Optical Quantum Computing

Overview Video – Learning Material – Exercises – Further Information & Literature – Quiz

If you are unsure about any terms, you can always check the glossary.

Overview Video

Learning Material

Scientists have high expectations for quantum computing. It is expected that quantum computers, and quantum technologies in general, will be able to solve major problems in today’s world, model molecules and perform some types of calculations that classical computers will never be able to do, or at least not in a reasonable time. But how is this possible? What is the fundamental difference between classical and quantum computing?

One of the two fundamental differences is quantum parallelism. Quantum computers are based on qubits. Like bits in classical computing, qubits hold the necessary information, store it and can be read out to access the information. Unlike bits, qubits can exist in a superposition of 0 and 1, and interference can occur and be used for explicit tasks. Further information on interference is described in Basic Rule 2:

Basic Rule 2: Interference of single quantum objects

Interference occurs if there are two or more “paths“ leading to the same experimental result. Even if these alternatives are mutually exclusive in classical physics, none of them will be “realised“ in a classical sense.

Let us discuss an example to help understand the basic rule and the notion of quantum parallelism. We compare a bit with a qubit: bits are the basis of a classical computer, qubits of a quantum computer. The measurement of both, the bit and the qubit, can result in two possible states: either 0 or 1. The difference is that a qubit can be in a superposition state. Before being measured, a bit is in a discrete state. It is either 0 or 1. You perform an operation and measure the bit. The result is one of the two possible states. In contrast, the qubit can be in a superposition state, existing simultaneously in both states 0 and 1. You perform an operation, measure the qubit and get one of the two possible states. So we need two classical bits, 0 and 1, to describe the superposition state of one qubit. But how many classical bits do we need to describe other multi-qubit systems?

Let us have a look at a 2-bit system. It can exist in four possible states, the 2-bit sequences describing these states are 00, 01, 10 and 11. So we would need four classical 2-bit sequences to describe the superposition state of a 2-qubit system. In a superposition state a 2-qubit system can simultaneously exist in the states 00, 01, 10 and 11.

Let us have a look at a 3-bit system. It can exist in eight possible states, the 3-bit sequences describing these states are 000, 001, 010, 100, 011, 101, 110, 111. So we would need eight classical 3-bit sequences to describe the superposition state of a 3-qubit system. In a superposition state a 3-qubit system can simultaneously exist in the states 000, 001, 010, 100, 011, 101, 110, 111.

Let us try to formulate a general statement: You can see that whenever we add a qubit to the multi-qubit system, the number of bit sequences doubles. This is because the number of possible states doubles, since the new states consist of all the previous states with an additional 0 and all the previous states with an additional 1. For example, let us look at the transition from one qubit to a 2-qubit system. We need two classical bits, 0 and 1, to describe the superposition state of one qubit. If we add an additional 0 to these states, we get 00 and 01. If we add an additional 1 to these states, we get 10 and 11. In total, we have four 2-bit sequences, 00, 01, 10, 11, which describe the superposition state of a 2-qubit system! In analogy, we do the same to describe the superposition state of a 3-qubit system, 4-qubit system and so on.

Finally, we would need $2^N$ classical states to describe the superposition state of an N-qubit system!

Dein Browser unterstützt dieses Bildformat nicht. Your browser does not support the image format. — In a superposition state a qubit can simultaneously exist in multiple states.

This is the quantum parallelism we mentioned at the beginning. While it may have almost no effect on one bit/qubit, it definitely distinguishes quantum computing from classical computing when considering a multitude of bits/qubits. You should keep in mind that we are speaking about superposition states. Like previously mentioned in Basic Rule 2 those states are not “realised“ in a classical sense! When measuring, there is a certain probability of measuring one of the states that are part of the superposition state.

However, this does not explain the quantum advantage at all. Even if qubits can simultaneously exist in multiple states, in the end you still need to measure one state. The quantum advantage comes from controlling interference processes, which makes it possible to control the output of qubit systems.

But what is interference? You may have heard of it in optics, when two waves overlap and create a pattern. Let us recreate an experiment that explains this phenomenon. For the experimental setup we need two beam splitters, two mirrors, two detectors, a single photon emitter and a phase shifter.

The emitted single photons have the state $\ket{0}$ . If detector A detects a photon, we interpret this as $\ket{0}$ , if detector B detects one, we interpret this as $\ket{1}$ . For simplicity, our phase shifter has only two states: either it is deactivated and the phase shift of the incoming waves is 0 (the wave remains the same), or it is activated and the phase shift of the incoming waves is $\pi$ (the wave is shifted by a half of its wavelength).

With this experimental setup, we are now able to perform calculations, in a sense we do optical quantum computing. Between the single-photon emitter and the detectors, the photon can be described as a wave that also has wavelike properties. This is the wave nature of quantum objects.

To potentially reach detector A, the photon has two possible paths. On both paths the photon is transmitted and reflected once! No matter which path the photon “takes”, the phase shift will be the same.

To possibly reach detector B, the photon also has two possible paths. This time the paths are different. In the first path the photon is transmitted twice, in the second path it is reflected once and transmitted once. This causes an invariable phase shift, which will be important for further considerations.

Let us recreate the paths through which the photon “walks”. In reality we cannot do this because the photon is in a state of superposition and we have already mentioned in Basic Rule 2 that in a state of superposition the alternatives are not realised in the classical way. But for the sake of simplicity, we will try to reconstruct what the photon “does”.

First, the photon is emitted from the single-photon emitter. From here it “walks” to the beam splitter. At the first beam splitter (BS1) there are two different paths. The photon now exists in a superposition of the transmission path and the reflection path. As mentioned in Basic Rule 2, the photon (quantum object) does not realise both paths in the classical sense. A measurement (by detectors in both paths) would detect the photon on one path, not on both paths! At the second beam splitter (BS2) there are again two different paths and the photon exists in a superposition of both. In the end, we have four possible paths (BS1 t + BS2 t; BS1 r + BS2 t; BS1 t + BS2 r; BS1 r + BS2 r).

Let us look at the possible paths to reach detector A. These are BS1 t + BS2 r and BS1 r + BS2 t, so two possible paths. In further considerations it is important to remember that we can describe the photon as a wave.

Our phase shifter is deactivated, the phase shift is 0. If we now emit single photons (state $\ket{0}$ ), we will always detect a $\ket{0}$ (i.e. detection at detector A). This is due to constructive interference. In the path between BS2 and detector A, the two possible paths overlap. Since we describe the photon as a wave, this means that two waves are superimposed. The phase shift is deactivated and the paths have the same conditions. All in all, both waves have the same phase shift, the maxima and minima of both waves are superimposed and amplified. The resulting wave has larger maxima and minima and the photon is detected at A.

Our phase shifter is activated, the phase shift is $\pi$ . If we now emit single photons (state $\ket{0}$ ), we will always detect a $\ket{1}$ (i.e. detection at detector B). This is due to destructive interference. In the path between BS2 and detector A, the two possible paths, i.e. waves, also overlap. But this time one of the waves has a phase shift of $\pi$ due to an activated phase shifter. As a result, the maxima of one wave are superimposed on the minima of the other wave and vice versa. The resulting “wave” is a constant line, without maxima and minima, and the photon is not detected at A.

Similar conclusions can be drawn for detector B. If the phase shifter is deactivated, detector B detects nothing. When the phase shifter is activated, detector B detects the photons.

The experimental setup described above is called a Mach-Zehnder interferometer. In terms of quantum computing, it could potentially be used for optical quantum computing, with a similar structure as shown in the figure below. By deliberately controlling the phase shift of the paths, interference can be used to control the outcome of a computation or a qubit sequence.

As in classical computing, we use gates to describe the structure of the computer. Quantum gates are difficult to realise. We will not go into the mathematical background or the technical realisation, they are not important at the moment and we want to concentrate on the qualitative narrative of quantum physics.

For our previous example we need a Hadamard gate and a phase shifter. A Hadamard gate can be thought of as a beam splitter. It puts the qubit in a superposition of the states $\ket{0}$ and $\ket{1}$ , just as the beam splitter puts the photon in a superposition of the “up” and “down” paths.

To realise the Mach-Zehnder interferometer in the sense of quantum computing, we need two Hadamard gates – analogous to two beam splitters – and the phase shifter. If we place the phase shifter between the two Hadamard gates, we have exactly the same situation as in the Mach-Zehnder interferometer. By controlling the phase shifter, we can again control the result.

Interestingly, the wavelike nature of quantum objects can be observed when we vary the phase shifter in a more flexible way – and not just distinguish between two states of phase shifting. The figure shows this. Here the counts are plotted as a function of the phase shifter, and we see a wave-like function. But we will not go into this here, we will look at it in the next topic!

Exercises

Exercise 1:

Name basic rule 2 and explain its meaning.

Exercise 2:

Repeat the experimental setup of our optical quantum computer and explain why and how the interference occurs.

Exercise 3:

Name any state of 4-bit sequences a 4-qubit system can simultaneously exist in. How many classical bit sequences are needed to describe the superposition state of a 4-qubit system?

Exercise 4:

A supercomputer has a capacity to exist in $0.5 \cdot 10^{15}$ states of bit sequences. Work out the number N of the N-qubit system that can exist in the same number of states as this supercomputer.

Solutions:

Further Information & Literature

Interference at Mach Zehnder Interferometer: Grangier, P., Roger, G. & Aspect, A. (1986). Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter: A New Light on Single-Photon Interferences. Europhysics Letters, 1(4), 173-179.

Mathematical explanation of single qubit interference: Youtube-Video & https://qubit.guide/2.4-single-qubit-interference.html

Quantum Gravimeter, a video from Atomionics: Youtube-Video

Quantum Gravimeter, an article: Menoret, V., Le Moigne, N., Bonvalot, S., Bouyer, P., Landragin, A. & Desruelle, B. (2018). Gravity measurements below 10^-9 g with a transportable absolute quantum gravimeter. Scientific Reports, 8:12300. DOI: 10.1038/s41598-018-30608-1.

Müller, R. & Greinert, F. (2024). Quantum Technologies: For Engineers. De Gruyter.

Qiang, X., Zhou, X., Wang, J., Wilkes, C.M., Loke, T., O’Gara, S., Kling, L., Marshall, G.D., Santagati, R., Ralph, T.C., Wang, J.B., O’Brien, J.L., Thompson, M.G. & Matthews, J.C.F. (2018). Large-scale silicon quantum photonics implementing arbitrary two-qubit processing. Nature Photo, 12. 534-539. doi: https://doi.org/10.1038/s41566-018-0236-y

(No author) (n.Y.). 2.4 Single qubit interference: Introduction. Online: https://qubit.guide/2.4-single-qubit-interference.html. [last access: 07-08-2023].

Quiz

Continue to Topic 3