Qubits/量子ビット

(Shrill of cicadas is one of Japanese Summer features.) So far, we have only talked about waves and superposition. We still have another half-dozen topics to introduce before we are fully ready to understand a “quantum computer.” These are all fundamental physical concepts independent of our desire to build a quantum computer but it’s convenient to talk in terms of quantum data so let’s pause to be a little bit quantum computer specific. In classical systems, the basic unit of the data is the “bit” and a group of bits together we call a “register”. In quantum systems, it’s similar. A basic unit of data is a “qubit” and a group of qubits together we call a “quantum register.”

First, let’s take a look at a single qubit. Classical bits can be anything that you can distinguish two amounts or two states of; the amount of electrical charge or the direction of the north and south poles of a magnet,for example. Quantum bits in contrast, Qubits, are quantum states that fit into the most fundamental equation of quantum mechanics, known as “Schrodinger’s equation.” Don’t worry, we’re not going to go into detail on the equation itself in this course. You’ve probably heard that one of the “mysteries” of quantum mechanics is that an electron or a photon can behave like either a particle or a wave.

In quantum computing, we’re going to take advantage of both of those characteristics but particularly focus on the wave aspect, which is why we have already spent so much effort talking about waves. So, we need to pick two quantum states to be our zero and one state. We just saw that standing waves in a box can be in different states, we can use two of those. At this point, we don’t need to worry about what those states are as long as we can tell them apart easily. Technically speaking, they need to be orthogonal. Later in the course, we will talk about some of the ways we can use photons, electrons or other phenomena as our qubits.

To distinguish our quantum bits from classical bits, we will write our zero state using what’s known as “Dirac’s ket notation” and we can draw it using two dials with a full vector on the upper one and nothing on the lower one. Our one state we will write as |1⟩ and draw it using two dials with full vector on the lower one and nothing on the upper one. Our quantum zero and one states are actually waves, just like the ones we have already studied. And just like regular waves, they have an amplitude and a phase, and they can be put into superposition.

We will use the vector inside the upper circle to represent the amount of the zero state we have, and the vector inside the lower circle to represent the amount of one state that we have. We can use this two-dial representation to keep track of superposition. For example, we can represent a state that is fifty percent zero and fifty percent one like this. The angle of our blue vector represents the phase. If the one state has a phase of 180 degrees, or pi, the vector will point down, like this. For the moment, when we write this state, we will indicate that phase by putting the angle in front of it like this.

By definition, the zero state always has zero phase, while the one state can have any phase. In fact, the phase of the one state is defined relative to the zero state. This condition will be revised slightly when we get to multi-qubit systems shortly. So, when we show you dials with amplitude and phase like this, you will only see one states with non-zero phase. These quantum waves are more than just regular waves, they are quantum probability amplitudes. They tell us what the probability is of us finding a particular state when we look at our qubit. We’ll talk about measurement next. Classical probabilities are real numbers between zero and one inclusive.

If you have a random variable, such as the roll of a single die (one dice), you will get each possible outcome with some probability. If your die is a good one, there is a 1/6th probability of coming up with each value, one through six. This gives us a total probability of one when we sum them all up. When you roll the die, something always happens, it’s never the case that nothing happens, and so the probabilities must total up to one or one hundred percent. In quantum mechanics, our quantum probability amplitudes are related to classical probabilities, except that the probability of a state is actually the absolute value of the square of the amplitude rather than just the bare amplitude.

Let’s write this down as an equation. I promise there won’t be a lot of equations in this course. We can write down the state of our qubit as α|0⟩+β|1⟩ so that alpha is the quantum probability amplitude for zero, and beta is the quantum probability amplitude for one. The probability for us being in the state zero is the absolute value of |α^2| and the probability for us being in the one state is the absolute value of |β^2| Now, our constraint that we always are in either the zero or one state means that the probabilities have to add up to one, absolute value of |α^2|+|β^2|=1. That process of making sure that they all add up to one is called “normalization.”

Now, you have seen a set of quantum states each with different phases. All of the states we have seen so far are fifty-fifty states of zero and one. We can also have states that

aren’t 50:50, such as 1/2|0⟩+√3/2|1⟩ where there is a 25% probability that the state is a zero and 75% probability that it is a one. Notice that the two vectors are now different lengths. In the article accompanying this video, there are a few additional details on the mathematics, including how imaginary numbers come into play, but throughout this course you will only need to understand this dial representation we have used in order to understand quantum computing. The quantum algorithms we are going to see later in this course manipulate the lengths of these vectors and the angle of the phase in order to use interference to solve problems.

Before we get to the algorithms though, we still have a few important quantum concepts to learn. In the next segment, we will learn how to measure qubits, and then shortly we will get to the single most mysterious aspect of quantum mechanics, known as “quantum entanglement.”