Abstract
This chapter introduces the fundamentals of quantum gates (operators) and describes generally used gates in designing quantum algorithms.
First I shall do some experiments before I proceed farther, because my intention is to cite experience first and then with reasoning show why such experience is bound to operate in such a way. And this is the true rule by which those who speculate about the effects of nature must proceed.
—Leonardo Da Vinci, C. 1513 (Quotation as cited in Fritjof Capra, The Science of Leonardo, Doubleday, New York, 2007.)
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Notes
- 1.
Dirac [8, p. 47].
- 2.
Dirac [8, p. 52].
- 3.
Landauer [11].
- 4.
Bennett [4].
- 5.
Toffoli [17].
- 6.
Fredkin and Toffoli [10].
- 7.
Deutsch [6, p. 107].
- 8.
Although we use the electron in an atom as an example here, one may, instead, use two different polarizations of a photon or the alignment of a nuclear spin in a uniform magnetic field or the two orthogonal spin states (up and down) of an electron, etc.
- 9.
Permutation: each of several possible ways in which a set or number of things can be ordered or arranged.
- 10.
A half-silvered mirror effects this transformation on a photon when it encounters the mirror.
- 11.
In modulo 2 arithmetic , the following addition rules for adding two binary numbers are quite obvious:
$$ 0 + 0 \equiv 0(\bmod 2),0 + 1 \equiv 1(\bmod 2),1 + 0 \equiv 1(\bmod 2),1 + 1 \equiv 0(\bmod 2). $$Analogously, the XOR (or Cnot) operation is:
$$ 0 \oplus 0 = 0,\quad 0 \oplus 1 = 1,\quad 1 \oplus 0 = 1,\quad 1 \oplus 1 = 0. $$ - 12.
See Barenco et al. [1].
- 13.
It is named after Tommaso Toffoli, who in 1980 showed that the classical version is universal for classical reversible computation. See Toffoli [17]. To know more about Toffoli, visit http://pm1.bu.edu/~tt/vita.pdf.
- 14.
Shinde and Markov [16].
- 15.
Named after Edward Fredkin. See Fredkin and Toffoli [10].
- 16.
- 17.
For a proof and implementation example, see Nielsen and Chuang [15, p. 176 and pp. 180–182].
- 18.
Muller [13].
- 19.
See, e.g., Nielsen and Chuang [15, pp. 189 and 191].
- 20.
Even though S = T2, i.e., the phase gate S is included because of its natural role in the fault-tolerant constructions. See Nielsen and Chuang [15, p. 189].
- 21.
DiVincenzo [9].
- 22.
Barenko [2].
- 23.
Lloyd [12].
- 24.
Deutsch et al. [7].
- 25.
Deutsch [6].
- 26.
Muthukrishnan [14].
- 27.
Muthukrishnan [14].
- 28.
Bennett [3].
- 29.
- 30.
Bennett [4].
- 31.
See, e.g., Landauer [11]; Bennett [3]. Bennett, in particular, showed that a Turing machine “may be made logically reversible at every step, while retaining their simplicity and their ability to do general computations.” He also discussed the biosynthesis of messenger RNA as a physical example of reversible computation.
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Bera, R.K. (2020). Quantum Gates. In: The Amazing World of Quantum Computing. Undergraduate Lecture Notes in Physics. Springer, Singapore. https://doi.org/10.1007/978-981-15-2471-4_7
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