Ahad Riaz

The world of quantum mechanics is weird and fuzzy, with lots of unanswered problems and intriguing challenges. What exactly happens to the isolated quantum system when we make a measurement using a macroscopic measurement device? What implications does loss of coherence have in the world of quantum computing? Quantum decoherence is a relatively modern upcoming theory that tends to formulate some sort of fundamental understanding of the problems mentioned above. The theory was introduced in 1970 by Heinz-Dieter Zeh in his paper “On the Interpretation of Measurement in Quantum Theory”, where he argued that “the probabilistic interpretation is compatible with an objective interpretation of the wavefunction”. [1]

Quantum decoherence studies the interaction between an isolated quantum system and the environment and how this leads to a loss of coherence in the system. The theory is also relevant to answer how a classical system emerges from the quantum world. The best way to visualize this concept is to delve into the wave nature of quantum objects. Knowing about the De Broglie waves in the quantum world [2], the most simplistic experiment to consider is the Young’s double slit experiment, where we pass electrons through both the slit rather than light waves. These electron waves combine to produce an intensity pattern of constructive and destructive interferences on the screen. The key driving force behind the occurrence of interference is the coherence of the incoming beams. In absence of coherence, there wouldn’t be any interference pattern and the electrons would be seen on the screen in form of a classical probability distribution. Instead of being an isolated system, if there was a detection made at one of the slits, we would observe that the interference between the two beams is no longer observed and the electron is found to be at one specific position on the screen. [3] This is shown in Fig. 1 . This quantum-classical transition is due to the wave function collapse and hence, the electron is found to be at a specific position with a unitary probability. Similarly, the interference pattern can also be lost in the case that some stray cosmic particles scatter off the incoming electrons. [4] In this case, these cosmic particles become entangled with the electrons. This leads to a quantum superposition and the coherence of the entire system exists only on the level of the larger system which accounts for both, the stray particles and the electrons. Hence, decoherence can be seen as a loss of information from the system when it interacts with an environment, which is the measurement device in this case. After being entangled with the environment, the loss of quantum behaviour due to decoherence is similar to the loss of energy from a classical system due to the presence of friction.

We can conclude that the environment’s interaction with the quantum system leads to suppression of the interference. In the process of making a measurement for a preferred set of states, the quantum system becomes entangled with the measurement device. Guido Bacciagalupi argues that “the states between which interference is suppressed are the ones which are least entangled with the environments under further interactions.” [4] Hence, for these preferred states, the information is stored in the environment and can be extracted by the observer without any further changes to the system.

We can see that decoherence is a vast theory that encapsulates the development of a quantum system under the presence of an interacting environment. We can extend this explanation for the double slit experiment to discuss how relevant is quantum decoherence to solve the measurement problem.

##### Decoherence in the world of Quantum Computing

Quantum computing is a non-classical computational model, which transforms the computational memory into a quantum superposition of classical states. Paul Benioff (1980) was one of the first physicists to propose a quantum model for the Turing machine, the traditional computing model. [13] Later on, Feynman (1984) suggested that these quantum computers can perform operations way faster than a regular computer. [14] Qubit, a two-level quantum mechanical system, is the basic unit of quantum information. A simple example of a qubit is that of electron’s spin with the levels being spin up and spin down. As opposed to the classical system, qubits can exist as a superposition of both of these levels simultaneously.

One of the biggest challenges in building these computers is to control or reduce the presence of decoherence, which is closely tied to the transfer of quantum information from a quantum system. Decoherence leads to the emergence of a classical system from a quantum system when it interacts with the environment and hence, causes the particles to lose their quantum properties. These losses are caused by lattice vibrations, temperature fluctuations, background thermonuclear spin, electromagnetic waves and other interactions with the environment. [15] The use of quantum gates for operation of the qubits can be a potential source of decoherence in the quantum computers. Fig. 2 shows how these quantum gates work. Unlike the classical logic gates, the gates must be reversible as retention of information is important in the system. If the information is lost after the process, it will affect the entangled qubits and lead to error in the computation.

So decoherence eventually affects any quantum system as it interacts with its environment. So, the lingering question is: how to build a computer that produces reliable results before being affected? There have been attempts to overcome or reduce decoherence in several ways before. In recent times, D-Wave (2013), a company making quantum computer, reduced the temperature of the qubits to 20 millikelvins to prevent the loss of coherence. [16] To combat the problem of noise from the environment that interacts with the qubit, there is the solution of quantum-classical algorithms. These run the key sections of a program on a quantum computer while running the major bulk of the program on a classical computer.

It is important to remember that there is a limit to the how large the system of qubits can be. This is because for larger qubit systems, there is greater degree of entanglement with the environment and hence, the system is more likely to be affected by decoherence. This limit on the length of the qubits is called the ‘Coherence Length’, which is a measure of how long the qubit can retain its quantum properties. [15] While the longest lasting qubit has stayed in the superposition for 39 minutes, maintaining the qubit system for complicated mathematical problems still remains a major challenge.

To increase the coherence length and mitigate the decoherence effects, the idea of “Quantum Error Correction” is used. This is needed by the quantum computers to deal with the noise in the stored quantum information as well as the presence of faulty quantum gates and measurements. Classically, classical error correction depends on redundancy to check for error. However, the no-cloning theorem prevents copying of quantum information. While information cannot be copied, it can still be spread from one qubit to highly entangled states of several qubit. This idea was discovered by Peter Shor (1995), who formulated a quantum error correcting code to store information of one qubit onto a highly entangled state of nine qubits. [17] The schematics are shown in Fig. 3. This keeps the quantum information in its encoded state but helps in retrieving the information about the error i.e. the corrupted qubit can be identified from the measurement of the highly entangled states. The only drawback of this error correction is that it consumes a lot of qubits, leaving relatively few qubits which ultimately reduces the size of the actual computation.

Putting all of these considerations into perspective, decoherence is perhaps the greatest challenge to the quantum computer. With the formulation of quantum error correction codes and quantum-classical algorithms, overcoming these effects of decoherence can be the gateway to introducing the world to the highly innovative world of universal quantum computers.

##### References

[1] Zeh, H. (1970). On the interpretation of measurement in quantum theory. Foundations of Physics, 1(1), pp.69-76.

[2] De Broglie, L. (1925). Recherches sur la théorie des Quanta. Annales de Physique, 10(3), pp.22-128.

[3] Feynman, R., Leighton, R. and Sands, M. (1964). The Feynman Lectures on Physics. Reading, MA [etc.]: Addison-Wesley, pp.1-11.

[4] Bacciagaluppi, G. (2003). The Role of Decoherence in Quantum Mechanics (Stanford Encyclopedia of Philosophy). [online] Plato.stanford.edu.

[5] Bartlett, S., Rudolph, T. and Spekkens, R. (2007). Reference frames, superselection rules, and quantum information. Reviews of Modern Physics, 79(2), pp.555-609.

[6] Burgos, M. (2015). The Measurement Problem in Quantum Mechanics Revisited. Selected Topics in Applications of Quantum Mechanics.

[7] Burgos, M. (2015). The Measurement Problem in Quantum Mechanics Revisited. Selected Topics in Applications of Quantum Mechanics.

[8] Price, H. (1997). Time’s arrow & Archimedes’ point. New York: Oxford University Press.

[9] Everett, H. (1957). “Relative State” Formulation of Quantum Mechanics. Reviews of Modern Physics, 29(3), pp.454-462.

[10] Everett, H. (1957). “Relative State” Formulation of Quantum Mechanics. Reviews of Modern Physics, 29(3), pp.454-462.

[11] Bohm, D. (1952). A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables. II. Physical Review, 85(2), pp.180-193.

[12] Allori, V., Duurr, D., Goldstein, S. and Zanghi, N. (2002). Seven steps towards the classical world. Journal of Optics B: Quantum and Semiclassical Optics, 4(4), pp.S482-S488.

[13] Benioff, P. (1980). The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines. Journal of Statistical Physics, 22(5), pp.563-591.

[14] Feynman, R. (1982). Simulating physics with computers. International Journal of Theoretical Physics, 21(6-7), pp.467-488.

[15] Bassan, T. (2018). Decoherence: Quantum Computer’s Greatest Obstacle. [online] Hackernoon.com.

[16] Jones, N. (2013). Computing: The quantum company. Nature, 498(7454), pp.286-288.

[17] Shor, P. (1995). Scheme for reducing decoherence in quantum computer memory. Physical Review A, 52(4), pp.R2493-R2496.

[18] Leggett, A. (2002). Testing the limits of quantum mechanics: motivation, state of play, prospects. Journal of Physics: Condensed Matter, 14(15), pp.R415-R451.

[19] J.S. Bell (1975). On the wave packet reduction in the Coleman-Hepp model. Helvetica Physica Acta, (48): pp. 93–98.

[20] Omnès, R. (1994). The Interpretation of Quantum Mechanics. Princeton University Press.

[21] Omnès, R. (1994). The Interpretation of Quantum Mechanics. Princeton University Press.

Title Image Source: Levine, A. (2018). [image] Available at: https://aeon.co/ideas/you-thought- quantum-mechanics-was-weird-check-out-entangled-time