Research Journey

I Chapter 1 Prologue — Half a Century as a Researcher

I was born in Kochi, Japan, and spent my years up through high school there. My earliest memory is of living on the second floor of a barn on a farm on the outskirts of the city. Later, my parents built a house nearby and we moved there. The area was surrounded by rice fields, and I spent my days playing in nature. I did not consciously aspire to science, but an interest in arithmetic and mathematics seems to have developed early. My father taught electricity at a technical high school, which may be why I felt no resistance to technical matters.

Through junior high and high school, my interest in mathematics grew stronger. I was drawn to its logical structure, where everything follows from basic definitions. By contrast, physics then seemed to consist of given laws to be applied, and I did not find it as appealing. I spent most of my time studying independently, working through problem books on my own.

I then entered the University of Tokyo. At first, I felt some discomfort with the atmosphere, but as I continued studying, my view of physics changed. University physics is a systematic discipline in which many results are derived from a few fundamental principles, and I came to feel it was closer in spirit to mathematics. This shift deepened my interest in physics, while mathematics, becoming more abstract, moved away from my interests.

In my third year, I studied subjects including statistical mechanics and quantum mechanics, and was particularly drawn to statistical mechanics, influenced by the lectures of Professor Masuo Suzuki. In my fourth year, I joined a statistical mechanics research group and began research for the first time. The initial problem was difficult, and only after much trial and error did I find a way forward by reformulating it. This experience strongly influenced my later research style.

In graduate school, I became interested in spin glass research, then rapidly developing worldwide. By applying the idea of duality, I arrived at an exact solution. The result was so simple for such a difficult problem that it was not readily accepted at first, but later proved to have important implications.

After completing my doctoral studies, I worked as a postdoctoral researcher at Carnegie Mellon University and Rutgers University, and then joined the Tokyo Institute of Technology. Starting as a research associate, I advanced to associate professor and then professor, engaging in education and research for many years, and also serving in administrative roles such as dean of the School of Science and as a member of the Institute Council. During these days, I continued working on spin glasses, quantum spin systems and quantum annealing with students and collaborators.

After retiring in March 2020, I have continued my research as a Specially-appointed Professor at the Institute of Science Tokyo. Counting from my graduate school years, I have been involved in statistical physics for nearly half a century. Looking back, I feel that the experiences described in the following chapters have accumulated and led to where I am today.

I had enjoyed arithmetic from elementary school onward, and when I entered middle school, I found mathematics even more interesting. I also developed an interest in English during my middle school years. My teacher’s instruction must have been excellent. In those days, tape recorders were in use, and the teacher would lend tapes recorded with native speakers to any student who wanted them. By taking those tapes home and listening to them repeatedly, I gradually developed a sense of speaking English naturally and directly, rather than first translating it from Japanese in my head. In this way, the psychological barrier I felt toward English, and more broadly toward other cultures, was lowered, and this may have had a significant influence on my later life as a researcher.

There is one passage from an English textbook that I still remember to this day: "Work hard and love God, and you will be a great man." Looking back on it now, it strikes me as a concise expression of what might be called the traditional moral foundation of America.

In any case, during this period I did not have any particularly strong interest in science. I did not dislike it, but I would not say that I was strongly drawn to it either.

When I entered high school, physics became more systematic than it had been in middle school. Yet at that time, physics still felt to me like something that was simply handed down from above. Various laws and formulas, such as Newton’s laws and Ohm’s law, were presented to us as given, and we memorized them and applied them to problems. Although this allowed us to explain natural phenomena, my strong impression at the time was that physics was a subject in which one memorizes and applies formulas and laws.

Mathematics, by contrast, felt more logical and possessed consistent elegance. As long as I understood the starting definitions, I could construct the entire structure simply by building up logical steps. If I had a firm grasp of the definition of trigonometric functions, for example, I could derive all the formulas myself even if I had forgotten them. In this sense, I felt that mathematics was more elegant and more interesting, and I found myself drawn deeper and deeper into it.

I spent my high school years heavily oriented towards study. I did not participate in any club activities, nor did I attend any cram schools, as many students did. Instead, I purchased reference books and problem sets on my own and worked through problem after problem at home. I attended classes diligently, but in practice I was almost entirely self-taught, living in a world that was complete in itself.

I chose the high school I attended mainly because it was close to home. It was a time when student movements were active, and under their influence I had developed something of a rebellious attitude. As a result, I felt hesitant to enrolling in what people called top schools, especially private high schools. Kochi Prefecture was somewhat unusual in Japan in that some private high schools had stronger reputations than public ones. The public school I attended was reasonably well regarded within the prefecture, but it was not the kind of place, unlike famous high schools in major cities, where students gathered with their sights set on entrance examinations for elite universities. It was, however, a school with a relatively calm atmosphere, and I was fortunate that, even as a somewhat unusual student, I had good class mates and teachers and was able to spend my time there without feeling too out of place.

After such a high school life, I entered the University of Tokyo. When I was young, I was too preoccupied with myself to think about it, but given that the university enrollment rate was still below 30% at the time in Japan, I now feel nothing but gratitude to my parents for sending me from the remote town of Kochi to Tokyo.

Before long after entering the University of Tokyo, I found myself feeling something unexpected. I had imagined that the University of Tokyo would be a place where people with strong intellectual passion and outstanding ability had gathered. But looking around at those nearby, I had the impression that this was not necessarily the case.

Their basic academic ability was certainly high, but many of them seemed to devote their energy to extracurricular activities and other pursuits outside of studying for most of the term, then switch gears as examinations approached, study past papers carefully, and manage to get through quite skillfully. What struck me most vividly was that shortly after entering, I encountered someone who were discussing the "trends and strategies" of that year's entrance examination, the very examination we had just taken. Since I had never studied in that way myself, I honestly did not quite understand what "trends and strategies" meant. I had simply worked through every problem set I could get my hands on, and had almost never been conscious of any particular strategy aimed at passing an entrance examination unlike most students.

There is one scene from classes that has stayed with me. In the very first class of first-year physics, the instructor said, "Rather than coming to a place like this, you should study on your own." At the time, I did not immediately understand what he meant.

Thinking about it afterward, he was probably trying to convey the idea that studying is something one is supposed to do independently. But to be honest, that instructor was not the type to motivate students through an enthusiastic teaching style. In fact, many of the instructors seemed to lecture at their own pace, without much concern for whether students were following or understanding, unfortunately.

As I continued studying on my own, my view of physics, which I had held through high school, changed considerably. In high school, physics had seemed like a subject in which one memorizes laws and formulas handed down from above and then applies them. Gradually, however, I came to understand that university-level physics is a far more systematic and logical discipline, in the sense that nearly everything can be derived from a very small number of starting points. I began to feel that university physics was, if anything, a natural extension of high school mathematics, and I found myself developing a strong interest in it.

University mathematics, on the other hand, left me with a somewhat different impression. Unlike the mathematics I had known up through high school, it had transformed into an abstract logical world built on the accumulation of theorems and proofs. Since I was the type who found more interest in calculation than in theorems and proofs, I gradually came to keep my distance from that direction.

Among those around me were several students of remarkable mathematical talent, and they have remained vivid in my memory. I visited one of them at his apartment on one occasion. After we had talked late into the night, I fell asleep there, but he continued studying mathematics throughout the night. The next morning I asked him how he could keep going like that, and he said, "There is no reason to stop." The way he engaged with mathematics as if it were a game, enjoying it as he went, left a deep impression on me. He approached mathematics the way a child becomes absorbed in a game.

Through experiences like these, it came to feel natural to me that the path I should take was not mathematics but physics.

At that time, the statistical mechanics group was not limited to Professor Suzuki's alone. It held joint seminars, called colloquia, together with the groups of Professor Ryogo Kubo and Professor Yasushi Wada. The total number of students was very large, making for an exceptionally lively environment. There were five or six students in each year, and a well-established pattern existed in which most of them advanced from the master's program directly into the doctoral program.

In such an atmosphere, it felt natural to me that I would continue in this field for the long term.

The atmosphere within the group of three professors was quite open and flat. Students discussed freely among themselves, and questions were exchanged without hesitation during the colloquia. The professors, of course, contributed their views, but the students also participated actively in the discussions. Looking back on it now, it was a remarkably fortunate environment, rare by any standard.

That said, Professor Ryogo Kubo would occasionally make very direct and pointed remarks. When I was preparing for my first presentation at an international conference and practicing my English, he said to me, "If your English were a little better, one might be able to understand what you are saying." I still remember those words clearly. That moment reminded me once again that English is not something one can manage casually; it only becomes effective through sufficient training.

There is another memory I have of Professor Kubo. He would sometimes fall asleep during a colloquium, which surprised me at first, but I gradually came to realize that it was not unusual. Toward the end of the presentation, he would wake up and say, "I'm afraid I fell asleep. Could you explain that last part again?" It was a disconcerting situation for a student, but it was also simply part of the character of that group.

It was in Professor Suzuki's group that I first engaged in what could truly be called research. Students were gathered, several candidate topics were presented, and each of us selected one to work on. I took up one of these topics, but in retrospect it was not a problem that could be solved.

Through this experience, I came to understand in a concrete way that research is not something that yields answers quickly. One of the first keywords I was given was "duality," but no explanation of the concept was provided. I had no choice but to investigate it and think it through on my own. I had always assumed that when there was something I did not understand, I would look it up myself, not asking around, and so I simply began working my way forward.

The problem I had been given was to formulate duality in a particular variant of the Heisenberg model and investigate its properties. However, I had no idea how to approach it, and after gaining a basic understanding of duality, I found myself thinking about the problem for a long time without progress. I was young then and had the concentration to match. I thought about the problem from morning until night, on the train, during meals, and at all other times. Yet, breakthrough was nowhere in sight. After months of trial and error, I began to feel that a way forward might open up if I modified the problem setting. By replacing the Heisenberg model with a more tractable model, the structure related to duality began to emerge. By turning my attention to a simplified system, what would now be called the Z_q model, I was able to move closer to the essence of the problem.

It was the moment when something like a clue finally began to emerge from a situation in which nothing had progressed at all. The original problem I had been given was, as I now understand, one that could not be solved. If I had fully understood duality from the outset, I would simply have reported to Professor Suzuki that "this is an unsolvable problem," and that would have been the end of it. By starting from a position of knowing nothing and groping forward on my own, I was able to arrive at a completely different perspective.

This experience had a profound influence on the direction of my research. The approach of not simply trying to solve a given problem as stated, but instead seeking its essence by engaging with it from a different angle, was one I would go on to employ repeatedly. One might even say that this experience functioned as the initial condition, in the sense of a differential equation, from which the subsequent course of my research was naturally determined.

Among the results I obtained at this time were properties that surprised even me. For example, when I calculated the specific heat in one dimension, the specific heat, which would normally be expected to have a single peak as a function of temperature, turned out to exhibit two peaks. As the temperature was raised from absolute zero, the specific heat would increase once, then decrease, then increase again before finally decreasing: quite an unusual behavior.

This kind of result was deeply striking to me at the time, and it was the first experience in which I felt, in a genuine way, that something interesting was happening. Looking back afterward, the model itself had already been known. However, at the time, the sense that I had discovered it myself was strong, and it became the experience that first gave me a real sense of what makes research interesting.

It was also the first time I had conducted research in the style of grasping the essence of a problem in a rough way, then turning inward and thinking about it independently, rather than thoroughly investigating what had already been done and considering what could be added on that basis.

When I entered graduate school in the latter half of the 1970s, it was a period of intense excitement in spin glass research. Triggered by the emergence of the Sherrington-Kirkpatrick model, research activity had expanded rapidly worldwide around the unusual properties of spin glasses, and many researchers in Japan were also actively engaged with the problem.

A spin glass is a state in which the orientations of spins in a magnetic material freeze into a random configuration. Rather than aligning in the regular, orderly manner seen in ordinary crystals, the spins are spatially random from site to site, yet remain temporally fixed in that configuration. This peculiar state of matter began to attract attention both theoretically and experimentally from around the mid-1970s, and the period that roughly coincided with the beginning of my graduate studies marked the peak of that activity.

Up to that point, I had been working on problems related to duality, and it occurred to me to consider what might happen if I combined this with the spin glass problem. The idea of applying the framework of duality to spin glasses had not yet been explored. By combining two topics that seemed at first glance to be unrelated, duality and spin glasses, I thought it might be possible to arrive at a new perspective.

This attempt, as it turned out, worked remarkably well. By introducing the idea of duality, the essence of frustration in spin glasses began to come into view. The concept of frustration itself had already been well defined and widely known at the time, but research aimed at understanding its properties in a systematic way had not yet progressed very far.

Within this framework, by making effective use of duality, I found that under certain conditions the spin glass problem could be solved exactly. I would say this result emerged from a fortunate convergence between the methods I had developed on my own up to that point and a new set of questions concerning spin glasses that came from outside.

I obtained this result at a relatively early stage after entering the doctoral program. My research at that time was carried out almost entirely with paper and pencil. The calculations were enormous in scale. Since I could not afford large quantities of notebooks or high-quality paper, I purchased inexpensive paper in bulk and filled it with writing as small and dense as possible. I would write calculations until I had used both sides of a sheet, then move on to the next one, repeating this process over and over.

To describe the computation somewhat more technically, by applying duality I discovered that even if the distribution of signs of the interactions in a spin glass differs, the partition function remains the same as long as the distribution of frustration is identical. This can now be derived easily using the gauge transformation technique, which I also later discovered on my own, though it had in fact already been known. At the time, I arrived at it by exploring applications of duality across various examples. This discovery became the key that allowed the calculations to proceed rapidly.

The result obtained after all that enormous calculation was something I could not immediately bring myself to believe. The spin glass problem is extraordinarily difficult, and even solving the simplified mean-field theory exactly had been considered a major challenge. In fact, the original solution of the mean-field theory by Sherrington and Kirkpatrick was not exact in the full sense. It was only through the later, rather enigmatic method proposed by Parisi that the correct solution was finally obtained.

Although even the relatively simplified mean-field problem was difficult to solve exactly, the exact solution I had obtained had a range of applicability that extended well beyond the mean-field framework. Of course, the conditions under which the solution was derived were restricted, but even so, the fact that an exact solution could be obtained at all came as a great surprise.

What was particularly striking was the form of the solution itself. The result showed that the energy E can be expressed by the extremely simple formula E = -N_B tanh(J/T), where tanh(J/T) is the hyperbolic tangent of the inverse temperature. As the answer to a complex problem, it appears almost too simple to be true.

This formula was valid in a region that includes the phase transition point. Normally, a quantity such as the energy exhibits singular behavior at the phase transition point, but no such singularity appeared in this exact solution. In other words, taken at face value, one would naturally think that something must be wrong.

I repeated the calculations again and again, continuing to check the result. However, no matter how many times I verified it, the same answer kept emerging. In desperation, I worked hard to teach myself the basics of computer programming and Monte Carlo simulation, areas in which I had no prior experience, and carried out numerical simulations as well. The results confirmed the correctness of the solution within the limits of statistical uncertainty.

In the end, I had no choice but to accept the result. This experience was an important one, through which I came to understand in a very direct way that the simplicity of a result does not necessarily reflect the difficulty of the problem.

Even after the paper was accepted, the results were not immediately widely recognized in Japan. However, when I obtained my doctorate and went to the United States as a postdoctoral researcher, I noticed that the situation was somewhat different. In the United States, my exact solution was already fairly well known. I was genuinely surprised by the difference in how it was received compared to Japan.

The way I came to apply for a postdoctoral position in the United States was itself somewhat accidental. Email did not yet exist in those days, and all information about preprints and job openings, both in Japan and abroad, arrived as printed materials delivered by post. The group had a notice board where such announcements were posted. Among them was a posting for a postdoctoral position with the group at Carnegie Mellon University led by Professor Robert Griffiths.

One day, I happened to be on the same subway train home as Professor Masuo Suzuki. Since it would have been awkward to remain silent, I mentioned the posting in passing. Professor Suzuki immediately showed interest and said, "Why don’t you apply?" Since I had brought up the subject myself, I could hardly refuse, and one thing led to another, and I ended up applying. A brief conversation on a train thus ended up determining what followed.

Probably because Professor Suzuki wrote me a strong letter of recommendation, I was offered the position and would be going to the United States. The contract, however, was for only one year. Professor Griffiths had plans to spend the following year on sabbatical in France, which meant the appointment was limited to one year with no possibility of extension. Looking back now, it was an extremely risky choice, but at the time I did not think about it very seriously. I simply thought, "Let me just go and see." It was the kind of decision that only youth, and a lack of experience, could produce. I could never make such a decision now.

The timing of my departure was also unusual. The U.S. academic year begins in September, so I needed to arrive accordingly. However, as a third-year doctoral student, I would not be able to complete the program until the following March, the end of a Japanese academic year. I consulted Professor Suzuki, who said, "If you submit your thesis in June, the examination will effectively be complete, so just go. I will take care of the rest." Following his advice, I departed while continuing to pay tuition for the remaining half year without actually being at the university. This would be considered quite unusual today, but at the time it caused no particular problems. I also asked the U.S. side whether they would hire me as a postdoctoral researcher, which is normally a position for someone who has already obtained a doctorate, before I had officially received mine. They replied, "If you have a track record equivalent to a doctorate, that will be fine." I was somewhat surprised by the flexibility of the U.S. system.

In this way, through a combination of coincidences and encouragement from those around me, I found myself heading to the United States.

After returning to Japan, I joined the group of Professor Satoru Miyake. His area of expertise was condensed matter theory, including superconductivity, which was somewhat different from my background in statistical mechanics. Nevertheless, he provided me with an extraordinarily generous environment in which I was free to pursue whatever topics interested me. I remain deeply grateful to Professor Miyake to this day.

I also came to interact frequently with Professor Takehiko Oguchi in the neighboring group, who specialized in statistical mechanics. He would often invite me to join his group for lunch, and through various conversations I had the opportunity to learn about quantum spin systems, which were attracting attention at the time. Spin systems in which quantum mechanical effects become important remain a major topic today, but research on them was already active then.

On one occasion, Professor Oguchi suggested, "Why not try some numerical calculations on quantum spin systems?" Up to that point, I had almost no experience with large-scale numerical computation. I had done little more than small Monte Carlo simulations. However, as I listened, I developed a sense that it would be interesting, and I decided to plunge in and give it a try.

Looking back, it was a somewhat unconventional decision in the context of Japanese academic culture, venturing on my own into a field quite different from that of the group to which I belonged, but the liberal atmosphere of the Miyake group made it possible.

One of the methods available at the time for studying quantum spin systems was the numerical diagonalization method, in which the Hamiltonian is directly diagonalized. By computing eigenvalues and eigenvectors numerically, one can investigate the properties of a system with almost no numerical error. However, the required memory becomes extremely large, which limits the size of the systems that can be studied.

Even so, the method had great appeal. If the program is written correctly, the computer does exactly what it is instructed to do, and one obtains a clear answer. Unlike stochastic methods such as Monte Carlo, which involve statistical fluctuations, it provides what I would describe as a "crisp, clear answer," and I found this very compelling. Since I had always enjoyed building things step by step through logical reasoning, I was strongly drawn to this field.

I studied books on numerical diagonalization and came to understand that an algorithm called the Lanczos method was well suited for this purpose. I wrote the program and began large-scale computations. Around that time, Japan's first supercomputer, a Hitachi HITAC S810, had just been installed at the Large-Scale Computing Center of the University of Tokyo, and the environment for using it had finally been established. The earliest supercomputers, developed in the United States, were cutting-edge machines primarily intended for defense-related computations and were not even installed at universities. The news that a supercomputer had been installed at the University of Tokyo is said to have come as a shock to the academic community in the United States. I heard that a campaign to bring supercomputers to university campuses was then launched under the leadership of Nobel Prize-winning physicist Kenneth Wilson.

It was not an era in which one could casually access such machines over a network as one can today. One had to go to the computer center in person, enter programs at a terminal near the supercomputer room, and receive the results on paper or magnetic tape, a cumbersome process by today's standards.

Even so, by making use of this machine, I was able to carry out computations using what was, at the time, an extremely large memory of 32 MB, not 32 GB. In particular, I was able to handle systems of up to 27 quantum spins, which was a world record at the time. With the cooperation of Hiizu Nakanishi of Keio University, and by exploiting symmetries to reduce the matrix size dramatically along with various other techniques, I was able to push the scale of the computations to its limit.

This work resulted in several papers and received a fair degree of recognition at the time. At the same time, however, I began to feel its limitations. What I truly wanted to understand was the behavior in the thermodynamic limit, that is, as the number of spins approaches infinity, which is the physically meaningful regime. However, extrapolating from around 27 spins to the infinite limit involved too large a gap.

Once I became aware of this gap, I came to feel that numerical computation based solely on diagonalization would not be sufficient to achieve a decisive understanding, and I decided to draw a line under this phase of my work.

Nevertheless, it seemed wasteful to simply abandon the computational techniques and programming innovations I had developed along the way. I therefore reorganized the programs I had written and rewrote them in a form that could be used by others for a wide range of problems. After carefully studying a book on programming methodology, I came to understand that a program that anyone can read is ultimately the most efficient one. I untangled the logic, added comments throughout to clarify what the program was doing, and released the result as a package that other researchers could use.

This program was named "TITPACK" after Tokyo Institute of Technology, and it went on to be widely used as a tool for numerical computation on quantum spin systems. The first version was developed together with Yoshihiro Taguchi, who was a student of Professor Oguchi at the time, but I rewrote the second version independently. The idea of distributing a library program together with an accessible manual was almost unheard of at the time. In that sense, our effort may be seen as a precursor to what is now called open source. It appears that various programs derived from and significantly extending TITPACK continue to be widely used today in various forms.

In this way, for some time after returning to Japan, I was absorbed in large-scale numerical computation on quantum spin systems and related topics. At the same time, however, I was becoming increasingly aware of the limitations of numerical computation, and the time was approaching when I would need to search for a new direction.

Around that time, in the mid-1980s, the neural network model proposed by John Hopfield, known as the Hopfield model, was attracting considerable attention. This model describes the mechanism of associative memory, addressing problems such as reconstructing a complete image from an incomplete one using the tools of physics.

What was particularly significant was that this model is deeply connected to spin glasses. The Hopfield model could be analyzed using methods from the statistical mechanics of spin glasses. A group of Israeli researchers extended this idea and carried out a thorough analysis of the model using the full machinery of spin glass theory.

Reading that paper came as a considerable shock, and I remember thinking, "How could anyone have come up with this?" The application of spin glass theory to problems in information processing went far beyond the scope I had previously considered.

While reading their paper, a new idea occurred to me. Their theory introduced classical temperature, or thermal fluctuations, into the Hopfield model, enabling a statistical mechanical analysis. I wondered, "What would happen if we introduced quantum fluctuations instead of temperature?"

This idea connected two lines of research I had been pursuing: spin glasses and quantum spin systems. I already had considerable knowledge and experience with quantum effects through my work on quantum spin systems, and I was also well acquainted with spin glasses and the Hopfield model. I thought that combining the two might open up a new direction.

More concretely, I considered a model in which quantum fluctuations were introduced into the Hopfield model through a transverse magnetic field. I worked on this problem together with Yoshihiko Nonomura, who was a postdoctoral researcher in my group at the time. The calculations turned out to be extremely involved.

Even analyzing the classical Hopfield model is far from straightforward, and adding quantum effects made the calculations even more cumbersome. It was an extremely painstaking process, involving nearly a hundred pages of handwritten calculations, checking each equation repeatedly as we proceeded.

The result we obtained was remarkably interesting. We found that quantum fluctuations and classical thermal fluctuations have nearly equivalent effects on the Hopfield model. It became clear that classical fluctuations due to temperature and quantum fluctuations play essentially the same role. This result came as a great surprise. At the same time, I felt some doubt: "Can that really be the case?" Given the sheer volume of calculations involved, rechecking them was not easy, but we continued to verify the results and eventually wrote them up as a paper.

This result would go on to have a major influence on subsequent research. It led to the idea of using quantum fluctuations in place of thermal fluctuations to explore and optimize the state of a system. This idea would later develop into the concept of quantum annealing.

When the idea of quantum annealing was first proposed, no actual device existed, and it attracted little attention even as a theoretical concept. Partly for that reason, I moved away from this topic for some time and continued working on other problems, such as spin glasses and the statistical mechanics of information processing.

In the 2010s, however, research on quantum computing began to advance rapidly. While work on gate-based quantum computing continued steadily, the influence of D-Wave also led to growing interest in quantum annealing, and related research projects began to emerge. In this environment, I found myself once again involved in the field.

Over roughly the past decade, I have continued research on quantum annealing. Beyond theoretical work, students and young researchers have played central roles in achieving significant results through numerical computations and experiments using actual machines. I feel that quantum annealing, which was once far ahead of its time, has now come to be recognized and more widely accepted.

Participating in DARPA and IARPA projects in the United States was also a valuable experience. The sheer number of talented researchers in the U.S., the breadth of research activities, and the strong leadership at the helm all conveyed a sense of power that made me think, "This is the strength of the United States." In Japan as well, substantial funding has recently been directed toward quantum research. However, I feel that unless we first focus on cultivating talent, rather than merely building hardware, it will be difficult to keep pace with the rest of the world, let alone lead it.

Looking back on my career, research often goes unrecognized even when one believes it to be important, and that may be the rule rather than the exception. More often than not, things do not go well, and failing to proceed as one hopes is simply the normal state of affairs. If one becomes discouraged by this, it is impossible to continue as a researcher.

There is a Japanese expression, senmitsu, which roughly means "three out of a thousand." Applied to research, it suggests that if you generate a thousand ideas, only about three will succeed. Among those three, only a very small fraction will stand the test of time. The important thing is therefore to accept that most attempts will not succeed and to continue thinking, generating ideas even in the face of repeated failure.

Ideas do not necessarily arise at special moments. In my case, I often did my thinking during my commute by train. The train ride provides an environment free from external distractions. When I focused my attention on a single problem in that setting, ideas would often emerge in unexpected ways.

In the end, the most important quality for a researcher may simply be the ability not to be discouraged by failure, to accept that things not going well is the normal state, and to move on to the next idea. I believe it is through this repeated cycle that meaningful results eventually emerge.

The habit of thinking independently is equally important. Rather than simply following fashionable topics, it is important to continue thinking about problems that you genuinely find interesting. I have done almost everything in a self-taught way since my student days, and I have carried that approach into my research as well.