The ion-trap quantum computer

The ion-trap quantum computer was introduced by Cirac and Zoller (1995) and since then many other po-tential and actual realizations of quantum computers have been pursued by many groups. The quantum hard-ware is the following: a qubit is a single ion held in a trap by laser cooling and the application of appropriate elec-tromagnetic fields; a quantum register is a linear array of ions; operations are effected by applying laser Rabi

pulses; information transmission is achieved as a result of the Coulomb interaction between ions and the ex-change of phonons from collective oscillations. We see again, at a very fundamental level, that information is physical. Using the Cirac-Zoller (CZ) technique, Mon-roe et al. (1995) were soon able to construct a single quantum gate.

The ion-trap proposal has several advantages: it calls for manipulation of quantum states that is already known from precision spectroscopy techniques; it has low decoherence rates due to the decay of excited states and the heating of the ionic motion; and it takes advan-tage of existing very efficient experimental methods for retrieving the information from the quantum computer, such as the mechanism of quantum jumps.

1. Experimental setup

The geometry of a radio-frequency (RF) ion trap or Paul trap is schematically shown in Fig. 44. An RF Paul trap uses static and oscillating electric potentials to con-fine particles within small (⬃1␮m) regions. To obtain a string of ions for forming the quantum register we need a quadrupole ion trap with a cylindrical geometry. The confining mechanism of ions is twofold:

(i) A strong radial confinement, achieved by RF po-tentials generally produced with four rod elec-trodes.

(ii) An axial confinement, achieved by applying a quadrupolar electrostatic potential through two end caps.

The ions lie along the trap axis and their oscillations are controlled by the axial potential. The collective os-cillations of the string center of mass are used as a sort of computational bus, transferring information from one ion to another by phonon exchange. The dimensions of the ion traps used by the Los Alamos group are typically 1 cm long and 1–2 mm wide (Hughes et al., 1998).

Before any computation takes place, the center of mass of the ion string must be set to its ground state.

This is accomplished by a laser cooling process that FIG. 43. Energy levels of a two-qubit spin system with Ising

interaction (units ប⫽1): On the left are the noninteracting Zeeman levels; on the right, the levels perturbed by the Ising term (when␻1⬍␻2⬍⫺J⬍0).

FIG. 44. Schematic geometry of a radio-frequency quadrupole linear ion trap. Laser beams address a string of ions in the middle of the setup with four linear rods and two end caps.

brings the ions to the ground state of their vibrational motion. The result is an ion string configuration as shown in Fig. 44, crystallizing into a linear array that makes it possible to address each ion individually by lasers. The inter-ion spacing can be controlled by balanc-ing the Coulomb repulsion of the ions and the axially confining potential (Wineland et al., 1998).

Several kinds of ions (Be^⫹, Ca^⫹, Ba^⫹, Mg^⫹, Hg^⫹, Sr^⫹) and qubit schemes have been proposed. The Cirac-Zoller qubit 兵^兩0典^,兩1典其 is built using some appropriate electronic ion states. For instance, the Los Alamos group (Hughes et al., 1998) has chosen Ca^⫹ ions, whose most relevant levels are shown in Fig. 45. The state qu-bits 兵兩0典^,兩1典其 and one extra auxiliary level 兩2典 ^{(to be} described below) are identified as follows (see Fig. 45):

兩0典⫽兩4²S_1/2,M_J⫽¹2典^, 兩1典⫽兩3²D_5/2,M_J⫽³2典^,

兩2典⫽兩3²D_5/2,M_J⫽⫺¹2典^{. (158)} The level (4 ²S_1/2,M_J⫽¹2) is the ground state, while (3 ²D_5/2,M_J⫽³2) is a metastable level with a long life-time (1.06 s). Both the electric dipole transition 4 ²S_1/2

→4 ²P_1/2at 397 nm wavelength and the electric quadru-pole transition 4 ²S_1/2→3 ²D_3/2 at 732 nm are suitable for Doppler and sideband laser cooling, respectively. In Doppler cooling the laser radiation pressure slows down the axial motion of the ions until temperatures T⬃a few mK. To further reduce the temperature (T⬃a few␮^K) until no phonons are present, one resorts to sideband cooling (Hughes, 1998).

The interaction between Cirac-Zoller qubits is achieved using two types of degrees of freedom: internal (the electronic states of the ions) and external (the vi-brational states of their collective excitations). Thus an active state for information processing is the tensor product of an electronic state and a quantum oscillator state of the axial potential, namely,

兩⌿典^⫽兩x典^兩␣典^{, x⫽0,1;} ␣⫽g,e, (159) where 兩x典 refer to the electronic levels and 兩g典^,兩e典 de-note the ground state and first excited state of the vibra-tional motion, respectively. In兩g典 there are no phonons present in the system, while there is one phonon in 兩e典 (see Fig. 46).

2. Laser pulses

With this structure of states one can apply two types of laser Rabi pulses to the ions in order to achieve quan-tum logic operations. These are called V and U pulses.

The V pulse implements one-qubit operations. Its fre-quency is tuned to resonate with the optical transition between the qubit states. It swaps the electronic states 兩0典↔兩1典 and leaves the vibrational mode in the ground state兩g典. The unitary evolution operator induced by this pulse is

V共␪^,␾兲ªe^⫺itHV/ប,

(160) H_Vª¹2ប⍀关e^⫺i␾兩1典具^0兩⫹ei␾兩0典具¹兩兴,

where ␪ª⍀t, HVis the V-pulse Hamiltonian, ⍀ is the Rabi frequency (proportional to the square root of the laser intensity), and␾is the laser phase. This pulse then produces the following action on the electronic states:

V共␪^,␾兲:

再

The U pulse is used to implement two-qubit opera-tions. The laser frequency is now adjusted to induce si-multaneously both an electronic and a vibrational tran-sition. To help perform the desired logic gates, an auxiliary electronic state 兩2典 (see Fig. 46) is available.

The time-evolution operator led by this pulse is U_xˆ共␬^,␾兲ªe^{⫺itHU(xˆ)/ប}, xˆ⫽1,2,

(162) H_U共xˆ兲ª¹2ប␩⍀关e^⫺i␾兩xˆ典具⁰兩a⫹e^i␾兩0典具xˆ兩a^†兴,

FIG. 45. Relevant energy levels in Ca^⫹ions.

FIG. 46. Schematic representation of the transitions generated by the V and U pulses.

where H_U is the U-pulse Hamiltonian,␬ª␩⍀t, ␩^{is the} Lamb-Dicke parameter,⁶³ and a^†,a are phonon creation and annihilation operators satisfying

a^†兩g典^⫽兩e典^{, a兩e}典^⫽兩g典^, 关a,a^†兴⫽1. (163) Several physical constraints on these parameters in a lin-ear ion trap must be fulfilled for it to function stably and as required (Cirac and Zoller, 1995).

The U pulse acts as follows:

U_xˆ共␬^,␾兲:

冦

3. Building logic gates

By controlling the duration of the laser pulses in Eqs.

(161) and (164) we can perform logic operations in a fashion akin to those for spin qubits with Rabi pulses.

The nice thing about the ion-trap quantum computer is that the same Rabi pulses can drive conditional logic when phonons are suitably put to work.

For instance, a CNOTgate can be constructed using a series of V and U pulses. To this end, we first reproduce a ␲ controlled-phase gate [Eq. (78)] between qubits at sites i,j as follows:

U_CPh^(i,j)共␲兲⫽U1

(i)共␲^,0兲U2

(j)共2␲^,0兲U1

(i)共␲^,0兲. (165) The explicit action of this sequence of operations is shown in Fig. 47. This two-bit gate is constructed only out of U pulses.

In order to construct CNOT from this gate [see Eq.

(79) and Fig. 25] we need to employ V pulses,

U_CNOT^(i,j) ⫽V^(j)共¹2␲^,12␲兲UCPh(i,j)共␲兲V^(j)共⫺¹2␲^,12␲兲, (166) where these V pulses correspond to Hadamard gates.

Other logic gates involving a larger number of qubits can be constructed similarly using these basic pulse op-erations (Cirac and Zoller, 1995).

Let us note that the 2␲auxiliary rotations in Eq. (165) do not produce any population of the auxiliary atomic levels nor of the center-of-mass levels. Thus a variation in the population of these levels when the gate is oper-ated would indicate a faulty experimental realization.

Upon completion of the quantum operations in the ion-trap quantum computer, we need to read out the result (see Sec. IX). This is done by measuring the state

of each qubit in the quantum register using the quantum-jump technique (Bergquist et al., 1986; Na-gourney et al., 1986; Sauter et al., 1986). For instance, for the Ca^⫹ qubits (158), the laser is tuned to the dipole transition 4 ²S_1/2→4 ²P_1/2at 397 nm (see Fig. 45). There are two possibilities for the ion being addressed with the laser: (i) if the ion radiates (fluoresces), this means that its state is 兩0典; (ii) if the ion does not radiate (remains dark), then it was in the 兩1典 state. Therefore just by observing which ions fluoresce and which remain dark we can retrieve the bit values of the register. Actually, there is a third possibility in which 4 ²P_1/2→3²D_3/2. In order to prevent this metastable level from being popu-lated, a pump-out laser is also required.

4. Further applications

The ion-trap technique has also found applications in the preparation of entangled states (Molmer and So-rensen, 1999). This has been experimentally realized by the NIST group (Sackett et al., 2000), who generated en-tangled states of two and four trapped ions. In Fig. 48 a four-qubit quantum register used in these experiments is shown.

Unavoidable errors impose computational limits on ion-trap quantum computers. Sources of these con-straints are the spontaneous decay of the metastable state, laser phase decoherence, ion heating, and other kinds of errors. Using simple physical arguments it is possible to place upper bounds on the number of laser pulses N_U sustained by the ion trap before it enters a decoherence regime (Hughes, James, et al., 1996),

N_UL^1.84⬍ 2Z共␶^{/1 s}兲

A^1/2F^3/2共␭/1 m兲^3/2, (167) where Z is the ion degree of ionization,␶is the lifetime of the metastable state, L is the number of ions, A is their atomic mass, F parametrizes the focusing

capabil-63This quantity is the ratio between the width of the ion os-cillation in the vibrational ground state of the register and the (reduced) laser wavelength ␭L/2␲: ␩ª(ប/2NMion␻z)^1/2

⫻(2␲/␭L), where N is the number of cold ions and␻zis the vibrational frequency of the register’s center of mass along the trap axis. The Lamb-Dicke criterion␩Ⰶ1 is required for Eq.

(162) to be a good approximation (Cirac and Zoller, 1995). For the Ca^⫹trap, with N⬃10, ␻z⬃100 kHz, then␩⬃0.2.

FIG. 47. Sequence of operations for a controlled-phase gate:

(a) Quantum circuit for the controlled-phase gate in an ion-trap quantum computer. We denote by 兩p(x1)典 ^{the phonon} states p(0)ªg,p(1)ªe. Note also that the overall final phase is (⫺1)^x1x2, as it corresponds to a controlled phase␾⫽␲. (b) Evolution of a state under the sequence of U pulses in Eq.

(165).

ity of the laser, and␭ is the laser wavelength. This bound depends on the ion parameters A and ␶, making some ion species more suitable than others.⁶⁴With this bound it is possible to estimate the number of ions needed to factorize a 438-bit number using ytterbium [with the transition 4f¹⁴6s ²S_1/2↔4f¹³6s^{2 2}F_1/2, which has a very long lifetime (1533 days) and a wavelength of 467 nm].

Around 2200 trapped ions and 4.5⫻10¹⁰pulses would be required to perform the desired factorization, in about 100 hours of computation time (Hughes, James, et al., 1996).

Scalability of the ion-trap quantum computer is a cen-tral issue if we want to have a useful machine for num-ber factoring and the like. With current techniques, it is believed that prospects for reaching a few tens of qubits are good (Hughes et al., 1998). Cirac and Zoller (2000) have proposed an ion-trap-based quantum computer with a two-dimensional array of independent ion traps and a different ion (head) that moves above this plane.

This setup is still conceptually simple and it is believed to be within reach of present experimental technologies.