Physics, Mathematics, and All that Quantum Jazz

The following sections are included:

• PREFACE
• CONTENTS

I will summarize my research carrier in the past 36 years and present future prospects of my research interest.

In this article, we first review some general mathematical framework for operator quantum error correction. Then implementation of these schemes on collective noise was discussed.

We show a systematic construction for implementing general measurements on a single qubit, including both strong (or projection) and weak measurements. We mainly focus on linear optical qubits. The present approach is composed of simple and feasible elements, i.e., beam splitters, wave plates, and polarizing beam splitters. We show how the parameters characterizing the measurement operators are controlled by the linear optical elements. We also propose a method for the implementation of general measurements in solid-state qubits.

The time-dependent matrix-product-state (TDMPS) simulation method has been known as one of fast simulation methods to study time-evolving quantum systems. Here, I report recent development of my open-source C++ library named ZKCM_QC designed for TDMPS simulations of quantum circuits with arbitrary floating-point precision. Simulation performance is reported for well-known quantum algorithms. In addition, it is numerically shown that a trustworthy simulation should be performed in multiprecision and should not involve truncations of nonzero Schmidt coefficients.

In this paper, we review the entanglement properties of a quantum lattice-gas model according to our previous paper [S. Tanaka, R. Tamura, and H. Katsura, Phys. Rev. A 86, 032326 (2012)]. The ground state of the model under consideration can be exactly obtained and expressed by the Rokhsar-Kivelson type quantum superposition. The reduced density matrices of the model on square and triangular ladders are related to the transfer matrices of the classical hard-square and hard-hexagon models, respectively. In our previous paper, we investigated the entanglement properties including the entanglement entropy, the entanglement spectrum, and the nested entanglement entropy. We found that the entanglement spectra are critical when parameters are chosen so that the corresponding classical model is critical. In order to further investigate the entanglement properties, we also considered the nested entanglement entropy. As a result, the entanglement properties of the model on square and triangular ladders are described by the critical phenomena of the Ising model and the three-state ferromagnetic Potts model in two dimension, respectively.

We naturally define the weak value on the basis of the probability theory in quantum mechanics. Furthermore, using this theory, we discuss the amplification process of a signal, and we have shown that there is no upper bound for the signal amplification.

Topologically protected fault-tolerant quantum computation on the surface code is one of the most promising routes toward the realization of scalable quantum computation. We review fault-tolerant quantum computation on the planer surface code in two dimensions and its measurement-based version, topologically protected measurement-based quantum computation in three dimensions. We also survey recent progress made based on these two models, such as topologically protected measurement-based quantum computation on the thermal states.

We discuss the efficiency of quantum annealing with antiferromagnetic fluctuations for mean-field models (the infinite-range ferromagnetic model and the Hopfield model) by analyzing those phase diagrams. The phase diagrams are obtained by using the Suzuki-Trotter formula, the static anzats, and the saddle-point method. The results for the ferromagnetic model show that the antiferromagnetic fluctuations let the system evolve only through second-order transitions when the order of interactions is a finite value greater than 3. The results for the Hopfield model with finite patterns are the same as those for the ferromagnetic model. In contrast, the antiferromagnetic fluctuations cannot avoid first-order transitions of the Hopfield model with many patterns, which is a spin-glass model. We thus conclude that the antiferromagnetic fluctuations are not effective for NP-hard problems.

In this paper, we review a way to change nature of phase transition with annealing methods in mind. Annealing methods are regarded as a general technique to solve optimization problems efficiently. In annealing methods, we introduce a controllable parameter which represents a kind of fluctuation and decrease the parameter gradually. Annealing methods face with a difficulty when a phase transition point exists during the protocol. Then, it is important to develop a method to avoid the phase transition by introducing a new type of fluctuation. By taking the Potts model for instance, we review a way to change the phase transition nature. Although the method described in this paper does not succeed to avoid the phase transition, we believe that the concept of the method will be useful for optimization problems.

Calculation was done about the optimized geometries of special pair consisting of “bacteriochlorophyll a” based on the PDB structural data (3zuw) by quantum simulation method. The quantum chemical calculation for the optimized geometries exhibited the electronic excitation transition with large oscillator strength at 860 nm, characterizing P870. Therefore, the optimized geometries were reasonably assigned to a model of real P870. The geometries of cationic dimer were also optimized by the above method. The charge separation process upon photo-excitation was concluded to be impossible by considering the difference between the heat of formation of the neutral and cation dimers and the ionization energy of the neutral dimer.

Quantum information science has rapidly grown with the promise to build up a quantum computer to solve many classically intractable problems by making use of properties of quantum particles, such as superposition and entanglement. Systems of trapped neutral atoms is one of the promising candidates for implementing a scalable quantum computer, since neutral atoms have an advantage of an intrinsically weaker interaction with the environment. We have discussed a design of a neutral atom quantum computer with a selective two-qubit gate operation. In this contribution, we propose a feasible experiment towards the selective two-qubit gate operation that is less demanding than our original proposal, although the gate operation is limited between neighboring atoms. We evaluate the process of a two-qubit gate operation that is applied on nearest neighboring trapped atoms and estimate the upper bound of the gate operation time and corresponding gate fidelity. This proposal can be demonstrated using current technology, which would be a starting point toward the realization of a fully controlled selective two-qubit gate operation using neutral atoms.

Note from Publisher: This article contains only abstract.

We propose a simple three-qubit OQEC scheme avoiding fully correlated noise. One data qubit is protected by encoding it with two arbitrary ancillae, which can be in the uniformly mixed state. We demonstrated our scheme experimentally with a three-qubit NMR quantum computer. Experimental results show that our scheme eliminates the effects of fully correlated noise.

This article attempts to give a brief overview of recent approaches to the black hole information loss puzzle by exploiting the gauge-gravity duality.

We study the finite-temperature behavior of a dilute Bose gas using projected Gross-Pitaevskii equation by comparing to that of quantum Monte Carlo calculation in a two dimensional homogeneous system. By investigating the behavior of the correlation function, obtained results indicate that projected Gross- Pitaevskii equation is effective model of the Bose atoms with the long wavelength at low temperatures including the superfluid transition temperature. As the demonstration in a two dimensional homogeneous system, we find that this model describe the characteristic of vortex excitation indicated the superfluid transition.

We investigate the effect of gauge symmetry breaking in an atomic quantum simulator of U(1) lattice gauge theory, finding that a system of cold atoms in an optical lattice with a general set of parameters can be implemented as a simulator of “U(1) lattice gauge-Higgs” model with asymmetric nearest-neighbor Higgs coupling. The gauge-Higgs coupling in the imaginary time direction naturally arises from the violation of the U(1) local gauge invariance of the simulators caused by the deviation from the fine-tuned system parameters. A general method to supply the Higgs coupling in all space-time directions may be realized by coupling atoms in an optical lattice to another particle reservoir filled with the Bose-condensed atoms via laser-induced transitions. Clarification of the dynamics of this gauge-Higgs model sheds some lights upon various unsolved problems including the inflation process of the early universe.

When the environmental noise acting on the system has certain symmetries, a subsystem of the total system can avoid errors. Encoding information into such a subsystem is advantageous since it does not require any error syndrome measurements, which may introduce further errors to the system. However, utilizing such a subsystem for large systems gets impractical with the increasing number of qudits. A recursive scheme offers a solution to this problem. Here, we review the recursive construct introduced in, which can asymptotically protect 1/d of the qudits in system against collective errors.

In NMR quantum computation and NMR experiments, there are many composite quantum gates to suppress one of two dominant errors. In this paper, we review some concatenated composite pulses (CCCPs) that is robust against two dominant errors simultaneously.

We formulate statistical mechanics based on a pure quantum state, which we call a “thermal pure quantum (TPQ) state”. A single TPQ state gives not only equilibrium values of mechanical variables, such as magnetization and correlation functions, but also those of genuine thermodynamic variables and thermodynamic functions, such as entropy and free energy. Among many possible TPQ states, we discuss the canonical TPQ state, the TPQ state whose temperature is specified. In the TPQ formulation of statistical mechanics, thermal fluctuations are completely included in quantum-mechanical fluctuations. As a consequence, TPQ states have much larger quantum entanglement than the equilibrium density operators of the ensemble formulation. We also show that the TPQ formulation is very useful in practical computations, by applying the formulation to a frustrated two-dimensional quantum spin system.

A brief review is provided on the recent developments on the theory of thermalization under unitary time evolution in isolated quantum systems. In particular, we focus on the theory of equilibration, which states the existence of the effective stationary state, and the eigenstate thermalization hypothesis, which is a scenario for the stationary state being regarded as the microcanonical ensemble. Then, we introduce other possible scenarios and compare them from the viewpoint of the finite-size scaling. We also discuss the relative importance among these scenarios.

I'd like to discuss generalizations of the second law of thermodynamics in the presence of quantum information processing such as quantum measurement and quantum feedback control. A quantum information content, referred to as QC-mutual information, is shown to play a crucial role. We also discuss the generalization of a quantum generalization of the Hatano- Sasa inequality for transitions between nonequilibrium steady states with quantum feedback control.

Note from Publisher: This article contains only abstract.