# Why are multiple quantum gates, in a quantum circuit, used?

After reading about quantum computers, for a high school research paper, a few questions arose concerning quantum logic gates.

• At first, it seems that quantum logic gates are in essence various forms of electromagnetic radiation that can change the energies of quantum particles, such as electrons, so that the pertaining quantum states change accordingly, although is this assertion correct?
• Moreover, if quantum gates are fundamentally described as EM-waves, then why are multiple steps of quantum gates in a quantum circuit required to manipulate the qubits; wouldn't it be easier to send a few EM-wave to the selected qubits so the desired quantum states are directly acquired, without using intermediate steps?
• Or would this be possible, but is it more effective to break up a certain problem into smaller steps through utilizing quantum gates?
• Lastly, if one wants to apply a quantum gate to four exemplary qubits, does this imply that each of those four qubits requires to be individually manipulated by the same quantum gate? In other words, send four identical EM-waves to each of the four qubits.

Hopefully, these questions are clearly formulated and do not cause any ambiguities. By the way, even though mathematics is a profound component of understanding quantum computing, I would rather prefer a more intuitive or casual explanation instead of mathematical expressions, since I simply do not possess the required mathematical abilities.

PS: Fortunately I am acquainted with the Dirac notation of quantum states, such as $$\frac{1}{4^{0.5}} (|00\rangle + |01\rangle + |10\rangle + |11\rangle).$$

I think you would benefit from realizing that quantum gates are an abstraction of the actual operations that we perform on qubits. Just as a qubit is an abstraction (a mathematical model to describe the state of a two-level quantum mechanical system) a quantum (logical) gate is a mathematical construct that we use in the study of quantum algorithms & quantum information science.

The way we implement these mathematical constructs varies. As you may be aware, there are multiple physical systems on which we implement the mathematical abstraction that is the qubit. This can be an electron's spin (which is either up or down, i.e. restricted to two levels), but there are myriad other physical implementations of qubits. If you are not familiar with the concept of spin, just think of it as an intrinsic rotation of the electron along an axis through its north- and south-pole. This can either be anti-clockwise (up) or clockwise (down).

At first it seems that quantum logic gates are in essence various forms of electromagnetic radiation that can change the energies of quantum particles, such as electrons, so that the pertaining quantum states change accordingly, although is this assertion correct?

The physical implementation of a quantum logic gate strongly depends on the physical implementation of the qubit which we are concerning. For qubits based on an electron's spin this implementation might indeed be by using electromagnetic waves, but other systems definitely do not always use EM waves. Moreover, electron-spin-qubits are also often operated on by applying a voltage to a conducting material very close to the electron.

The currently most used implementation of a qubit, the transmon qubit, is addressed by applying EM radiation (in the X-ray-regime). A transmon is essentially an altered LC-circuit, (a circuit with a capacitor (dutch: condensator) and an inductor (dutch: spoel).) The LC circuit is altered so that it only has two available quantum states, which are the $$|0\rangle$$ and $$|1\rangle$$ states.

Moreover, if quantum gates are fundamentally described as EM-waves, then why are multiple steps of quantum gates in a quantum circuit required to manipulate the qubits, wouldn't it be easier to send a few EM-wave to the selected qubits so the desired quantum states are directly acquired, without using intermediate steps?

There exist literally uncountable many operations that one can perform on a qubit, so it is virtually impossible to be able to implement all of them explicitly. However, if you restrict yourself to a (relatively small) subset of all operation, you can approach every possible gate very, very well. (Such a subset, by the way, is called an universal gate set). It is much, much easier to engineer only an implementation of a subset of all the quantum gates.

Or would this be possible, but is it more effective to break up a certain problem into smaller steps through utilizing quantum gates? Essentially, yes. In both engineering and study of algorithms it helps tremendously to limit to a subset. This is not to say that this is without problems though. The subset that you use might be different than for instance I - so to implement your hypothetical algorithm on my hypothetical quantum computer we might need to translate your 'code' or 'instructions' to something that my quantum computer can read. Since you're familiar with computer programming, you might see this as the usage of different, non-compatible assembly languages. In quantum computing, we often call these translation steps 'transpilation'.

Sometimes it is indeed more useful to think of some very elaborate quantum operation as something on its own, instead of a series of smaller, easier gates. A clear example is the quantum Fourier transform, which uses many smaller gates 'under the hood', but still has a clear defined action. A high-level quantum programmer might want to use the quantum Fourier transform as-is in his or her code, whereas the an implementation on actual hardware will need to be 'dumbed down' to all its smaller parts - this process is known as 'compilation'.

Lastly, if one wants to apply a quantum gate to four exemplary qubits, does this imply that each of those four qubits requires to be individually manipulated by the same quantum gate? In other words, send four identical EM-waves to each of the four qubits.

In a quantum computer, we want to be able to control separate qubits separately. That is to say, maybe we want to flip one qubit, without affecting the other. For this individual control, different qubits are addressed by different operations. For instance, two electron-spin qubits are addressed by applying a voltage to two different physical 'things' - that way, you know that you apply the gate to only the qubit you want. When gates are implemented using EM radiation, different qubits are addressed by EM radiation of different wavelength - the qubits only couple to the EM radiation with the 'correct' wavelength. So to apply the same gate to different qubits means to physically doing separate things.

Note that in our mathematical abstraction (not considering the implementations) the order of application of gates do not matter if they are on separate qubits. For instance, if we have four qubits and we want to perform a bit-flip on all of them, this is represented by the operation $$XXXX$$ - the bit-flip or $$X$$ operation on all four separate qubits. However, this does need to be performed in parallel. Performing nothing on a qubit is denoted by the Identity or $$I$$ operation. So flipping only the first qubit is the operation $$XIII$$. Then flipping only the second is denoted by the operation $$IXII$$, etc. So the succession of operations $$XIII$$, $$IXII$$, $$IIXI$$ and $$IIIX$$ will indeed give the operation $$XXXX$$.

• Thanks for your exhaustive and quick answer. In addition please allow me to ask a few more questions in reference to your answer. Firstly, you stated that there is an infinite number of qubit operations possible, what does this specifically mean? Regular quantum gates such as the CNOT, X-gate, H-gate, etc. merely alter the quantum of a qubit in a certain way. And what does it by the way mean to rotate the quantum state of a qubit along the X or Y axis? Why doesn't the Z-axis only suffice? Thanks for your patience and I look forward hearing form you soon. – Jelle 3.0 Dec 2 '19 at 20:43
• Please ask these questions as separate questions; continued questions within the comments of a different question is generally unfavourable, especially if these questions are perfectly valid on their own, as these onces are. I'll try to answer them subsequently! – JSdJ Dec 2 '19 at 21:23
• Thanks for the nuance, I'll ask these questions as a seperate thread. My apologies for not correctly formatting the questions. – Jelle 3.0 Dec 3 '19 at 11:29
• Don't mention it - you're doing great. You can ask those questions in one thread, as the explanation of the two is somewhat related. – JSdJ Dec 3 '19 at 11:39