A fundamental challenge of quantum computation is being able to scale up a large number of high fidelity quantum gates while noise and error are affecting the gate’s physical control parameters. This dissertation focuses on the fidelity of single-qubit quantum gates constructed by a change in quantum geometric phase, while the control parameters are affected by random noise and systematic error. A unified model of geometric quantum computation is developed, in which a qubit state is controlled by a composite Hamiltonian, resulting in a multiple-segment rotation of the quantum state and allowing characterization of evolution paths depending on the associated geometric and dynamic phase. The fidelity of the quantum gates in the presence of different noise error is compared for pu rely geometric, hybrid (having both geometric and dynamic phase), and conventional dynamic quantum gates built on single Hamiltonians. Results showed hybrid quantum gates had the highest fidelities, followed by geometric gates, and conventional dynamic gates had the lowest fidelities. In addition, there was indication in some cases higher fidelities result from gates created from a larger number of segments in the quantum state rotation. These results can be understood by the relation of the control parameters with the evolution path geometry. By translating between control parameters, our model can be applied to different systems for quantum computation, including: the laser manipulation of a two-level atom, laser manipulation of trapped ions, nuclear magnetic resonance, polarization states of photons, superconducting qubits in cavity QED, and quantum dots.
A fundamental challenge of quantum computation is being able to scale up a large number of high fidelity quantum gates while noise and error are affecting the gate’s physical control parameters. This dissertation focuses on the fidelity of single-qubit quantum gates constructed by a change in quantum geometric phase, while the control parameters are affected by random noise and systematic error. A unified model of geometric quantum computation is developed, in which a qubit state is controlled by a composite Hamiltonian, resulting in a multiple-segment rotation of the quantum state and allowing characterization of evolution paths depending on the associated geometric and dynamic phase. The fidelity of the quantum gates in the presence of different noise error is compared for pu rely geometric, hybrid (having both geometric and dynamic phase), and conventional dynamic quantum gates built on single Hamiltonians. Results showed hybrid quantum gates had the highest fidelities, followed by geometric gates, and conventional dynamic gates had the lowest fidelities. In addition, there was indication in some cases higher fidelities result from gates created from a larger number of segments in the quantum state rotation. These results can be understood by the relation of the control parameters with the evolution path geometry. By translating between control parameters, our model can be applied to different systems for quantum computation, including: the laser manipulation of a two-level atom, laser manipulation of trapped ions, nuclear magnetic resonance, polarization states of photons, superconducting qubits in cavity QED, and quantum dots.
Universal robust quantum control is essential for large-scale quantum computation. The geometric phase, as a key element with intrinsic error-resilient features, can be well integrated into a quantum control process to enhance control robustness. However, due to the inability to optionally avoid some trajectory segments seriously affected by systematic errors, the robust universality of current geometric quantum control based on some special geometric evolution trajectories is still lacking, resulting in unsatisfactory performance of arbitrary geometric gate. Here, we propose another scheme for universal robust geometric control based on geometric trajectory correction, where enough available evolution parameters are introduced, without more experimental demands, to ensure the effective reduction of sensitivity to systematic errors. Numerical simulation shows that our implemented arbitrary geometric gates have error-resilient features better than conventional ones. In addition, we also verify the feasibility of the high-fidelity physical implementation of our scheme on superconducting quantum circuits. Therefore, our theoretical work is expected to offer an attractive avenue for realizing practical quantum computation in existing experimental platforms.
Abstract Quantum computation based on nonadiabatic geometric phases has attracted a broad range of interests, due to its fast manipulation and inherent noise resistance. However, it is limited to some special evolution paths, and the gate-times are typically longer than conventional dynamical gates, resulting in weakening of robustness and more infidelities of the implemented geometric gates. Here, we propose a path-optimized scheme for geometric quantum computation (GQC) on superconducting transmon qubits, where high-fidelity and robust universal nonadiabatic geometric gates can be implemented, based on conventional experimental setups. Specifically, we find that, by selecting appropriate evolution paths, the constructed geometric gates can be superior to their corresponding dynamical ones under different local errors. Numerical simulations show that the fidelities for single-qubit geometric phase, π /8 and Hadamard gates can be obtained as 99.93%, 99.95% and 99.95%, respectively. Remarkably, the fidelity for two-qubit control-phase gate can be as high as 99.87%. Therefore, our scheme provides a new perspective for GQC, making it more promising in the application of large-scale fault-tolerant quantum computation.
Recently, nonadiabatic geometric quantum computation has been received great attentions, due to its fast operation and intrinsic error resilience. However, compared with the corresponding dynamical gates, the robustness of implemented nonadiabatic geometric gates based on the conventional single-loop geometric scheme still has the same order of magnitude due to the requirement of strict multi-segment geometric controls, and the inherent geometric fault-tolerance characteristic is not fully explored. Here, we present an effective geometric scheme combined with a general dynamical-corrected technique, with which the super-robust nonadiabatic geometric quantum gates can be constructed over the conventional single-loop geometric and two-loop composite-pulse geometric strategies, in terms of resisting the systematic error, i.e., σx error. In addition, combined with the decoherence-free subspace (DFS) coding, the resulting geometric gates can also effectively suppress the σz error caused by the collective dephasing. Notably, our protocol is a general one with simple experimental setups, which can be potentially implemented in different quantum systems, such as Rydberg atoms, trapped ions and superconducting qubits. These results indicate that our scheme represents a promising way to explore large-scale fault-tolerant quantum computation.
Universal single-qubit gates are constructed from a basic Bloch rotation operator realized through nonadiabatic Abelian geometric phase. The driving Hamiltonian in a generic two-level model is parameterized using controllable physical variables. The fidelity of the basic geometric rotation operator is investigated in the presence of systematic error in control parameters, such as the driving pulse area and frequency detuning. Compared to a conventional dynamic rotation, the geometric rotation shows improved fidelity.
High-fidelity manipulation is key to the physical realization of fault-tolerant quantum computation. Here, we present a protocol to realize universal nonadiabatic geometric gates for silicon-based spin qubits. We find that the advantage of geometric gates over dynamical gates depends crucially on the evolution loop for the construction of the geometric phase. Under appropriate evolution loops, all geometric single- and two-qubit gates can outperform their dynamical counterparts for both systematic and off-resonance noises. We also perform randomized benchmarking using noise amplitudes consistent with experiments in silicon. For the static noise model, the averaged fidelities of geometric gates are around 99.90% or above, while for the time-dependent 1/f-type noise, the fidelities are around 99.98% when only the off-resonance noise is present. We also show that the improvement in fidelities of the geometric gates over dynamical ones typically increases with the exponent $\ensuremath{\alpha}$ of the 1/f noise, and the ratio can be as high as 4 when $\ensuremath{\alpha}\ensuremath{\approx}3$. Our results suggest that geometric gates with judiciously chosen evolution loops can be a powerful way to realize high-fidelity quantum gates.
Quantum manipulation based on geometric phases provides a promising way towards robust quantum gates. However, in the current implementation of nonadiabatic geometric phases, operational and/or random errors tend to destruct the conditions that induce geometric phases, thereby smearing their noise-resilient feature. In a recent experiment [Y. Xu et al., Phys. Rev. Lett. 124, 230503 (2020)], high-fidelity universal geometric quantum gates have been implemented in a superconducting circuit, which are robust to different types of errors under different configurations of the geometric evolution paths. Here, we apply the path-design strategy to explain in detail why both configurations can realize universal quantum gates in a single-loop way. Meanwhile, we purposefully induce our geometric manipulation by selecting the path configuration that is robust against the qubit-frequency-drift-induced error, which is the dominant error source on realistic superconducting circuits and has not been deliberately addressed. Moreover, our proposal can further integrate with the composite scheme to enhance the gate robustness, which is verified by numerical simulations. Therefore, our scheme provides a promising way towards practical realization of high-fidelity and robust nonadiabatic geometric quantum gates.