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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 Mølmer-Sørensen (MS) 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 so that the laser amplitude may be represented as a general continuous function. Varying the Fourier coefficients used to generate the pulse produces trade-offs between the laser power, gate time, and fidelity. We specifically study gate-timing errors, and we have shown that the sensitivity of the fidelity to these errors can be improved without a significant increase in the average laser power or the gate time. We plot atomic population vs. time for both the traditional MS protocol and the protocol with AM, highlighting the increased robustness of the AM gates. Our central result is that we improve the leading order dependence on gate timing errors from O(∆t2 ) to O(∆t6 ), and the protocol allows for arbitrarily high orders of scaling to be achieved in principle.

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 Molmer-Sorensen (MS) 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 \\(\\mathcal{O}(\\Delta t^2)\\) to \\(\\mathcal{O}(\\Delta t^6)\\) with the addition of one linear constraint on the Fourier coefficients and to \\(\\mathcal{O}(\\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.

Quantum computers and simulators promise to enable the study of strongly correlated quantum systems. Yet, surprisingly, it is hard for them to compute ground states. They can, however, efficiently compute the dynamics of closed quantum systems. We propose a method to study the quantum thermodynamics of strongly correlated electrons from quantum dynamics. We define time-averaged classical shadows (TACS) and prove it is a classical shadow(CS) of the von Neumann ensemble, the time-averaged density matrix. We then show that the diffusion maps, an unsupervised machine learning algorithm, can efficiently learn the phase diagram and phase transition of the one-dimensional transverse field Ising model both for ground states using CS \\emph{and state trajectories} using TACS. It does so from state trajectories by learning features that appear to be susceptibility and entropy from a total of 90,000 shots taken along a path in the microcanonical phase diagram. Our results suggest a low number of shots from quantum simulators can produce quantum thermodynamic data with a quantum advantage.