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Scientific Reports

Superconductor qubits hamiltonian approximations effect on quantum state evolution and control

Javad Sharifi

Published Jun 17, 2021• 2 citations
Publisher

Abstract

Microwave IQ-mixer controllers are designed for the three approximated Hamiltonians of charge, phase and flux qubits and the controllers are exerted both on approximate and precise quantum system models. The controlled qubits are for the implementation of the two quantum-gates with these three fundamental types of qubits, Quantum NOT-gate and Hadamard-gate. In the charge-qubit, for implementation of both gates, in the approximated and precise model, we observed different controlled trajectories. But fortunately, applying the controller designed for the approximated system over the precise system leads to the passing of the quantum state from the desired state sooner that the expected time. Phase-qubit and flux qubit have similar behaviour under the control system action. In both of them, the implementation of NOT-gate operation led to same trajectories which arrive at final goal state at different times. But in both of those two qubits for implementation of Hadamard-gate, desired trajectory and precise trajectory have some angle of deviation, then by exerting the approximated design controller to precise system, it caused the quantum state to approach the goal state for Hadamard gate implementation, and since the quantum state does not completely reach the goal state, we can not obtain very high gate fidelity.

DOI
10.1038/s41598-021-92290-0

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Microwave IQ-mixer controllers are designed for the three approximated Hamiltonians of charge, phase and flux qubits and the controllers are exerted both on approximate and precise quantum system models. The controlled qubits are for the implementation of the two quantum-gates with these three fundamental types of qubits, Quantum NOT-gate and Hadamard-gate. In the charge-qubit, for implementation of both gates, in the approximated and precise model, we observed different controlled trajectories. But fortunately, applying the controller designed for the approximated system over the precise system leads to the passing of the quantum state from the desired state sooner that the expected time. Phase-qubit and flux qubit have similar behaviour under the control system action. In both of them, the implementation of NOT-gate operation led to same trajectories which arrive at final goal state at different times. But in both of those two qubits for implementation of Hadamard-gate, desired trajectory and precise trajectory have some angle of deviation, then by exerting the approximated design controller to precise system, it caused the quantum state to approach the goal state for Hadamard gate implementation, and since the quantum state does not completely reach the goal state, we can not obtain very high gate fidelity.
Quantum state on Bloch sphere for superconducting charge qubit, phase qubit and flux qubit for all time in absence of external drive is stable to initial state. By driving the qubits, approximation of charge and flux Hamiltonian lead to quantum state rotation in Bloch sphere around an axis completely differ from rotation vector of exact Hamiltonian. The trajectory of quantum state for phase qubit for approximated and exact Hamiltonian is the same but the expectation of quantum observable has considerable errors as two other qubits. microwave drive control is designed for approximated Hamiltonian and exerted on actual systems and shows completely different trajectory with respect to desired trajectory. Finally a nonlinear control with external μμμV voltage control and nA current control is designed for general qubit which completely stabilizes quantum state toward a desired state.
Abstract Quantum state on Bloch sphere for superconducting charge qubit, phase qubit and flux qubit for all time in absence of external drive is stable to initial state. By driving the qubits, approximation of charge and flux Hamiltonian lead to quantum state rotation in Bloch sphere around an axis completely differ from rotation vector of exact Hamiltonian. The trajectory of quantum state for phase qubit for approximated and exact Hamiltonian is the same but the expectation of quantum observable has considerable errors as two other qubits. microwave drive control is designed for approximated Hamiltonian and exerted on actual systems and shows completely different trajectory with respect to desired trajectory. Finally a nonlinear control with external µV voltage control and nA current control is designed for general qubit which completely stabilizes quantum state toward a desired state.
Superconducting devices as quantum bits (qubits) are one of the most promising candidates for implementing a practical quantum computer. Over the last decade, quantum coherence times of superconducting qubits have been continuously improved. Operating qubits in noise insensitive points, eliminating the participation of the lossy materials, and better isolation from environment increased the decoherence times from nanoseconds to tens of microseconds. One recent approach on the qubit design is to shunt the Josephson junctions with large capacitors to reduce the sensitivity to noise and to redistribute the stored electromagnetic energy in a well-engineered area. This approach was first applied on charge qubits and later adopted to other types of superconducting qubits. \n \nIn this thesis, we present the design and the characterization of a three-Josephson junction superconducting circuit with three large shunt capacitors. We present experiments on decoherence in the circuit used as a qubit. The qubit relaxation time T₁ is reaching as high as 47 μs and the spin-echo dephasing time T₂ = 9.4 μs. In addition, we present spectroscopy experiments and decoherence characterization in the qutrit subspace formed by the lowest three energy levels. The spectroscopy data is in excellent agreement with the complete circuit model based on the system capacitance matrix. At the flux symmetry point, the circuit has a large anharmonicity, defined as the difference between the 1-2 and 0-1 transition frequencies, approaching 2π × 3.7 GHz, which is enabling fast single qubit operations. We performed randomized benchmarking with the qubit gates with a duration of 1.62 ns and an average gate fidelity of 99.9%. \n \nMoreover, we present experiments with multi-level control in qutrit subspace, formed by the lowest three energy levels. We demonstrate a single step experimental implementation of a generalized Walsh-Hadamard gate. We use a decomposition of the quantum gate into two unitary operations, one implemented by an off-diagonal Hamiltonian and the other implemented by a diagonal Hamiltonian. The off-diagonal Hamiltonian is obtained by the simultaneous driving of the transitions between levels 0-1, 1-2, and 0-2, with the latter being a two-photon process. The diagonal Hamiltonian is effectively implemented by appropriately shifting the phases of adjacent pulses. The gate is characterized using tomography process and the average fidelity exceeds 90%, in good agreement with numerical simulations that take into account the multi-level structure of the system.
We present theoretical schemes for performing high-fidelity one- and two-qubit pulsed gates for a superconducting flux qubit. The 'IBM qubit' consists of three Josephson junctions, three loops and a superconducting transmission line. Assuming a fixed inductive qubit–qubit coupling, we show that the effective qubit–qubit interaction is tunable by changing the applied fluxes, and can be made negligible, allowing one to perform high-fidelity single qubit gates. Our schemes are tailored to alleviate errors due to 1/ f noise; we find gates with only 1% loss of fidelity due to this source, for pulse times in the range of 20–30 ns for one-qubit gates ( Z rotations, Hadamard) and 60 ns for a two-qubit gate (controlled- Z ). Our relaxation and dephasing time estimates indicate a comparable loss of fidelity from this source. The control of leakage plays an important role in the design of our shaped pulses, preventing shorter pulse times. However, we have found that imprecision in the control of the quantum phase plays a major role in the limitation of the fidelity of our gates.
Despite progress towards achieving low error rates with superconducting qubits, error-prone two-qubit gates remain a bottleneck for realizing large-scale quantum computers. Therefore, a systematic framework to design high-fidelity gates becomes imperative. One type of two-qubit gate in superconducting qubits is the controlled-phase (CPHASE) gate, which utilizes a conditional interaction between higher energy levels of the qubits controlled by a baseband flux pulse on one of the qubits or a tunable coupler. In this work, we study an adiabatic implementation of CPHASE gates and formulate the design of the control trajectory for the gate as a pulse-design problem. We show in simulation that the Chebyshev-based trajectory can, in certain cases, enable gates with gate infidelity lower by an average of 23.3% when compared to the widely used Slepian-based trajectory.
We present a novel method to realize a multi-target-qubit controlled phase gate with one microwave photonic qubit simultaneously controlling n - 1 target microwave photonic qubits. This gate is implemented with n microwave cavities coupled to a superconducting flux qutrit. Each cavity hosts a microwave photonic qubit, whose two logic states are represented by the vacuum state and the single photon state of a single cavity mode, respectively. During the gate operation, the qutrit remains in the ground state and thus decoherence from the qutrit is greatly suppressed. This proposal requires only a single-step operation and thus the gate implementation is quite simple. The gate operation time is independent of the number of the qubits. In addition, this proposal does not need applying classical pulse or any measurement. Numerical simulations demonstrate that high-fidelity realization of a controlled phase gate with one microwave photonic qubit simultaneously controlling two target microwave photonic qubits is feasible with current circuit QED technology. The proposal is quite general and can be applied to implement the proposed gate in a wide range of physical systems, such as multiple microwave or optical cavities coupled to a natural or artificial Λ-type three-level atom.