Microwave Journal
www.microwavejournal.com/articles/34557-understanding-quantum-computing

Understanding Quantum Computing

September 14, 2020

“I think I can safely say that NOBODY understands quantum mechanics.”

- Physicist Richard Feynman, winner of the 1965 Nobel Prize for his groundbreaking theory of quantum electrodynamics.

Research into quantum physics applications such as computing, communications, simulation and sensing is moving at a frantic pace. Its promise for orders of magnitude advances in cryptanalysis, secure communications, prediction of materials properties and spectroscopy has caught the attention of many national governments and private investors eager to fund further research.

So, what is the big deal?

A regular digital computer performs data processing tasks by manipulating bits; each bit can have a value of one or zero. A quantum physicist would say this is a “classical” implementation of computing. A “quantum” implementation of a computer manipulates quantum bits (qubits). Qubits can have a value of one, zero or both simultaneously. When the bit is simultaneously a one and a zero, the bit is said to be in a state of superposition. Moreover, the state of one qubit can influence another qubit, even if they are separated by great distance, in this case, the states are said to be entangled. Superposition and entanglement are at the heart of quantum computing and provide capabilities that can speed the types of calculation required for cryptanalysis from years to minutes.

To understand how a quantum computer works, we need to understand the properties of an electron, how electrons behave in the presence of electromagnetic (EM) fields and how this is used within a qubit. We can then understand the implementation of the qubit and how to control and measure its state. Finally, we will address how qubits interact with each other and how, when put all together, they can perform computing. Putting Richard Feynman’s comments regarding quantum physics to one side, we can more easily understand how to control and manipulate data and perform computations with a fairly basic understanding of RF/microwave signal behavior, especially transmission lines and the inductor/capacitor (LC) tuned circuit.

HOW ATOMS AND ELECTRONS WORK

Quantum computing is built on the work of the great physicists: Planck for saying energy is not contiguous but quantized, Einstein for discovering the photoelectric effect, Bohr and Rutherford for applying Planck’s quantized energy rules to electron orbits, Louis de Broglie for proposing that electron particles also have wave properties and Schrödinger for introducing probabilities into the energy states of an electron.

Figure 1

Figure 1 The frequency of applied EM energy causes electrons to move from one energy state to another.

Each atomic orbital is represented by an energy level measured in electron volts (eV), with the lowest orbit called the ground state. As a particle can also be a wave, its energy level has a frequency equal to the energy level in eV divided by Planck’s constant (the quantization constant). Consider Figure 1. If we want the electron to move to a higher energy state, we apply EM energy at a frequency equal to the desired energy level minus the current energy level, divided by Planck’s constant. The electron will absorb the energy and jump to the next quantum energy level. Once the energy is removed, it will fall back to its original level, emitting the energy at the frequency previously absorbed. The frequency of the stimulus - not the amplitude - is key. Increasing the amplitude will not cause the electron to move to a higher energy level, increasing the frequency will. With this understanding, if we can constrain the energy levels to two, we have the fundamental building blocks for manipulating ones and zeros with a single electron.



Electrons also possess a type of angular momentum called spin. As the electron moves from one energy level to another, the spin momentum changes. At the lower energy level, the momentum is pointing down, called the “spin-down.” When EM energy is applied, the spin changes until the momentum is pointing upwards as the electron achieves the next energy level. This is the “spin-up” state. When the electron state can be defined like this, it is said to possess an eigenstate, as both the position and momentum are known and can be quantified through measurement.

Schrödinger postulated the probability that an electron can be in neither a spin-up or spin-down state, rather between. As the electron is not at one energy state or the other and not oscillating between the two, it is in both states at the same time or a superposition of the two states. Another way to say this is: when two disturbances occupy the same space at the same time, the resulting disturbance is the sum of two disturbances, like standing waves in a transmission line. We know that each energy level is proportional to frequency, and since a particle is a wave, the state of superposition is simply the vector addition of the upper and lower states.

While superposition is fundamental to the operation of a quantum computer, we have what is referred to as the “measurement problem.” A state of superposition only can exist if you don’t “observe” it - the idea behind the popularized Schrödinger’s cat example. In a quantum system, observation is synonymous with measurement. By applying a measurement frequency pulse to a qubit in superposition, the state of that qubit collapses or snaps back to one of the two quantized energy levels. As a physicist would say, the measurement causes the particle to be projected to one of its eigenstates.

WHAT IS A QUBIT AND HOW DO I MAKE ONE?

A qubit is an artificial implementation of an atom. As described, two energy states are associated with the electron orbits of an atom, and applying the appropriate frequency can affect the energy level and spin of the electron to create a logical one state, a logical zero state or something in between, i.e., superposition.

There are many implementations of quantum bits, ranging from solid-state superconducting qubits to photon-based systems using lasers and modified crystals. To illustrate quantum computing, we’ll look at the transmon solid-state implementation. A transmon, which is short for transmission line shunted plasma oscillation qubit, is a type of solid-state qubit. Fundamentally, a transmon is a tuned LC circuit connected to a transmission line, which resonates when an appropriate frequency is applied. In many cases, the frequency is sub-10 GHz, although some systems use higher resonant frequencies. This type of resonant circuit and transmission line is ideal, as the resonant frequency equates to the equivalent of an electron energy state.

Figure 2

Figure 2 Schematic representation of a qubit.

Figure 2 represents an actual qubit circuit. As the spin and energy level change with the application of certain frequencies, a qubit is fundamentally based on several tuned circuits connected via transmission lines, for both qubit control and measurement. Note the inductive component in the qubit has been replaced with a Josephson junction. While still fundamentally an LC structure, this modifies the inductive properties of the circuit so only two resonant or energy states can occur, since the system must be constrained to two levels. The circuit is cooled to about -450°F to obtain superconducting properties and exhibit the behavior of an electron, following the rules of quantum physics. To control the qubit, we apply the appropriate frequency for a certain duration to set the qubit to a one, zero or state of superposition. This pulse of RF energy can have different durations and different time domain shapes, depending on the control required.

Quantum states or spins are sometimes visualized by physicists on a unitary circle diagram. Spin-up and spin-down are opposites. If a base state is defined on a unit circle as the horizontal vector (1, 0), after applying energy the orthogonal spin-up state would be at the vector (0, 1) or π/2 on the unitary circle. This is a simplification, as these are complex numbers, i.e., a spin-down is represented by the vector (1 + j0, 0 + j0) and spin-up as (0 + j0, 1 + j0). Adding the two vectors give the superposition vectors (1/√ 2,1/√ 2). Another method of visualizing the quantum spin state is the three-dimensional model called the Bloch sphere. The unitary circle and Bloch sphere are mathematically related; however, instead of using cartesian coordinates, the Bloch sphere uses polar coordinates and defines spin as the difference in angle between the horizontal and vertical base states. A spin-up is represented by a vertical arrow pointing upwards within the sphere, i.e., to the north pole; with the downward spin, the arrow points to the south pole. Positions along the equator represent the spin in superposition. Both visualization methods are shown in Figure 3.

Figure 3

Figure 3 Quantum gate programming and visualization methods.

HELLO QUANTUM WORLD

With a qubit supercooled, its states can be controlled and measured by applying specific pulses of RF energy. Another fundamental: a quantum computer is not an independent computer with an operating system and programming language. It has no operating system or programming language, it is a set of bits manipulated from a classical computer (see Figure 4), with a classical computing bit or register usually mapped to a quantum bit. When the classical register is set high, a value of one, an appropriate RF frequency pulse is applied to the qubit circuit, which causes the electron to move to the next energy state, orthogonally changing the magnetic spin on the unitary circle or rotating the vector by 180 degrees on the Bloch sphere, setting the quantum bit to a value of one. This can be verified by reading the classical register, which means another RF pulse is applied and a measurement made to determine the state of the bit. The value of the measured phase or frequency defines whether the system has snapped to a logical one or a logical zero. In most cases, multiple measurements are required to ensure the answer is correct. Making 100 measurements, the probability that a value of one will be read back will be very high; however, the answer can never be 100 percent, as there will always be some level of interference affecting the measurement. We can do the same with zero, again, with the probability of the qubit returning a zero after several measurements will be very high.

Figure 4

Figure 4 Quantum computing system hardware.

QUANTUM LOGIC GATES

We can use gates with the qubit and cover three types of circuits: the NOT, Hadamard and controlled NOT (C-NOT). Each gate is implemented by applying an appropriate RF pulse to the qubit circuit. All the gates are shown in Figure 3.

A NOT gate means changing the energy state of the electron and changing the spin. Starting with a bit in a zero or ground electron energy state, application of the appropriate frequency will cause the electron to move to the next quantized energy state, visualized with a unitary circle or Bloch sphere diagram. Applying a Hadamard gate puts the qubit into a state of superposition - two energy levels at the same time - until it is observed. When we make a measurement, the quantum state will collapse and the qubit will return to a classical quantized energy state representing a one or zero. Making 100 measurements, the chance of returning a logical one 50 percent of the time and a logical zero the other 50 percent are very high. Two perfect qubits and Hadamard gates on each bit would create a 2-bit random number generator that generates the values zero, one, two and three, each 25 percent of the measurements. With interference present, the probabilities will be less even. Outside or unintended RF interference can seriously affect the probability of a correct answer, so designers of quantum systems need to ensure noise and interference are minimized. However, one form of interference can be used to the computer’s advantage. We call this entanglement.

WHAT IS ENTANGLEMENT?

When two electrons become entangled they will no longer exhibit independent behaviors. If one electron is measured and has a clockwise spin, the second will have an anti-clockwise spin. So why is this useful?

Electrically, the superconducting, very cold qubits are part of a cavity oscillator. As such, they are close enough to share EM fields and resonate, enabling the phenomena of quantum entanglement. A practical example of a type of entanglement is the C-NOT gate. The C-NOT connects two bits together, i.e., “entangles” them. Logically, the C-NOT gate works as follows: if the control port is set to a one, the output is the inversion of the input; if the control port is set to zero, the output of the gate equals the input.

The power of the Hadamard and C-NOT gates can be understood when guessing the value of an unknown number. Not knowing the value of a bit and wanting to guess with high probability that it is a logical one, we would use two bits to perform this operation. Both bits would be set into superposition, except one of the Hadamard gates would have its basis state out of phase by placing a NOT gate before it. If we collapse the quantum state and measure the output, one would be a one, and the other a zero. Entangling the qubits by connecting a C-NOT between the bits means the probability of the output being a one with a single guess is high.

While this seems fairly benign when guessing whether a single bit is set to a one or a zero - after all, you can guess a single bit number with a maximum of two (21) attempts: it is either a one or a zero. A 2-bit number would require 22 guesses and an 8-bit number would require 28 or 256 guesses. The number of guesses increases exponentially as the power of two increases. Guessing a number with 72-bits would require up to 4,722,366,482,869,645,213,696 guesses - which would potentially take a supercomputer hundreds of years to crack.

A good 8-bit analogy: eight coins spinning together hold 256 possible states in limbo. We know one of these states represents the right answer. But which one? The problem is the act of measuring the qubits will cause the superposition to collapse, like banging a fist on the table will cause the coins to fall to either heads or tails. We can increase the probability that each coin will fall heads or tails to give the correct answer with a collection of gates, or quantum algorithms, which loads the probability of the qubits to make each more likely to fall on the correct side and give the right answer. The quantum algorithm effectively weighs the superposition on the unitary circle, increasing the probability that it will snap to a one, rather than a 50/50 chance of snapping to a one or a zero.

CONCLUSION

A classical computer uses groups of transistors to form NAND gates that perform the logical functions which enable data processing. A quantum computer uses groups of RF/microwave pulses with different shapes and durations, acting on supercooled semiconducting resonators to create different logical operations, such as NOT, Hadamard and C-NOT. Putting these together, we have demonstrated some simple quantum circuits and shown how superposition and entanglement can be used to guess numbers.

Wave-particle duality, the photoelectric effect, magnetic resonance and probability are key elements. The manipulation of these phenomena to create various electron or photon spins can ultimately help crack codes and facilitate secure communications, although there is still a long way to go. Most quantum computing machines reside in research labs, with scientists trying to solve problems such as how long a state of superposition can last and reducing interference so more qubits can work together.

The industry has two benchmarks: 1) quantum advantage, the demonstration that a quantum device can solve a problem faster than classical computers; and 2) quantum supremacy, the ability that a programmable quantum device can solve a problem that classical computers practically cannot. Skeptics say that a large scale quantum device may never be realized, however, the research continues to move forward, achieving new breakthroughs every year. In October 2019, Google published an article in Nature stating they achieved quantum supremacy with their benchmarking, claiming the equivalent task for a state-of-the-art classical supercomputer would take approximately 10,000 years.1

Reference

  1. Frank Arute, Kunal Arya, et al., Quantum Supremacy Using a Programmable Superconducting Processor, Nature, October 23, 2019, www.nature.com/articles/s41586-019-1666-5.