Quantum Science Philippines
Quantum Science Philippines

What quantum computers may tell us about quantum mechanics

Carlo Paul P. Morente

Even if quantum mechanics occupies a unique role in the course of history, its foundations are often questioned due to the difficulty of reconciling it with classical laws of physics. While today’s technology leads to the device being miniature towards atomic scale, quantum effects are beginning to confront this trend such as unnecessary quantum tunneling of electrons and large signal fluctuations. While many of these effects inhabit the continued miniaturization, another opportunities are arising such as quantum information processing which not only provide a way faster devices in terms of its performance but may also eclipse our existing technology. Instead of shrinking further the size of chip components, this new field takes advantage of different physical principles underlying these components.

Quantifying information began in mid century from the discovery of binary digit or bits by Claudde Shannon. Impressive growth in the technology of processing information speed and computing power is described exponentially where the chip components shrink in size. This is exponential growth is described in Moore’s law. New information sciences arise then as the limit of classical bits are meet, such as quantum information processing. Whereas Shannon’s bits can be 0 or 1, quantum bits, which is the simplest mechanical unit of information known as qubits, on the other hand can store superposition of 0 and 1. A single qubit is represented by the quantum state:

[eq]\psi = \alpha\left | 0 \right\rangle + \beta\left | 1 \right\rangle[/eq]

Where α and β are complex amplitudes of the superposition.

Hints of the power of quantum computing arise from the fact that in general for N qubits, it stores a superposition of 2N binary numbers. 2N are possibilities of measurement, and according to quantum mechanics the measurement yields only one answer out of these possibilities which makes it hard in designing useful quantum computing algorithm. The trick behind a useful quantum computer is the phenomenon called quantum interference. As the complex amplitude evolves in the wave equation it is made to interfere with each other, and in the end these amplitudes will cancel out leaving only few or one answer. In some case this implies exponential speed up over what can be obtained classically.

In order to understand the nature of quantum computing or we need to be familiar with one of its major properties: quantum entanglement. Quantum entanglement is combination of two properties in quantum mechanics – superposition and measurement – that are themselves unremarkable, but taken together cause all the usual interpretative conundrums of quantum mechanics. Yet quantum entanglement seemed to be the most misunderstood concept in quantum mechanics. There are several definitions of quantum entangle with their own supporting assumptions.

The hunt for the way to measure or quantify how much entangled a given quantum state is leads to the formulation of two major definitions.

First: an entangled state is one that is not separable, where highly quantum-efficient measurements are performed on one constituent without affecting the others. This definition arises from the fact that there is a correlation between the subsystems of the entangled state, wherein one cannot measure the constituent state without affecting the others. Therefore when measuring the state there should be high detector quantum efficiency that reflects a measurement of any previously prepared quantum state. It is required that the probability distribution of measurement should results accurately reflect the amplitude of the original quantum states.

Second is a more strict definition of entanglement which is somehow derived from above definition having considered the bells inequality. It states that: An entangled state is one that is not separable, where highly quantum measurements are performed on one constituent without affecting the others and where the constituents are space-like separated during the measurement time. This would rule any possibility of interaction between constituents during a measurement requiring that the two subsystems must be separated by a space-like interval. In general, there is no known measure of how much entanglement a given quantum state has.

These definitions comprise the reference in building a quantum computer. One must have arbitrary and controlled unitary operators to launch a pure initial quantum state. This will require that the qubits must be very isolated from the environment and preserve the superposition character, yet it must interact strongly with one another in order to become entangled. On the other hand it also requires that there must be a strong interaction with the environment to be switch on at will in order to measure qubits, which confronts physicist with the problem of quantum measurement. This will require high quantum efficiency.  Thus the most attractive physical candidate for quantum information processors are said to be “exotic” physical systems offering a high degree of quantum control.

Physicists today continue to probe the foundational aspects of quantum mechanics trying to meld it to quantum measurement. Theories like bohemian mechanics, many-worlds interpretations, transactional interpretation and the quantum decohernce theory, all attempts to meld quantum mechanics and quantum measurement with their supporting assumptions and theories yet non address the quantum measurement problems. There at least one alternative to quantum mechanics that is testable called “spontaneous wave function collapse”. This theory attempts to meld quantum measurement and quantum mechanics by adding nonlinear stochastic driving field to quantum mechanics that randomly localizes or collapses the wave function. What makes it remarkable is that this theory is testable, yet this is still far beyond the realization of a quantum processor.

This journey towards quantum computers yields at least three possible results, in which two are tantalizing: either a full blown large-scale quantum computer will be built, or the theory of quantum mechanics will be found incomplete. The third possibility would be that we can never reach to have the first possibility due to economic constraints.

*A summary of Christopher Monroe’s “What Quantum Computers May Tell Us About Quantum Mechanics”

Science and Ultimate Reality, eds. J. D. Barrow, P. C. W. Davies and C. L. Harper Jr. Published by Cambridge University Press. Cambridge University Press 2004.

Carlo Paul P. Morente is a graduate student in Physics of Mindanao State University-Iligan Institute of technology (MSU-IIT) Iligan City Philippines.

In order to understand the nature of quantum computing or we need to be familiar with one of its major properties: quantum entanglement. Quantum entanglement is combination of two properties in quantum mechanics – superposition and measurement – that are themselves unremarkable, but taken together cause all the usual interpretative conundrums of quantum mechanics. Yet quantum entanglement seemed to be the most misunderstood concept in quantum mechanics. There are several definitions of quantum entangle with their own supporting assumptions.

The hunt for the way to measure or quantify how much entangled a given quantum state is leads to the formulation of two major definitions.

First: an entangled state is one that is not separable, where highly quantum-efficient measurements are performed on one constituent without affecting the others. This definition arises from the fact that there is a correlation between the subsystems of the entangled state, wherein one cannot measure the constituent state without affecting the others. Therefore when measuring the state there should be high detector quantum efficiency that reflects a measurement of any previously prepared quantum state. It is required that the probability distribution of measurement should results accurately reflect the amplitude of the original quantum states.

Second is a more strict definition of entanglement which is somehow derived from above definition having considered the bells inequality. It states that: An entangled state is one that is not separable, where highly quantum measurements are performed on one constituent without affecting the others and where the constituents are space-like separated during the measurement time. This would rule any possibility of interaction between constituents during a measurement requiring that the two subsystems must be separated by a space-like interval. In general, there is no known measure of how much entanglement a given quantum state has.

These definitions comprise the reference in building a quantum computer. One must have arbitrary and controlled unitary operators to launch a pure initial quantum state. This will require that the qubits must be very isolated from the environment and preserve the superposition character, yet it must interact strongly with one another in order to become entangled. On the other hand it also requires that there must be a strong interaction with the environment to be switch on at will in order to measure qubits, which confronts physicist with the problem of quantum measurement. This will require high quantum efficiency. Thus the most attractive physical candidate for quantum information processors are said to be “exotic” physical systems offering a high degree of quantum control.

Physicists today continue to probe the foundational aspects of quantum mechanics trying to meld it to quantum measurement. Theories like bohemian mechanics, many-worlds interpretations, transactional interpretation and the quantum decohernce theory, all attempts to meld quantum mechanics and quantum measurement with their supporting assumptions and theories yet non address the quantum measurement problems. There at least one alternative to quantum mechanics that is testable called “spontaneous wave function collapse”. This theory attempts to meld quantum measurement and quantum mechanics by adding nonlinear stochastic driving field to quantum mechanics that randomly localizes or collapses the wave function. What makes it remarkable is that this theory is testable, yet this is still far beyond the realization of a quantum processor.

This journey towards quantum computers yields at least three possible results, in which two are tantalizing: either a full blown large-scale quantum computer will be built, or the theory of quantum mechanics will be found incomplete. The third possibility would be that we can never reach to have the first possibility due to economic constraints.

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Entanglement, Decoherence and the Quantum/Classical Boundary*

Christine Marie T. Ceblano

Since 1996 to 1998, great advancement has been made in the entangling experiments, like ions in trap and atoms in high Q-cavities which are disparate in techniques and yet similar in a way that they both comprehend a simple situation in which a two-level atom is coupled to a quantized harmonic oscillator. The Hamiltonian of this system, though simple, describes a great variety of interesting situations. With such system, proposals have been made to incorporate the Schrödinger’s cat where its role would be played by an excited harmonic oscillator.

In the ion trap experiment of the National Institute of Standards and Technology (NIST) group in Boulder, Colorado headed by Monroe and Wineland, a single beryllium ion is monitored by using sequence of laser pulses which later on split the ion’s Gaussian wavefunction into two wave packets, correlated to the [eq]\mid +>[/eq] and [eq]\mid ->[/eq]  hyperfine states, a Schrödinger’s cat situation, then finally recombine the two hyperfine states.

The experiment was repeated with different values of phase [eq]\varphi[/eq]. When it approaches zero, interference fringes were observed in  [eq]\mid +>[/eq] fluorescence signal. This shows the coherent superposition of the ion’s states of motion. The interesting part of the experiment is the fact that the packets remain Gaussian and do not disperse in time, which is somehow useful for decoherence studies.

Another entangling experiment is the atom-cavity performed by Serge Haroche group in Paris where the role of the cat is played by a field oscillator consisting of a few photons stored in high Q-cavity. Later on a rubidium atom (quantum mouse) is sent across the cavity probing the cat state to detect the oscillator quantum coherence. After interaction, the field oscillates with two different phases at once, a Schrodinger’s cat situation. By varying the delay time between the two atoms, decoherence is observed and is faster with large phase separation between cavity field components. This is an implication of the fragility of coherences between two macroscopically distinct states. Somehow, this experiment gives us a glimpse from the quantum mechanical (with the cat both dead and alive) to the classical realm transition (with the cat is either dead or alive).

Overall, both experiments show a statistical distribution of outcomes over many repetitions of experiments. And both can also be useful to various interesting applications like quantum information processing machines and quantum teleportation.

*This is a summary of “ENTANGLEMENT, DECOHERENCE AND THE QUANTUM/CLASSICAL BOUNDARY” by Serge Haroche

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A Controlled-NOT Quantum Logic Gate

by Majvell Kay G. Odarve

A quantum computer is device for computation that uses the phenomena of quantum mechanics to perform operations on data. Because of the quantum mechanical phenomena, such as entanglement and superposition of states, quantum computers have great offers in the field of computations and data handling. The distinctive feature of a quantum computer lies on its ability to store and process superposition of numbers.  This potential for parallel computing points out that some problems can be efficiently solved using quantum computers compared to the classical one. Shor’s algorithm, for example, which solves the problem that ‘Given an integer N, find its prime factors’, shows that quantum computer should be able to efficiently factor large numbers. The field appears to be of great interest since most data encryption schemes (in cryptography-science of information security) relies on the inability of the classical computers to factor large numbers.

To have an experimental realization of a quantum computer, there is a requirement of an isolated quantum system which will act as qubits and the presence of controlled unitary interactions between the qubits that allow construction of a controlled-NOT (CN) quantum logic gate. Quantum logic gates are building blocks of quantum circuits which operate on small numbers of quantum bits (qubits).  Quantum logic gates, unlike the classical logic gates, are reversible.  A CN quantum logic gate is one of the commonly used logic gates which operate on two qubits (we label the qubits as [eq]\epsilon_1[/eq] and [eq]\epsilon_2[/eq]).   The CN gate transforms the state of the two qubits from [eq]|\epsilon_1>|\epsilon_2> [/eq] to [eq]|\epsilon_1>|\epsilon_1\oplus\epsilon_2>[/eq] where [eq]\oplus[/eq] is an addition modulo 2. The CN gate represents a computation at the most fundamental level, that is, a certain ‘target’ qubit [eq]\epsilon_2[/eq] is flipped depending on the state of a ‘control’ qubit [eq]\epsilon_1[/eq].

Christopher Monroe and his team from National Institute of Standards and Technology (NIST) Laboratory in Boulder, Colorado, who are working on ion-trapped quantum computers, have been able to demonstrate the operation of a two-bit controlled-NOT quantum logic gate operating on prepared quantum states.  In their experiment, the two qubits comprise two internal (hyperfine) states and two external (quantized motional harmonic oscillator) states of a single trapped atom using a single beryllium ion (Be+). The trapped ions are first laser cooled to zero-point energy for them to stay in the ground state.   In the trapped-ion architecture, the qubits are associated with the internal states of the ions and information is transferred between the qubits through a shared motional degree of freedom. With this configuration, decoherence can be small so it will be easier to extend the idea to large registers and the qubit readout will have a nearly unit efficiency.

To realize the CN gate, three sequential pulses of the Raman beams is applied to the ion, namely the π/2 pulse applied to the carrier transition, a 2π pulse applied on the blue side band transition and a π/2 pulse applied to the carrier transition with a π phase shift relative to the first pulse. The truth table for a CN gate operation is given as:

Input State        [eq]\longrightarrow[/eq]     Output State

[eq]|0>|\downarrow> [/eq]  [eq]\longrightarrow[/eq]  [eq]|0>|\downarrow>[/eq]

[eq]|0>|\uparrow> [/eq]  [eq]\longrightarrow[/eq]  [eq] |0>|\uparrow>[/eq]

[eq]|1>|\downarrow> [/eq] [eq]\longrightarrow[/eq]  [eq]|1>|\uparrow>[/eq]

[eq]|1>|\uparrow> [/eq]  [eq]\longrightarrow[/eq]  [eq]|1>|\downarrow>[/eq]

In their experiment, the key features of the CN gate was demonstrated by verifying that the populations of the register follow the truth table and by demonstrating the conditional quantum dynamics associated with the CN operation. On the results of the experiments of Monroe, et. al., decoherence were still present. The decoherence were caused by instabilities of the laser beam power, the position of the ions relative to the laser beams, the fluctuation of external magnetic fields, instabilities in the trap drive frequency and voltage amplitude, dissipation of the ion motion and some spontaneous emission caused by off-resonant transitions. The single ion quantum register in the experiment comprises only two qubit which is not useful for computations. The next step of the researchers then is to apply their operation techniques to many ions cooled at a state of collective motion for the possibility of implementing computations on larger quantum registers.

Reference:

1. C. Monroe, D.M. Meekhof, B.E. King, W.M. Itano, and D.J. Wineland, Physical Review Letters, Vol. 75, Num. 25, December 1995.

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Reality and Quantum theory

Patrick Alvin M. Alcantara

Einstein maintained that quantum mechanics entails “spooky actions at a distance” (the interaction of two objects which are separated in space with no known mediator of the interaction); experiments have now shown that what bothered Einstein is not a debatable point but the observed behavior of the real world. He called this “spooky action at a distance” because he didn’t know about decoherence, so it seemed spooky to him.

In May 1935, Albert Einstein, Boris Podolsky and Nathan Rosen published the EPR Paper, an argument that quantum mechanics fails to provide a complete description of physical reality. The theoretical and experimental work it inspired remain remarkable for the vivid illustration they provide of one of the most bizarre aspects of the world revealed to us by the quantum theory. Their work describes a situation ingeniously to force the quantum theory into asserting that properties in space-time region B are the result of an act of measurement in another region A; so far from B that there is no possibility of the measurement in A exerting an influence on region B by any known dynamical mechanism. Under these conditions, Einstein maintained that the properties in A must have existed all along. The fundamental result that they were trying to show in their paper was not that quantum mechanics is wrong. They did, in fact, acknowledge that quantum mechanics could be used to make highly accurate statistical predictions about experiments. They were interested mainly in what the fundamental properties of reality are.

Their paper involves a paradox — a thought experiment which challenged long-held ideas about the relation between the observed values of physical quantities and the values that can be accounted for by a physical theory. According to its authors the EPR experiment yields a dichotomy. Either:

  1. The result of a measurement performed on one part A of a quantum system has a non-local effect on the physical reality of another distant part B, in the sense that quantum mechanics can predict outcomes of some measurements carried out at B.
  2. Quantum mechanics is incomplete in the sense that some element of physical reality corresponding to B cannot be accounted for by quantum mechanics (that is, some extra variable is needed to account for it).

An enormous set of data, generated out from the apparatus used in the said experiment, by many, many runs. Thus, as Einstein partly said on his letter to Max Born: …I am therefore inclined to believe that the description of quantum mechanics…has to be regarded as an incomplete and indirect description of reality…

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On Paul Dirac’s Theory of Electrons and Positrons

Hananish Joy G. Odarve

Paul Adrien Maurice Dirac was born on 8th August, 1902, at Bristol, England. He was educated at the Merchant Venturer’s Secondary School, Bristol, and then went on to Bristol University where he studied and obtained B.Sc. in Electrical Engineering degree. He also studied mathematics for two years at Bristol University, later going on to St. John’s College, Cambridge, as a research student in mathematics. He received his Ph.D. degree in 1926. He became a Fellow of St. John’s College and also held the position Lucasian Professor of Mathematics at Cambridge. Dirac was given The Nobel Prize in Physics 1933 together with Erwin Schrödinger for their discovery of new productive forms of atomic theory. He then gave a lecture regarding matter and antimatter specifically on electrons and protons on the Nobel Lecture he delivered on December 12, 1933.

In his lecture, Dirac emphasizes that the procedure he came up with is successful in the case of electrons and positrons and that he hoped that in the future some such procedure will be found for the case of the other particles. He considered the electron and positron because in their case, the theory has been developed further. He outline the method for electrons and positrons, showing how one can deduce the spin properties of the electron, and then how one can infer the existence of positrons with similar spin properties and with the possibility of being annihilated in collisions with electrons.

The general quantum mechanics at Dirac’s time describe the motion of any kind of particle, no matter what their properties are. However, it is only valid when the particles have small velocities and fail when the effect of relativity comes in. Basically, Dirac started with an equation connecting the kinetic energy [eq]W[/eq] and momentum [eq]p_r[/eq] and let this act on a wave function [eq]\Psi[/eq] since we can view [eq]W[/eq] and [eq]p_r[/eq] as operators. The equation is not linear in the kinetic energy and momentum. Now, according to the general requirement of quantum mechanics, the wave equation should be linear in the operator [eq]W[/eq] and in order that the equation may have relativistic invariance, it must also be linear in [eq]p_r[/eq]. Thus, new variables where introduced which give rise to the spin of the electron and give rise to some rather unexpected phenomena concerning the motion of the electron. In practice, the kinetic energy of a particle is always positive however the equation allows two kinds of motion. Only one motion is familiar. The other corresponds to electrons with a very peculiar motion. The faster they move, the less energy they have, and one must put energy into them to bring them to rest called the positron which corresponds to the motion of an electron with a positive charge instead of the usual negative one. We can then look at the process of annihilation where an ordinary electron, with positive energy, drops into a hole, fill up this hole and electromagnetic radiation is liberated. On the other hand, creation of an electron and a positron from electromagnetic radiation should also be observed.

Also, he added that if we accept the view of complete symmetry between positive and negative electric charge so far as concerns the fundamental laws of Nature, we can also get negative protons. However, the process will be more rigorous since protons are more complicated and the theory would require reliable basis which was not yet discovered at that time.

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