We have implemented a Walsh-Hadamard gate, which performs a quantum Fourier transform, in a superconducting qutrit. The qutrit is encoded in the lowest three energy levels of a capacitively shunted flux device, operated at the optimal flux-symmetry point. We use an efficient decomposition of the Walsh-Hadamard gate into two unitaries, generated by off-diagonal and diagonal Hamiltonians, respectively. The gate implementation utilizes simultaneous driving of all three transitions between the three pairs of energy levels of the qutrit, one of which is implemented with a two-photon process. The gate has a duration of 35 ns and an average fidelity over a representative set of states, including preparation and tomography errors, of 99.2%, characterized with quantum-state tomography. Compensation of ac-Stark and Bloch-Siegert shifts is essential for reaching high gate fidelities.
We have implemented a Walsh-Hadamard gate, which performs a quantum Fourier transform, in a superconducting qutrit. The qutrit is encoded in the lowest three energy levels of a capacitively shunted flux device, operated at the optimal flux-symmetry point. We use an efficient decomposition of the Walsh-Hadamard gate into two unitaries, generated by off-diagonal and diagonal Hamiltonians, respectively. The gate implementation utilizes simultaneous driving of all three transitions between the three pairs of energy levels of the qutrit, one of which is implemented with a two-photon process. The gate has a duration of 35 ns and an average fidelity over a representative set of states, including preparation and tomography errors, of 99.2%, characterized with quantum-state tomography. Compensation of ac-Stark and Bloch-Siegert shifts is essential for reaching high gate fidelities.
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 characterize control of a qutrit implemented in the lowest three energy levels of a capacitively shunted flux-biased superconducting circuit.Randomized benchmarking over the qutrit Clifford group yields an average fidelity of 98.89 ± 0.05%.For a selected subset of the Clifford group, we perform quantum process tomography and observe the behavior of repeated gate sequences.Each qutrit gate is generated using only two-state rotations via a method applicable to any unitary.We find that errors are due primarily to decoherence and have a significant contribution from level shifts.This work demonstrates high-fidelity qutrit control and outlines avenues for future work on the optimal control of superconducting qudits.
Superconducting devices have emerged as leading candidates for building practical quantum computers, owing to their scalability, designability, and ease of control. Development of superconducting devices has been accompanied by increases in coherence times from nanoseconds to tens of microseconds, and a multitude of new types of superconducting qubits. In turn, these improvements in coherence, and study of superconducting qubits has produced models for these devices accurate and trustworthy enough to enable design decisions to be determined prior to device fabrication. Recent work by the superconducting quantum devices (SQD) group has produced a variant of the capacitively-shunted flux qubit that features fast control over the lowest three levels, allowing the device to be used as a qutrit. This thesis presents a model and characterization results for this device. \n \nThis thesis also presents experimental results characterizing control fidelity for a qutrit using randomized benchmarking (RB). Quantum process tomography and application of repeated gate sequences are also used to support the randomized benchmarking results. The method of synthesizing the gates in the qutrit Clifford group can decompose any gate into two-state rotations. The fidelity of 99.0 +/- 0.2% obtained by randomized benchmarking establishes the device as a useful qutrit, and outlines interesting directions in using higher levels to improve qubit-based quantum computers, and in using higher levels to pack more states into fewer devices in quantum computers. Analysis of experimental results indicates errors are dominated by ac-Stark and Bloch-Siegert shifts during qutrit control. \n \nThis thesis then presents two new devices meant to study the impact of changing design parameters on the suitability of flux qubits for multilevel control. The first device test individual qubits, with the goal of further improving the coherence time, and improving qutrit control. The second device couples two devices together, so that two-qubit gates can be studied. The available design parameters and their impact on qubit properties are discussed, and design variations are selected for the single and two-qubit devices. The result is a set of devices ready for manufacture and experimental characterization.
Although only two quantum states of a physical system are often used to encode quantum information in the form of qubits, many levels can in principle be used to obtain qudits and increase the information capacity of the system. To take advantage of the additional levels, a parameterization of unitary transformations in terms of experimentally realizable operations is needed. Many parameterizations of unitary 3 * 3 matrices (U(3)) exist. One decomposition of a general unitary matrix can be expressed as the product of an exponential of a diagonal matrix and an exponential of an off-diagonal matrix. This decomposition is relevant for controlling superconducting qutrits using fixed-frequency resonant control pulses. This decomposition is numerically confirmed to allow the parameterization of any element in U(3). It is shown that a simple setting of parameter ranges of parameters can easily lead to an over-parameterization, in the sense that several different sets of values for the parameters produce the same element in U(3). This fact is demonstrated using the Walsh-Hadamard (WH) matrix as an example, which is also a special qutrit gate of practical interest. The different decompositions are shown to be related, and the relationships between them are presented using general methods. The shortest path needed for the implementation of a qutrit gate is found. Other parameterizations obtained by other analytic means, which can be advantageous for various reasons, are also discussed.
High-dimensional quantum systems, such as qutrits (quantum three-level systems), have multiple accessible energy levels beyond the two-level qubits. Therefore, qutrits can offer a larger state space to improve the efficiency of quantum computation. Here, we demonstrate a high-fidelity iSWAP-like gate operation on a frequency-tunable superconducting qutrits system. The superconducting quantum system consists of two qutrits that are coupled via a resonator with fixed qutrit-resonator coupling strengths. Through designing the frequency pulse profile and optimizing the parameter values, the gate error can be suppressed below 1.5 × 10−3. To bear out the feasibility of the proposal, we have conducted our study with experimentally accessible parameters. As the resonator can mediate the interaction between the irrelevant qutrits, the presented approach can also be used to couple multiple qutrits together, providing a good platform for quantum information processing.
Compared to a qubit, a qutrit (a three-level or three-state quantum system) possesses a larger Hilbert space to process and store quantum information. On the other hand, large-scale qutrit-based hybrid quantum computing usually requires performing hybrid multi-qutrit quantum gates with diverse qutrits, different in their nature or in their encoding format. In this work, we consider two types of qutrits, i.e., superconducting (SC) qutrits and cat-state qutrits. We propose to implement a hybrid controlled-SUM gate with one SC qutrit simultaneously controlling multiple-target cat-state qutrits. The gate is implemented in a circuit-QED system, which is composed of an SC ququart and multiple microwave cavities. The SC ququart here refers to a four-level quantum system, with the three lowest levels forming a qutrit and an auxiliary higher energy level utilized for the coherent state manipulation. The gate implementation does not require applying a classical pulse. Because the auxiliary higher energy level of the SC ququart is only virtually excited during the gate operation, decoherence from this level is greatly suppressed. The gate is deterministic, as it requires no measurement of the cavity or SC ququart states. Moreover, the gate operational time is independent of the number of qutrits, thus it does not increase with the number of qutrits. As an application of this gate, we further discuss the generation of a hybrid maximally entangled state of one SC qutrit and multiple cat-state qutrits. We also numerically analyze the experimental feasibility of creating the hybrid entangled state of one SC qutrit and two cat-state qutrits in a circuit QED system. This proposal may be extended to accomplish the same task in other physical systems, such as a four-level artificial atom (e.g., a quantum dot, an NV center, a magnon, etc.) coupled to multiple optical or microwave cavities.