# Quantum Information and Quantum Optics with Superconducting Circuits - Exercise Solutions (Chapter 9)

9.1 (1) So, by setting , we have: where (2) We apply an adiabatic change to . On the other hand, Hence, from the Schroedinger equation , we have: From the definition of , we have: So, --- (2-1)By differentiating the both sides of , So, fro…

# Quantum Information and Quantum Optics with Superconducting Circuits - Exercise Solutions (Chapter 8)

8.1 By expanding the tensor product, we have the all possible binary numbers.As discussed in 6.1.4, for a single qubit, Hadamard gate can be implemented as: using the coherent drive: that's on resonance and .To apply it to identical flux q…

# Derivation of Input-Output relations for waveguide-QED

Target system The system is similar to the following one.enakai00.hatenablog.comHere we restrict the states with only one excitation (either qubit or a propagating photon). --- (7.13)The Heisenberg equations are: --- (7.14)Using the same a…

# Detailed derivation of the Input-Output theory

Basic definitions Think of a system consisted of: A long waveguide with length accommodating a photon field . A small resonator at accommodating an electric oscillation . that has the RWA Hamiltonian (5.60): --- (5.60)where we use the unit…

# Quantum Information and Quantum Optics with Superconducting Circuits - Exercise Solutions (Chapter 7)

7.1 Using the unit system and adding constant to the total energy, the Hamiltonian is: The anzatz is: where Then, Hence, from the Schroedinger equation , we have: 7.2 From (5.61), we have a Langevin equation for the qubit's amplitude. We i…

# Quantum Information and Quantum Optics with Superconducting Circuits - Exercise Solutions (Chapter 6) : Part2

6.10 (1) Define as: Then, So, the interaction Hamiltonian is: Since: We have: By dropping the non-RWA terms such as , we have: (2) By dropping the third state, we have: For Hence, from the Schroedinger equation: , we have: --- (1) --- (2)B…

# Quantum Information and Quantum Optics with Superconducting Circuits - Exercise Solutions (Chapter 6) : Part1

6.1 Unitary operation for a rotation around axis is: An operator to swap x and y is: An operator to swap x and z is: 6.2 The general solution to the linear interaction case is given by (5.10) as: Note that this is valid not only for , but …

# Quantum Information and Quantum Optics with Superconducting Circuits - Exercise Solutions (Chapter 5)

5.1 You can redefine with the same commutation relation .So you can set without losing the generality of the discussion.Now, and, Hence, By scaling back with , For and , use the relationship 5.2 gist.github.com 5.3 Hence, the coherent stat…

# Quantum Information and Quantum Optics with Superconducting Circuits - Exercise Solutions (Chapter 4)

4.1 For the thermal state : For the pure state : 4.2 By mathematical induction, you can prove: Hence: 4.4 Current conservation: Branch currents: Hence: Without the register: With the register: 4.5 4.6 This is very rough apporximation. I'm …

# Derivation of the first and second Josephson relation

In the following discussion, we assume that the quantum states are quasi-static. We solve the time-independent Schrödinger equation supposing that external potentials are independent of time . When we change them slowly enough, the corresp…

# Derivation of the gauge-independent relation between the phase and the electric field

The effective wavefunction and the charge current are given as: --- (3.4) ---(3.13)The wavefunction follows the Schrödinger equation: --- (3.5)Without losing the generality, we can take the Coulomb gauge: --- (1)Now, we assume that the cha…

# Study notes on "Quantum Information and Quantum Optics with Superconducting Circuits"

Quantum Information and Quantum Optics with Superconducting Circuits作者:García Ripoll, Juan JoséCambridge University PressAmazon Derivation of the Lindblad master equation (2.26) I found the following paper is useful to understand the gen…

# Gauge Invariance of superconducting circuits wavefunction model

Effective wavefunction describing the flow of superconducting elections (Cooper pairs): Schrödinger equation for : --- (3.5)I will show that this model is invariant under the gauge transformation: [Proof](3.5) is equivalent to --- (1)A sim…

# (Informal) Errata of "A short introduction to the Lindblad Master Equation"

arxiv.orgp.8 Equation (24) p.10 Equation (30) p.11 Equation (31) p.13 Derivation of (45) from (44)First, in (44), we change the integral variable from (without any approximation) to get: Then, we assume that the kernel in the integration i…

# 強化学習（DQN）に Explainable AI のテクニックを応用してみる

Explainable AI とは 学習済みのディープラーニングのモデルをリバースエンジニアリング的に分析して、モデルがどのようなロジックで推論しているのかを明らかにする手法です。特定の決まった技術があるわけではなく、モデルの種類に応じてさまざまなテクニ…

# マイクロサービスに関する参考書籍

Microservices in Action作者:Bruce, Morgan,Pereira, Paulo A.Manning PublicationsAmazonマイクロサービスの全体像を把握するのに最適 Microservices Patterns: With examples in Java (English Edition)作者:Richardson, ChrisManningAmazonマイクロサー…