Simulating Cognition with Quantum Computers Hongbin Wang, Jack W. Smith, Yanlong Sun {hwang, jwsmith, ysun}@medicine.tamhsc.edu Biomedical Informatics, College of Medicine, Texas A&M University 2121 W. Holcombe Blvd, Houston, TX 77030 USA Abstract finding that in recent years cognitive science has been more and more dominated by empirical psychology (and less There are inherent limits in classical computation for it to serve as an adequate model of human cognition. In particular, computational explorations) (Gentner, 2010). In a sense, we non-commutativity, while ubiquitous in physics and are forced to re-examine the relationship between cognition psychology, cannot be sufficiently dealt with. We propose and computation and answer a similar question asked by that we need a new mathematics that is capable of expressing Feynman in 1981: Can we simulate cognition by a more complex mathematical structures in order to tackle those computer? If so, what kind of computer are we going to hard X-problems in cognitive science. A quantum approach is use? advocated and we propose a way in which quantum computation might be realized in the brain. The issue is also of particular importance in the light of the recent resurgence of AI. Unprecedented capacities in Keywords: Computation, cognition, quantum mind, various areas including machine vision, natural language entanglement, non-commutativity processing, and game playing have been recently gained thanks to the advances in deep learning methods. Can we Introduction realize human-level intelligence by deep mining big data In 1981, Richard Feynman delivered a keynote speech in the using deep learning with more powerful computers? Can first conference on “physics and computation” held at MIT. deep learning technology finally lead to intuition and In the speech, entitled “simulating physics with computers”, consciousness in artificial systems? Feynman asked if physics can be simulated by a classical Here we intend to provide some general arguments universal computer. Given the fact that at that time towards answering these questions. In particular, we argue computers have already been extensively used for physical that human cognition involves more complex mathematical simulation and computation at that time the question might structures (e.g., non-commutativity) that are fundamentally sound strange. However, equipped with some profound new beyond the expressive power of classical Turing computers. understanding regarding Turing computation and quantum Consequentially, in order to develop a simultaneously mechanics, Feynman answered the question with a clear normative and descriptive theory of cognition one has to go “no”. He concluded, “Nature isn't classical, dammit, and if beyond classical set-theoretical computation. We you want to make a simulation of Nature, you'd better make demonstrate how formalisms in quantum theory affords it quantum mechanical, and by golly it's a wonderful structures necessary for us to model interesting cognitive problem, because it doesn't look so easy.” (Feynman, 1982, phenomena. Finally, we address a common criticism of the p.486). quantum brain approach and suggest a way where quantum We face a similar problem in cognitive science today. The behavior might be realized in a warm, wet, and noisy brain. tenet of cognition as computation directly catalyzed the We conclude by advocating that we need to go beyond cognitive revolution in the 1950s and has since become one classical computers for cognitive modeling in order to gain of the pillars of cognitive science (Miller, 2003). brand new insights about how the brain and the mind work. Undoubtedly, an impressive body of new results have been obtained, including the magic number seven two (Miller, The Inadequacy of Classical Computation for 1956) and the emergence of various executable cognitive Cognition architectures (e.g., Newell, 1990). On the other hand, Although the limits of the classical theory of computation unfortunately, these advances are in stark contrast with the have been well-known since its inception, computers can do lack of progress in answering some of the long-standing so much nowadays that it’s easy to forget that they were tough questions in cognitive science. Theoretically, for invented for what they could not do. In his 1937 paper, “On example, what are the implications of Gödel’s computable numbers, with an application to the incompleteness theorem on seeking a computational theory Entscheidungs problem,” Alan Turing defined the notion of of cognition (e.g., Penrose, 1989)? Empirically, some of the a universal digital computer (a Turing machine), and his fundamental psychological phenomena, including goal was to show that there were tasks that even the most consciousness, intuition, and various so-called “cognitive powerful computing machine could not perform (for biases” continue to defy a satisfactory computational example, the halting problem). These problems are simply description. Such theoretical and empirical dilemmas hint at beyond computation, in accordance with the now-famous “the unreasonable ineffectiveness of mathematics in Turing-Church thesis. cognitive science” (Poli, 1999) and partially explains the A similar result was obtained at about the same time by used a mathematics that is not powerful enough to capture Kurt Gödel. In 1931, through an ingenious device known as all the complexity of human cognition. Gödel numbering, Gödel found a way to assign natural There have been efforts to extend the classical theory of numbers in a unique way to the statements of arithmetic computation – its history is almost as long as the classical themselves, effectively turning numbers into statements that computational theory itself. Various models of computation talk about numbers. This permitted him to prove an that can compute functions not effectively computable in the incompleteness theorem, which basically says that there are Church-Turing thesis sense, often called hypercomputation true statements of mathematics (theorems) which we can or super-Turing computation, have been suggested (see never formally know to be true. https://en.wikipedia.org/wiki/Hypercomputation). It is interesting that although both Turing and Gödel One notable hypercomputational model is the Blum- proved that the complete body of human knowledge cannot Shub-Smale (BSS) machine (Blum et al., 1997), also called be acquired by formal computation alone given the a real computer. As we know, a classical computer depends method’s inherent limits (see Figure 1), they appear to offer on discrete symbols (e.g., 0s and 1s) to encode information different reasons for why the human mind is able to achieve and presupposes that all the underlying sets are countable the feat. According to Turing, it is unfair to compare a (one-to-one correspondence to natural numbers N). A real Turing machine and a human mind – the former runs computer is able to handle real numbers (R, a continuum) algorithmically and never makes mistakes and the latter and therefore can answer questions about subsets which are does “trial-and-error” and makes wild guesses all the time. uncountable (e.g., “is the Mandelbrot set decidable?”). “If a machine is expected to be infallible, it cannot also be While it has been shown that real computation can be intelligent”. And a machine can become intelligent and directly applied to problems in numerical analysis and human-like only if it makes no pretense at infallibility. scientific computing, it is not clear if it helps reduce the Gödel, on the other hand, did not want to give up on the discrepancy shown in Figure 1 in any fundamental way. We consistency of human knowledge. He suggested that “it argue that the extension from N to R, as significant as it remains possible that there may exist (and even be may seem, remains inadequate to handle some of the empirically discovered) a theorem-proving machine which toughest problems (see below) in cognitive science. in fact is equivalent to mathematical intuition, but cannot be proved to be so, nor can be proved to yield only correct X-problems and Non-Commutative theorems of finitary number theory”. Observables Following Penrose’s practice in physics (Penrose, 1997), it is helpful to distinguish two classes of problems in cognitive science that long for answers. One class can be called Z- problems (for puZZle), which refer to those empirical findings that are puzzling but somewhat explainable in classical computational terms. Examples of Z-problem include the distinction of short-term memory and long-term memory, the concept of working memory capacity, skill acquisition by forming and tuning if-then production rules, and attention through bottom-up and top-down controls. Another class of problems can be called X-problems (for paradoXes), referring to those empirical findings that are so Figure 1. A hypothetical problem complexity space. mysterious that they seem to defy classical mathematical descriptions. Examples of this class include consciousness In the end of his now famous 1960 article entitled “The and awareness, intuition, feeling, gestalt phenomena in unreasonable effectiveness of mathematics in the natural visual perception, and various so-called “cognitive biases” sciences”, Eugene Wigner wondered “if we could, some in human judgment and decision-making, to name a few. day, establish a [mathematical] theory of the phenomena of Discrediting them as ephemeral and unworthy, or simply consciousness, or of biology, which would be as coherent labeling them as “human biases and heuristics”, or and convincing as our present theories of the inanimate suggesting ad-hoc patched explanations, is inadequate. world”. Half a century later, Wigner’s hope apparently These phenomena are functions of the human brain/body, hasn’t been fulfilled. The list of “cognitive biases” on which resulted from millions of years of evolution and Wikipedia is getting longer and longer. Human adaption. They deserve more rigorous and more systematic “irrationality” seems everywhere. Of course irrationality in treatments and it is fair to say that cognitive science so far all these cases is defined as human deviation from classical has fallen short in this regard.