The quasi-analog formulation of duality computers is provided, inspired from the actual path integral understanding of the double-slit experiment and the inadequacies of the standard formulation of duality computers in explaining the experiment. In (analog) quantum mechanics, position and momentum are not typically discrete variables, but it is feasible to take the qubit (digital) approximation by selecting a set of approximately orthogonal states as computational basis for the qubits, such as Gaussian states with sufficient distances. The qubit approximation as an effective representation cannot be used for addressing the question of whether the computational system evolves unitarily without violating linearity, and fine-grained dynamics can respect linearity when effective dynamics do not. A general limiting procedure for determining qubit states regardless of effective unitarity violation is provided. The new formulation resolves the zero wavefunction paradox naturally out of the box and provides a method for detecting the unnormalized zero wavefunction. The resulting formulation of duality computers does not require additional postselection or non-unitary operators over unitary quantum mechanics and has realistic potentials to provide additional computational advantage over ordinary quantum computers.
The quasi-analog formulation of duality computers is provided, inspired from the actual path integral understanding of the double-slit experiment and the inadequacies of the standard formulation of duality computers in explaining the experiment. In (analog) quantum mechanics, position and momentum are not typically discrete variables, but it is feasible to take the qubit (digital) approximation by selecting a set of approximately orthogonal states as computational basis for the qubits, such as Gaussian states with sufficient distances. The qubit approximation as an effective representation cannot be used for addressing the question of whether the computational system evolves unitarily without violating linearity, and fine-grained dynamics can respect linearity when effective dynamics do not. A general limiting procedure for determining qubit states regardless of effective unitarity violation is provided. The new formulation resolves the zero wavefunction paradox naturally out of the box and provides a method for detecting the unnormalized zero wavefunction. The resulting formulation of duality computers does not require additional postselection or non-unitary operators over unitary quantum mechanics and has realistic potentials to provide additional computational advantage over ordinary quantum computers.
At the heart of the puzzlement induced in many by the quantum theory lies a tension between reality and knowledge, between facts and information. Do the basic entities of the theory, quantum states or their wavefunctions, directly correspond to something in the real world so that something—wavefunction stuff—actually does pass through both slits in a two-slit interference experiment? If so, then either we are faced with the puzzle of how something real can suffer abrupt changes in response to faraway events or, if we banish wavefunction collapse from the theory, we are faced with a reality that absurdly evolves to encompass myriads of alternative histories growing ever more unalike. These puzzles melt away if the basic entities of the theory are merely representations of knowledge, only to be replaced by others. Whose knowledge? Knowledge of what? The new gedanken technology of quantum computation provides an unfamiliar perspective on such vexing questions, by using the quantum theory, not to expand our understanding and control of the physical world, but to exploit the quantum behavior of the physical world as a novel way to encode and process information. The information is primary; the underlying physical system only matters as a vehicle for that information. Quantum computer scientists view a set of n interacting spins-½ not for the insight it offers into the nature of magnetic materials, but as a way to represent and manipulate integers, through their n -bit binary representations as orthogonal states in a “computational basis” that specifies whether each individual spin is up (1) or down (0). Quantum computation differs in several crucial ways from ordinary classical computation: • The states of a quantum bit (which I here call a Qbit, in quixotic defiance of the fashionable but orthographically preposterous “qubit”) are not restricted to 0 or 1, as in a classical computer, but can be in arbitrary superpositions of 0 and 1. • Even more foreign to classical intuition, the state of the entire computer can be what Schrödinger called verschränkt (entangled), with individual Qbits having no (pure) states of their own at all. While a classical computation flips classical Cbits between 0 and 1, a quantum computation subjects the Qbits to the much more general unitary transformations that specify the time evolution of quantum states.
Analog quantum algorithms are formulated in terms of Hamiltonians rather than unitary gates and include quantum adiabatic computing, quantum annealing, and the Quantum Approximate Optimization Algorithm (QAOA).These algorithms are promising candidates for near-term quantum applications, but they often require fine tuning via the annealing schedule or variational parameters.In this work we connect all these algorithms to the optimal analog procedure.Notably, we explore how the optimal procedure approaches a smooth adiabatic procedure but with a superposed oscillatory pattern that can be explained in terms of the interactions between the ground state and first excited state.Furthermore, we provide numeric and analytic evidence that QAOA emulates this optimal procedure with the length of each QAOA layer equal to the period of the oscillatory pattern.Furthermore, the ratios of the QAOA bangs is determined by the smooth, non-oscillatory part of the optimal procedure.We provide arguments for these phenomena in terms of the product formula expansion of the optimal procedure.All of this shows that these analog algorithms are all different limits and approximations of the same optimal quantum protocol.In the appendix, we present a new algorithm for better approximating the optimal protocol using the insights from the main body.
One of the yet unsolved, simplest yet fundamental problems in physics is the question of the speed ofgravitational action. Suppose that the Sun disappeared suddenly. The question is: will gravitational pull of the Sun on Earth disappear instantaneously or with a delay of the speed of light? Based onastronomical observations, some experts have argued that the direction of gravitational pull of the Sun on Earth now is directed towards the position of the Sun now, implying instantaneous transmission. Moreover, they have argued that finite speed of gravity would result in instability of planetary systems. However, another line of reasoning by this author has also shown that this literal view of instantaneous transmission is also wrong. This paradox leads us to a profound conclusion that the speed of gravity has dual nature: light speed and instantaneous. How can this be? The new finding is that causality will be (apparently or actually) violated in gravitational phenomenon as follows. Suppose that the Sun disappeared at time t = 0. The strange thing is that the Sun will anticipate its disappearance and will emit zero gravitational field toward the Earth 8.3 minutes before its actual disappearance! This signal travels at the speed of light and reach the Earth after 8.3 minutes. This means that the signal of zero gravitational field will reach the Earth at time t = 0. The arrival of zero gravitational field at the Earth will coincide exactly in time with the disappearance of the Sun, implying instantaneous transmission. Actually, the discovery of this idea of causality violation did not emerge from investigation of the gravitational phenomenon discussed above. It emerged from a line of reasoning during investigation of quantum phenomena, such as particle interference patterns in double-slit experiments, quantum entanglement, “which way” and quantum erasure experiments. From a new insight about the internal structure and dynamics of quantum particles such as photons and electrons, a conclusion has been reached that the exact point on the detector screen of a double-slit experiment where a photon will land is completely predetermined by fine tuning the initial conditions of the photon and the electron during emission. This means that where exactly a photon lands on the detecting screen is completely predetermined by its initial conditions, just as where exactly a ball lands on the ground is predetermined by its initial conditions. This means that in principle an interference pattern or any arbitrary pattern could be formed on the detector screen by (almost) infinitely fine-tuning the initial conditions of the photon during emission, regardless of whether one or both slits are open, regardless of the distance between the slits, regardless of the distance of the detecting screen. The conclusion is that aiming a photon to a particular atom on the screen requires (almost) infinite fine-tuning during emission. This raises the burning question: WHO fine-tunes the photons during emission so that they form an interference pattern or a Gaussian pattern? The emitting atom? The light source conspiring with the detecting screen? Atoms have infinitely small intelligence as to do the infinitely complex task of fine-tuning a photon. The conclusion is that a super natural being is fine-tuning and, literally, aiming each and every photon emitted not only from the light source of a double slit experiment, but also all photons emitted from all light sources in the universe. The overwhelming evidence of divine intervention is that photons and electrons cannot by themselves behave both as waves and as particles simultaneously. LITERALLY, God has been behind all confusing light speed experiments and observations made over centuries. It has been discovered that the correct model of the speed of light is a novel, seamless fusion of ether theory and emission theory ( Apparent Source Theory ). Photons cannot by themselves exhibit this dual nature. Literally, all dualities observed so far in light, electromagnetism and gravity are possible ONLY by DIRECT divine intervention.
In this paper, we firstly briefly review the duality quantum computer. Distinctly, the generalized quantum gates, the basic evolution operators in a duality quantum computer are no longer unitary, and they can be expressed in terms of linear combinations of unitary operators. All linear bounded operators can be realized in a duality quantum computer, and unitary operators are just the extreme points of the set of generalized quantum gates. A d-slits duality quantum computer can be realized in an ordinary quantum computer with an additional qudit using the duality quantum computing mode. Duality quantum computer provides flexibility and clear physical picture in designing quantum algorithms, serving as a useful bridge between quantum and classical algorithms. In this review, we will show that duality quantum computer can simulate quantum systems more efficiently than ordinary quantum computers by providing descriptions of the recent efficient quantum simulation algorithms of Childs et al [Quantum Information & Computation, 12(11-12): 901-924 (2012)] for the fast simulation of quantum systems with a sparse Hamiltonian, and the quantum simulation algorithm by Berry et al [Phys. Rev. Lett. 114, 090502 (2015)], which provides exponential improvement in precision for simulating systems with a sparse Hamiltonian.
We briefly review what a quantum computer is, what it promises to do for us and why it is so hard to build one. Among the first applications anticipated to bear fruit is the quantum simulation of quantum systems. While most quantum computation is an extension of classical digital computation, quantum simulation differs fundamentally in how the data are encoded in the quantum computer. To perform a quantum simulation, the Hilbert space of the system to be simulated is mapped directly onto the Hilbert space of the (logical) qubits in the quantum computer. This type of direct correspondence is how data are encoded in a classical analogue computer. There is no binary encoding, and increasing precision becomes exponentially costly: an extra bit of precision doubles the size of the computer. This has important consequences for both the precision and error-correction requirements of quantum simulation, and significant open questions remain about its practicality. It also means that the quantum version of analogue computers, continuous-variable quantum computers, becomes an equally efficient architecture for quantum simulation. Lessons from past use of classical analogue computers can help us to build better quantum simulators in future.
This paper introduces several new classes of mathematical structures that have close connections with physics and with the theory of dynamical systems. The most general of these structures, called indivisible stochastic processes, collectively encompass many important kinds of stochastic processes, including Markov chains and random dynamical systems. This paper then states and proves a new theorem that establishes a precise correspondence between any indivisible stochastic process and a unitarily evolving quantum system. This theorem therefore leads to a new formulation of quantum theory, alongside the Hilbert-space, path-integral, and quasi-probability formulations. The theorem also provides a first-principles explanation for why quantum systems are based on the complex numbers, Hilbert spaces, linear-unitary time evolution, and the Born rule. In addition, the theorem suggests that by selecting a suitable Hilbert space, together with an appropriate choice of unitary evolution, one can simulate any indivisible stochastic process on a quantum computer, thereby potentially opening up an extensive set of novel applications for quantum computing.