Numerical optimization of amplitude-modulated pulses in microwave-driven entanglement generation
Markus Duwe, Giorgio Zarantonello, Nicolas Pulido-Mateo, H. Mendpara, Ludwig Krinner, A. Bautista-Salvador, Nikolay V. Vitanov, Klemens Hammerer, Reinhard F. Werner, C. Ospelkaus
Abstract Microwave control of trapped ions can provide an implementation of high-fidelity two-qubit gates free from errors induced by photon scattering. Furthermore, microwave conductors may be embedded into a scalable trap structure, providing the chip-level integration of control that is desirable for scaling. Recent developments have demonstrated how amplitude modulation of the gate drive can permit a two-qubit entangling operation to become robust against motional mode noise and other experimental imperfections. Here, we discuss a method for the numerical optimization of the microwave pulse envelope to produce gate pulses with noise resilience, considerably faster operation and high energy efficiency.
Abstract Microwave control of trapped ions can provide an implementation of high-fidelity two-qubit gates free from errors induced by photon scattering. Furthermore, microwave conductors may be embedded into a scalable trap structure, providing the chip-level integration of control that is desirable for scaling. Recent developments have demonstrated how amplitude modulation of the gate drive can permit a two-qubit entangling operation to become robust against motional mode noise and other experimental imperfections. Here, we discuss a method for the numerical optimization of the microwave pulse envelope to produce gate pulses with noise resilience, considerably faster operation and high energy efficiency.
Trapped ions, together with superconducting qubits, are one of the two leading hardware platforms for scalable quantum information processing. The development of quantum computers represents a major technological breakthrough comparable to the introduction of classical computing. The benefits of this technology are currently limited by the technical capability to perform high fidelity entangling operations on the qubits. When gate fidelities surpass the fault-tolerance threshold it becomes possible, through error correction, to increase the system size to an arbitrary number of qubits. In this cumulative thesis we address some of the issues in the scalability of the trapped-ion quantum computer based on microwave near-fields. In this approach, gate operations on one or multiple ions are driven by an oscillating magnetic field generated by a current flowing through a conductor. In the first part of this work we discuss the design of traps toward the implementation of large scale systems. We introduce the basic design of a surface-electrode ion trap with embedded microwave conductors. The oscillating magnetic field required to perform the operations is generated by a single optimized conductor. We discuss the simulation and characterization of the magnetic field pattern, which is fixed by the microwave conductor design. In addition, we demonstrate the capability to simulate, fabricate and characterize a multilayer surface ion trap. Multilayer traps are a key aspect for scalability since they are necessary to achieve large system sizes, with many `ion registers', where the registers are interconnected by physically transporting ions between them. In the second part of this thesis we demonstrate the implementation of a two-qubit entangling gate and we explore the possibilities offered by quantum control methods to improve its fidelity. We perform an entangling gate on two 9Be+ ions and measure a Bell state fidelity of 98.2(1.2)%. Error characterization shows that the gate result is limited by technical issues connected to the ions' motional states. To reduce these errors we apply amplitude modulation of the gate microwave drive. After stabilization of the ions' radial modes, we obtain an amplitude modulated gate with infidelity in the 10^-3 range. The result is confirmed by analyzing the data using three different methods. Using additional dynamic decoupling techniques, these results could bring microwave near-field gates past the fault-tolerance threshold.
Microwave trapped-ion quantum logic gates avoid spontaneous emission as a fundamental source of decoherence. However, microwave two-qubit gates are still slower than laser-induced gates and hence more sensitive to fluctuations and noise of the motional mode frequency. We propose and implement amplitude-shaped gate drives to obtain resilience to such frequency changes without increasing the pulse energy per gate operation. We demonstrate the resilience by noise injection during a two-qubit entangling gate with ^{9}Be^{+} ion qubits. In the absence of injected noise, amplitude modulation gives an operation infidelity in the 10^{-3} range.
Almost every quantum circuit is built with two-qubit gates in the current stage, which are crucial to the quantum computing in any platform. The entangling gates based on Mølmer-Sørensen schemes are widely exploited in the trapped-ion system, with the utilization of the collective motional modes of ions and two laser-controlled internal states, which are served as qubits. The key to realize high-fidelity and robust gates is the minimization of the entanglement between the qubits and the motional modes under various sources of errors after the gate operation. In this work, we propose an efficient numerical method to search high-quality solutions for phase-modulated pulses. Instead of directly optimizing a cost function, which contains the fidelity and the robustness of the gates, we convert the problem to the combination of linear algebra and the solution to quadratic equations. Once a solution with the gate fidelity of 1 is found, the laser power can be further reduced while searching on the manifold where the fidelity remains 1. Our method largely overcomes the problem of the convergence and is shown to be effective up to 60 ions, which suffices the need of the gate design in current trapped-ion experiments.
Towards the goal of a large-scale quantum computer based on trapped ions, near-field microwaves represent a promising approach to perform the techni- cally challenging key operation of a two-qubit entangling gate. In this thesis we present the first microwave-driven two-qubit gate in 9Be+ ions employing a first-order field-independent qubit transition and a scalable surface-electrode ion trap at room temperature. We test the quality of the gate operation by producing a maximally entangled state and measuring the resulting state preparation fidelity in a reduced tomography procedure. For the best two- qubit gate achieved in the system we find this fidelity to be F = 98.2 ± 1.2 %. Following a comprehensive error analysis based on numerical simulations and experimentally determined input parameters, we identify current infidelity con- tributions of the apparatus. Here, the natural error source of the microwave near-field approach, namely fluctuating AC Zeeman shifts, could be reduced to the 10^−4 level due to the optimized design of the employed microwave con- ductor. As we find that the three largest errors can all be reduced upon purely technical improvements, higher fidelities are feasible in the future. Besides the gate realization, the thesis also comprises the initial characterization of the employed ion trap as well as the design and construction of a Raman laser at ∼ 313 nm which is utilized to perform near ground state cooling of radial modes of a single- and two-ion crystal. Here, ensuing heating rate measure- ments on a single-ion’s radial modes show good agreement with the electric- field noise spectral density expected from literature given the chosen mode frequency and nearest ion-to-electrode distance of 70 µm. Finally, we present the operation of a first multi-layer ion trap whose electrode layout features a 3-dimensional microwave conductor intended for performing microwave-driven two-qubit gates. Following a characterization of the resulting near-field pat- tern using a single ion as a local field probe, we find the multi-layer conductor to have significantly better field properties when compared to an equivalent single-layer design. Given these promising results, future work will focus on the integration of similar microwave circuitry in a multi-zone trap array as envisioned by the QCCD architecture.
Abstract Robust two-qubit gates are essential for scalable silicon spin qubits, but their fidelities are limited by microwave-control imperfections and charge-noise-induced exchange fluctuations. We propose a practical scheme for robust controlled-Z (CZ) and controlled-NOT (CNOT) gates in a silicon double quantum dot. By applying a transverse microwave drive, the effective Hamiltonian can be decomposed into two driven subspaces. In the subspace mainly controlled by the microwave field, a Broadband 1 (BB1) composite pulse is used to suppress errors in the microwave amplitude. In the exchange-assisted subspace, the microwave phases and durations are kept fixed, while a few piecewise-constant exchange amplitudes are optimized by a GRAPE-like procedure. This design reduces the sensitivity to both microwave-amplitude and exchange-amplitude errors while keeping the control waveform simple and experimentally friendly. Numerical simulations show that the optimized CNOT gate reaches a fidelity above 99.99% under experimentally relevant noise strengths, and the CZ gate also maintains a fidelity above 99.9%. These results indicate that combining composite pulses with optimized exchange control provides an experimentally feasible route to robust two-qubit gates in silicon spin qubits.
Trapped-ion systems are a promising route toward the realization of both near-term and universal quantum computers. However, one of the pressing challenges is improving the fidelity of two-qubit entangling gates. These operations are often implemented by addressing individual ions with laser pulses using the M\o{}lmer-S\o{}rensen protocol. Amplitude modulation (AM) is a well-studied extension of this protocol, where the amplitude of the laser pulses is controlled as a function of time. We present an analytical study of AM, using a Fourier series expansion to maintain the generality of the laser amplitude's functional form. We then apply this general AM method to gate-timing errors by imposing conditions on these Fourier coefficients, producing trade-offs between the laser power and fidelity at a fixed gate time. The conditions derived here are linear and can be used, in principle, to achieve arbitrarily high orders of insensitivity to gate-timing errors. Numerical optimization is then employed to identify the minimum-power pulse satisfying these constraints. Our central result is that the leading-order dependence on gate-timing errors is improved from $O(\mathrm{\ensuremath{\Delta}}{t}^{2})$ to $O(\mathrm{\ensuremath{\Delta}}{t}^{6})$ with the addition of one linear constraint on the Fourier coefficients and to $O(\mathrm{\ensuremath{\Delta}}{t}^{10})$ with two linear constraints without a significant increase in the average laser power. The increase approaches zero as more Fourier coefficients are included. In further studies, this protocol can be applied to other error sources and used in conjunction with other error-mitigation techniques to improve two-qubit gates.