Physical |?|2 + |?|2 = 1. Then, a

Physical realization of large-scale quantum informationprocessors is still beyond the scope of our currently avail-able technology. As we have seen, quantum informationis encoded in quantum bits or qubits which takes the form|?i = ? |0i + ? |1i, where |?|2 + |?|2 = 1. A quantumcomputer should be composed of at least 102 qubits to beable to perform algorithms with more efficiency than itsclassical counterparts. Even though it has been alreadyaccomplished a great deal in physical implementation ofquantum computering, it is still a challenging task weneed to deal with.But, what do we need to construct a full capable quan-tum computer? How is it done? First of all we will talkabout the so-called Di Vincenzo criteria, the necessaryconditions that any physical system has to fulfill to bea candidate for a viable quantum computer. Later on,we will talk about what we consider the most interestingproposals of physical implementation of quantum com-putering: Trapped Ions and Josephson Junction Qubits.A. DiVincenzo CriteriaNot any physical system is suitable for the imple-mentation of quantum computing. How can we knowthen if a concrete proposal is useful for the constructionof a quantum computer? If so, is there any criteriawe can use to know how suitable a proposal is? Theanswer is yes; these are the DiVincenzo criteria whichestablish minimum requisites that any physical systemmust satisfy to be a viable quantum computer.In his article 12 DiVincenzo proposed five necessaryconditions any physical system must accomplish to bea valid candidate for quantum computing. Let’s have aclose look at them:1. A scalable physical system with well-characterizedqubits.First we need to be capable of storing information, justas we do in classical computers. The way of doing thisis by a quantum register conformed by many qubits. Wecan obtain a qubit using any two level quantum-systemsuch as an electron (or any 1/2 spinned particle) or bytwo mutually orthogonal polarization states of a singlephoton. It is also possible to employ a two-dimensionalsubspace of a Hilbert space, such as the ground andthe first excited state of the atomic energy levels. Theimportant idea is to identify the basis vectors of theHilbert space, {|0i, |1i}, with any two states such asthose we have mentioned so the general state for a singlequbit can be represented in the form |?i = ? |0i + ? |1i,where |?|2 + |?|2 = 1. Then, a multiqubit state will berepresented in terms of the tensor product of the basisvectors. To sum up, the system must be extendable to alarge number of qubits in which we can store informationand each of them must be separately addresable.2. The ability to initialize the state of the qubits to asimple fiducial state, such as |00…0i.Once again, this is an important feature concerningboth quantum and classical computation in order toobtain a reliable output out of the computation processeswe are performing. So, as important as the operationswe realize over it is the possibility of preparing thedesired input over which perform those operations so wecan get the desired results. In fact, we just need to beable to prepare the n qubits we are working with intothe |00…0i state and them apply any unitary transformto get the desired input state. Input state preparationdepends on which physical system are we working with.In many physical realization we can get the systeminitialized by cooling it to the ground state. However,this won’t be precise for systems in which the differenceof energies between the energy levels (such as the groundand the first excited state) are significant enough so thatKBT << ?E.3. Long decoherence times, much longer than the gateoperation time.One of the main problems which arises in the con-struction of a quantum computer is the appearance ofdecoherence due to external disturbance which finallytranslates into the loss of information enconded in thesystem. Decoherence is a consequence of the interactionof the system with its environment which causes analteration of the state of the system. We can seedecoherence as the time required for a pure state?0 = (? |0i + ? |1i)(??h0| + ??h1|)to change into the mixed state? ? |?|2|0i h0| + |?|2|1i h1|13in which individual qubits are no longer addressable.That is, the time needed for the coherent terms ???tobe negligible.However, one should notice that decoherence time byitself is not what really matters but in relation with thegate operation time. That is, we need the system toavoid decoherence long enough to perform the necessarygate operations before the quantum state decays.4. A "universal" set of quantum gates.A quantum algorithm can be described as a se-quence of unitary transformations U1, U2, U3..., whichact over, typically, two qubits. However, arbitraryquantum gates acting on n number of qubits canbe constructed from single qubit gates and CNOTgates as an arbitrary unitary matrix U on a Hilbertspace of dimension d can be decomposed into aproduct of d two level matrices U = U1U2..Ud. Wecan visualize this by identifying Hamiltonians whichcould generate the unitary transformations such asU1 = eiH1t/~, U2 = eiH2t/~, .., Ud = eiHdt/~; we thenshould be able to activate and deactivate these Hamil-tonians the in an orderly way and the necessary timeto perform the computation. Obviously, reality is quitemore complicated than this.5. A qubit-specific measurement capability.In contrast to classical computation, the readout of theresults of the computation process is non-trivial in quan-tum computation and it strongly depends on the physicalsystem we are working with. To get to the results of theexecution of a quantum algorithm we must measure thestate of the system after its application. For most phys-ical realizations projective measurements are employedwhile in some cases, in which proyective measurement isnot possible, one must turn to averaged measurementsto extract the outcome of the computation. We can un-derstand the measurement process if we think about aqubit coupled to a classical system so that we can knowthe state of the qubit by the state of the classical sys-tem. This way, a qubit state ? |0i + ? |1i, where |0i and|1i represent the ground and excited state of a two-levelatom, can be measured using a single photon detector todetermine whether a photon have been emitted duringthe qubit transition from its excited to its ground state,collapsing the qubit state to its ground state if the photonis detected.B. Trapped IonsOne of the most interesting proposals for the physicalrealization of a quantum computer is to use ions trappedin an external potential as qubits. This proposal wasfirst carried out by Cirac and Zoller in 1995 where theyshowed how this system could be used to perform acontrolled-NOT quantum gate.Controlling motion- Trapping ionsElectron and nuclear spins are good candidates forqubits, however the energy difference between differentspin states is too small compared with the kineticenergy of the atoms at a standard temperature so theyare difficult to observe and control. However, one canreduce its kinetic energy by isolating and trapping themin electromagnetic traps and reducing its temperatureenough so the spin energy contribution begins beingsignificant in comparison. Ions in the trap need to beapproximately stationary, which means they need tobe cooled down to very low temperatures. The wayof obtaining this result is by employing laser coolingtechnologies. The idea is rather simple. Lets consideran ion moving along the x axis with certain velocityand two laser beams also moving int he x and the -xdirection. Due to the Doppler effect, the ion will seedifferent frequencies depending of its relative motionin relation to the laser beam. If the ion is movingtowards the laser beam, we will say the laser frequencyis blue-shifted, which means that its frequency seemsslightly higher for the ion than it really is. On the otherhand, if the ion is moving in the same direction, thefrequency of the laser will appear slightly lower and wesay the laser is red-detuned. If one can set up the laserfrequency to be just below the resonance frequency ofthe ion the laser photons will only be absorbed if theion is moving towards the laser beam so its frequencywill be shifted up by the Doppler effect. As the photonsabsorbed by the ion are moving in the opposite direc-tion, its momentum is opposite to the momentum ofthe ion and as a consequence it will eventually slow down.Paul TrapsThese traps are constructed by combining a constantand a alternating electric field (since from Earnshaw'sTheorem a point charge cannot be confined in a stableequilibrium by just static fields) to immerse the ions inan harmonic potentialV (x, y, z) = 12M(?2xx2 + ?2yy2 + ?2zz2)which we can approximate by V (z) = 12M(?2zz2) (anion moving along the z axis) if we consider ?2z << ?2x,y.Lets consider an ion as a two-level system representedby the ground state (|gi ? |0i) and an excited state(|ei ? |1i) trapped in this potential and located in anoscillating electric field of the formE~ = E1xcos ˆ (?t ? kz ? ?)In this situation we will find two kinds of Hamiltonian:the first one, which we will denote by H0 is the Hamilto-14nian in absence of an electric field; the second one wouldbe the interaction Hamiltonian. The Hamiltonians canbe described as follows:H0 = ?~2?0?z + ~?za†a.Where?z =1 00 ?1is the Pauli matrix corresponding to the z axisand a, a† are the annihilation and creation operators,respectively, defined as followsa =qM?z2~(z +ipzM?z)a† =qM?z2~(z ?ipzM?z)This Hamiltonian presents two internal states: |0i withenergy E0 = ?~?0/2 and |1i with energy E1 = ~?0/2.On the other hand, the interaction with the electricfield is described by the Hamiltonian:Hint = ?~2?1?+ + ??ei(?t??)e?ikz + e?i(?t??)eikzWith ?± = (?x ± i?y)/2:?+ =0 10 0?? =0 01 0We have denoted by ?1 the Rabi frequency whichis the angular frequency of the Rabi oscillations, theperiodic exchange of energy between a light field and atwo-level system.We can also separate Hint by expanding e±ikz in series:e±ikz ' 1 ± ikzThis expansion is valid if k?z0 << 1 , where?z0 =q ~2M?zis the spread of the wave functioncorresponding to the ground state, and ? = k?z0 iscalled the Lamb–Dicke parameter.Substituting e±ikz ' 1 ± ikz in the interactionHamiltonian, we obtain two contributions:H1 = ?~2?1?+ + ??ei(?t??) + e?i(?t??)which represents a rotational term, andH2 = ?i~??12a + a†?+ + ??ei(?t??) + e?i(?t??)which takes account of the vibrational motion.Lets go back to H0 for a while. The temporal evolutionof the operators we are using under this Hamiltonian, inthe Heisenberg Picture of motion, will be:?+(t) = eiH0t/~?+e?iH0t/~ = ?+e?i?0t.Doing so with the rest of the operators we are workingwith, we obtain:??(t) = ??ei?0t.a(t) = ae?i?zt.a†(t) = a†ei?zt.Applying this results to H1, we obtain the temporalevolution for the Hamiltonian:H1 = ?~2?1?+e?i?0t+??ei?0tei(?t??)+e?i(?t??) '?~2?1?+ei(???0)t?i? + ??e?i(???0)t?i?.Where the last term is obtained by ignoring the con-tribution of the rapid oscillations defined by e±i(?+?0)tin the so-called rotating-wave approximation. The termswhose contributions we have neglected oscillate too fastand when we make an average their contribution willbe zero. However, we can justify the approximationin another way: those terms ??a† and ?+a lead toprocesses that are really strange such as the systemgoing to its ground state and absorbing a phoon so wecan obvious them.If we set the tuning frequency of the electricfield to the resonance frequency, ? = ?0 we getH1 = ?~2?1?+e?i? + ??ei? and the unitary operatorwhich defines the temporal evolution then isU(t) = e?i?1t/2?+e?i?+??ei? =e?i?1t/2?x cos ?+?y sin ?which defines a rotation of the spin by an angle? = ??1t about the axis defined by the directionnˆ = (cos ?,sin ?, 0). So, this way, the spin can be madeto rotate by a given angle by adjusting the duration t ofthe interaction allowing us to handle the ion states.On the other hand, the temporal evolution of H2 is:H2(t) = ?i~??12?+ae?i(?0+?z)t + ?+a†e?i(?0??z)t +??aei(?0??z)t + ??a†ei(?0+?z)txei(?t??) ? e?i(?t??)By tuning the laser frequency to ? = (?0 + ?z), sothe laser is blue-detuned as we introduced in the verybeginning of this section, we obtain for H2 (once wehave introduced the rotating wave approximation):H2(t) = ?i~??12?+ae?i? ? ??a†ei?15which allow us to force transitions between the states|0, 0i and |1, 1i as?+a |1, 1i = |0, 0i and??a†|0, 0i = |1, 1iWhere we have expressed the ion state in the form|n, mi, being n = 0,1 the internal state and m = 0,1 thevibrational state of the harmonic oscillator.If we red-detune the laser by choosing its frequencyto be ? = (?0 ? ?z), then we can induce transitionsbetween the |1, 0i and |0, 1i states as follows:?+a†|1, 0i = |0, 1i and??a |0, 1i = |1, 0iFigure 7: Energy levels of the qubitC. Superconducting qubitsEssentially different from other physical realizationof quantum computing, its main characteristic is themacroscopic degrees of freedom carried out by the qubitswhich means that the parameters of the system can befixed by human fabrication modifying the dimensionsof the circuit. In this physical implementation, qubitsare represented by the current in a superconductingcircuit which contains one or more Josephson. Theuse of superconducting materials allows to reduce agreat deal the losses in the system and the Josephsonjunctions introduce the necessary non-linear behavior.The energy levels of a regular superconducting LCcircuit are equally spaced by a difference of energy of~?0 so it is not possible to control transitions betweenthe ground and the first excited levels without inducingat the same time transitions to higher energy levelsmaking impossible a two-level system. As the equalspacing of energy levels in an harmonic oscillator isrelated to its linearity, the necessity of the introductionof non-linearities becomes evident. By the introductionof non-linearities allows to obtain quantum energy levelsnot equally spaced which permits the isolation of twoenergy levels which provide the basis states for a qubit:|0i and |1i.A Josephson junction consist of two superconductingelectrodes separated by a thin layer of an insulat-ing material, thing enough to permit a quantumprocess called tunelling in which the wave functionof a particle is capable of traveling through a barrierthat classically would not be allowed to due to its energy.The Hamiltonian of a circuit built with one Josephsonjunction, an inductor L and a capacitor C, whose totalmagnetic flux ? is composed of an external flux, denotedby ?ex and the flux due to the inductance (? ? ?ex), isH =12Cq2 +12L(? ? ?ex)2 ?I0?02?cos 2???0The term ?I0?02?cos 2???0is characteristic of a Joseph-son junction and represents the energy stored in theinductor, EJ , with current IJ = I0 sin(?). The termI0 is called critical current and it depends on thejunction and ? =2???0is the phase difference between thewave functions of the Cooper pairs, coupled electronswhich present the same wave function, generated in thesuperconductors at each side of the junction.The quantized version of the Hamiltonian isH =12C(Q2) + 12L(? ? ?ex)2 ? EJ cos 2???0=12C(?~2 ?2??2 ) + U(?)Where we have identified the operators Q = i~?/??and ? = ?.Flux qubitsOne of the circuits proposed to serve as a supportfor qubits corresponds to the case EJ >> q2C /2C,where qC is the charge of the Cooper pair, two timesthe charge of the electron. A circuit that presents thischaracteristic is the support for the so-called flux qubits.The potential present in the Hamiltonian of the circuitU(?) = 12L(? ? ?ex)2 ? EJ cos 2???0is symmetric for thevalue of the external flux ?ex = ?0/2 and its structureis the one of a double well, thus presenting degenerationfor the ground states as the potential minima of the twowells are found in the same energy. If we denote by |0ithe state in which the current flows in the circuit in theclockwise direction and by |1i the state in which thecurrent flows counter-clockwise, the possible states forthe system are|±i = ?12(|0i ± |1i)Due to the tunneling effect between the two wells,the two states are coupled. If we denote by c0 theprobability amplitude of finding the system in the state|0i and by c1 the probability amplitude of finding the16system in the state |1i, the dynamics of the two stateswill bei~dc0dt = E0c0 ? Ac1i~dc1dt = E0c1 ? Ac0with energy E± = E0 ? A, where A is the so-calledtunneling amplitude.However, the symmetry of the potential U(?) isbroken for values ?ex 6= ?0/2. As we increase the valueof |?ex|, the minima of the wells begin to separate bya distance that can be experimentally controllable aspresents a linear dependence of the external flux applied:?U ??202L(?ex?0?12). As we increase the relative deepnessof the two wells, the possible states in which we can findthe system approach to the |0i and |1i states. In the |±ibasis, the Hamiltonian can be written as:H = ?A?z + 2?U?xand its possible energies are E± = ±pA2 + (2?U)2.As we can see, for ?U = 0, corresponding to ?ex = ?0/2,the possible values for the energy are E± = ±A, whilefor |?U| >> A, this values are, approximately,E± = ±?2|?U|.The readout of the qubit state is done by a mag-netometer, constructed with Josephson juctions, wichmeasures the direction of the magnetic field and thusthe sense of rotation of the current.

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