v~ . (2.24)
5. Multilevel Landau-Zener problems
Unlike the two level Landau-Zener problem, multilevel Landau-Zener problems do not not have a known exact analytic solution in a general case. The earliest works from Demkov and Osherov [34,35] discussed a single level with an energy linearly varying in time interacting with a set of parallel, non-degenerate levels, with time independent energies. It was shown that in this case the probabilities att→ ∞ are in accordance with a simple semi-classical model.
Furthermore, Carroll and Hioe [36] showed that a three level Landau-Zener crossing is solvable up to an inverse Fourier transform for the components of the wavefunction. Damski and Zurek [37] devised a treatment of the two level Landau-Zener problem, treating the evolution in an adiabatic manner everywhere except close to the anti-crossings where the evolution is non-adiabatic. Following on their work Rangelov, Pillo and Vitanov [38] expanded the formalism to treat three level Landau-Zener processes in the case when a single level with an energy linearly varying in time intersects with two non-degenerate levels with constant energies.
6. Fock’s adiabatic theorem
A common approach of treating adiabatic time-dependent problems is approxi-mating them within the Fock’s adiabatic theorem, stating that the system remains in the instantaneous eigenstate of the Hamiltonian if the Hamiltonian is slowly varying in time. In the case of Landau-Zener processes where n time dependent levels with an energy linearly varying in time vnt intersect with m levels, with a constant interaction matrix elementτnm, the adiabatic requirement is
v1~, v2~, ...vn~τ1,12 , ...τ1,m2 , ...τn,12 ..., τn,m2 . (2.25)
2.3. Quantum computers
In classical computation all information is stored into classical bits having two distinct values 0 and 1. In early 1980s Benioff [39], Manin [40], Feynman [41] and
Deutsch [42] concluded that computing abiding the laws of quantum mechanics would provide advantages over computing abiding the laws of classical mechanics.
In quantum computing, the basic unit of information is a quantum bit (qubit).
In contrast to a classical bit, other than the pure |0i and |1i states a qubit can also be in a quantum superposition of these states (|0i ± |1i)/√
2. Furthermore, two or more quantum particles can be correlated in such a way that one cannot distinguish between the quantum states of the particles. This phenomenon is called entanglement and it has been described as “spooky action at a distance”
by Albert Einstein. Einstein, Podolski and Rosen devised a thought experiment proving that quantum mechanics violates local realism and is therefore incomplete [43].
In 1964 John S. Bell derived a theorem stating that quantum mechanics violates either locality or realism [44]. Locality means that distant objects cannot influence one another on timescales shorter than the time it takes a photon to propagate between them. The context of realism was defined a bit differently compared to modern philosophy. Realism in the context of quantum mechanics means that all experimental outcomes possible prior to the experiment could have occurred as an experimental outcome. In his seminal paper Bell also proposed an experiment that would prove his theorem was correct and that quantum mechanics indeed violates either locality or realism. Since then many researchers have conducted experiments proving that quantum mechanics indeed violates locality or realism [45–50]. Latest cutting edge experiments with spins in nitrogen vacancies [51]
and entangled photons [52, 53] have simultaneously closed loopholes in the Bell experiments, proving the Bell theorem below the statistical error margin.
Quantum computing exploits superposition and entanglement to create quan-tum processors comprising of correlated qubits. One of the most common used protocols to encrypt data is the RSA protocol. The security of the RSA proto-col is based on the fact that factoring products of prime numbers with classical computers is a difficult, time demanding process. On the other hand, Shor’s quan-tum algorithm would allow factoring of integers in polynomial times, significantly outperforming sub-exponential times of the best known classical algorithm [54].
Another quantum algorithm outperforming classical algorithms is the Grover’s search algorithm [55]. Although it provides only quadratic speedup it can still significantly outperform classical algorithms when the number of entries is large.
1. Single qubit gates
A quantum gate is a device performing operations on qubits. All quantum gates can be represented with unitary matrices and visualized as rotations on the Bloch sphere [22,56].
The Haddamard gate is extremely important for quantum computing
If the input state was|0ithe Haddamard gate yields (|0i+|1i)/√
2. Furthermore, if the input state was |1i the Haddamard gate H yields (|0i − |1i)/√
2. The X gate rotates the state of the qubit for an angle ofπ around thexaxis of the Bloch sphere
Likewise, theY gate rotates the state of the qubit for an angle ofπ around the y axis of the Bloch sphere
Y =
0 −i i 0
. (2.28)
The Z gate rotates the state of the qubit for an angle of π around the z axis of the Bloch sphere
The phase shift gate Rφ adds a phase of local φ to one of the qubit states while it leaves the other qubit state unchanged, for instance |0i → |0i and
|1i → exp (iφ)|1i. It corresponds to a rotation of the qubit state on the Bloch sphere for an azimuthal angle φ
Rφ=
Two qubit gates simultaneously operates on two qubits [22,56]. The SWAP gate swaps the state of two qubits
SWAP =
in the {|0i|0i,|1i|0i,|0i|1i,|1i|1i} basis. A universal gate is such a gate which, combined with single qubit gates, performs any two qubit operation. SWAP gate is not universal as it has too much symmetry, however performing square root of a SWAP operation is an universal operation when combined with single qubit gates.
in the{|0i|0i,|1i|0i,|0i|1i,|1i|1i}basis.
Another important gate for quantum computing is the controlled-NOT gate CNOT. To understand the operation of the CNOT gate one of the qubits will be addressed to as the control qubit and the other will be addressed as the target qubit. If the control qubit was in the |0i state the state of the target qubit is unchanged. However, if the state of the control qubit is|1ithe state of the target qubit is changed. On operation like this is described with the following unitary matrix
Common alternatives to the CNOT and √
SWAP gates are the CSIGN gate, also known as CPHASE or CZ [56]
CSIGN =
the iSWAP gate, implemented commonly in superconducting qubits,
√
the SWAPα, occurring commonly in spintronic quantum computation,
SWAPα =
and the Berkley B gate
B=
It should be noted that any two qubit gate can be implemented by a combination of any other two qubit gate and single qubit gates.
3. The DiVincenzo criteria
In his groundbreaking paper, David DiVincenzo proposed a set of 5 criteria, necessary in order to achieve computing abiding the laws of quantum mechanics [57].
1) Qubits must be well characterized and a must be scalable. A subspace defin-ing the qubit states must be well defined. Lets assume a Hilbert space containdefin-ing N > 2 states exists. One must be able to identify the 2 states comprising the qubit subspace. Furthermore, the possibility to build a quantum processor having many interacting qubits must exist.
2) The state of the qubit must be reliably initialized.
3) All quantum bits must be isolated from coruptive couplings to their envi-ronments. In semiconductor system this requirement is quite difficult achieve due to the coupling of qubits to charge noise [58], nuclear spin noise [59] and interface defects [17]. A noisy environment coupled to the qubit causes decoherence i.e., the qubit starts behaving like a classical mixture and the quantum mechanical
nature of the qubit is irreversibly lost.
4) A universal set of single and two qubit gates, necessary for quantum comput-ing must exist for a given qubit. As isolatcomput-ing the qubit fully from the environment completely is impossible this requirement is connected with the previous one. In the sense that the time in which single and two qubit gates are performed much be much shorter than the typical time in which the qubit decoheres.
5) The state of the qubit must be reliably measured after the manipulation of the qubit.