Correction to: International Symposium on Mathematics, Quantum Theory, and Cryptography

The original version of the book was published with incorrect part names in the website for second part FM now updated with correct title. This has been corrected in the updated version.

Verified Numerical Computations and Related Applications

The author has been engaged in the study of numerical computations with result verification starting from 1990.

Quantum Optics with Giant Atoms—the First Five Years

In quantum optics, it is common to assume that atoms can be approximated as point-like compared to the wavelength of the light they interact with. However, recent advances in experiments with artificial atoms built from superconducting circuits have shown that this assumption can be violated. Instead, these artificial atoms can couple to an electromagnetic field at multiple points, which are spaced wavelength distances apart. In this chapter, we present a survey of such systems, which we call giant atoms. The main novelty of giant atoms is that the multiple coupling points give rise to interference effects that are not present in quantum optics with ordinary, small atoms. We discuss both theoretical and experimental results for single and multiple giant atoms, and show how the interference effects can be used for interesting applications. We also give an outlook for this emerging field of quantum optics.

Emerging Ultrastrong Coupling Between Light and Matter Observed in Circuit Quantum Electrodynamics

The strength of the coupling between an atom and a single electromagnetic field mode is defined as the ratio of the vacuum Rabi frequency to the Larmor frequency, and is determined by a small dimensionless physical constant, the fine structure constant $$\alpha =Z_{vac} / 2R_{K}$$. On the other hand, the quantum circuit including Josephson junctions behaving as artificial atoms and it can be coupled to the electromagnetic field with arbitrary strength (Devoret et al. 2007). Therefore, the circuit quantum electrodynamics (circuit QED) is extremely suitable for studying much stronger light-matter interaction.

Number Theoretic Study in Quantum Interactions

The quantum interaction models, with the quantum Rabi model as a distinguished representative, are recently appearing ubiquitously in various quantum systems including cavity and circuit quantum electrodynamics, quantum dots and artificial atoms, with potential applications in quantum information technologies including quantum cryptography and quantum computing (Haroche and Raimond 2008; Yoshihara et al. 2018). In this extended abstract, based on the contents of the talk at the conference, we describe shortly certain number theoretical aspects arising from thenon-commutative harmonic oscillators (NCHO: see Parmeggiani and Wakayama 2001; Parmeggiani 2010) and quantum Rabi model (QRM: see Braak 2011 for the integrability) through their respective spectral zeta functions.

Extended Divisibility Relations for Constraint Polynomials of the Asymmetric Quantum Rabi Model

The quantum Rabi model (QRM) is widely regarded as one of the fundamental models of quantum optics. One of its generalizations is the asymmetric quantum Rabi model (AQRM), obtained by introducing a symmetry-breaking term depending on a parameter $$\varepsilon \in \mathbb {R}$$ to the Hamiltonian of the QRM. The AQRM was shown to possess degeneracies in the spectrum for values $$\epsilon \in 1/2\mathbb {Z}$$ via the study of the divisibility of the so-called constraint polynomials. In this article, we aim to provide further insight into the structure of Juddian solutions of the AQRM by extending the divisibility properties and the relations between the constraint polynomials with the solution of the AQRM in the Bargmann space. In particular we discuss a conjecture proposed by Masato Wakayama.

Quantum Random Numbers Generated by a Cloud Superconducting Quantum Computer

A cloud quantum computer is similar to a random number generator in that its physical mechanism is inaccessible to its users. In this respect, a cloud quantum computer is a black box. In both devices, its users decide the device condition from the output. A framework to achieve this exists in the field of random number generation in the form of statistical tests for random number generators. In the present study, we generated random numbers on a 20-qubit cloud quantum computer and evaluated the condition and stability of its qubits using statistical tests for random number generators. As a result, we observed that some qubits were more biased than others. Statistical tests for random number generators may provide a simple indicator of qubit condition and stability, enabling users to decide for themselves which qubits inside a cloud quantum computer to use.

A Review of Secret Key Distribution Based on Bounded Observability

Secret key distribution is a technique for a sender and a receiver to share a secret key, which is not known by any eavesdropper, when they share no common secret information in advance. By using this technique, the sender and the receiver can transmit a message securely in the sense that the message remains secret from any eavesdropper. We introduced a secret key distribution based on the Bounded Observability (Muramatsu et al. 2010, 2013, 2015), which provides a necessary and sufficient condition for the possibility of secret key distribution. This condition describes limits on the information obtained by observation of a random object, and models the practical difficulty of completely observing random physical phenomena.

From the Bloch Sphere to Phase-Space Representations with the Gottesman–Kitaev–Preskill Encoding

In this work, we study the Wigner phase-space representation of qubit states encoded in continuous variables (CV) by using the Gottesman–Kitaev–Preskill (GKP) mapping. We explore a possible connection between resources for universal quantum computation in discrete-variable (DV) systems, i.e. non-stabilizer states, and negativity of the Wigner function in CV architectures, which is a necessary requirement for quantum advantage. In particular, we show that the lowest Wigner logarithmic negativity corresponds to encoded stabilizer states, while the maximum negativity is associated with the most non-stabilizer states, H-type and T-type quantum states.

A Survey of Solving SVP Algorithms and Recent Strategies for Solving the SVP Challenge

Recently, lattice-based cryptography has received attention as a candidate of post-quantum cryptography (PQC). The essential security of lattice-based cryptography is based on the hardness of classical lattice problems such as the shortest vector problem (SVP) and the closest vector problem (CVP). A number of algorithms have been proposed for solving SVP exactly or approximately, and most of them are useful also for solving CVP. In this paper, we give a survey of typical algorithms for solving SVP from a mathematical point of view. We also present recent strategies for solving the Darmstadt SVP challenge in dimensions higher than 150.