Seeking a quantum advantage with trapped-ion quantum simulations of condensed-phase chemical dynamics.

Simulating the quantum dynamics of molecules in the condensed phase represents a longstanding challenge in chemistry. Trapped-ion quantum systems may serve as a platform for the analog-quantum simulation of chemical dynamics that is beyond the reach of current classical-digital simulation. To identify a 'quantum advantage' for these simulations, performance analysis of both analog-quantum simulation on noisy hardware and classical-digital algorithms is needed. In this Review, we make a comparison between a noisy analog trapped-ion simulator and a few choice classical-digital methods on simulating the dynamics of a model molecular Hamiltonian with linear vibronic coupling. We describe several simple Hamiltonians that are commonly used to model molecular systems, which can be simulated with existing or emerging trapped-ion hardware. These Hamiltonians may serve as stepping stones towards the use of trapped-ion simulators for systems beyond the reach of classical-digital methods. Finally, we identify dynamical regimes in which classical-digital simulations seem to have the weakest performance with respect to analog-quantum simulations. These regimes may provide the lowest hanging fruit to make the most of potential quantum advantages.

Quantum simulation of conical intersections using trapped ions.

Conical intersections often control the reaction products of photochemical processes and occur when two electronic potential energy surfaces intersect. Theory predicts that the conical intersection will result in a geometric phase for a wavepacket on the ground potential energy surface, and although conical intersections have been observed experimentally, the geometric phase has not been directly observed in a molecular system. Here we use a trapped atomic ion system to perform a quantum simulation of a conical intersection. The ion's internal state serves as the electronic state, and the motion of the atomic nuclei is encoded into the motion of the ions. The simulated electronic potential is constructed by applying state-dependent optical forces to the ion. We experimentally observe a clear manifestation of the geometric phase using adiabatic state preparation followed by motional state measurement. Our experiment shows the advantage of combining spin and motion degrees for quantum simulation of chemical reactions.

Quantum Simulation of Polarized Light-Induced Electron Transfer with a Trapped-Ion Qutrit System.

Electron transfer within and between molecules is crucial in chemistry, biochemistry, and energy science. This study describes a quantum simulation method that explores the influence of light polarization on electron transfer between two molecules. By implementing precise and coherent control among the quantum states of trapped atomic ions, we can induce quantum dynamics that mimic the electron-transfer dynamics in molecules. We use three-level systems (qutrits), rather than traditional two-level systems (qubits), to enhance the simulation efficiency and realize high-fidelity simulations of electron-transfer dynamics. We treat the quantum interference between the electron coupling pathways from a donor with two degenerate excited states to an acceptor and analyze the transfer efficiency. We also examine the potential error sources that enter the quantum simulations. The trapped-ion systems have favorable scalings with system size compared to those of classical computers, promising access to richer electron-transfer simulations.

Crosstalk Suppression in Individually Addressed Two-Qubit Gates in a Trapped-Ion Quantum Computer.

Crosstalk between target and neighboring spectator qubits due to spillover of control signals represents a major error source limiting the fidelity of two-qubit entangling gates in quantum computers. We show that in our laser-driven trapped-ion system coherent crosstalk error can be modeled as residual Xσ[over ^]_{ϕ} interaction and can be actively canceled by single-qubit echoing pulses. We propose and demonstrate a crosstalk suppression scheme that eliminates all first-order crosstalk utilizing only local control of target qubits, as opposed to an existing scheme which requires control over all neighboring qubits. We report a two-qubit Bell state fidelity of 99.52(6)% with the echoing pulses applied after collective gates and 99.37(5)% with the echoing pulses applied to each gate in a five-ion chain. This scheme is widely applicable to other platforms with analogous interaction Hamiltonians.

Determination of Multimode Motional Quantum States in a Trapped Ion System.

Trapped atomic ions are a versatile platform for studying interactions between spins and bosons by coupling the internal states of the ions to their motion. Measurement of complex motional states with multiple modes is challenging, because all motional state populations can only be measured indirectly through the spin state of ions. Here we present a general method to determine the Fock state distributions and to reconstruct the density matrix of an arbitrary multimode motional state. We experimentally verify the method using different entangled states of multiple radial modes in a five-ion chain. This method can be extended to any system with Jaynes-Cummings-type interactions.

Optimizing Stabilizer Parities for Improved Logical Qubit Memories.

We study variants of Shor's code that are adept at handling single-axis correlated idling errors, which are commonly observed in many quantum systems. By using the repetition code structure of the Shor's code basis states, we calculate the logical channel applied to the encoded information when subjected to coherent and correlated single qubit idling errors, followed by stabilizer measurement. Changing the signs of the stabilizer generators allows us to change how the coherent errors interfere, leading to a quantum error-correcting code which performs as well as a classical repetition code of equivalent distance against these errors. We demonstrate a factor of 3.78±1.20 improvement of the logical T2^{*} in a distance-3 logical qubit implemented on a trapped-ion quantum computer. Even-distance versions of our Shor-code variants are decoherence-free subspaces and fully robust to identical and independent coherent idling noise.

Fault-tolerant control of an error-corrected qubit.

Quantum error correction protects fragile quantum information by encoding it into a larger quantum system1,2. These extra degrees of freedom enable the detection and correction of errors, but also increase the control complexity of the encoded logical qubit. Fault-tolerant circuits contain the spread of errors while controlling the logical qubit, and are essential for realizing error suppression in practice3-6. Although fault-tolerant design works in principle, it has not previously been demonstrated in an error-corrected physical system with native noise characteristics. Here we experimentally demonstrate fault-tolerant circuits for the preparation, measurement, rotation and stabilizer measurement of a Bacon-Shor logical qubit using 13 trapped ion qubits. When we compare these fault-tolerant protocols to non-fault-tolerant protocols, we see significant reductions in the error rates of the logical primitives in the presence of noise. The result of fault-tolerant design is an average state preparation and measurement error of 0.6 per cent and a Clifford gate error of 0.3 per cent after offline error correction. In addition, we prepare magic states with fidelities that exceed the distillation threshold7, demonstrating all of the key single-qubit ingredients required for universal fault-tolerant control. These results demonstrate that fault-tolerant circuits enable highly accurate logical primitives in current quantum systems. With improved two-qubit gates and the use of intermediate measurements, a stabilized logical qubit can be achieved.

Between Shor and Steane: A Unifying Construction for Measuring Error Syndromes.

Fault-tolerant quantum error correction requires the measurement of error syndromes in a way that minimizes correlated errors on the quantum data. Steane and Shor ancilla are two well-known methods for fault-tolerant syndrome extraction. In this Letter, we find a unifying construction that generates a family of ancilla blocks that interpolate between Shor and Steane. This family increases the complexity of ancilla construction in exchange for reducing the rounds of measurement required to fault tolerantly measure the error. We then apply this construction to the toric code of size L×L and find that blocks of size m×m can be used to decode errors in O(L/m) rounds of measurements. Our method can be applied to any Calderbank-Shor-Steane code and presents a new direction for optimizing fault-tolerant quantum computation.

Dipole-phonon quantum logic with alkaline-earth monoxide and monosulfide cations.

Dipole-phonon quantum logic (DPQL) leverages the interaction between polar molecular ions and the motional modes of a trapped-ion Coulomb crystal to provide a potentially scalable route to quantum information science. Here, we study a class of candidate molecular ions for DPQL, the cationic alkaline-earth monoxides and monosulfides, which possess suitable structure for DPQL and can be produced in existing atomic ion experiments with little additional complexity. We present calculations of DPQL operations for one of these molecules, CaO+, and discuss progress towards experimental realization. We also further develop the theory of DPQL to include state preparation and measurement and entanglement of multiple molecular ions.

High-Fidelity Two-Qubit Gates Using a Microelectromechanical-System-Based Beam Steering System for Individual Qubit Addressing.

In a large scale trapped atomic ion quantum computer, high-fidelity two-qubit gates need to be extended over all qubits with individual control. We realize and characterize high-fidelity two-qubit gates in a system with up to four ions using radial modes. The ions are individually addressed by two tightly focused beams steered using microelectromechanical system mirrors. We deduce a gate fidelity of 99.49(7)% in a two-ion chain and 99.30(6)% in a four-ion chain by applying a sequence of up to 21 two-qubit gates and measuring the final state fidelity. We characterize the residual errors and discuss methods to further improve the gate fidelity towards values that are compatible with fault-tolerant quantum computation.

Graphene reinforced carbon fibers.

The superlative strength-to-weight ratio of carbon fibers (CFs) can substantially reduce vehicle weight and improve energy efficiency. However, most CFs are derived from costly polyacrylonitrile (PAN), which limits their widespread adoption in the automotive industry. Extensive efforts to produce CFs from low cost, alternative precursor materials have failed to yield a commercially viable product. Here, we revisit PAN to study its conversion chemistry and microstructure evolution, which might provide clues for the design of low-cost CFs. We demonstrate that a small amount of graphene can minimize porosity/defects and reinforce PAN-based CFs. Our experimental results show that 0.075 weight % graphene-reinforced PAN/graphene composite CFs exhibits 225% increase in strength and 184% enhancement in Young's modulus compared to PAN CFs. Atomistic ReaxFF and large-scale molecular dynamics simulations jointly elucidate the ability of graphene to modify the microstructure by promoting favorable edge chemistry and polymer chain alignment.

Photon-mediated charge exchange reactions between 39K atoms and 40Ca+ ions in a hybrid trap.

We present experimental evidence of charge exchange between laser-cooled potassium 39K atoms and calcium 40Ca+ ions in a hybrid atom-ion trap and give quantitative theoretical explanations for the observations. The 39K atoms and 40Ca+ ions are held in a magneto-optical (MOT) and a linear Paul trap, respectively. Fluorescence detection and high resolution time of flight mass spectra for both species are used to determine the remaining number of 40Ca+ ions, the increasing number of 39K+ ions, and 39K number density as functions of time. Simultaneous trap operation is guaranteed by alternating periods of MOT and 40Ca+ cooling lights, thus avoiding direct ionization of 39K by the 40Ca+ cooling light. We show that the K-Ca+ charge-exchange rate coefficient increases linearly from zero with 39K number density and the fraction of 40Ca+ ions in the 4p 2P1/2 electronically-excited state. Combined with our theoretical analysis, we conclude that these data can only be explained by a process that starts with a potassium atom in its electronic ground state and a calcium ion in its excited 4p 2P1/2 state producing ground-state 39K+ ions and metastable, neutral Ca (3d4p 3P1) atoms, releasing only 150 cm-1 equivalent relative kinetic energy. Charge-exchange between either ground- or excited-state 39K and ground-state 40Ca+ is negligibly small as no energetically-favorable product states are available. Our experimental and theoretical rate coefficients are in agreement given the uncertainty budgets.

Quantum Error Correction Decoheres Noise.

Typical studies of quantum error correction assume probabilistic Pauli noise, largely because it is relatively easy to analyze and simulate. Consequently, the effective logical noise due to physically realistic coherent errors is relatively unknown. Here, we prove that encoding a system in a stabilizer code and measuring error syndromes decoheres errors, that is, causes coherent errors to converge toward probabilistic Pauli errors, even when no recovery operations are applied. Two practical consequences are that the error rate in a logical circuit is well quantified by the average gate fidelity at the logical level and that essentially optimal recovery operators can be determined by independently optimizing the logical fidelity of the effective noise per syndrome.

Spectroscopy of Molecular Ions in Coulomb Crystals.

In this Perspective, we examine the use of laser-cooled atomic ions and sympathetically cooled molecular ions in Coulomb crystals for molecular spectroscopy. Coulomb crystals are well-isolated environments that provide localization and long storage times for sensitive measurements of weak signals and cold temperatures for precise spectroscopy. Coulomb crystals of molecular and atomic ions enable the detection of single-photon molecular ion transitions at a range of wavelengths by a change in atomic ion fluorescence at visible wavelengths. We give an overview of the state of the art from action spectroscopy to quantum logic spectroscopy for a wide range of molecular transitions from rotational sublevels separated by 10-7 cm-1 to rovibronic transitions at 25 000 cm-1. We emphasize how this system allows for unparalleled control of the molecular ion state for precision spectroscopy with applications in astrochemistry and fundamental physics. We conclude with an outlook of the use of this control in cold molecular ion reactions.

Rovibronic Spectroscopy of Sympathetically Cooled 40CaH.

We measure the rovibronic transitions X 1Σ+, v″ = 0, J″ → A 1Σ+, v' = 0-3, J' of CaH+ and obtain rotational constants for the A 1Σ+ state. The spectrum is obtained using two-photon photodissociation of CaH+ cotrapped with Doppler cooled Ca+. The excitation is driven by a mode-locked, frequency-doubled Ti:Sapph laser, which is then pulse shaped to narrow the spectral bandwidth. The measured values of the rotational constants are in agreement with ab initio theory.

Robust 2-Qubit Gates in a Linear Ion Crystal Using a Frequency-Modulated Driving Force.

In an ion trap quantum computer, collective motional modes are used to entangle two or more qubits in order to execute multiqubit logical gates. Any residual entanglement between the internal and motional states of the ions results in loss of fidelity, especially when there are many spectator ions in the crystal. We propose using a frequency-modulated driving force to minimize such errors. In simulation, we obtained an optimized frequency-modulated 2-qubit gate that can suppress errors to less than 0.01% and is robust against frequency drifts over ±1 kHz. Experimentally, we have obtained a 2-qubit gate fidelity of 98.3(4)%, a state-of-the-art result for 2-qubit gates with five ions.

Stabilizer Slicing: Coherent Error Cancellations in Low-Density Parity-Check Stabilizer Codes.

Coherent errors are a dominant noise process in many quantum computing architectures. Unlike stochastic errors, these errors can combine constructively and grow into highly detrimental overrotations. To combat this, we introduce a simple technique for suppressing systematic coherent errors in low-density parity-check stabilizer codes, which we call stabilizer slicing. The essential idea is to slice low-weight stabilizers into two equally weighted Pauli operators and then apply them by rotating in opposite directions, causing their overrotations to interfere destructively on the logical subspace. With access to native gates generated by three-body Hamiltonians, we can completely eliminate purely coherent overrotation errors, and for overrotation noise of 0.99 unitarity we achieve a 135-fold improvement in the logical error rate of surface-17. For more conventional two-body ion trap gates, we observe an 89-fold improvement for Bacon-Shor-13 with purely coherent errors which should be testable in near-term fault-tolerance experiments. This second scheme takes advantage of the prepared gauge degrees of freedom, and to our knowledge is the first example in which the state of the gauge directly affects the robustness of a code's memory. This Letter demonstrates that coherent noise is preferable to stochastic noise within certain code and gate implementations when the coherence is utilized effectively.

Fault-tolerant quantum error detection.

Quantum computers will eventually reach a size at which quantum error correction becomes imperative. Quantum information can be protected from qubit imperfections and flawed control operations by encoding a single logical qubit in multiple physical qubits. This redundancy allows the extraction of error syndromes and the subsequent detection or correction of errors without destroying the logical state itself through direct measurement. We show the encoding and syndrome measurement of a fault-tolerantly prepared logical qubit via an error detection protocol on four physical qubits, represented by trapped atomic ions. This demonstrates the robustness of a logical qubit to imperfections in the very operations used to encode it. The advantage persists in the face of large added error rates and experimental calibration errors.

Vibronic Spectroscopy of Sympathetically Cooled CaH.

We report the measurement of the 11 Σ→21 Σ transition of CaH+ by resonance-enhanced photodissociation of CaH+ that is co-trapped with laser-cooled Ca+ . We observe four resonances that we assign to transitions from the vibrational v=0 ground state to the v'=1-4 excited states based on theoretical predictions. A simple theoretical model that assumes instantaneous dissociation after resonant excitation yields results in good agreement with the observed spectral features except for the unobserved v'=0 peak. This discrepancy is attributed to an insufficient understanding of the dissociation process, and further experimental and theoretical studies are required to confirm the assignment. The resolution of our experiment is limited by the mode-locked excitation laser, but this survey spectroscopy enables future rotationally resolved studies with applications in astrochemistry and precision measurement.

Observation of vibrational overtones by single-molecule resonant photodissociation.

Molecular ions can be held in a chain of laser-cooled atomic ions by sympathetic cooling. This system is ideal for performing high-precision molecular spectroscopy with applications in astrochemistry and fundamental physics. Here we show that this same system can be coupled with a broadband laser to discover new molecular transitions. We use three-ion chains of Ca(+) and CaH(+) to observe vibrational transitions via resonance-enhanced multiphoton dissociation detected by Ca(+) fluorescence. On the basis of theoretical calculations, we assign the observed peaks to the transition from the ground vibrational state, ν=0 to ν=9 and 10. Our method allows us to track single-molecular events, and it can be extended to work with any molecule by using normal mode frequency shifts to detect the dissociation. This survey spectroscopy serves as a bridge to the precision spectroscopy required for molecular ion control.