Quantum cellular automaton explained

A quantum cellular automaton (QCA) is an abstract model of quantum computation, devised in analogy to conventional models of cellular automata introduced by John von Neumann. The same name may also refer to quantum dot cellular automata, which are a proposed physical implementation of "classical" cellular automata by exploiting quantum mechanical phenomena. QCA have attracted a lot of attention as a result of its extremely small feature size (at the molecular or even atomic scale) and its ultra-low power consumption, making it one candidate for replacing CMOS technology.

Usage of the term

In the context of models of computation or of physical systems, quantum cellular automaton refers to the merger of elements of both (1) the study of cellular automata in conventional computer science and (2) the study of quantum information processing. In particular, the following are features of models of quantum cellular automata:

• The computation is considered to come about by parallel operation of multiple computing devices, or cells. The cells are usually taken to be identical, finite-dimensional quantum systems (e.g. each cell is a qubit).
• Each cell has a neighborhood of other cells. Altogether these form a network of cells, which is usually taken to be regular (e.g. the cells are arranged as a lattice with or without periodic boundary conditions).
• The evolution of all of the cells has a number of physics-like symmetries. Locality is one: the next state of a cell depends only on its current state and that of its neighbours. Homogeneity is another: the evolution acts the same everywhere, and is independent of time.
• The state space of the cells, and the operations performed on them, should be motivated by principles of quantum mechanics.

Another feature that is often considered important for a model of quantum cellular automata is that it should be universal for quantum computation (i.e. that it can efficiently simulate quantum Turing machines,^{[1] [2]} some arbitrary quantum circuit^[3] or simply all other quantum cellular automata^{[4] [5]}).

Models which have been proposed recently impose further conditions, e.g. that quantum cellular automata should be reversible and/or locally unitary, and have an easily determined global transition function from the rule for updating individual cells.^[2] Recent results show that these properties can be derived axiomatically, from the symmetries of the global evolution.^{[6] [7] [8]}

Models

Early proposals

In 1982, Richard Feynman suggested an initial approach to quantizing a model of cellular automata.^[9] In 1985, David Deutsch presented a formal development of the subject.^[10] Later, Gerhard Grössing and Anton Zeilinger introduced the term "quantum cellular automata" to refer to a model they defined in 1988,^[11] although their model had very little in common with the concepts developed by Deutsch and so has not been developed significantly as a model of computation.

Models of universal quantum computation

The first formal model of quantum cellular automata to be researched in depth was that introduced by John Watrous.^[1] This model was developed further by Wim van Dam,^[12] as well as Christoph Dürr, Huong LêThanh, and Miklos Santha,^{[13] [14]} Jozef Gruska.^[15] and Pablo Arrighi.^[16] However it was later realised that this definition was too loose, in the sense that some instances of it allow superluminal signalling.^{[6] [7]} A second wave of models includes those of Susanne Richter and Reinhard Werner,^[17] of Benjamin Schumacher and Reinhard Werner,^[6] of Carlos Pérez-Delgado and Donny Cheung,^[2] and of Pablo Arrighi, Vincent Nesme and Reinhard Werner.^{[7] [8]} These are all closely related, and do not suffer any such locality issue. In the end one can say that they all agree to picture quantum cellular automata as just some large quantum circuit, infinitely repeating across time and space. Recent reviews of the topic are available here.^{[18] [19]}

Models of physical systems

Models of quantum cellular automata have been proposed by David Meyer,^{[20] [21]} Bruce Boghosian and Washington Taylor,^[22] and Peter Love and Bruce Boghosian^[23] as a means of simulating quantum lattice gases, motivated by the use of "classical" cellular automata to model classical physical phenomena such as gas dispersion.^[24] Criteria determining when a quantum cellular automaton (QCA) can be described as quantum lattice gas automaton (QLGA) were given by Asif Shakeel and Peter Love.^[25]

Quantum dot cellular automata

A proposal for implementing classical cellular automata by systems designed with quantum dots has been proposed under the name "quantum cellular automata" by Doug Tougaw and Craig Lent,^[26] as a replacement for classical computation using CMOS technology. In order to better differentiate between this proposal and models of cellular automata which perform quantum computation, many authors working on this subject now refer to this as a quantum dot cellular automaton.

Notes and References

1. .
2. C. Pérez-Delgado and D. Cheung, "Local Unitary Quantum Cellular Automata", Phys. Rev. A 76, 032320, 2007. See also arXiv:0709.0006 (quant-ph)
3. D.J. Shepherd, T. Franz, R.F. Werner: Universally programmable Quantum Cellular Automaton. Phys. Rev. Lett. 97, 020502 (2006)
4. P. Arrighi, R. Fargetton, Z. Wang, Intrinsically universal one-dimensional quantum cellular automata in two flavours, Fundamenta Informaticae Vol.91, No.2, pp.197-230, (2009). See also (quant-ph)
5. P. Arrighi, J. Grattage, A quantum Game of Life, Proceedings of JAC 2010, Turku, December 2010. TUCS Lecture Notes 13, 31-42, (2010). See also (quant-ph) and (Companion Website)
6. B. Schumacher and R. Werner, "Reversible quantum cellular automata", quant-ph/0405174
7. Pablo Arrighi, Vincent Nesme, Reinhard Werner, One-dimensional quantum cellular automata over finite, unbounded configurations. See also (quant-ph)
8. Pablo Arrighi, Vincent Nesme, Reinhard Werner, N-dimensional quantum cellular automata. See also (quant-ph)
9. R. Feynman, "Simulating physics with computers", Int. J. Theor. Phys. 21, 1982: pp. 467–488.
10. D. Deutsch, "Quantum theory, the Church-Turing principle and the universal quantum computer" Proceedings of the Royal Society of London A 400 (1985), pp. 97–117.
11. G. Grossing and A. Zeilinger, "Quantum cellular automata", Complex Systems 2 (2), 1988: pp. 197–208 and 611–623.
12. W. van Dam, "Quantum cellular automata", Master Thesis, Computer Science Nijmegen, Summer 1996.
13. C. Dürr and M. Santha, "A decision procedure for unitary linear quantum cellular automata", quant-ph/9604007 .
14. C. Dürr, H. LêTanh, M. Santha, "A decision procedure for well-formed linear quantum cellular automata", Rand. Struct. Algorithms 11, 1997: pp. 381–394. See also cs.DS/9906024.
15. J. Gruska, "Quantum Computing", McGraw-Hill, Cambridge 1999: Section 4.3.
16. Pablo Arrighi, An algebraic study of unitary one dimensional quantum cellular automata, Proceedings of MFCS 2006, LNCS 4162, (2006), pp122-133. See also quant-ph/0512040
17. S. Richter and R.F. Werner, "Ergodicity of quantum cellular automata", J. Stat. Phys. 82, 1996: pp. 963–998. See also cond-mat/9504001
18. P. Arrighi, An overview of quantum cellular automata, arXiv:1904.12956
19. Terry Farrelly, A review of Quantum Cellular Automata arXiv:1904.13318
20. D. Meyer, "From quantum cellular automata to quantum lattice gases", Journal of Statistical Physics 85, 1996: pp. 551–574. See also quant-ph/9604003.
21. D. Meyer, "On the absence of homogeneous scalar unitary cellular automata'", Physics Letters A 223, 1996: pp. 337–340. See also quant-ph/9604011.
22. B. Boghosian and W. Taylor, "Quantum lattice-gas model for the many-particle Schrödinger equation in d dimensions", Physical Review E 57, 1998: pp. 54–66.
23. P. Love and B. Boghosian, "From Dirac to Diffusion: Decoherence in Quantum Lattice Gases", Quantum Information Processing 4, 2005, pp. 335–354.
24. B. Chophard and M. Droz, "Cellular Automata modeling of Physical Systems", Cambridge University Press, 1998.
25. When is a quantum cellular automaton (QCA) a quantum lattice gas automaton (QLGA)?. Journal of Mathematical Physics. 2013-09-01. 0022-2488. 092203. 54. 9. 10.1063/1.4821640. Asif. Shakeel. Peter J.. Love. 1209.5367 . 2013JMP....54i2203S . 2351651.
26. P. Tougaw, C. Lent, "Logical devices implemented using quantum cellular automata", J. Appl. Phys. 75, 1994: pp. 1818–1825