Physical implementation of quantum gates acting on qubits does not achieve a perfect fidelity of 1. The actual output qubit may not match the targeted output of the desired gate. According to theoretical estimates, intrinsic gate fidelities >99.99% are necessary so that error correction codes can be used to achieve perfect fidelity. Here we test what fidelity can be accomplished for a CNOT gate executed by a shaped ultrafast laser pulse interacting with vibrational states of the molecule SCCl(2). This molecule has been used as a test system for low-fidelity calculations before. To make our test more stringent, we include vibrational levels that do not encode the desired qubits but are close enough in energy to interfere with population transfer by the laser pulse. We use two complementary approaches: optimal control theory determines what the best possible pulse can do; a more constrained physical model calculates what an experiment likely can do. Optimal control theory finds pulses with fidelity >0.9999, in excess of the quantum error correction threshold with 8 × 10(4) iterations. On the other hand, the physical model achieves only 0.9992 after 8 × 10(4) iterations. Both calculations converge as an inverse power law toward unit fidelity after >10(2) iterations/generations. In principle, the fidelities necessary for quantum error correction are reachable with qubits encoded by molecular vibrations. In practice, it will be challenging with current laboratory instrumentation because of slow convergence past fidelities of 0.99.

Readout of the final states of qubits is a crucial step towards implementing quantum computation in experiment. Although not scalable to large numbers of qubits per molecule, computational studies show that molecular vibrations could provide a significant (factor 2-5 in the literature) increase in the number of qubits compared to two-level systems. In this theoretical work, we explore the process of readout from vibrational qubits in thiophosgene molecule, SCCl 2 , using quantum beat oscillations. The quantum beats are measured by first exciting the superposition of the qubit-encoding vibrational states to the electronically excited readout state with variable time-delay pulses. The resulting oscillation of population of the readout state is then detected as a function of time delay. In principle, fitting the quantum beat signal by an analytical expression should allow extracting the values of probability amplitudes and the relative phases of the vibrational qubit states. However, we found that if this procedure is implemented using the standard analytic expression for quantum beats, a non-negligible phase error is obtained. We discuss the origin and properties of this phase error, and propose a new analytical expression to correct the phase error. The corrected expression fits the quantum beat signal very accurately, which may permit reading out the final state of vibrational qubits in experiments by combining the analytic fitting expression with numerical modelling of the readout process. The new expression is also useful as a simple model for fitting any quantum beat experiments where more accurate phase information is desired. © 2014 AIP Publishing LLC.

In the context of molecular quantum computation the optimal control theory (OCT) is used to obtain shaped laser pulses for high-fidelity control of vibrational qubits. Optimization is done in time domain and the OCT algorithm varies values of electric field in each time step independently, tuning hundreds of thousands of parameters to find one optimal solution. Such flexibility is not available in experiments, where pulse shaping is done in frequency domain and the number of "tuning knobs" is much smaller. The question of possible experimental interpretations of theoretically found OCT solutions arises. In this work we analyze very accurate optimal pulse that we obtained for implementing quantum gate CNOT for the two-qubit system encoded into the exited vibrational states of thiophosgene molecule. Next, we try to alter this pulse by reducing the number of available frequency channels and intentionally introducing systematic and random errors (in frequency domain, by modifying the values of amplitudes and phases of different frequency components). We conclude that a very limited number of frequency components (only 32 in the model of thiophosgene) are really necessary for accurate control of the vibrational two-qubit system, and such pulses can be readily constructed using OCT. If the amplitude and phase errors of different frequency components do not exceed ±3% of the optimal values, one can still achieve accurate transformations of the vibrational two-qubit system, with gate fidelity of CNOT exceeding 0.99.

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