◦ Quantum Computing with Superconductors
2. Computing with Qubits
Very interestingly, it’s possible to much faster solve some problems which is practically troublesome with classical algorithms with using quantum computation.
For example, factorization of large numbers is the best instance to show effectiveness of a quantum algorithm proposed by P.Shor.
Shor showed that exponential steps are needed for factorization of large numbers through classical computers, meanwhile only polynomial steps are required for solving such a problem with quantum computers.
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We explain what it means with a modern workstation cluster. 10＾10 years are required to perform a factorization of a number N with L ＝ 400 digits. This time is larger than the age of the universe.
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But, it’s possible to reduce the time for the task to below 3 years through a single hypothetical (≒ virtual) quantum computer ！
Shor’s factorizing algorithm works with quantum computation to quickly determine the period of the function F ( x ) ＝ a＾x mod N .
※ In this case, a is a randomly chosen small number with no factors in common with N .
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Techniques developed in the number theory can be used to factorize N from this period with high probability.
Two main algorithmic methods, namely, modular exponentiation and the inverse quantum Fourier transform take only L＾3 operations.
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Prime factorization is an essential part of modern public key cryptographic protocols, it’s most important for privacy & security in the electronic world.
As quantum computation can theoretically factorize numbers in exponentially fewer steps than classical computation. Besides, they can be used to crack any modern cryptographic protocol.
Another problem is called sorting, it can be treated very efficiently by quantum computation.
It’s possible to search databases in 〜√N queries rather than 〜N queries with using quantum computers.
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We briefly discuss the basic computational operations with a spin systems of qubits as an example.
Manipulations of spin systems have been widely studied, physicists about NMR ( Nuclear Magnetic Resonance ) can prepare the spin system in any state and let it evolve to any other state nowadays.
一
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Controlled evolution between the two states |0〉and |1〉is obtained by applying resonant microwaves to the system, but state control can also be achieved with a fast DC pulse of high amplitude.
If the appropriate pulse widths are chosen, the NOT operation ( spin flip ) can be established as
|0〉→ |1〉; |1〉→ |0〉 ( 1 )
or the Hadamard transformation ( preparation of a superposition ) can hold as
|0〉→ （|0〉＋ |1〉）/ √2 ; |1〉→ （|0〉− |1〉）/ √2 ( 2 )
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It has been impossible to make a quantum computer with only these unitary ‘single bit’ operations yet ( about 2003 ).
Furthermore, it’s equal to this subject in importance to perform ‘two-bit’ quantum operation ( or to control the unitary e-volution of entangled states ).
Thus, both one & two-qubit gates are needed to build an universal quantum computer.
An example for an universal two-qubit gate is the controlled-NOT operation :
|00〉→ |00〉; |01〉→ |01〉; |10〉→ |11〉; |11〉→ |10〉
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