Quantum theory, the Church–Turing principle and the universal quantum computer

1985 ◽  
Vol 400 (1818) ◽  
pp. 97-117 ◽  
Keyword(s):  
Quantum Theory ◽  
Complexity Theory ◽  
Turing Machine ◽  
Physical System ◽  
Quantum Computer ◽  
Physical Principle ◽  
Universal Turing Machine ◽  
Universal Quantum ◽  
Computing Machines ◽  
The Church

It is argued that underlying the Church–Turing hypothesis there is an implicit physical assertion. Here, this assertion is presented explicitly as a physical principle: ‘every finitely realizible physical system can be perfectly simulated by a universal model computing machine operating by finite means’. Classical physics and the universal Turing machine, because the former is continuous and the latter discrete, do not obey the principle, at least in the strong form above. A class of model computing machines that is the quantum generalization of the class of Turing machines is described, and it is shown that quantum theory and the 'universal quantum computer’ are compatible with the principle. Computing machines resembling the universal quantum computer could, in principle, be built and would have many remarkable properties not reproducible by any Turing machine. These do not include the computation of non-recursive functions, but they do include ‘quantum parallelism’, a method by which certain probabilistic tasks can be performed faster by a universal quantum computer than by any classical restriction of it. The intuitive explanation of these properties places an intolerable strain on all interpretations of quantum theory other than Everett’s. Some of the numerous connections between the quantum theory of computation and the rest of physics are explored. Quantum complexity theory allows a physically more reasonable definition of the ‘complexity’ or ‘knowledge’ in a physical system than does classical complexity theory.

Sand Reckoning: Whose Information is It, Anyway?

2018 ◽  
Author(s):  
Vlatko Vedral
Keyword(s):  
Quantum Computer ◽  
Classical Mechanics ◽  
Single Atom ◽  
Second Law ◽  
Finish Line ◽  
Universal Turing Machine ◽  
Universal Quantum ◽  
Time And Energy ◽  
Microsoft Windows ◽  
The Universe

In Chapter 9 we discussed the idea of a universal Turing machine. This machine is capable of simulating any other machine given sufficient time and energy. For example, we discussed how your fridge microprocessor could be programmed to run Microsoft Windows, then we described Moore’s logic, that computers are becoming faster and smaller. Therefore, one day, a single atom may be able to simulate fully what a present day PC can do. This leads us to the fascinating possibility that every little constituent of our Universe may be able to simulate any other, given enough time and energy. The Universe therefore consists of a great number of little universal quantum computers. But this surely makes the Universe itself the largest quantum computer. So how powerful is our largest quantum computer? How many bits, how many computational steps? What is the total amount of information that the computer can hold? Since our view is that everything in reality is composed of information, it would be useful to know how much information there is in total and whether this total amount is growing or shrinking. The Second Law already tells us that the physical entropy in the Universe is always increasing. Since physical entropy has the same form as Shannon’s information, the Second Law also tells us that the information content of the Universe can only ever increase too. But what does this mean for us? If we consider our objective to be a full understanding of the Universe then we have to accept that the finish line is always moving further and further away from us. We define our reality through the laws and principles that we establish from the information that we gather. Quantum mechanics, for example, gives us a very different reality to what classical mechanics told us. In the Stone Age, the caveman’s perception of reality and what was possible was also markedly different from what Newton would have understood. In this way we process information from the Universe to create our reality. We can think of the Universe as a large balloon, within which there is a smaller balloon, our reality.

scholarly journals Turing Incomputable Computation

10.29007/x5g2 ◽  
2018 ◽  
Author(s):  
Michael Stephen Fiske
Keyword(s):  
Turing Machine ◽  
Active Element ◽  
Computational Procedure ◽  
Random Input ◽  
Firing Patterns ◽  
Reverse Engineer ◽  
New Class ◽  
Computing Model ◽  
Universal Turing Machine ◽  
Computing Machines

A new computing model, called the active element machine (AEM), is presented that demonstrates Turing incomputable computation using quantum random input. The AEM deterministically executes a universal Turing machine (UTM) program η with random active element firing patterns. These firing patterns are Turing incomputable when the AEM executes a UTM having an unbounded number of computable steps. For an unbounded number of computable steps, if zero information is revealed to an adversary about the AEM’s representation of the UTM’s state and tape and the quantum random bits that help determine η’s computation and zero information is revealed about the dynamic connections between the active elements, then there does not exist a “reverse engineer” Turing machine that can map the random firing patterns back to the sequence of UTM instructions. This casts a new light on Turing’s notion of a computational procedure. In practical terms, these methods present an opportunity to build a new class of computing machines where the program’s computational steps are hidden. This non-Turing computing behavior may be useful in cybersecurity and in other areas such as machine learning where multiple, dynamic interpretations of firing patterns may be applicable.

Quantum information distance

2021 ◽  
Author(s):  
Songsong Dai
Keyword(s):  
Quantum Information ◽  
Turing Machine ◽  
Kolmogorov Complexity ◽  
Transition Program ◽  
Information Distance ◽  
Universal Turing Machine ◽  
Classical Setting ◽  
Universal Quantum ◽  
Definition Of ◽  
Quantum Turing Machine

In this paper, we give a definition for quantum information distance. In the classical setting, information distance between two classical strings is developed based on classical Kolmogorov complexity. It is defined as the length of a shortest transition program between these two strings in a universal Turing machine. We define the quantum information distance based on Berthiaume et al.’s quantum Kolmogorov complexity. The quantum information distance between qubit strings is defined as the length of the shortest quantum transition program between these two qubit strings in a universal quantum Turing machine. We show that our definition of quantum information distance is invariant under the choice of the underlying quantum Turing machine.

Entanglement

2017 ◽  
Author(s):  
Richard Healey
Keyword(s):  
Quantum Theory ◽  
Quantum State ◽  
Physical Condition ◽  
Physical System ◽  
Polarization State ◽  
Quantum Systems ◽  
Ghz State ◽  
Mathematical Representation ◽  
Entangled State ◽  
Physical Relation

Often a pair of quantum systems may be represented mathematically (by a vector) in a way each system alone cannot: the mathematical representation of the pair is said to be non-separable: Schrödinger called this feature of quantum theory entanglement. It would reflect a physical relation between a pair of systems only if a system’s mathematical representation were to describe its physical condition. Einstein and colleagues used an entangled state to argue that its quantum state does not completely describe the physical condition of a system to which it is assigned. A single physical system may be assigned a non-separable quantum state, as may a large number of systems, including electrons, photons, and ions. The GHZ state is an example of an entangled polarization state that may be assigned to three photons.

scholarly journals Simulating molecules on a cloud-based 5-qubit IBM-Q universal quantum computer

2021 ◽  
Vol 4 (1) ◽  
Author(s):  
S. Leontica ◽  
F. Tennie ◽  
T. Farrow
Keyword(s):  
Quantum System ◽  
Energy Transport ◽  
Quantum Computer ◽  
Complex Molecule ◽  
Dissipative System ◽  
Quantum Simulation ◽  
Quantum Systems ◽  
Molecular Structures ◽  
Unitary Evolution ◽  
Universal Quantum

AbstractSimulating the behaviour of complex quantum systems is impossible on classical supercomputers due to the exponential scaling of the number of quantum states with the number of particles in the simulated system. Quantum computers aim to break through this limit by using one quantum system to simulate another quantum system. Although in their infancy, they are a promising tool for applied fields seeking to simulate quantum interactions in complex atomic and molecular structures. Here, we show an efficient technique for transpiling the unitary evolution of quantum systems into the language of universal quantum computation using the IBM quantum computer and show that it is a viable tool for compiling near-term quantum simulation algorithms. We develop code that decomposes arbitrary 3-qubit gates and implement it in a quantum simulation first for a linear ordered chain to highlight the generality of the approach, and second, for a complex molecule. We choose the Fenna-Matthews-Olsen (FMO) photosynthetic protein because it has a well characterised Hamiltonian and presents a complex dissipative system coupled to a noisy environment that helps to improve the efficiency of energy transport. The method can be implemented in a broad range of molecular and other simulation settings.

scholarly journals Undecidability in quantum thermalization

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Naoto Shiraishi ◽  
Keiji Matsumoto
Keyword(s):  
Turing Machine ◽  
Thermal Equilibrium ◽  
General Theorem ◽  
Nearest Neighbour ◽  
Many Body ◽  
Initial State ◽  
One Dimensional ◽  
Universal Turing Machine ◽  
Many Body Systems ◽  
Neighbour Interaction

AbstractThe investigation of thermalization in isolated quantum many-body systems has a long history, dating back to the time of developing statistical mechanics. Most quantum many-body systems in nature are considered to thermalize, while some never achieve thermal equilibrium. The central problem is to clarify whether a given system thermalizes, which has been addressed previously, but not resolved. Here, we show that this problem is undecidable. The resulting undecidability even applies when the system is restricted to one-dimensional shift-invariant systems with nearest-neighbour interaction, and the initial state is a fixed product state. We construct a family of Hamiltonians encoding dynamics of a reversible universal Turing machine, where the fate of a relaxation process changes considerably depending on whether the Turing machine halts. Our result indicates that there is no general theorem, algorithm, or systematic procedure determining the presence or absence of thermalization in any given Hamiltonian.

Sign in / Sign up

Export Citation Format

Share Document