The physical constraints underlying the concept of quantum circuit are considered. In particular it is shown that the point of departure for their modeling starts from the interconnection of the components into a classical network, followed by quantization of the latter, and not by the interconnection of already quantized components. The procedure is straightforward for lossless networks but cannot be worked out in presence of resistors for the impossibility of constructing a Lagrangian function. However the difficulty is circumvented by distinguishing thermal from radiative resistors, the former being the usual ones, the latter being realized by semi-infinite LC transmission lines, for which the Lagrangian exists. In the complex plane s = σ + jω the impedance of a thermal resistor is Z(s) = R in the entire plane while that of a radiative resistor is Rsign(σ), so that the latter is lossless and does not dissipate energy but conveys it to the infinity. Comparison with Lindblad approach shows that the resistor fits into it in the RHP. Radiative resistors make evolution reversible, which is not true for a physical system including thermal.
The physical constraints underlying the concept of quantum circuit are considered. In particular it is shown that the point of departure for their modeling starts from the interconnection of the components into a classical network, followed by quantization of the latter, and not by the interconnection of already quantized components. The procedure is straightforward for lossless networks but cannot be worked out in presence of resistors for the impossibility of constructing a Lagrangian function. However the difficulty is circumvented by distinguishing thermal from radiative resistors, the former being the usual ones, the latter being realized by semi-infinite LC transmission lines, for which the Lagrangian exists. In the complex plane s = σ + jω the impedance of a thermal resistor is Z(s) = R in the entire plane while that of a radiative resistor is Rsign(σ), so that the latter is lossless and does not dissipate energy but conveys it to the infinity. Comparison with Lindblad approach shows that the resistor fits into it in the RHP. Radiative resistors make evolution reversible, which is not true for a physical system including thermal.
Abstract Superconducting circuits are one of the leading quantum platforms for quantum technologies. With growing system complexity, it is of crucial importance to develop scalable circuit models that contain the minimum information required to predict the behaviour of the physical system. Based on microwave engineering methods, divergent and non-divergent Hamiltonian models in circuit quantum electrodynamics have been proposed to explain the dynamics of superconducting quantum networks coupled to infinite-dimensional systems, such as transmission lines and general impedance environments. Here, we study systematically common linear coupling configurations between networks and infinite-dimensional systems. The main result is that the simple Lagrangian models for these configurations present an intrinsic natural length that provides a natural ultraviolet cutoff. This length is due to the unavoidable dressing of the environment modes by the network. In this manner, the coupling parameters between their components correctly manifest their natural decoupling at high frequencies. Furthermore, we show the requirements to correctly separate infinite-dimensional coupled systems in local bases. We also compare our analytical results with other analytical and approximate methods available in the literature. Finally, we propose several applications of these general methods to analogue quantum simulation of multi-spin-boson models in non-perturbative coupling regimes.
Abstract The quantization of systems composed of transmission lines connected to lumped circuits poses significant challenges, arising from the interaction between continuous and discrete degrees of freedom. A widely adopted strategy, based on the pioneering work of Yurke and Denker, entails representing the lumped circuit contributions using Lagrangian densities that incorporate Dirac δ-functions. However, this approach introduces complications, as highlighted in the recent literature, including divergent momentum densities, necessitating the use of regularization techniques. In this work, we introduce a δ-free Lagrangian formulation for a transmission line capacitively coupled to a lumped circuit without the need for a discretization of the transmission line or mode expansions. This is achieved by explicitly enforcing boundary conditions at the line ends in the principle of least action. In this framework, the quantization and the derivation of the Heisenberg equations of the network are straightforward. We obtain a reduced model for the lumped circuit in the quantum Langevin form, which is valid for any coupling strength between the line and the lumped circuit. We apply our approach to an analytically solvable network consisting of a semi-infinite transmission line capacitively coupled to an LC circuit and study the behavior of the network as the coupling strength varies.
During the last 30 years, stimulated by the quest to build superconducting quantum processors, a theory of quantum electrical circuits has emerged, which is called circuit quantum electrodynamics or circuit-QED. The goal of the theory is to provide a quantum description of the most relevant degrees of freedom. The central objects to be derived and studied are the Lagrangian and the Hamiltonian governing these degrees of freedom. Central concepts in classical network theory such as impedance and scattering matrices can be used to obtain the Hamiltonian and Lagrangian description for the lossless (linear) part of the circuits. Methods of analysis, both classical and quantum, can also be developed for nonreciprocal circuits. These lecture notes aim at giving a comprehensive, theoretically oriented, overview of this subject for Master or PhD students in physics and electrical engineering.
Except for the NOT gate, all the other familiar classical gates are irreversible in the sense that we cannot uniquely reconstruct the input states from the output states. For example, OR, AND, NAND, NOR, etc. are irreversible. They all map a two-bit input state into a single bit output state. Thus one bit is erased during operation of each of these gates, and according to Landauer’s principle that requires dissipation of a minimum amount of energy. Now a simple question arises in our curious mind: Is it possible to avoid this loss of energy? The answer is yes. If we don’t erase any bit then we can circumvent this energy loss. Thus we need to map n bit input states to n bit output states. In addition, if a one-toone correspondence exists between the input states and the output states then only we will be able to uniquely reconstruct the input states from the output states. In such a case the gate is called reversible. For example, if we have f(00) = 00, f(01) = 10, f(10) = 01, f(11) = 11 then f represents a reversible gate. Quantum evolution operators are unitary so for every operator U we have an inverse operator U−1 = U†. Therefore, quantum evolution operators are the natural choice for the construction of energy efficient reversible gates. However, it is not the only choice. We can have classical reversible gates, too. Usually by reversible gates we refer to classical reversible gates, and quantum gates are specifically referred to as quantum gates. Reversible gates and quantum gates are similar but there exists a fundamental difference that reversible gates cannot accept superposition states (e.g. α|0〉 + β|1〉) as input states, whereas quantum gates can. Thus all quantum gates are essentially reversible but the converse is not true.
In quantum theory, a wave function expresses the state of matter waves and is not considered to physically exist in general. Equations that are satisfied by a wave function are called wave equations. In classical physics, wave equations are well known and are derived from the Maxwell equations. However, the Maxwell equations do not define important physical properties, such as the lossless property of circuits. The lossless property is represented by the cascade matrix defined in analog circuit theory. The definition of the lossless property in wave equations became possible owing to the telegrapher's equations introduced by Heaviside. The cascade matrix can be determined from uniform transmission lines that can be obtained from the telegrapher's equations and those extended using the Riccati differential equation.
Abstract The importance of dissipation engineering ranges from universal quantum computation to non‐equilibrium quantum thermodynamics. In recent years, more and more theoretical and experimental studies have shown the relevance of this topic for circuit quantum electrodynamics, one of the major platforms in the race for a quantum computer. This article discusses how to simulate thermal baths by inserting resistive elements in networks of superconducting qubits. Apart from pedagogically reviewing the phenomenological and microscopic models of a resistor as thermal bath with Johnson–Nyquist noise, the paper introduces some new results in the weak coupling limit, showing that the most common examples of open quantum systems can be simulated through capacitively coupled superconducting qubits and resistors. The aim of the manuscript, written with a broad audience in mind, is to be both an instructive tutorial about how to derive and characterize the Hamiltonian of general dissipative superconducting circuits with capacitive coupling, and a review of the most relevant and topical theoretical and experimental works focused on resistive elements and dissipation engineering.