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IEEE International Conference on Quantum Computing and Engineering (QCE)

Developing SLH Theory for Use with Microwave Circuits

Antonio Cobarrubia, K. M. Sundqvist

Published Oct 1, 2021• 0 citations
Publisher

Abstract

Circuit modeling and simulation techniques are necessary to the development of advanced microwave architectures. In particular, microwave circuits used in the quantum regime can produce a variety of single-photon phenomena related to qubits such as unitary gate operations, quantum illumination, and quantum interferometry. However, classical microwave formalism is incomplete to sufficiently describe these systems at this level. Noisy environments, for instance, destroy internal quantum information at the single-photon level, which makes it difficult to reliably control a quantum state. A theoretical infrastructure to account for this problem has been well-established in quantum optics as quantum input-output network (QION) theory. In this work, we consider an extension to QION theory, called SLH theory, to expand quantum stochastic differential equations to multi-port networks. A mathematical object composed of a network’s scattering parameters (S), coupling vector (L), and internal energy (H), known as an SLH triple, can fully describe its dynamical evolution at the quantum level. We show how microwave components can be converted into an SLH framework that retains the mathematical formalism of classical S-parameter techniques at the quantum limit. This makes SLH theory applicable to many interdisciplinary subfields within physics and engineering. We describe how a dynamical circuit model with universal quantum multi-port networks can be devised within SLH formalism. This can be applied to microwave circuit designs in the quantum limit.

DOI
10.1109/qce52317.2021.00077

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•Jan 1, 2009•0 citations
Circuit modeling and simulation techniques are necessary to the development of advanced microwave architectures. In particular, microwave circuits used in the quantum regime can produce a variety of single-photon phenomena related to qubits such as unitary gate operations, quantum illumination, and quantum interferometry. However, classical microwave formalism is incomplete to sufficiently describe these systems at this level. Noisy environments, for instance, destroy internal quantum information at the single-photon level, which makes it difficult to reliably control a quantum state. A theoretical infrastructure to account for this problem has been well-established in quantum optics as quantum input-output network (QION) theory. In this work, we consider an extension to QION theory, called SLH theory, to expand quantum stochastic differential equations to multi-port networks. A mathematical object composed of a network’s scattering parameters (S), coupling vector (L), and internal energy (H), known as an SLH triple, can fully describe its dynamical evolution at the quantum level. We show how microwave components can be converted into an SLH framework that retains the mathematical formalism of classical S-parameter techniques at the quantum limit. This makes SLH theory applicable to many interdisciplinary subfields within physics and engineering. We describe how a dynamical circuit model with universal quantum multi-port networks can be devised within SLH formalism. This can be applied to microwave circuit designs in the quantum limit.
Classical microwave circuit theory is incapable of representing some phenomena at the quantum level. To include quantum statistical effects, various theoretical treatments can be employed. Quantum input-output network (QION) theory is one such treatment. Another formalism, called SLH theory, incorporates scattering matrices (S), coupling vectors (L), and system Hamiltonians (H). These theoretical treatments require a reformulation of classical microwave theory. To make these topics comprehensible to an electrical engineer, we demonstrate some underpinnings of microwave quantum optics in terms of microwave engineering. For instance, we equate traveling-wave phasors in a transmission line (V0+) directly to bosonic field operators. Furthermore, we extend QION to include a state-space representation and a transfer function for a single port quantum network. This serves as a case study to highlight how microwave methodologies can be applied in open quantum systems. Although the same conclusion could be found from a full SLH theory treatment, our method was derived directly from first principles of QION.
Classical microwave circuit theory is incomplete in that it is incapable of fully modeling some phenomena at the quantum level. Various theoretical frameworks can be employed to incorporate single-photon statistical effects in the treatment of microwave networks. Such methods include quantum input-output network (QION) theory and SLH theory. A synthesis of the quantum and classical circuit treatments requires a description of second quantization within classical microwave theory. In order to make these topics understandable to an electrical engineer, we demonstrate some underpinnings of quantum optics in terms of microwave engineering. For example, we relate traveling-wave phasors for transmission-lines, such as voltage and current, to bosonic field operators. The second quantization and necessary components for a quantum treatment of microwave circuit theory are summarized in a table that maps microwave scattering parameters to bosonic operators in a transmission-line. To illustrate the need for second quantization, we use the results of second quantization along with first principles of quantum input-output network theory to determine a state-space representation and a transfer function of a single port quantum network. The same results could be obtained from SLH theory, and this serves as a case study for applying microwave theory to open quantum systems. The results of this work could be used to treat NISQ and other superconducting hardware. Additionally, these results could be incorporated into an applied curriculum, assisting in the engineering education of the future quantum workforce.
Many emerging quantum technologies demand precise engineering and control over networks consisting of quantum mechanical degrees of freedom connected by propagating electromagnetic fields, or quantum input-output networks. Here we review recent progress in theory and experiment related to such quantum input-output networks, with a focus on the SLH framework, a powerful modeling framework for networked quantum systems that is naturally endowed with properties such as modularity and hierarchy. We begin by explaining the physical approximations required to represent any individual node of a network, e.g. atoms in cavity or a mechanical oscillator, and its coupling to quantum fields by an operator triple (S,L,H). Then we explain how these nodes can be composed into a network with arbitrary connectivity, including coherent feedback channels, using algebraic rules, and how to derive the dynamics of network components and output fields. The second part of the review discusses several extensions to the basic SLH framework that expand its modeling capabilities, and the prospects for modeling integrated implementations of quantum input-output networks. In addition to summarizing major results and recent literature, we discuss the potential applications and limitations of the SLH framework and quantum input-output networks, with the intention of providing context to a reader unfamiliar with the field.
The precise engineering of electromagnetic couplings is paramount for constructing scalable and high-fidelity superconducting quantum processors. While essential for orchestrating qubit operations, these couplings also present significant design challenges, including the mitigation of crosstalk and the management of environmental decoherence. A clear and unified theoretical framework is therefore crucial for the design, simulation, and analysis of these complex quantum circuits. This paper presents a comprehensive theoretical treatment of the fundamental electromagnetic coupling mechanisms in superconducting devices. Starting from first principles, we formulate the equations of motion and derive the input-output relations for canonical systems, including a single resonator coupled to a multi-port microwave network, interacting resonators, and coupled transmission lines. We review rigorous definitions for key parameters such as the energy decay rate (κ\kappaκ) and the dimensionless coupling coefficient (ζ\zetaζ) and connect these formalisms to practical methods of parameter extraction from electromagnetic simulations. This work provides a rigorous and pedagogical foundation for understanding and modeling linear electromagnetic interactions, serving as a vital resource for the development of advanced superconducting quantum hardware.
Quantum Feedback Network Theory (also known colloquially as the SLH-framework) is a powerful tool for modeling systems in quantum optics. This paper describes a method for discretising SLH-networks such that they can be implemented on a quantum computer. Building on a discretisation theorem for quantum stochastic differential equations, we establish strong convergence, uniformly on compact time intervals, of the discretised system to the continuous SLH-network as the discretisation mesh goes to zero.
In this thesis, experimentally relevant aspects and open questions of quantum information processing and measurement in circuit quantum electrodynamics have been investigated theoretically. Circuit quantum electrodynamics is a relatively young field combining superconducting transmission line resonators on-chip with superconducting quantum bits which serve as artificial atoms. Remarkable experiments have underlined the prospects of circuit QED as a possible architecture for quantum information processing as well as a framework within which quantum optics experiments can be performed. In contrast to their optical counterparts, these experiments reach the ultra-strong coupling limit and allow e.g. the generation of Fock states, the study of decoherence and the observation of quantum jumps. We present the physics of superconducting circuits based on the Josephson junction as a non-linear circuit element. We discuss the various types of quantum bits (qubits) before turning to an introduction on cavity QED and conclude the introduction with a brief presentation of the most relevant experiments and a short introduction to the principles and prospects of quantum information processing. Building on the pioneering theoretical and experimental work in which up to three qubits have been integrated on a chip and successfully coupled via a cavity mode, we propose a scheme to overcome the natural limits concerning scalability in these systems. Usually, scalability is severely limited by the resonance width of the qubit transitions and the attainable frequency range for both qubit transition and cavity resonance frequencies. This limits the present setups to at most ten qubits. As a possible solution, we propose and investigate a cross-bar grid layout of cavities with qubits at the intersections. It turns out that this setup, allowing for up to about 1000 qubits, offers significant advantages over e.g. nearest-neighbor coupling schemes. In addition, it can serve as a building block for a truly scalable scheme for fault-tolerant quantum information processing with superconducting circuits by combining 7x7 arrays in a staircase manner. Furthermore, we discuss and simulate different possibilities to improve upon the suggested scheme by e.g. using resonant gates and optimal control theory. Another focus of the present thesis is measurement physics. In order to investigate e.g. the possibilities of using a circuit QED system to dispersively detect single microwave photons or to generate multi-qubit entangled states, we introduce in a pedagogical manner the basics of stochastic master equations which generate quantum trajectories before turning to concrete applications. Quantum trajectories have proven to be a state-of-the-art tool to analyze measurement situations in a very realistic manner by giving access to both the measurement record as well as the internal quantum dynamics conditioned on this record. Using the method of quantum trajectories as generated by stochastic master equations, we discuss a scheme to efficiently generate multi-qubit entangled states in a very flexible way. The method is based on a collective, dispersive qubit readout. By this measurement, a suitable product state of N qubits can be reliably converted into e.g. an N-qubit Greenberger-Horne-Zeilinger or W state. We propose and discuss the scheme and investigate the effects of decoherence and parameter spread as they would be encountered in experiment. Last but not least, we propose and fully analyze a scheme to dispersively detect single itinerant microwave photons. While experiments to generate single microwave photons have already been successfully performed, their single-shot quantum non-demolition detection has not been possible so far. The scheme presented in this thesis closes this gap in the circuit QED toolbox and reaches detection efficiencies of 30% only limited by fundamental quantum mechanics in this case the Quantum Zeno effect. This detection scheme could be used in quantum communication or quantum cryptography applications.