Satoshi Iriyama, Masanori Ohya, Igor Volovich
Ohya and Volovich have proposed a new quantum computation model with chaotic amplification to solve the SAT problem, which went beyond usual quantum algorithm. In this paper, we generalize quantum Turing machine, and we show in this general quantum Turing machine (GQTM) that we can treat the Ohya-Volovich (OV) SAT algorithm.
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We investigate a 2-spin quantum Turing architecture, in which discrete local rotations α_m of the Turing head spin alternate with quantum controlled NOT-operations. We demonstrate that a single chaotic parameter input α_m leads to a chaotic dynamics in the entire Hilbert-space.
We study a 2-spin quantum Turing architecture, in which discrete local rotations α_m of the Turing head spin alternate with quantum controlled NOT-operations. We show that a single chaotic parameter input α_m leads to a chaotic dynamics in the entire Hilbert space. The instability of periodic orbits on the Turing head and `chaos swapping' onto the Turing tape are demonstrated explicitly as well as exponential parameter sensitivity of the Bures metric.
We study a two-spin quantum Turing architecture, in which discrete local rotations α_m of the Turing head spin alternate with quantum controlled NOT-operations. Substitution sequences are known to underlie aperiodic structures. We show that parameter inputs α_m described by such sequences can lead here to a quantum dynamics, intermediate between the regular and the chaotic variant. Exponential parameter sensitivity characterizing chaotic quantum Turing machines turns out to be an adequate criterion for induced quantum chaos in a quantum network.
We show that the lambda-q calculus can efficiently simulate quantum Turing machines by showing how the lambda-q calculus can efficiently simulate a class of quantum cellular automaton that are equivalent to quantum Turing machines. We conclude by noting that the lambda-q calculus may be strictly stronger than quantum computers because NP-complete problems such as satisfiability are efficiently solvable in the lambda-q calculus but there is a widespread doubt that they are efficiently solvable by quantum computers.
Ordinary approach to quantum algorithm is based on quantum Turing machine or quantum circuits. It is known that this approach is not powerful enough to solve NP-complete problems. In this paper we study a new approach to quantum algorithm which is a combination of the ordinary quantum algorithm with a chaotic dynamical system. We consider the satisfiability problem as an example of NP-complete problems and argue that the problem, in principle, can be solved in polynomial time by using our new quantum algorithm.
In honor of Alan Turing's hundredth birthday, I unwisely set out some thoughts about one of Turing's obsessions throughout his life, the question of physics and free will. I focus relatively narrowly on a notion that I call "Knightian freedom": a certain kind of in-principle physical unpredictability that goes beyond probabilistic unpredictability. Other, more metaphysical aspects of free will I regard as possibly outside the scope of science. I examine a viewpoint, suggested independently by Carl Hoefer, Cristi Stoica, and even Turing himself, that tries to find scope for "freedom" in the uni…
Bit flips, saturation, and quantum chaos in dissipative cat qubits
通过量子轨迹模拟、半经典分析和Liouvillian谱方法，研究耗散猫量子比特中比特翻转的动力学，揭示反射对称性破缺导致饱和，并发现混沌爆发机制。
We study numerically and analytically the time dependence and saturation of out-of-time ordered correlators (OTOC) in chaotic few-body quantum-mechanical systems: quantum Henon-Heiles system (weakly chaotic), BMN matrix quantum mechanics (strongly chaotic) and Gaussian random matrix ensembles. The growth pattern of quantum-mechanical OTOC is complex and nonuniversal, with no clear exponential regime at relevant timescales in any of the examples studied (which is not in contradiction to the exponential growth found in the literature for many-body systems, i.e. fields). On the other hand, the pl…
Quantum Kolmogorov complexity and quantum correlations in deterministic-control quantum Turing machines
本文扩展 dcq-TM 以处理混合态并定义量子态 Kolmogorov 复杂度，据此分析经典表示、不可复制性与量子关联，提出满足信息对称性的算法互信息。
In maximally chaotic quantum systems, a class of out-of-time-order correlators (OTOCs) saturate the Maldacena-Shenker-Stanford (MSS) bound on chaos. Recently, it has been shown that the same OTOCs must also obey an infinite set of (subleading) constraints in any thermal quantum system with a large number of degrees of freedom. In this paper, we find a unique analytic extension of the maximally chaotic OTOC that saturates all the subleading chaos bounds which allow saturation. This extremally chaotic OTOC has the feature that information of the initial perturbation is recovered at very late tim…
We investigate why the Lyapunov exponents $λ$ of out-of-time-ordered correlators (OTOCs) satisfy a universal bound $λ< 2 πk_B T/{\hbar}$ by probing imaginary-time path-integral space using ring-polymer molecular dynamics (RPMD) for a barrier-crossing model. We find that the RPMD OTOC satisfies the same bound as the quantum OTOC, which is caused by the stability of quantum thermal fluctuations around the instanton on the barrier. We expect that similar instantons (or other delocalised structures) will be found in many other systems with exponentially growing OTOCs.
We prove the following facts about the language recognition power of quantum Turing machines (QTMs) in the unbounded error setting: QTMs are strictly more powerful than probabilistic Turing machines for any common space bound $ s $ satisfying $ s(n)=o(\log \log n) $. For "one-way" Turing machines, where the input tape head is not allowed to move left, the above result holds for $s(n)=o(\log n) $. We also give a characterization for the class of languages recognized with unbounded error by real-time quantum finite automata (QFAs) with restricted measurements. It turns out that these automata ar…
Quantum Random Self-Modifiable Computation
该研究提出可执行图灵机指令、量子随机指令与元指令的ex-machine模型，证明其可计算图灵不可计算语言且存在能逐步判定停机问题的演化路径，为希尔伯特的有限过程构想提供了新的实现思路。
Spread and Spectral Complexity in Quantum Spin Chains: from Integrability to Chaos
本文研究混合场伊辛模型和 XXZ 自旋链次近邻变形中传播复杂度与谱复杂度的行为，发现峰值前饱和是混沌特征，饱和值受哈密顿量对称性和维度约束，并指出谱复杂度饱和由最小能量差决定，同时论证热场双态适合探测量子多体混沌。
We conjecture a chaos energy bound, an upper bound on the energy dependence of the Lyapunov exponent for any classical/quantum Hamiltonian mechanics and field theories. The conjecture states that the Lyapunov exponent $λ(E)$ grows no faster than linearly in the total energy $E$ in the high energy limit. In other words, the exponent $c$ in $λ(E) \propto E^c \,(E\to\infty)$ satisfies $c\leq 1$. This chaos energy bound stems from thermodynamic consistency of out-of-time-order correlators (OTOC's) and applies to any classical/quantum system with finite $N$ / large $N$ ($N$ is the number of degrees…
Spread complexity and quantum chaos for periodically driven spin chains
本文利用Arnoldi迭代研究周期性踢动Ising自旋链的传播复杂度，发现混沌动力学中Arnoldi系数涨落受抑制且复杂度饱和值更大，并考察了驱动频率的影响。
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