◦ 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〉
( 3 )
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It has been shown that the single-bit operations and the controlled-NOT operation are sufficient to implement optional algorithms on a quantum computer.
And, ‘quantum computers can be regarded as programmable quantum interferometers’ .
First, the initial state is prepared in a superposition of the possible inputted states with the Hadamard gate ( 2 ), and then, the computation evolves in parallel along all possible paths, e-volutions along those paths interfere constructively towards the desired output state.
This intrinsic parallelism in the e-volution of quantum systems allows us to realize an exponentially more efficient way for performing computations.
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Detailed contents are not shown here because of space limitation, in a word, two registers of n ＝ 2［ log 2 N ］and m ＝ ［ log 2 N ］qubits are used in Shor’s algorithm.
The algorithm is realized by five major computation steps, namely,
( 1 ) Initialization of both registers by preparing their initial states
( 2 ) Applications of a Hadamard transformation to the first n qubits
( 3 ) Multiplying the second register by a＾x mod N for some random a ＜ N without common factors with N
( 4 ) Performing the inverse quantum Fourier transformation ( based on two-qubit controlled phase rotation operator ) on the first register
( 5 ) Measurements of the qubits in the first register
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§ Qubits : how to realize them
It’s common to adopt the spin 1/2 particle language for describing quantum algorithms.
Quantum theory predicts that if such a system is strongly coupled to the environment, it remains localized in one state and therefore behaves classically.
Thus, it’s very important that the quantum system is separated from the rest of the world.
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Weak coupling with the environment damps the coherent oscillations between the states discussed in the above. The damping rate vanishes as the coupling to the environment goes to 0 .
The inverse of the damping rate is often called ‘decoherence time τ dec .
This time is essentially the quantum memory of the system, namely, after a long enough time t ＞ τ dec , the system ‘forgets’ its initial quantum state and it’s no longer coherent with the memory.
In the ideal case, it’s expected that the decoherence time τ dec → ∞ can be defined to use a quantum system as qubit.
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There are at least five important criteria that must be satisfied by possible hardware for a quantum computer. They are as follows.
1. Identifiable qubits and the ability to scale them in number. This means that even if only a few qubits can be constructed, it’s not sufficient to realize useful quantum computation. Very many qubits are required in some controlled and reliable way for realizing practical quantum computation.
2. Ability to prepare the initial state of the whole system. All qubits first must be prepared in some state and only after that, quantum computation can be started.
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3. Low decoherence 〜 the key issue, many candidate systems for quantum hardwares are ruled out by this character. In general, quantum-coherent oscillations are caused at τ dec Δ / h ≫ 1 .
An approximate benchmark for sufficiently low decoherence is less than 10＾4 per elementary quantum gate operation in faithful loss.
4. Quantum gates. The universal set of gates is needed in order to control the system ( for example, Hamiltonian ). After preparing an optional state, we have to be able to switch an interaction between them on & off to make qubits act together and make computation useful.
5. Measurement ; the final requirement for quantum computation is the ability to perform quantum measurements on the qubits for obtaining a calculated result. Such readout transfers information to the external world to make the information useful.
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Any candidates for quantum computing hardware should be assessed against this ‘DiVincenzo checklist’ .
A number of two-level systems have been examined as candidates for qubits and quantum computation for recent few years.
These include ions in an electromagnetic trap, atoms in beams interacting with cavities, electronic states and spin states in quantum dots, nuclear spins in molecules or solids, charged states of nano-scale superconductors, flux states of circuits at superconducting condition, quantum Hall systems, electrons in superfluid helium and nano-scale magnetic particles.
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Though these systems fulfill some points of the above checklist, some open questions remain.
There have never been clear favorite points for quantum computation which are similar to a transistor for silicon-based classical computation for now ( about 2003 ).
New candidates for quantum computation should be explored to further work them on existing systems.
It’s a major challenge for practical quantum computation to maintain the coherence of a quantum device throughout calculation.
The device should be maximally suppressed interferences with the environment to avoid decoherence and the loss of quantum information.
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§ Why Superconductors？
A main merit of micro-quantum systems such as atoms, spins, photons, etc is that they can be easily isolated from the environment. Accordingly, they can reduce decoherence.
Meanwhile, the demerit is that integration of many qubits into a more complex circuit is required to build a practical computer.
From the viewpoint, quantum systems offer much flexibility to design a quantum computer with standard integrated circuit technologies.
Already proposed macro-qubits are based on nano-structured electronic circuits, they may be composed of quantum dots or ‘superconducting Josephson junctions’ .
