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The Crown Jewels of Quantum Algorithms

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The Amazing World of Quantum Computing

Part of the book series: Undergraduate Lecture Notes in Physics ((ULNP))

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Abstract

This chapter provides detailed descriptions of the most intellectually valued algorithms in quantum computing, including Peter Shor’s factoring algorithm and Lov Grover search algorithm, among others. An attempt is made to explain the subtle aspects of the algorithms and why such algorithms are valued.

Programming is the art of algorithm design and the craft of debugging errant code.

—Ellen Ullman

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Notes

  1. 1.

    Cleve et al. [12].

  2. 2.

    The reader may refer to Nielsen and Chuang [34], pp. 238–240 to learn more.

  3. 3.

    The reader may refer to Nielsen and Chuang [34], pp. 240–242 to learn more.

  4. 4.

    A number is an abstract idea. The symbolic representation of a number (such as 1, 2, …, or I, II, …, or i, ii, …, etc.) is called a numeral. However, in common usage, the word number is used for both the idea and the symbol. Our notion of a number has enlarged over many centuries to include such numbers as negative numbers, rational and irrational numbers, and complex numbers. The notion of a negative number took several centuries before it found acceptance by mathematicians.

    The notion of complex numbers, which require taking the square root of negative numbers, took place in the sixteenth century. It was originally used as a mathematical artifice to enable square roots to be taken with impunity. Its remarkable properties were later discovered by a bunch of talented mathematicians. It is difficult to imagine the creation of quantum mechanics without complex numbers. The secrets of nature lie hidden in the numbers.

  5. 5.

    Gauss [17].

  6. 6.

    Coinciding exactly when the remainders are compared.

  7. 7.

    See, e.g., Weisstein, Eric W., “Congruence.” From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/Congruence.html.

  8. 8.

    From the Latin integer, which means with untouched integrity, whole, entire. The symbol Z comes from the German word Zahlen, which means numbers.

  9. 9.

    Deutsch [14].

  10. 10.

    Bernstein and Vazirani [7].

  11. 11.

    Tarantola [41]. See also: http://www.dwavesys.com/, Website of D-Wave.

  12. 12.

    The D-Wave 2000Q™ System: Technology Overview. https://www.dwavesys.com/sites/default/files/D-Wave%202000Q%20Tech%20Collateral_0117F.pdf.

  13. 13.

    Kelly [29].

  14. 14.

    Intel [25].

  15. 15.

    IBM [24].

  16. 16.

    See, e.g., Quantum Computing Market Forecast 2020–2025. Market Research Media, 16 May 2019. https://www.marketresearchmedia.com/?p=850; and ScienceDaily.com at https://www.sciencedaily.com/ for new developments.

  17. 17.

    The quantum version was worked out by Coppersmith [13] and Deutsch (1994) [unpublished] independently. See also: Ekert and Jozsa [15] and Barenko [3].

  18. 18.

    To keep the discussions simple, we shall not consider the case when N ≠ 2n. However, we do remark that the larger the power of 2 used as a base for the transform, the better is the approximation. See Rieffel and Polak [36], p. 318.

  19. 19.

    These are a Hadamard gate and n − 1 conditional rotations on the first qubit , followed by a Hadamard gate and n − 2 conditional rotations on the second qubit , and so on. In all there will be \( n + \left( {n - 1} \right) + \cdots + 1 = n\left( {n + 1} \right)/2 \) gates. In addition, there are at most n/2 swaps to be done where each swap can be accomplished using three controlled-not gates.

  20. 20.

    This, of course, does not mean that QFT can be used in such applications as speech recognition or other signal processing applications. This is because the amplitudes in a quantum computer cannot be directly accessed by measurement. Thus, there is no way of determining the Fourier transformed amplitudes of the original state. Worse still, there is, in general, no way to efficiently prepare the original state to be Fourier transformed (see Nielsen and Chuang [34], p. 220).

  21. 21.

    Cleve et al. [12], p. 6.

  22. 22.

    Two integers are relatively prime if they do not share common positive factors (divisors) except 1.

  23. 23.

    For applications in quantum computing, it is convenient to allow \( a_{0} = 0 \) as well.

  24. 24.

    For a good description of continued fractions see http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.html#euclidsAlg.

  25. 25.

    Shor [37]. An expanded version of the paper is available in Shor [38].

  26. 26.

    The RSA system was invented by Ronald Rivest, Adi Shamir, and Leonard Adleman in 1978. It rules e-commerce and pops up in countless security applications. They received the Turing Award (2002) for their contributions to public key cryptography .

  27. 27.

    Coprime: Two integers a and b are coprime (or relatively prime) if the greatest common divisor of a and b is 1. For example, 55555 and 7811 are coprime even though neither number is itself a prime. One may use Euclid’s algorithm to determine if x and N are coprime.

  28. 28.

    Vandersypen et al. [45]. They designed and made a molecule that has seven nuclear spins—the nuclei of five fluorine and two carbon atoms—which can interact with each other as qubits , be programmed by radio frequency pulses and be detected by nuclear magnetic resonance (NMR) instruments. They controlled a vial of a billion-billion (1018) of these molecules as they executed Shor’s algorithm and correctly identified 3 and 5 as the factors of 15.

  29. 29.

    Lucero et al. [32].

  30. 30.

    Monz et al. [33]. See also: Johnston [27].

  31. 31.

    Smolin et al. [39, 40].

  32. 32.

    See, e.g., Johansson and Larsson [26].

  33. 33.

    See, e.g., Geller and Zhou [18].

  34. 34.

    See, e.g., Nielsen and Chuang [34], pp. 221–226, Cleve et al. [12], Kitaev [30], Jozsa [28] and Vazirani [46].

  35. 35.

    Grover [20, 21]. A popularized version of the algorithm appears in Grover [22]. See also: Grover [23].

  36. 36.

    Boyer et al. [8].

  37. 37.

    Chuang et al. [11]. See also: Gershenfeld and Chuang [19].

  38. 38.

    Vandersypen et al. [44].

  39. 39.

    Figgatt et al. [16].

  40. 40.

    Bennett et al. [4].

  41. 41.

    Modal value: the most frequently occurring value.

  42. 42.

    Bennett and Wiesner [6].

  43. 43.

    Bennett et al. [5].

  44. 44.

    Torlina et al. [42]. See also: ANU [1].

  45. 45.

    Peres [35].

  46. 46.

    ASRC [2].

  47. 47.

    Turing [43].

  48. 48.

    Landauer [31].

  49. 49.

    Brandt et al. [9].

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    Rajendra K. Bera

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Bera, R.K. (2020). The Crown Jewels of Quantum Algorithms. In: The Amazing World of Quantum Computing. Undergraduate Lecture Notes in Physics. Springer, Singapore. https://doi.org/10.1007/978-981-15-2471-4_10

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