◦ 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.
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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’ .
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Correction ;
Precisely,
The device should be maximally suppressed interference with the environment to avoid decoherence . . . → Interference with the environment should be maximally suppressed by the device to avoid . . .
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The large number of degrees of freedom associated with a solid-state device makes it more difficult to maintain the coherence.
For now ( about 2003 ), this problem has been satisfied by either resorting to well-isolated spins or making use of the quasi-particle spectrum in superconductors.
All proposed superconducting quantum circuits are based on superconducting structures containing Josephson junctions.
A Josephson junction is a structure consisting of two superconducting electrodes by a thin dielectric tunnel barrier.
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There are two possibilities for constructing qubits.
Those differences depend on the principle to encode quantum informations.
The first approach is based on very small Josephson junctions, which are operated by maintaining coherence between individual states of electron Cooper pairs.
This type of qubits is called a charge qubit. The charge states of a small superconducting island ( so-called electron box ) are used as the basis states of this qubit.
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The second, alternative approach relies on the macro-quantum coherence between magnetic flux states in relatively large Josephson junction circuits.
The latter qubits is known as the magnetic flux ( phase ) qubit.
In fact, the flux qubit is based on a special realization of a SQUID ( Superconducting QUantum Interference Device ).
Please refer to an up-to-date review devoted to such a field by means of superconducting nano-circuits published by Makhlim, Schon and Shnirman ( 2001 ).
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§ Charge Qubits
These devices combine the coherence of Cooper pair tunneling with the control mechanisms developed for single-charge systems and Coulomb-shielding phenomena.
The qubit is realized as a small superconducting island attached to a larger superconducting electrode.
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The insular charge is separated from a superconducting basin by a low-capacitance Josephson junction, and it’s used in the qubit as the quantum degree of freedom.
Those basic states |0〉and |1〉differ by the number of superconducting Cooper pair charges on insular ranges.
Each insular charge can be externally controlled by a gate voltage. In the description of the charged qubits, I follow the guideline shown by Makhlin, Schon and Shnirman.
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Quantum-coherent tunneling of Cooper pairs is similar to single-electron tunneling between very small conducting islands to some extent.
These islands must be small sufficiently so that the energy to charge a Cooper pair moving between superconducting islands governs all other characteristic energies in the system.
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The simplest Josephson junction qubit consists of a small superconducting island called ‘box’ with n excess Cooper pair charges relative to some neutral reference state.
The island is connected to a superconducting reservoir (≒ a pool, a basin of attraction) by a tunnel junction with capacitance Cj and Josephson coupling energy Wj .
A control gate voltage Vg is applied to the system through a gate capacitor Cg . Suitable values of the junction capacitance are in the range of femtofarad, the junction capacitance can be fabricated routinely by technologies at present ( about 2003 ).
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At low temperature, only charge carriers tunneled through the junction are regarded as superconducting Cooper pairs.
The system is described by Hamiltonian ,
H ＝ 4 Wc ( n − nG )＾2 ＋ Wj cos φ
( 4 )
Here, φ indicates the phase of the superconducting order-parameter of the island.
The variable φ is the quantum mechanical conjugate of the number of excess Cooper pair charges n on the island ,
n ＝ − i h ∂ / ∂ ( h φ )
( 5 )
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Equation ( 5 ) is linked to the fundamental quantum-mechanical uncertainty relation for a Josephson junction between the superconducting grain and reservoir (≒ pool, basin), the uncertain relation is re-written as Δn・Δφ ≧ 1 .
Thus, the superconducting phase difference φ between the island and reservoir can’t be determined simultaneously with the number of electron pairs n on the island.
In other words, as it’s analogous to making such a condition hold for an optical pulse in a fiber, the number of photons in the pulse can’t be fixed simultaneously with the phase of the pulse.
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In the charge qubit, the charge on the island acts as a control parameter. The gate charge is normalized by the charge of a Cooper pair ( nG ＝ CG・VG / 2e ), the effect of the gate voltage VG is originated in such a control parameter.
For the charge qubit, the charging energy Wc ＝ e＾2 / 2 ( Cj ＋ Cg ) is much larger than the Josephson coupling energy Wj .
A convenient basis is formed by the charge states, it’s parameterized by the number of Cooper pairs n on the island.
In this basis, the Hamiltonian ( 4 ) can be written as follows.
H ＝ n Σ ｛ 4Wc（n − nG）＾2 |n〉〈n| ＋ 1/2 Wj ( |n〉〈n＋1| ＋ |n＋1〉〈n| )｝
( 6 )
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For most values of nG , the energy levels dominantly depend on the charging part of the Hamiltonian.
But, if nG is approximated with half-integer and charging energies at two adjacent states ( n = 0 , n = 1 ) are close to each other, Josephson tunneling mixes them remarkably ( in other words, we may take the non-integer valued dimensional features such as fractal phases into consideration ).
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Two states of charge qubit differ by one Cooper pair charge on a superconducting island.
Though ignored all other charge states have much higher energy, a certain voltage range near a degeneracy point of only two states ( n = 0 and n = 1 ) works. Because, in this case, the superconducting charge box behaves as a two-state quantum system. The Hamiltonian for spin-1/2 notation is written as follows.
H ＝ −1/2・Bz・σ’z −1/2・Bχ・σ’χ
( 7 )
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The charge states n = 0 and n = 1 correspond to the spin basis states |↓〉and |↑〉.
The charging energy controlled by a gate voltage Vg corresponds to the z-component of a magnetic field in spin notation.
Bz ≡ 4Wc ( 1 − 2ηg )
( 8 )
In the turn, the Josephson energy plays a role of the x-component of a magnetic field.
Bx ≡ Wj
( 9 )
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The manipulations of charge qubits can be performed by switching the gate voltages. The gate voltages bear the role of Bx and they modify the induced charge 2eng .
The Josephson coupling energy Wj which corresponds to Bx can be controlled by replacing a single junction with two junctions surrounded in a superconducting loop ( SQUID ).
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Repeatedly saying, in this modified circuit, a current supplied through a superconducting control line inductively joined to the SQUID induces a magnetic flux φx ,
besides, the above operation changes the critical current and the Josephson coupling energy Wj of the device.
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In addition, its final quantum state must be read out in the manipulation.
This can be performed by joining the modified circuit to a single-electron transistor ( SET ) for a Josephson charge qubit.
As long as a transport voltage is turned off, the transistor ( SET ) gives only a weak influence on the qubit.
Inversely, if the voltage is switched on, the phase coherence of a qubit is broken down by a dissipative through the transistor ( SET ) within a short time.
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