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Part 3 of 3 Part Series

The Essence of Quantum Computing
Part 3 of 3 Part Series

By contrast, Albert Einstein’s theory of relativity rests on two intuitively simple and physically understandable principles: (1) the speed of light is constant, and (2) the laws of physics are the same for two observers moving at constant speed relative to one another. From these the rest of the theory follows. Physicists yearn to be able to state quantum theory with similar, intuitively understandable simplicity, perhaps without discarding its unique probabilistic aspect, but nevertheless explains the contradictory presence of superposition and wave function collapse in a measurement. Figure 3.1 in Part I provided a tantalizing, partial glimpse of such a possibility. Measurement is the mystery. a quantum object generally offers more encoded, revelatory options for measurement than can be seen in practice. It is this aspect that we are concerned with here. In short, our attempt is to speak about the “underlying reality” that creates those measurement probabilities.

Note that a mathematical theory is abstract. It describes any system by some indexed list of properties and their possible values and the rules by which the properties can change. It is devoid of semantics. Our approach therefore neither addresses nor alters any underlying physics measurable at the quantum mechanical level. It just suggests a mechanism for relating inputs with outputs at the subquantum level. In a sense, we are assuming that quantum mechanics approximates a deeper theory. At the level of quantum computing and the no go theorems, it is obvious that quantum superposition and entanglement enable us to encode and preserve information and retrieve it from space and time provided all processes are unitary operations.

While the formalism of quantum mechanics is widely accepted, it does not have a unique interpretation due to the incompatibility between postulates 2 and 3 (see Part I). Indeed, without postulate 3 telling us what we can observe, the equations of quantum mechanics would be just pure mathematics devoid of physical meaning. Any interpretation can come only after an investigation of the logical structure of the postulates of quantum mechanics is made. For example, Newtonian mechanics does not define the structure of matter. Therefore, how we model the structure of matter is largely an independent issue. However, given the success of Newtonian mechanics, any model we propose is expected to be compatible with Newton’s laws of motion in the realm where it rules. Since Newton, our understanding of the structure of matter has undergone several changes without affecting Newton’s laws of motion. A question such as whether a particular result deduced from Newton’s laws of motion is deducible from a given model of material structure is therefore not relevant.

Likewise, as long as our interpretation (or model) of superposition, entanglement, and measurement does not require the postulates of quantum mechanics to be altered, all the predictions made by quantum mechanics will remain compatible with our interpretation. This assertion is important because we make no comments on the Hamiltonian, which captures the detailed dynamics of a quantum system. Quantum mechanics does not tell us how to construct the Hamiltonian. In fact, real life problems seeking solutions in quantum mechanics need to be addressed in detail by physical theories built within the framework of quantum mechanics. The postulates of quantum mechanics provide only the scaffolding around which detailed physical theories are to be built. This gives us an opportunity to speculate about the abstract state vector |ψ〉 at the sub-Planck level without affecting the postulates of quantum mechanics. The sub-Planck scale gives us the freedom to construct mechanisms for our interpretation that need not be bound by the laws of quantum mechanics because they are not expected to rule in that scale. The high point of the interpretation is that it explains the measurement postulate as the inability of a classical measuring device to measure at a precisely predefined time.

Following the principle of Occam’s razor that “entities should not be multiplied unnecessarily” or the law of parsimony, the interpretation adopted by Bera and Menon20 posits that the sub-Planck scale structure of the state vector is such that its eigenstates are dynamically time-division multiplexed. To this is added a probabilistic measurement model which determines only the instantaneous eigenstate of the system at the instant of measurement. The instant of measurement is chosen randomly by the classical measurement apparatus, once activated, within a small interval. The measured result is regarded as the joint product of the quantum system and the macroscopic classical measuring apparatus. Measurement is complete when the wave function assumes the measured state.

In the dynamic time-division multiplexing, superposed states appear as time-sliced in a cyclic manner such that the time spent by an eigenstate in a cycle is related to the complex amplitudes appearing in the state vector. Entangled states binding two or more particles appear in this interpretation as the synchronization of the sub-Planck level oscillation of the participating particles. Unlike the Copenhagen interpretation, in this interpretation it is not meaningless to ask about the state of the system in the absence of a measuring system. Indeed, the interpretation can be related to the macroscopic world we live in and hence appears more intuitive to the human mind than the other interpretations in the literature.

Consider a qubit with the state vector |ψ〉 = α|0〉 + β|1〉. Its hypothesized structure in the sub-Planck scale is illustrated in Figure 13.1.


Figure 13.1 A single particle system in superposition state.

In a cycle time of Tc the qubit oscillates between the eigenstates |0〉 and |1〉. We assume Tc to be much smaller than Planck time (<< 10-43 sec) to allow us to interpret the state vector independently of the Schrödinger equation. (The implicit assumption here is that in some averaged sense, perhaps with some additional information, these oscillations will represent the state vector |ψ〉, say, analogous to the case of a volume of gas in classical mechanics, where random molecular motions, appropriately averaged, represent classical pressure, temperature, and density of a volume of gas.) It is not necessary for us to know the value of Tc. We only assert that it is a universal constant. Within a cycle, the time spent by the qubit in state |0〉 is T0 = |α|2 Tc and in state |1〉 is T1 = |β|2Tc so that Tc = T0 + T1. Superposition is interpreted here as the deterministic linear sequential progression of the qubit’s states |0〉 and |1〉. A measurement on this qubit will return the instantaneous pre-measurement state of the qubit and collapse the qubit to the measured state.

20Bera & Menon (2009).

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