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arXiv:1411.0157 · Physical Review A

Multilevel superconducting circuits as two-qubit systems: Operations, state preparation, and entropic inequalities

Evgeniy O. Kiktenko, Aleksey K. Fedorov, Olga V. Man’ko, Vladimir I. Man’ko

Published Apr 10, 2015• 88 citations
arXivHTMLPDF

Abstract

We theoretically study operations with a four-level superconducting circuit as a two-qubit system. Using a mapping on a two-qubit system, we show how to implement iswap gates and Hadamard gates through pulses on transitions between particular pairs of energy levels. Our approach allows one to prepare pure two-qubit entangled states with desired form of reduced density matrices of the same purity and, in particular, arbitrary identical reduced states of qubits. We propose using schemes for the Hadamard gate and two-qubit entangled states with identical reduced density matrices in order to verify $logN$ inequalities for Shannon and R\'enyi entropies for the considered noncomposite quantum system.

DOI
10.1103/physreva.91.042312

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We theoretically study operations with a four-level superconducting circuit as a two-qubit system. Using a mapping on a two-qubit system, we show how to implement iswap gates and Hadamard gates through pulses on transitions between particular pairs of energy levels. Our approach allows one to prepare pure two-qubit entangled states with desired form of reduced density matrices of the same purity and, in particular, arbitrary identical reduced states of qubits. We propose using schemes for the Hadamard gate and two-qubit entangled states with identical reduced density matrices in order to verify logNlogNlogN inequalities for Shannon and R'enyi entropies for the considered noncomposite quantum system.
Herein, a two‐qubit system is experimentally simulated through a four‐level superconducting circuit. By mapping the energy levels of the two‐qubit system to a noncomposite system, some two‐qubit gates with designed microwave pulses, such as Hadamard and CNOT gates, are implemented. Furthermore, the Shannon entropy of the simulated two‐qubit system is measured and the log 2 uncertainty relation is verified, which describes that the noncomposite single‐qudit system possesses hidden quantum correlations.
In article number 2100500, Xinsheng Tan, Yang Yu, and co-workers use a four-level superconducting quantum loop to simulate a two-qubit quantum system. By mapping the energy levels of the two-qubit system to a non-composite system, simulations of several quantum gate operations, including the CNOT gate and the Hadamard gate, are completed in a convenient way. Furthermore, the authors generate entangled states to verify the log2 entropic inequality. The results indicate that the superconducting qudit is indeed a huge quantum platform for quantum information.
We study a class of entropic inequalities obtained for noncomposite quantum system realized by a many-level superconducting circuit. By using a mapping on bipartite quantum states, we consider possible realization schemes of various quantum logic gates for noncomposite quantum systems. In this framework, we discuss logN entropic inequalities for Shannon and R eniy entropies based on the quantum Fourier transform. Implementation of the quantum Fourier transform on a quantum processor based on many-level superconducting circuits opens a way for experimental verication of logN entropic inequalities.
We show that the density-matrix states of noncomposite qudit systems satisfy entropic and information relations like the subadditivity condition, strong subadditivity condition, and Araki-Lieb inequality, which characterize hidden quantum correlations of observables associated with these indivisible systems. We derive these relations employing a specific map of the entropic inequalities known for density matrices of multiqudit systems to the inequalities for density matrices of single-qudit systems. We present the obtained relations in the form of mathematical inequalities for arbitrary Hermitian N × N -matrices. We consider examples of superconducting qubits and qudits. We discuss the hidden correlations in single- qudit states as a new resource for quantum technologies analogous to the known resource in correlations associated with the entanglement in multiqudit systems.
Compared with a qubit, a qudit (i.e., a ddd-level or ddd-state quantum system) provides a larger Hilbert space to store and process information. On the other hand, qudit-based hybrid quantum computing usually requires performing hybrid quantum gates with qudits that are different in their nature or in their encoding format. In this work, we consider the qutrit case, i.e., the case for a qudit with d=3d=3d=3. We propose a simple method to realize a hybrid quantum controlled-sum gate with one superconducting (SC) qutrit and a cat-state qutrit. This gate plus single-qutrit gates form a universal set of ternary logic gates for quantum computing with qutrits. Our proposal is based on circuit QED and operates essentially by employing a SC ququart (a four-level quantum system) dispersively coupled to a microwave cavity. The gate implementation is quite simple because it only requires a single basic operation. Neither classical pulse nor measurement is needed. The auxiliary higher energy level of the SC ququart is virtually excited during the gate operation, thus decoherence from this level is greatly suppressed. As an application of this gate, we discuss the generation of a hybrid maximally entangled state of one SC qutrit and one cat-state qutrit. We further analyze the experimental feasibility of creating such a hybrid entangled state in circuit QED. This proposal is quite general and can be extended to accomplish the same task in a wide range of physical system, such as a four-level natural or artificial atom coupled to an optical or microwave cavity.
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.