01834nas a2200217 4500008004100000245007700041210006900118260001400187520122500201100001101426700001301437700001601450700001501466700001201481700001401493700001501507700002501522700001701547700001501564856003701579 2023 eng d00aNon-equilibrium critical scaling and universality in a quantum simulator0 aNonequilibrium critical scaling and universality in a quantum si c9/19/20233 a

Universality and scaling laws are hallmarks of equilibrium phase transitions and critical phenomena. However, extending these concepts to non-equilibrium systems is an outstanding challenge. Despite recent progress in the study of dynamical phases, the universality classes and scaling laws for non-equilibrium phenomena are far less understood than those in equilibrium. In this work, using a trapped-ion quantum simulator with single-ion resolution, we investigate the non-equilibrium nature of critical fluctuations following a quantum quench to the critical point. We probe the scaling of spin fluctuations after a series of quenches to the critical Hamiltonian of a long-range Ising model. With systems of up to 50 spins, we show that the amplitude and timescale of the post-quench fluctuations scale with system size with distinct universal critical exponents. While a generic quench can lead to thermal critical behaviour, we find that a second quench from one critical state to another (i.e. a double quench) results in critical behaviour that does not have an equilibrium counterpart. Our results demonstrate the ability of quantum simulators to explore universal scaling beyond the equilibrium paradigm.

1 aDe, A.1 aCook, P.1 aCollins, K.1 aMorong, W.1 aPaz, D.1 aTitum, P.1 aPagano, G.1 aGorshkov, Alexey, V.1 aMaghrebi, M.1 aMonroe, C. uhttps://arxiv.org/abs/2309.1085602060nas a2200217 4500008004100000245006500041210006400106260001400170520145900184100001501643700001201658700001501670700002001685700001301705700002001718700001501738700001201753700002501765700001501790856003701805 2021 eng d00aObservation of Stark many-body localization without disorder0 aObservation of Stark manybody localization without disorder c2/14/20213 a

Thermalization is a ubiquitous process of statistical physics, in which details of few-body observables are washed out in favor of a featureless steady state. Even in isolated quantum many-body systems, limited to reversible dynamics, thermalization typically prevails. However, in these systems, there is another possibility: many-body localization (MBL) can result in preservation of a non-thermal state. While disorder has long been considered an essential ingredient for this phenomenon, recent theoretical work has suggested that a quantum many-body system with a uniformly increasing field -- but no disorder -- can also exhibit MBL, resulting in `Stark MBL.' Here we realize Stark MBL in a trapped-ion quantum simulator and demonstrate its key properties: halting of thermalization and slow propagation of correlations. Tailoring the interactions between ionic spins in an effective field gradient, we directly observe their microscopic equilibration for a variety of initial states, and we apply single-site control to measure correlations between separate regions of the spin chain. Further, by engineering a varying gradient, we create a disorder-free system with coexisting long-lived thermalized and nonthermal regions. The results demonstrate the unexpected generality of MBL, with implications about the fundamental requirements for thermalization and with potential uses in engineering long-lived non-equilibrium quantum matter.

1 aMorong, W.1 aLiu, F.1 aBecker, P.1 aCollins, K., S.1 aFeng, L.1 aKyprianidis, A.1 aPagano, G.1 aYou, T.1 aGorshkov, Alexey, V.1 aMonroe, C. uhttps://arxiv.org/abs/2102.0725001620nas a2200193 4500008004100000245006300041210006200104260001400166520106800180100001201248700001401260700001901274700002101293700002101314700001801335700001501353700002101368856003701389 2020 eng d00aProbing many-body localization on a noisy quantum computer0 aProbing manybody localization on a noisy quantum computer c6/22/20203 a

A disordered system of interacting particles exhibits localized behavior when the disorder is large compared to the interaction strength. Studying this phenomenon on a quantum computer without error correction is challenging because even weak coupling to a thermal environment destroys most signatures of localization. Fortunately, spectral functions of local operators are known to contain features that can survive the presence of noise. In these spectra, discrete peaks and a soft gap at low frequencies compared to the thermal phase indicate localization. Here, we present the computation of spectral functions on a trapped-ion quantum computer for a one-dimensional Heisenberg model with disorder. Further, we design an error-mitigation technique which is effective at removing the noise from the measurement allowing clear signatures of localization to emerge as the disorder increases. Thus, we show that spectral functions can serve as a robust and scalable diagnostic of many-body localization on the current generation of quantum computers. 

1 aZhu, D.1 aJohri, S.1 aNguyen, N., H.1 aAlderete, Huerta1 aLandsman, K., A.1 aLinke, N., M.1 aMonroe, C.1 aMatsuura, A., Y. uhttps://arxiv.org/abs/2006.1235501912nas a2200265 4500008004100000245007900041210006900120260001500189520117500204100001601379700001501395700001201410700001501422700002001437700001101457700001301468700001901481700002001500700001701520700001501537700001701552700002501569700001501594856003701609 2019 eng d00aObservation of Domain Wall Confinement and Dynamics in a Quantum Simulator0 aObservation of Domain Wall Confinement and Dynamics in a Quantum c12/23/20193 a

Confinement is a ubiquitous mechanism in nature, whereby particles feel an attractive force that increases without bound as they separate. A prominent example is color confinement in particle physics, in which baryons and mesons are produced by quark confinement. Analogously, confinement can also occur in low-energy quantum many-body systems when elementary excitations are confined into bound quasiparticles. Here, we report the first observation of magnetic domain wall confinement in interacting spin chains with a trapped-ion quantum simulator. By measuring how correlations spread, we show that confinement can dramatically suppress information propagation and thermalization in such many-body systems. We are able to quantitatively determine the excitation energy of domain wall bound states from non-equilibrium quench dynamics. Furthermore, we study the number of domain wall excitations created for different quench parameters, in a regime that is difficult to model with classical computers. This work demonstrates the capability of quantum simulators for investigating exotic high-energy physics phenomena, such as quark collision and string breaking

1 aTan, W., L.1 aBecker, P.1 aLiu, F.1 aPagano, G.1 aCollins, K., S.1 aDe, A.1 aFeng, L.1 aKaplan, H., B.1 aKyprianidis, A.1 aLundgren, R.1 aMorong, W.1 aWhitsitt, S.1 aGorshkov, Alexey, V.1 aMonroe, C. uhttps://arxiv.org/abs/1912.1111701709nas a2200241 4500008004100000245007100041210006900112260001500181520103600196100001501232700002101247700001801268700001801286700002501304700001301329700001401342700001201356700001501368700001701383700001401400700001601414856003701430 2019 eng d00aProgrammable Quantum Simulations of Spin Systems with Trapped Ions0 aProgrammable Quantum Simulations of Spin Systems with Trapped Io c12/17/20193 a

Laser-cooled and trapped atomic ions form an ideal standard for the simulation of interacting quantum spin models. Effective spins are represented by appropriate internal energy levels within each ion, and the spins can be measured with near-perfect efficiency using state-dependent fluorescence techniques. By applying optical fields that exert optical dipole forces on the ions, their Coulomb interaction can be modulated in ways that give rise to long-range and tunable spin-spin interactions that can be reconfigured by shaping the spectrum and pattern of the laser fields. Here we review the theoretical mapping of atomic ions to interacting spin systems, the experimental preparation of complex equilibrium states, and the study of dynamical processes of this many-body interacting quantum system. The use of such quantum simulators for studying spin models may inform our understanding of exotic quantum materials and shed light on interacting quantum systems that cannot be modeled with conventional computers. 

1 aMonroe, C.1 aCampbell, W., C.1 aDuan, L., -M.1 aGong, Z., -X.1 aGorshkov, Alexey, V.1 aHess, P.1 aIslam, R.1 aKim, K.1 aPagano, G.1 aRicherme, P.1 aSenko, C.1 aYao, N., Y. uhttps://arxiv.org/abs/1912.0784501764nas a2200289 4500008004100000245007400041210006900115260001500184520095900199100001501158700001401173700001501187700002001202700001101222700001701233700001901250700002001269700001601289700002901305700001801334700001801352700001201370700001501382700002501397700001501422856003701437 2019 eng d00aQuantum Approximate Optimization with a Trapped-Ion Quantum Simulator0 aQuantum Approximate Optimization with a TrappedIon Quantum Simul c06/06/20193 a

Quantum computers and simulators may offer significant advantages over their classical counterparts, providing insights into quantum many-body systems and possibly solving exponentially hard problems, such as optimization and satisfiability. Here we report the first implementation of a shallow-depth Quantum Approximate Optimization Algorithm (QAOA) using an analog quantum simulator to estimate the ground state energy of the transverse field Ising model with tunable long-range interactions. First, we exhaustively search the variational control parameters to approximate the ground state energy with up to 40 trapped-ion qubits. We then interface the quantum simulator with a classical algorithm to more efficiently find the optimal set of parameters that minimizes the resulting energy of the system. We finally sample from the full probability distribution of the QAOA output with single-shot and efficient measurements of every qubit. 

1 aPagano, G.1 aBapat, A.1 aBecker, P.1 aCollins, K., S.1 aDe, A.1 aHess, P., W.1 aKaplan, H., B.1 aKyprianidis, A.1 aTan, W., L.1 aBaldwin, Christopher, L.1 aBrady, L., T.1 aDeshpande, A.1 aLiu, F.1 aJordan, S.1 aGorshkov, Alexey, V.1 aMonroe, C. uhttps://arxiv.org/abs/1906.0270001463nas a2200229 4500008004100000245006800041210006700109520082000176100001500996700001701011700001901028700001601047700001701063700001501080700002001095700001401115700001901129700002201148700001101170700001501181856003701196 2018 eng d00aCryogenic Trapped-Ion System for Large Scale Quantum Simulation0 aCryogenic TrappedIon System for Large Scale Quantum Simulation3 a

We present a cryogenic ion trapping system designed for large scale quantum simulation of spin models. Our apparatus is based on a segmented-blade ion trap enclosed in a 4 K cryostat, which enables us to routinely trap over 100 171Yb+ ions in a linear configuration for hours due to a low background gas pressure from differential cryo-pumping. We characterize the cryogenic vacuum by using trapped ion crystals as a pressure gauge, measuring both inelastic and elastic collision rates with the molecular background gas. We demonstrate nearly equidistant ion spacing for chains of up to 44 ions using anharmonic axial potentials. This reliable production and lifetime enhancement of large linear ion chains will enable quantum simulation of spin models that are intractable with classical computer modelling.

1 aPagano, G.1 aHess, P., W.1 aKaplan, H., B.1 aTan, W., L.1 aRicherme, P.1 aBecker, P.1 aKyprianidis, A.1 aZhang, J.1 aBirckelbaw, E.1 aHernandez, M., R.1 aWu, Y.1 aMonroe, C. uhttps://arxiv.org/abs/1802.0311801356nas a2200181 4500008004100000245006000041210005900101260001500160490000700175520083200182100001801014700002401032700002301056700002201079700001501101700002101116856003701137 2018 eng d00aMachine learning assisted readout of trapped-ion qubits0 aMachine learning assisted readout of trappedion qubits c2018/05/010 v513 a

We reduce measurement errors in a quantum computer using machine learning techniques. We exploit a simple yet versatile neural network to classify multi-qubit quantum states, which is trained using experimental data. This flexible approach allows the incorporation of any number of features of the data with minimal modifications to the underlying network architecture. We experimentally illustrate this approach in the readout of trapped-ion qubits using additional spatial and temporal features in the data. Using this neural network classifier, we efficiently treat qubit readout crosstalk, resulting in a 30\% improvement in detection error over the conventional threshold method. Our approach does not depend on the specific details of the system and can be readily generalized to other quantum computing platforms.

1 aSeif, Alireza1 aLandsman, Kevin, A.1 aLinke, Norbert, M.1 aFiggatt, Caroline1 aMonroe, C.1 aHafezi, Mohammad uhttps://arxiv.org/abs/1804.0771801943nas a2200169 4500008004100000245007600041210006900117520143500186100001601621700001801637700001801655700002101673700001201694700001501706700001501721856003701736 2018 eng d00aParallel Entangling Operations on a Universal Ion Trap Quantum Computer0 aParallel Entangling Operations on a Universal Ion Trap Quantum C3 a

The circuit model of a quantum computer consists of sequences of gate operations between quantum bits (qubits), drawn from a universal family of discrete operations. The ability to execute parallel entangling quantum gates offers clear efficiency gains in numerous quantum circuits as well as for entire algorithms such as Shor's factoring algorithm and quantum simulations. In cases such as full adders and multiple-control Toffoli gates, parallelism can provide an exponential improvement in overall execution time. More importantly, quantum gate parallelism is essential for the practical fault-tolerant error correction of qubits that suffer from idle errors. The implementation of parallel quantum gates is complicated by potential crosstalk, especially between qubits fully connected by a common-mode bus, such as in Coulomb-coupled trapped atomic ions or cavity-coupled superconducting transmons. Here, we present the first experimental results for parallel 2-qubit entangling gates in an array of fully-connected trapped ion qubits. We demonstrate an application of this capability by performing a 1-bit full addition operation on a quantum computer using a depth-4 quantum circuit. These results exploit the power of highly connected qubit systems through classical control techniques, and provide an advance toward speeding up quantum circuits and achieving fault tolerance with trapped ion quantum computers.

1 aFiggatt, C.1 aOstrander, A.1 aLinke, N., M.1 aLandsman, K., A.1 aZhu, D.1 aMaslov, D.1 aMonroe, C. uhttps://arxiv.org/abs/1810.1194801952nas a2200229 4500008004100000245009200041210006900133260001500202300001200217490000800229520128300237100001401520700001501534700001701549700002001566700001501586700001501601700002501616700001801641700001501659856004801674 2017 eng d00aObservation of a Many-Body Dynamical Phase Transition with a 53-Qubit Quantum Simulator0 aObservation of a ManyBody Dynamical Phase Transition with a 53Qu c2017/11/29 a601-6040 v5513 a

A quantum simulator is a restricted class of quantum computer that controls the interactions between quantum bits in a way that can be mapped to certain difficult quantum many-body problems. As more control is exerted over larger numbers of qubits, the simulator can tackle a wider range of problems, with the ultimate limit being a universal quantum computer that can solve general classes of hard problems. We use a quantum simulator composed of up to 53 qubits to study a non-equilibrium phase transition in the transverse field Ising model of magnetism, in a regime where conventional statistical mechanics does not apply. The qubits are represented by trapped ion spins that can be prepared in a variety of initial pure states. We apply a global long-range Ising interaction with controllable strength and range, and measure each individual qubit with near 99% efficiency. This allows the single-shot measurement of arbitrary many-body correlations for the direct probing of the dynamical phase transition and the uncovering of computationally intractable features that rely on the long-range interactions and high connectivity between the qubits.

1 aZhang, J.1 aPagano, G.1 aHess, P., W.1 aKyprianidis, A.1 aBecker, P.1 aKaplan, H.1 aGorshkov, Alexey, V.1 aGong, Z., -X.1 aMonroe, C. uhttps://www.nature.com/articles/nature2465402040nas a2200193 4500008004100000245007800041210006900119260001500188300001000203490000800213520145000221100001601671700001801687700001601705700002101721700001501742700001501757856007401772 2016 eng d00aDemonstration of a small programmable quantum computer with atomic qubits0 aDemonstration of a small programmable quantum computer with atom c2016/08/04 a63-660 v5363 a

Quantum computers can solve certain problems more efficiently than any possible conventional computer. Small quantum algorithms have been demonstrated on multiple quantum computing platforms, many specifically tailored in hardware to implement a particular algorithm or execute a limited number of computational paths. Here, we demonstrate a five-qubit trapped-ion quantum computer that can be programmed in software to implement arbitrary quantum algorithms by executing any sequence of universal quantum logic gates. We compile algorithms into a fully-connected set of gate operations that are native to the hardware and have a mean fidelity of 98 %. Reconfiguring these gate sequences provides the flexibility to implement a variety of algorithms without altering the hardware. As examples, we implement the Deutsch-Jozsa (DJ) and Bernstein-Vazirani (BV) algorithms with average success rates of 95 % and 90 %, respectively. We also perform a coherent quantum Fourier transform (QFT) on five trappedion qubits for phase estimation and period finding with average fidelities of 62 % and 84 %, respectively. This small quantum computer can be scaled to larger numbers of qubits within a single register, and can be further expanded by connecting several such modules through ion shuttling or photonic quantum channels.

1 aDebnath, S.1 aLinke, N., M.1 aFiggatt, C.1 aLandsman, K., A.1 aWright, K.1 aMonroe, C. uhttp://www.nature.com/nature/journal/v536/n7614/full/nature18648.html01724nas a2200181 4500008004100000245005800041210005800099260001500157520121100172100001801383700001801401700002101419700001601440700001601456700001801472700001501490856003701505 2016 eng d00aExperimental demonstration of quantum fault tolerance0 aExperimental demonstration of quantum fault tolerance c2016/11/213 a

Quantum computers will eventually reach a size at which quantum error correction (QEC) becomes imperative. In order to make quantum information robust to errors introduced by qubit imperfections and flawed control operations, QEC protocols encode a logical qubit in multiple physical qubits. This redundancy allows the extraction of error syndromes and the subsequent correction or detection of errors without destroying the logical state itself through direct measurement. While several experiments have shown a reduction of high intrinsic or artificially introduced errors in logical qubits, fault-tolerant encoding of a logical qubit has never been demonstrated. Here we show the encoding and syndrome measurement of a fault-tolerant logical qubit via an error detection protocol on four physical qubits, represented by trapped atomic ions. This demonstrates for the first time the robustness of a fault-tolerant qubit to imperfections in the very operations used to encode it. This advantage persists in the face of large added error rates and experimental calibration errors.

1 aLinke, N., M.1 aGutierrez, M.1 aLandsman, K., A.1 aFiggatt, C.1 aDebnath, S.1 aBrown, K., R.1 aMonroe, C. uhttps://arxiv.org/abs/1611.0694602050nas a2200205 4500008004100000245008900041210006900130260001500199520143900214100001801653700001401671700001601685700001401701700001701715700001701732700001801749700002501767700001501792856003701807 2016 eng d00a{O}bservation of {P}rethermalization in {L}ong-{R}ange {I}nteracting {S}pin {C}hains0 aO bservation of P rethermalization in L ong R ange I nteracting  c2016/08/023 a

Statistical mechanics can predict thermal equilibrium states for most classical systems, but for an isolated quantum system there is no general understanding on how equilibrium states dynamically emerge from the microscopic Hamiltonian. For instance, quantum systems that are near-integrable usually fail to thermalize in an experimentally realistic time scale and, instead, relax to quasi-stationary prethermal states that can be described by statistical mechanics when approximately conserved quantities are appropriately included in a generalized Gibbs ensemble (GGE). Here we experimentally study the relaxation dynamics of a chain of up to 22 spins evolving under a long-range transverse field Ising Hamiltonian following a sudden quench. For sufficiently long-ranged interactions the system relaxes to a new type of prethermal state that retains a strong memory of the initial conditions. In this case, the prethermal state cannot be described by a GGE, but rather arises from an emergent double-well potential felt by the spin excitations. This result shows that prethermalization occurs in a significantly broader context than previously thought, and reveals new challenges for a generic understanding of the thermalization of quantum systems, particularly in the presence of long-range interactions.

1 aNeyenhuis, B.1 aSmith, J.1 aLee, A., C.1 aZhang, J.1 aRicherme, P.1 aHess, P., W.1 aGong, Z., -X.1 aGorshkov, Alexey, V.1 aMonroe, C. uhttps://arxiv.org/abs/1608.00681