Quantum Computing Theory and Qubit Explanations


and then combine them with the set of POTENTIAL GEOMETRY POSTS

Excerpt from > Life on the Edge: The Coming of Age of Quantum Biology  — Author ??

Computing with qubits

When we think of a computer today we mean any electronic device capable of carrying out instructions to manipulate and process information via a collection of electrical switches that can either be ON or OFF – each capable of encoding a binary digit (or bit) as a 1 or a 0.  A collection of such switches can be arranged to build circuits that perform logic instructions, which can be combined and used to carry out arithmetical operations such as addition and subtraction or indeed the opening and closing of gates that we described for neurons.  The great advantage of this electrical digital computer is that it is very much faster than any manual way of performing the same kind of task, whether by counting on fingers, mental arithmetic, or using a pen and paper.

But while electronic computer may be extraordinarily fast at doing sums, even they cannot keep track of the complexity of the quantum world with its multitude of overlapping probabilities.  To overcome this problem, the Nobel Prize –  winning physicist Richard Feynman came up with a possible solution.  He suggested performing calculations in the quantum world, with a quantum computer.

To see how quantum computer might work, it will be useful first to represent the “bit” of a classical computer as a kind of spherical compass whose needle may point at either 1 (north pole) or 0 (south pole) and is capable of turning through 180 degrees to switch between these two states (figure 8.4a LINKED BELOW).  The central processing unit (CPU) of a computer consists of many millions of these one-bit switches, so the entire computational process can be envisaged as the application of a complex set of switching rules (algorithms) that can flip lots and lots of spheres by 180 degrees.

The quantum computing equivalent of the bit is called a qubit.  The is similar to the classical sphere (Bloch sphere), but its movement is not limited to a 180 degree flip.  Instead, it can rotate through any arbitrary angle in space and, being quantum mechanical, it can also point in many directions simultaneously in a quantum coherent superposition (figure 8.4b LINKED BELOW).  This increased flexibility allows a qubit to encode more information than a classical bit.  But the real boost to computing power comes when you put qubits together.

Whereas the state of one classical bit has no influence on its neighbors, qubits may also be quantum entangled.  You may remember from chapter 6 that entanglement is a quantum step up from coherence whereby quantum particles lose their individuality, so that what happens to one affects them all, instantaneously.  From the perspective of quantum computing, entanglement can be visualized as each qubit sphere being connected by elastic strings(in reality the strings represent the mathematical relationship between the phase and amplitude of the entangled qubits instantiated in the Schrodinger equation) to every other qubit (figure 8.4c LINKED BELOW).  Now let us imagine that we rotate just one of the spheres.  Without entanglement, the rotation will not affect neighboring qubits.  But if our qubit is entangled with other qubits, then the rotation changes the tensions in all the connecting strings between these connected qubits.  The computational resource of all those entanglement strings increases exponentially with the number of qubits, which means that it increases very rapidly indeed.

To get a feeling for the exponential growth, you maye have heard the fable about the Chinese emperor who was so pleased by the invention of chess that he promised to reward its inventor with a price of his own choosing.  The canny inventor asked for just one grain of rice for the first square on the chessboard, two grains of rice for the second, four for the third and so on, doubling the number of grains with each successive square until he reached the sixty-fourth square.  The emperor, considering this to be a modest request, eagerly agreed and ordered his servants to bring out the rice.  But, when the rice grains were counted out, he soon discovered his error.  The first row of squares amassed only 255 grains (2^8 – 1) and even by the end of the second row of squares he had to find only 32,768 grains, just less than a kilogram of rice.  But as the kilograms begin to multiply on subsequent squares, the emperor was dismayed to discover that by the end of the third row he had to hand over half a ton of rice.  Reaching even the end of the fifth row would have bankrupt the kingdom!  In fact, to reach the end of the chessboard would have required 9,223,372,036,854,775,808 (2^64 – 1) grains of rice, or 230,584,300,921 tons, which is roughly equivalent to the entire world’s rice harvest through the history of humankind.

The problem for the emperor was his failure to realize that double a number again and again leads to exponential growth – which is another way of saying that the increase from one number to the next is proportional to the size of the previous number.  Exponential growth is explosive growth, as the emperor discovered to his cost.  And just as the rice grains in the fable increased exponentially with the number of chessboard squares, so the power of a quantum computer scales exponentially with its number of qubits.

This is very different from a classical computer, whose power increases only linearly with the number of bits.  For example, adding one more bit to an 8-bit classical computer will increase its power by a factor of one-eighth; to double its power, the number of bits will have to be doubled.  But simply adding one qubit to a quantum computer will double its power, leading to the same kind of exponential increase in power that the emperor saw running away with his rice grains.  In fact, if a quantum computer could maintain coherence and entanglement within just 300 qubits, which could potentially involve just 300 atoms, it could outperform, on certain tasks, a classical computer the size of the entire universe.



Information content in a single qubit

  • infinite number of qubit states
  • but single measurement reveals only 0 or 1 with probabilities or – measurement will collapse state vector on basis state –
  • to determine and an infinite number of measurements has to be made
  • but, if not measured qubit contains ‘hidden’ information




http://physics.mq.edu.au/~jcresser/Phys301/Chapters/Chapter10.pdf   STATE SPACES OF INFINITE DIMENSION

Physicists discover an infinite number of quantum speed limits


quote sphere parts …

https://phys.org/news/2014-06-mathematician-unleashes-results-geometric-analysis.html  REVIEW FOR “infinitely many” phrase use

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