Multiport Impedance Quantization

With the increase of complexity and coherence of superconducting systems made using the principles of circuit quantum electrodynamics, more accurate methods are needed for the characterization, analysis and optimization of these quantum processors. Here we introduce a new method of modelling that can be applied to superconducting structures involving multiple Josephson junctions, high-Q superconducting cavities, external ports, and voltage sources. Our technique, an extension of our previous work on single-port structures [1], permits the derivation of system Hamiltonians that are capable of representing every feature of the physical system over a wide frequency band and the computation of T1 times for qubits. We begin with a black box model of the linear and passive part of the system. Its response is given by its multiport impedance function Zsim(w), which can be obtained using a finite-element electormagnetics simulator. The ports of this black box are defined by the terminal pairs of Josephson junctions, voltage sources, and 50 Ohm connectors to high-frequency lines. We fit Zsim(w) to a positive-real (PR) multiport impedance matrix Z(s), a function of the complex Laplace variable s. We then use state-space techniques to synthesize a finite electric circuit admitting exactly the same impedance Z(s) across its ports; the PR property ensures the existence of this finite physical circuit. We compare the performance of state-space algorithms to classical frequency domain methods, justifying their superiority in numerical stability. The Hamiltonian of the multiport model circuit is obtained by using existing lumped element circuit quantization formalisms [2, 3]. Due to the presence of ideal transformers in the model circuit, these quantization methods must be extended, requiring the introduction of an extension of the Kirchhoff voltage and current laws

Blackbox Quantization of Superconducting Circuits using exact Impedance Synthesis

We propose a new quantization method for superconducting electronic circuits involving a Josephson junction device coupled to a linear microwave environment. The method is based on an exact impedance synthesis of the microwave environment considered as a blackbox with impedance function Z(s). The synthesized circuit captures dissipative dynamics of the system with resistors coupled to the reactive part of the circuit in a non-trivial way. We quantize the circuit and compute relaxation rates following previous formalisms for lumped element circuit quantization. Up to the errors in the fit our method gives an exact description of the system and its losses

Multi-qubit parity measurement in circuit quantum electrodynamics

We present a concept for performing direct parity measurements on three or more qubits in microwave structures with superconducting resonators coupled to Josephson-junction qubits. We write the quantum-eraser conditions that must be fulfilled for the parity measurements as requirements for the scattering phase shift of our microwave structure. We show that these conditions can be fulfilled with present-day devices. We present one particular scheme, implemented with two-dimensional cavity techniques, in which each qubit should be coupled equally to two different microwave cavities. The magnitudes of the couplings that are needed are in the range that has been achieved in current experiments. A quantum calculation indicates that the measurement is optimal if the scattering signal can be measured with near single photon sensitivity. A comparison with an extension of a related proposal from cavity optics is presented. We present a second scheme, for which a scalable implementation of the four-qubit parities of the surface quantum error correction code can be envisioned. It uses three-dimensional cavity structures, using cavity symmetries to achieve the necessary multiple resonant modes within a single resonant structure

Analysis and Synthesis of Multi-Qubit, Multi-Mode Quantum Devices

In this thesis we propose new methods in multi-qubit multi-mode circuit quantum electrodynamics (circuit-QED) architectures. In Chapter 2 we describe a direct parity measurement method for three qubits, which can be realized in 2D circuit-QED with a possible extension to four qubits in a 3D circuit-QED setup for the implementation of the surface code. In Chapter 3 we show how to derive Hamiltonians and compute relaxation rates of the multi-mode superconducting microwave circuits consisting of single Josephson junctions using an exact impedance synthesis technique (the Brune synthesis) and applying previous formalisms for lumped element circuit quantization. In the rest of the thesis we extend our method in Chapter 3 to multi-junction (multi-qubit) multi-mode circuits through the use of state-space descriptions which allows us to quantize any multiport microwave superconducting circuit with a reciprocal lossy impedance response

Analysis and Synthesis of Multi-Qubit, Multi-Mode Quantum Devices

In this thesis we propose new methods in multi-qubit multi-mode circuit quantum electrodynamics (circuit-QED) architectures. In Chapter 2 we describe a direct parity measurement method for three qubits, which can be realized in 2D circuit-QED with a possible extension to four qubits in a 3D circuit-QED setup for the implementation of the surface code. In Chapter 3 we show how to derive Hamiltonians and compute relaxation rates of the multi-mode superconducting microwave circuits consisting of single Josephson junctions using an exact impedance synthesis technique (the Brune synthesis) and applying previous formalisms for lumped element circuit quantization. In the rest of the thesis we extend our method in Chapter 3 to multi-junction (multi-qubit) multi-mode circuits through the use of state-space descriptions which allows us to quantize any multiport microwave superconducting circuit with a reciprocal lossy impedance response

Simple Impedance Response Formulas for the Dispersive Interaction Rates in the Effective Hamiltonians of Low Anharmonicity Superconducting Qubits

For superconducting quantum processors consisting of low anharmonicity qubits such as transmons, we give a complete microwave description of the system in the qubit subspace. We assume that the qubits are dispersively coupled to a distributed microwave structure such that the detunings of the qubits from the internal modes of the microwave structure are stronger than their couplings. We define “qubit ports” across the terminals of the Josephson junctions and “drive ports” where transmission lines carrying drive signals reach the chip and we obtain the multiport impedance response of the linear passive part of the system between the ports. We then relate interaction parameters in between qubits and between the qubits and the environment to the entries of this multiport impedance function; in particular, we show that the exchange coupling rate J between qubits is related in a simple way to the off-diagonal entry connecting the qubit ports. Similarly, we relate couplings of the qubits to voltage drives and lossy environment to the entries connecting the qubits and the drive ports. Our treatment takes into account all the modes (possibly infinite) that might be present in the distributed electromagnetic structure and provides an efficient method for the modeling and analysis of the circuits