Śrīdhara Brāhmaṇa: Imaginary Numbers and Electron Spin

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This post continues my commentary (Śrīdhara Brāhmaṇa, श्रीधरब्राह्मण) on Quantum Integer Programming (QuIP).

For ease of access, here is the previous post:

Numbers and Magnets.

Imaginary Numbers

First there were positive whole numbers, quickly followed by negative numbers. Then, there were ratios, or fractions, or rational numbers. Zero joined them. Then came irrational numbers. All of these are called real numbers. It is this continuum (in 3 space or (one time plus 3 space =) 4 dimensions) that forms the basis of a lot of classical physics.

The square root of -1 (denoted by i, for imaginary, as it is not a real number) was conceived by the Italians in the 1500s. Numbers of the form a + bi are called complex numbers where a and b are real numbers.

Originally viewed with deep suspicion, it became clear that these complex numbers possess great power. 

An initial surprise was that it was possible to show relationships between two algebraic expressions consisting only of real numbers more elegantly and swiftly by passing through imaginary numbers.

That is: we deliberately introduce some imaginaries, manipulate these imaginaries, and when done, the imaginaries disappear, and the required result among real numbers appears.

An illustrative example that caught my attention as a teenager is at the top of this post.

Electron Spin

The word spin unfortunately conjures up an image of something like a rotating earth (around its axis) or a spinning top. 

In quantum mechanics, it is not that. 

To begin with, when measured, spin can take on only two values, that we can call up and down, or +1 and -1.

Quantum Mechanics works at two levels, in two different mathematical worlds, a conceptual world (pre-measurement) and a physical world (post measurement):

Pre-measurement, we are in conceptual world and any thingy (entity)  – say an electron —  has a wavelike representation.

This is not the physical wave of water in a pond or even the wave of light in Young or Maxwell’s representation (that are in the physical world we live in).

If there is only one electron, the conceptual world has 4 dimensions (one for time and three for space), but should not to be confused with the physical world we human beings live in, just because it sounds so similar.

If there are 2 electrons, pre-measurement, we are in 7-dimensional conceptual world, one for time and each electron will have three dimensions for space.

If there are 20 electrons, the dimension of the conceptual world is 61 (= 1 + 3*20). You get the point.

In this conceptual world, the wave representation of the electron (or any thingy) will have complex variables as amplitudes. (Mathematically, this is called a Hilbert Space.)

Post-measurement, we are in our physical world of 3 physical dimensions and one time dimension, regardless of how many electrons are there.

Any electron, post-measurement, is a point particle (that is, it has no volume, and so there is no axis to rotate about!) occupying a position in the 4-dimensional continuum of real numbers. 

An electron has a negative charge. It has mass. It has spin.

Before its measurement, the spin of an electron can take one of an infinity of values that are superpositions of the two values (up and down); that is, it can take any value that is a complex combination of these two values such that the square of these two complex coefficients add up to one. It lives in the conceptual world as part of the electron wave function.

So: quantum mechanics studies the evolution of the spin of the electron (say in an electromagnetic field), pre-measurement, using complex numbers (and differential equations and matrices) in a conceptual world (Hilbert Space). 

When we actually measure its value, in physical space (where we live), at some time, the spin will take on one of the two (real) values with a probability that is proportional to the square of the complex coefficient of the superposition state (at this time) that it had (just before the measurement) in the conceptual world. 

When folks casually say “wave-particle” duality, this is what they mean, and not that a thingy is both a particle and a wave in our physical world at the same time. That would be absurd. A thingy is a wave in the conceptual world before measurement and a particle in the physical world after measurement.

This is quantum mechanics in a nutshell.

Now to computing.

In Classical computing, we deal with manipulation of zeroes and ones using bits. A physical bit can either hold a value of zero or one, but not both at the same time.

In Quantum computing, we manipulate the superposition state of spin using qubits (quantum bits). A physical qubit, pre-measurement, is in a superposition state (represented by a complex number). 

Quantum computing is the manipulation of these superposition states, pre-measurement, until we are ready to measure.

A good quantum algorithm is one that:

  • Prepares an initial superposition spin state;
  • Manipulates the superposition state in a (finite) series of steps (Unitary transformations) without measurement, ending up at a final superposition state;
  • When this final superposition is measured, the (measured) states of the qubits (that can take on only one of two values each) provide the desired answer (in +1 or -1 terms) with sufficiently high probability; and
  • The steps of manipulation needed before the measurement are not too many.

So: searching for a good real solution for a real problem using a real (physical) quantum computer (this hardware is an analog device), it is not that different, in my view, than of using imaginary numbers to prove equations that contain only real numbers more elegantly and faster.

QuIP: Finding integer solutions to integer problems by preparing, manipulating and measuring spin in the physical world.

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