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Time-varying Hamiltonians can be reconstructed experimentally if the variation takes place in a single basis, but more general cases are complicated. Here, the authors present an approach to estimate the general time-dependent Hamiltonian of a single spin-qubit and apply it to a trapped-ion transported through a static laser beam.

The noncommutativity of position and momentum observables is a hallmark feature of quantum physics. However, this incompatibility does not extend to observables that are periodic in these base variables. Such modular-variable observables have been suggested as tools for fault-tolerant quantum computing and enhanced quantum sensing. Here, we implement sequential measurements of modular variables in the oscillatory motion of a single trapped ion, using state-dependent displacements and a heralded nondestructive readout. We investigate the commutative nature of modular variable observables by demonstrating no-signaling in time between successive measurements, using a variety of input states. Employing a different periodicity, we observe signaling in time. This also requires wave-packet overlap, resulting in quantum interference that we enhance using squeezed input states. The sequential measurements allow us to extract two-time correlators for modular variables, which we use to violate a Leggett-Garg inequality. Signaling in time and Leggett-Garg inequalities serve as efficient quantum witnesses, which we probe here with a mechanical oscillator, a system that has a natural crossover from the quantum to the classical regime.

The selective number-dependent arbitrary phase gates form a powerful class of quantum gates, imparting arbitrarily chosen phases to the Fock states of a cavity. However, for short pulses, coherent errors limit the performance. Here, we demonstrate in theory and experiment that such errors can be completely suppressed, provided that the pulse times exceed a specific limit. The resulting shorter gate times also reduce incoherent errors. Our approach needs only a small number of frequency components, the resulting pulses can be interpreted easily, and it is compatible with fault-tolerant schemes.

We implement direct readout of the symmetric characteristic function of quantum states of the motional oscillation of a trapped calcium ion. Suitably chosen internal state rotations combined with internal state-dependent displacements, based on bichromatic laser fields, map the expectation value of the real or imaginary part of the displacement operator to the internal states, which are subsequently read out. Combining these results provides full information about the symmetric characteristic function. We characterize the technique by applying it to a range of archetypal quantum oscillator states, including displaced and squeezed Gaussian states as well as two and three component superpositions of displaced squeezed states. For each, we discuss relevant features of the characteristic function and Wigner phase-space quasiprobability distribution. The direct reconstruction of these highly nonclassical oscillator states using a reduced number of measurements is an essential tool for understanding and optimizing the control of oscillator systems for quantum sensing and quantum information applications.

The stable operation of quantum computers will rely on error correction, in which single quantum bits of information are stored redundantly in the Hilbert space of a larger system. Such encoded qubits are commonly based on arrays of many physical qubits, but can also be realized using a single higher-dimensional quantum system, such as a harmonic oscillator 1-3 . In such a system, a powerful encoding has been devised based on periodically spaced superpositions of position eigenstates 4-6 . Various proposals have been made for realizing approximations to such states, but these have thus far remained out of reach 7-11 . Here we demonstrate such an encoded qubit using a superposition of displaced squeezed states of the harmonic motion of a single trapped 40 Ca + ion, controlling and measuring the mechanical oscillator through coupling to an ancillary internal-state qubit 12 . We prepare and reconstruct logical states with an average squared fidelity of 87.3 ± 0.7 per cent. Also, we demonstrate a universal logical single-qubit gate set, which we analyse using process tomography. For Pauli gates we reach process fidelities of about 97 per cent, whereas for continuous rotations we use gate teleportation and achieve fidelities of approximately 89 per cent. This control method opens a route for exploring continuous variable error correction as well as hybrid quantum information schemes using both discrete and continuous variables 13 . The code states also have direct applications in quantum sensing, allowing simultaneous measurement of small displacements in both position and momentum 14,15 .

Quantum error correction is essential for realizing the full potential of large-scale quantum information processing devices 1,2 . Fundamental to its experimental realization is the repetitive detection of errors via projective measurements of quantum correlations among qubits, as well as corrections using conditional feedback 3 . Repetitive application of such tasks requires that they neither induce unwanted crosstalk nor impede further control operations, which is challenging owing to the need to dissipatively couple qubits to the classical world for detection and reinitialization. For trapped ions, state readout involves scattering large numbers of resonant photons, which increases the probability of stray light causing errors on nearby qubits and leads to undesirable recoil heating of the ion motion. Here we demonstrate up to 50 sequential measurements of correlations between two beryllium ion microwave qubits using an ancillary optical qubit in a calcium ion, and implement feedback that allows us to stabilize two-qubit subspaces as well as Bell states, a class of maximally entangled states. Multi-qubit mixed-species gates are used to transfer information within the register from the qubit to the ancilla, enabling readout with negligible crosstalk to the data qubits. Heating of the ion motion during detection is mitigated by recooling all three ions using light that interacts with only the calcium ion, known as sympathetic cooling. A key element of our experimental setup is a powerful classical control system that features flexible in-sequence processing for feedback control. The methods employed here provide essential tools for scaling trapped-ion quantum computing, quantum-state control and entanglement-enhanced quantum metrology 4 .

We demonstrate control of a trapped-ion quantum harmonic oscillator in a squeezed Fock state basis, using engineered Hamiltonians analogous to the Jaynes-Cummings and anti-Jaynes-Cummings forms. We demonstrate that for squeezed Fock states with low n the engineered Hamiltonians reproduce the sqrt[n] scaling of the matrix elements which is typical of Jaynes-Cummings physics, and also examine deviations due to the finite wavelength of our control fields. Starting from a squeezed vacuum state, we apply sequences of alternating transfer pulses which allow us to climb the squeezed Fock state ladder, creating states up to excitations of n=6 with up to 8.7 dB of squeezing, as well as demonstrating superpositions of these states. These techniques offer access to new sets of states of the harmonic oscillator which may be applicable for precision metrology or quantum information science.

We directly observe the quantum interference between two well-separated trapped-ion mechanical oscillator wave packets. The superposed state is created from a spin-motion entangled state using a heralded measurement. Wave packet interference is observed through the energy eigenstate populations. We reconstruct the Wigner function of these states by introducing probe Hamiltonians which measure Fock state populations in displaced and squeezed bases. Squeezed-basis measurements with 8 dB squeezing allow the measurement of interference for Δα=15.6, corresponding to a distance of 240 nm between the two superposed wave packets.