal Fermi gas, however these systems have a nonuniform density due to the confining potential. We show that the effect of the density variation, which typically washes out any semblance of a Fermi surface step in the momentum distribution, can be mitigated by selectively probing atoms near the center of a trapped gas. With this approach, we have directly measured a Fermi surface in momentum space for a nearly ideal gas, where the average density and temperature of the probed portion of the gas can be determined from the location and sharpness of the Fermi surface.

As experimental platforms for quantum information processing continue to mature, characterization of the quality of unitary gates that can be applied to their quantum bits (qubits) becomes essential. Eventually, the quality must be sufficiently high to support arbitrarily long quantum computations. Randomized benchmarking already provides a platform-independent method for assessing the quality of one-qubit rotations. Here we describe an extension of this method to multi-qubit gates. We provide a platform-independent protocol for evaluating the performance of experimental Clifford unitaries, which form the basis of fault-tolerant quantum computing. We implemented the benchmarking protocol with trapped-ion two-qubit phase gates and one-qubit gates and found an error per random two-qubit Clifford unitary of $0.162 \pm 0.008$, thus setting the first benchmark for such unitaries. By implementing a second set of sequences with an extra two-qubit phase gate at each step, we extracted an error per phase gate of $0.069 \pm 0.017$. We conducted these experiments with movable, sympathetically cooled ions in a multi-zone Paul trap - a system that can in principle be scaled to larger numbers of ions., Comment: Corrected description of parallel single-qubit benchmark experiment. Results unchanged

Wave-vector resolved radio frequency (rf) spectroscopy data for an ultracold trapped Fermi gas are reported for several couplings at Tc, and extensively analyzed in terms of a pairing-fluctuation theory. We map the evolution of a strongly interacting Fermi gas from the pseudogap phase into a fully gapped molecular Bose gas as a function of the interaction strength, which is marked by a rapid disappearance of a remnant Fermi surface in the single-particle dispersion. We also show that our theory of a pseudogap phase is consistent with a recent experimental observation as well as with Quantum Monte Carlo data of thermodynamic quantities of a unitary Fermi gas above Tc., Comment: 9 pages, 9 figures. Substantially revised version (to appear in Phys. Rev. Lett.)

Ultracold atomic Fermi gases present an opportunity to study strongly interacting Fermi systems in a controlled and uncomplicated setting. The ability to tune attractive interactions has led to the discovery of superfluidity in these systems with an extremely high transition temperature, near T/T_F = 0.2. This superfluidity is the electrically neutral analog of superconductivity; however, superfluidity in atomic Fermi gases occurs in the limit of strong interactions and defies a conventional BCS description. For these strong interactions, it is predicted that the onset of pairing and superfluidity can occur at different temperatures. This gives rise to a pseudogap region where, for a range of temperatures, the system retains some of the characteristics of the superfluid phase, such as a BCS-like dispersion and a partially gapped density of states, but does not exhibit superfluidity. By making two independent measurements: the direct observation of pair condensation in momentum space and a measurement of the single-particle spectral function using an analog to photoemission spectroscopy, we directly probe the pseudogap phase. Our measurements reveal a BCS-like dispersion with back-bending near the Fermi wave vector k_F that persists well above the transition temperature for pair condensation.

Many-body fermion systems are important in many branches of physics, including condensed matter, nuclear, and now cold atom physics. In many cases, the interactions between fermions can be approximated by a contact interaction. A recent theoretical advance in the study of these systems is the derivation of a number of exact universal relations that are predicted to be valid for all interaction strengths, temperatures, and spin compositions. These equations, referred to as the Tan relations, relate a microscopic quantity, namely, the amplitude of the high-momentum tail of the fermion momentum distribution, to the thermodynamics of the many-body system. In this work, we provide experimental verification of the Tan relations in a strongly interacting gas of fermionic atoms. Specifically, we measure the fermion momentum distribution using two different techniques, as well as the rf excitation spectrum and determine the effect of interactions on these microscopic probes. We then measure the potential energy and release energy of the trapped gas and test the predicted universal relations., Comment: 11 pages, 4 figures

Ultracold atom gases provide model systems in which many-body quantum physics phenomena can be studied. Recent experiments on Fermi gases have realized a phase transition to a Fermi superfluid state with strong interparticle interactions. This system is a realization of the BCS-BEC crossover connecting the physics of BCS superconductivity and that of Bose-Einstein condensation (BEC). While many aspects of this system have been investigated, it has not yet been possible to measure the single-particle excitation spectrum, which is a fundamental property directly predicted by many-body theories. Here we show that the single-particle spectral function of the strongly interacting Fermi gas at T ~ Tc is dramatically altered in a way that is consistent with a large pairing gap. We use photoemission spectroscopy to directly probe the elementary excitations and energy dispersion in the Fermi gas of atoms. In these photoemission experiments, an rf photon ejects an atom from our strongly interacting system via a spin-flip transition to a weakly interacting state. We measure the occupied single-particle density of states for an ultracold Fermi gas of 40-potassium atoms at the cusp of the BCS-BEC crossover and on the BEC side of the crossover, and compare these results to that for a nearly ideal Fermi gas. Our results probe the many-body physics in a way that could be compared to data for high-Tc superconductors. This new measurement technique for ultracold atom gases, like photoemission spectroscopy for electronic materials, directly probes low energy excitations and thus can reveal excitation gaps and/or pseudogaps. Furthermore, this technique can provide an analog to angle-resolved photoemission spectroscopy (ARPES) for probing anisotropic systems, such as atoms in optical lattice potentials.

We have produced and detected molecules using a p-wave Feshbach resonance between 40K atoms. We have measured the binding energy and lifetime for these molecules and we find that the binding energy scales approximately linearly with magnetic field near the resonance. The lifetime of bound p-wave molecules is measured to be 1.0 +/- 0.1 ms and 2.3 +/- 0.2 ms for the m_l = +/- 1 and m_l = 0 angular momentum projections, respectively. At magnetic fields above the resonance, we detect quasi-bound molecules whose lifetime is set by the tunneling rate through the centrifugal barrier.

We present a measurement of the potential energy of an ultracold trapped gas of $^{40}$K atoms in the BCS-BEC crossover and investigate the temperature dependence of this energy at a wide Feshbach resonance, where the gas is in the unitarity limit. In particular, we study the ratio of the potential energy in the region of the unitarity limit to that of a non-interacting gas, and in the T=0 limit we extract the universal many-body parameter $\beta$. We find $\beta = -0.54^{+0.05}_{-0.12}$; this value is consistent with previous measurements using $^{6}$Li atoms and also with recent theory and Monte Carlo calculations. This result demonstrates the universality of ultracold Fermi gases in the strongly interacting regime.