What happens physically when we run a "gate" on an ion trap quantum hardware?
In this post, I focus on running a Hadamard Gate using a pair of laser beams, how to flip the qubit and take the next step in the running an algorithm using an ion-trapping computer. It is similar to building assembly language chaining primitive steps together in classical computers. In Quantum computing, in this case, we use ions for quantum calculation, and gate/circuit building.
This post covers how gate operations work in an ion trap quantum machine. Other ion-trapping Quantum computers perform similar functions, using primitive steps chaining together, assembling algorithms, and processing data. This post aims to demonstrate these operations' physical and logical nature, covers the Hadamard gate covering 2D and Sphere perspectives. Most of the data found in public documentation from available online quantum computer companies mentioned in my posts.
The critical Hadamard gate is a fundamental step in the famous Shor's algorithm. Looking at popular algorithms, the Hadamard gate is used in the Deutsch–Jozsa*, Simon's**, Bernstein-Vazirani *** algorithms, and Grover's algorithm.
To understand quantum computers, it is necessary to know the basics of qubit frequency and just about all practical quantum objects, including these ions or oscillators - I think the clock face is a good analogy. They're like the hands of a clock that sweep around in a circle once a day. The difference is they go around probably a billion times faster. Qubits have strange behavior -they can exist in a superposition of the zero state and the one state with unequal probabilities.
The picture is one of two clocks; the hour hands of the two clocks can point to different times, and this difference is the phase of the qubit.
In this example, the probability of getting a zero is two out of three, and the probability/likelihood of getting a one is one out of three; the face is 90 degrees.
There is another way to visualize a qubit using the Bloch sphere. I have a few post about it already and you can find them here and here. Latitude is the angular distance of a place north or south of the earth's Equator or a celestial object north or south of the celestial equator, usually expressed in degrees and minutes, for example, at a latitude of 51° N.
The word longitude is the angular distance of a place east or west of the meridian at Greenwich, England, or west of the standard meridian of a celestial object, usually expressed in degrees and minutes. For example, at a longitude of 2° W.
From that point of view, here's how our qubit looks: the latitude is about 75 degrees south of the North Pole, corresponding to two-thirds probability, and the phase is 90 degrees.
See the X line from the center of the sphere to the point labeled X, which is the positive x-axis, a line from the center to the point labels, y, which is the positive y-axis, and the line from the center to the North Pole is the positive Z-axis.
There's also a corresponding negative axis going in the opposite direction for every position.
Let's see an example:
We'll apply a Hadamard gate to our qubit. The result is shown in the second part of this figure. Can you see what happened? It's a bit difficult to visualize, but the point is still on the sphere's surface. So it didn't change the length of the vector - the Hadamard gate does a rotation. We can use this primitive operation in many places in many quantum algorithms.
Here is the math (Wikipedia):
When you start with the qubit zero, it points to the North Pole. It moves the North Pole to the point where the x-axis touches the Equator. In the second part of the figure, we start at the South Pole. It moved the South Pole to a point where the negative x-axis touches the Equator. In the third part of the illustration, we start at the point label x, which moves the qubit to the North Pole. Now it's a little easier to see what this gate does; it does a 90-degree rotation about the y axis followed by a 180-degree rotation about the x-axis.
Because the combination of two rotations is again a rotation - the Hadamard gate is equivalent to a single rotation about some axis. It turns out that any single-qubit gate operation is equivalent to a rotation to a single rotation about some axis. The most fundamental gate "up rotates" about the x, y, and z axis.
Briefly, In the ion trapping machines, machine rotations "about the X and Y axis," or "any axis" lying on the Equator, is accomplished by shining a "pair of laser beams" on the trapped ion. That is two laser beams at exactly 12.5 gigahertz.
The two beams must differ in frequency by one part and 65,000 translating to the cubit frequency 12.6 gigahertz. So what is the reason we need this frequency difference?
Let's talk about the old physics, the BFO.
The Beat Frequency Oscillator (BFO) is used for CW and SSB reception (requires frequency stability.)
The oscillator generates a signal mixed with the output of the IF amplifier in the detector to produce an AM signal that can then be demodulated by the detector/sensor.
So the reason has to do with the phenomenon familiar from electronics, "Beat Frequency Oscillation," the combination of two frequencies that are close together produces a signal whose amplitude oscillates at a different frequency called the beat frequency.
This figure shows an oscillation at 10 Hertz in the first trace.
We have an oscillation at 11 hertz with a 90-degree phase lag in the second trace. In the third trace, we have the sum of the two. As you can see, you get an oscillation of about 10 hertz with an amplitude that varies at a rate of about one hertz, the phase of the amplitude oscillation is again 90 degrees. When you superimpose two waves with similar frequency, you get Beats whose frequency is equal to the difference in the frequencies of the two waves you started "with," and whose phase is equal to the phase difference of those two waves.
In the ion trap QC (Honeywell, and perhaps ionq?) the Beats in the laser beam whose frequency matches the qubit resonate with the ionic state causing a rotation in the Bloch Sphere. If the two beams are in phase with one another, the axis of rotation is just the x-axis, zero degrees longitude. If the two beams are out of phase with one another, the axis of rotation is shifted from x by an angle corresponding to the phase difference.
See the image above. Look at the third row. Here's two beams with a 90-degree phase shift that rotates about the y axis. So you can get a rotation about an axis in the equatorial plane using a pair of laser beams.
Of course, interesting, valuable algorithms don't just involve operations on a single qubit. They involve one qubit interacting with another, often one qubit interacting with many. For example, if the control bit is set, the controlled-NOT gate flips the data bit.
Based on what I read, in the Honeywell device, this is accomplished by moving one trapped ion close to another one so that there is electrostatic repulsion between them -under the influence of a pair of laser beams which influences their motion. It is a similar method used in ionq quantum platform as explained in their documentation.
The ions are cool to 12 degrees above absolute zero so that their vibrational motion concerning one another is almost nil. In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities[1] asserting a fundamental limit to the accuracy with which the values for certain pairs of physical quantities of a particle, such as position, x, and momentum, p, can be predicted from initial conditions. The Heisenberg uncertainty principle prevents the motion from ceasing altogether, and the result is that you get quantized vibrations.
Just as a quantized electromagnetic oscillation is called a photon, the quantized vibrations are called phonons. The prefix phone refers to sound you may have heard of photons in solid-state physics. As they did for single-qubit gates, ion trapping QC uses a pair of laser beams with slightly different frequencies. The frequency difference "the Beat Frequency" is matched to the natural frequency of the ionic motion vibrations - causing the two qubits to change their state in a coordinated fashion.
The vibration frequency and the beat frequency of the laser beams can be 25 megahertz. Now the exact effect of this operation is not a controlled "NOT" gate, but you can build up a controlled-NOT gate by combining these "two two-qubit gate" with some other single-qubit operations. As you see, a pair of a laser beams can build and operate gates, the fundamental steps in quantum computing.
It all looks like magic, but it is not!.
Similar posts:
How to run an algorithm on IBM quantum computer:
qubit control and measurements:
SUMMARY KEYWORDS
qubit, gate, rotation, frequency, axis, degrees, Honeywell, phase, laser beams, algorithm, operations, oscillation, ion, rotate, north pole, single qubit, clocks, beams, point, amplitude
Resources:
Coherent laser company
Uncertainty principle:
*The Deutsch–Jozsa algorithm is a deterministic quantum algorithm proposed by David Deutsch and Richard Jozsa in 1992 with improvements by Richard Cleve, Artur Ekert, Chiara Macchiavello, and Michele Mosca in 1998.[1][2] Although of little current practical use, it is one of the first examples of a quantum algorithm that is exponentially faster than any possible deterministic classical algorithm.
**In computational complexity theory and quantum computing, Simon's problem is a computational problem that is proven to be solved exponentially faster on a quantum computer than on a classical (that is, traditional) computer. The quantum algorithm solving Simon's problem, usually called Simon's algorithm, served as the inspiration for Shor's algorithm.[1] Both problems are special cases of the abelian hidden subgroup problem, which is now known to have efficient quantum algorithms.
*** The Bernstein–Vazirani algorithm, which solves the Bernstein–Vazirani problem is a quantum algorithm invented by Ethan Bernstein and Umesh Vazirani in 1992.[1] It is a restricted version of the Deutsch–Jozsa algorithm where instead of distinguishing between two different classes of functions, it tries to learn a string encoded in a function.[2] The Bernstein–Vazirani algorithm was designed to prove an oracle separation between complexity classes BQP and BPP.[1]
****In quantum computing, Grover's algorithm, also known as the quantum search algorithm, refers to a quantum algorithm for unstructured search that finds with high probability the unique input to a black box function that produces a particular output value, using just {\displaystyle O({\sqrt {N}})} evaluations of the function, where {\displaystyle N} is the size of the function's domain. It was devised by Lov Grover in 1996.[1]
Hadamard Gate:
Edit: Fixed a few things
Another Edit: Fixed bad spelling
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