Still, while it might be optimal from the point of view of memory formation, it means not using some of the techniques that games, books, and movies use to keep you interested. How can you get the most out of reading in this new mnemonic medium? The ideal is to do an initial read, followed by a few short review sessions over the next few days prompted by our notifications to help you internalize the ideas. Then optionally a followup read, where you can more deeply understand the material.
And finally, some sessions of followup review to ensure you remember for the long term:. This is a larger commitment than traditional reading. But for a small factor in effort, you will understand the material far more deeply, and remember it for more than 10x as long. Trust the reminders, and this will all happen as a matter of course. Many more states are possible, and those extra states endow qubits with capabilities not available to ordinary classical bits.
In general, remember, a quantum state is a two-dimensional vector. In this example, the state 0. In the usual vector notation that means the state is:. Of course, the example just shown has real entries, as do the computational basis states. But for a general quantum state the entries can be complex numbers.
So, for instance, another quantum state is the vector. Still, visualizing it as a plane is sometimes a handy way of thinking. But why is that true?
Why a complex vector space, and how should we think about the complex numbers? These are good questions. But they do take some time to answer. Many Nobel prizes were won for milestones along the way. In fact, the idea of describing a simple quantum system using a complex vector in two dimensions summarizes much of what was learned over that 25 years. One of the most common terms is superposition. People will say a state like 0. Another common term is amplitude. An amplitude is the coefficient for a particular state in superposition.
For instance, in the state 0. The constraint is this: the sums of the squares of the amplitudes must be equal to 1 1 1.
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So, for example, for the state 0. The constraint is now that the sum of the squares of the amplitudes is 1 1 1 , i. Summing up all these ideas in one sentence: the quantum state of a qubit is a vector of unit length in a two-dimensional complex vector space known as state space.
Of course, we will explore the definition further, deepening our understanding, but it will always come back to that basic fact. You can think of the 0 0 0 or 1 1 1 states of a bit as corresponding to two very different but stable states of a physical system. For instance, a 0 0 0 or 1 1 1 can be indicated by the presence or absence of a hole at some location in a punch card. And so a single punch card can be used to store hundreds or thousands of bits:. Source: Gwern Most ways of storing classical bits are variations on this idea.
For instance, the dynamic random access memory RAM inside your computer is based on the idea of having two tiny metal plates separated by a miniscule gap. Electric charge is stored on those plates, setting up an electric field between them. The 0 0 0 and 1 1 1 states of the bit correspond to two different configurations of charge on the plates.
So, how should we think about quantum states? Indeed, in some proposals they may correspond to different charge configurations, similar to dynamic RAM.
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But by and large you can think of the computational basis states as representing a physical system in one of two well-defined configurations. How should we think about such superposition states? At least in popular media accounts, a very common description is that a state like 0. Or, even if you reject that, to get hung up worrying about what a quantum state is. But the trouble is that there is enormous disagreement amongst physicists themselves about how to think about the quantum state. Indeed, many active researchers are trying to understand what the correct way of thinking about the quantum state is, exploring multiple approaches in great depth.
You start to theorize about the best ways of solving it, how to understand it, and so on. But you never actually play around with it, getting familiar with how it behaves. Often the best way to understand something is to first use it, to get comfortable, to play a lot, and to do lots of informal experiments. As you build familiarity you understand why things are the way they are. At that point, you can go back and better understand the meaning of the basics.
Well, we can think of quantum computing and quantum mechanics as an especially complicated type of puzzle! In Part I we learned about the state of a qubit. We need to be able to do things with them! We do that using quantum logic gates. A quantum logic gate is simply a way of manipulating quantum information, that is, the quantum state of a qubit or a collection of qubits.
In particular, much as any classical computation can be built up using AND, OR, and NOT gates, the quantum gates we describe over the next few sections suffice to do any quantum computation. But a few are different. Those differences appear innocuous, almost trivial.
But in those differences lies the power of quantum computation, and the possibility for quantum computers to be vastly superior to classical computers. Part II is, frankly, a bit of a slog. Still, the spaced-repetition testing should make this basic memory work much easier than is ordinarily the case.
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And it will prepare you well for some of the more conceptual issues discussed in Part III, where we return to high-level questions about the ultimate nature of computation, and what quantum computers are good for. But for historical reasons people working on quantum computing usually use a different notation, the notation X X X.
And so the above may be rewritten:. This operation later became of interest to people working on quantum computing. In a quantum circuit we depict an X X X gate as follows:. A quantum wire represents a single qubit. But it's often helpful to instead think of left-to-right as representing the passage of time.
So the initial segment of wire is just the passage of time, with nothing happening to the qubit. Then the X X X gate is applied. And then the quantum wire continues, leaving the desired output. It is, in fact, our first quantum computation. A simple computation, involving just a single qubit and a single gate, but a genuine quantum computation nonetheless! And so we have:. By the way, regarding the X X X gate as a matrix clarifies what may have been a confusing point earlier.
The reason is that for linear functions, i. The simplest possible quantum circuit does nothing at all. This circuit is just a single qubit being preserved in time.
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Mathematically, this circuit is trivial. In many physical systems, the quantum wire is actually one of the hardest quantum computations to implement! The reason is that quantum states are often incredibly fragile. And so while quantum wires are mathematically trivial, they can be one of the hardest elements to build in real systems. That said, there are some systems where quantum wires are easy to implement. If you store your qubit in a neutrino then the state will actually be pretty well preserved.
The reason is that neutrinos barely interact with other forms of matter at all — a neutrino can easily pass through a mile of lead without being disturbed. But if the qubits only interact weakly with other systems then that also makes it hard to manipulate the qubits. Thus, systems which makes good quantum wires are often hard to build quantum gates for.
Much of the art of designing quantum computers is about finding ways to navigate this tension.