Optical Quantum Computer

Optical Quantum Computer

Recall from Sect. 8.4 that photons with their two orthogonal polarization states are natural candidates for representing qubits. Photons have the ad- vantage of being very robust and versatile carriers of quantum information, as they can propagate quickly over long distances in optical ?bers, without un- dergoing much absorption and decoherence. Here we thus present an envisaged quantum computer with photonic qubits.
A schematic representation of an optical quantum computer is shown in Fig. 10.5. In the initialization section of the computer, deterministic sources of single photons generate single-photon pulses with precise timing and well- de?ned polarization and pulse–shapes. A collection of such photons constitutes the quantum register. The qubit basis states are represented by the vertical
| ‡) ? |0) and horizontal | ?) ? |1) polarization states of the photons. The preparation of an initial state of the register and the execution of the program according to the desired quantum algorithm is implemented by the quantum processor. This amounts to the application of a certain sequence of single-qubit U and two-qubit W unitary transformations, whose physical realization is described below. Finally, the result of computation is read-out by a collection of e?cient polarization–sensitive photon detectors.




Fig. 10.5. Schematic representation of a quantum computer with photonic qubits. Qubit initialization is realized by deterministic single-photon sources (SPhS). Infor- mation processing is implemented by the quantum processor with single-qubit U and two-qubit W logic gates. Read-out of the result of computation is accomplished by e?cient single-photon detectors (SPhD).


According to Sect. 8.4.2, for the photon-polarization qubit |?1) = ? | ‡) + ? | ?), the combination of the polarization rotation R(?) and phase-shift operations T (?) can realize any single-qubit unitary transformation U given


by (8.31). Together with a two-photon conditional operation realizing either the cnot or the equivalent cz gate, we can then implement the Universal set of quantum gates.


(a)



1


(b)




FR

R(?)



PBS


T(?)





U ? 1


PBS


? A 1
AB A B

?
XPM

WCZ ?1 ?1



B PBS

Fig. 10.6. Physical implementations of quantum logic gates. (a) Single-qubit logic gates U are implemented with a sequence of two linear–optics operations: R(?)— Faraday rotation (FR) of photon polarization by angle ? about the propagation direction; T (?)—relative phase-shift ? of the photon’s | ?) and | ‡) polarized com- ponents due to their optical paths di?erence. (b) Two-qubit controlled-Z (cz) gate Wcz is realized using polarizing beam-splitters (PBS) and ? cross-phase modulation (XPM).


In Fig. 10.6(a) we show the possible implementations of the polarization rotation of a photon by angle ? about the propagation direction, R(?), and the relative phase-shift ? of the | ‡) and | ?) polarized components of the pho- ton, T (?). Both operations are easy to realize with standard linear optical ele- ments. Recall from classical optics that linear polarization can be decomposed into left- and right-circular polarization components. In a circular–birefringent medium, these two orthogonal polarizations propagate with di?erent phase velocities due to the di?erent refraction indexes nL ƒ= nR. Then, upon pass- ing through the medium of length L, the left- and right-circular polarization components acquire a phase di?erence ?? = 2?L?n/?, which translates into the rotation of the linear polarization by the angle ? = 1 ??. Thus, the ro- tation angle ? can be controlled by the di?erence of the refraction indices ?n ? nR ? nL, which in a medium susceptible to the Zeeman e?ect can be manipulated by a longitudinal magnetic ?eld B = zˆB leading to ?n ? B. This is the essence of the magneto-optical Faraday e?ect. In turn, the T (?) operation is nothing more than the phase shift ? = ?s/? due to the optical


path di?erence ?s = s? ? s‡ between the photon’s | ?) and | ‡) polarized components

A possible realization of the cz logic gate W AB

between two photonic

qubits A and B is shown in Fig. 10.6(b). There, after passing through a po-
larizing beam-splitter, the vertically polarized component | ‡) of each photon is transmitted, while the horizontally polarized component | ?) is directed
into the active medium, wherein the two-photon state |?in) = A)| ?B)
acquires a conditional phase-shift ?c = ?. This is possible when the Hamil-
tonian governing the evolution of the two optical ?eld modes in the active medium has the form
H = ?k? a† aA a† aB , (10.35)
A B
where ? is the so-called cross-phase modulation (XPM) coe?cient, while a† and aj are the creation and annihilation operators of the corresponding mode j = A,B. Then, during the interaction, the input state evolves according to

i A B

|?(t)) = e? k Ht |?in) = ei?t |? )| ?

) , (10.36)


where we have used the fact that there is only one photon in each mode j, therefore a†aj |1j ) = |1j ). At the exit from the medium of length L, the accumulated conditional phase shift is then ?c = ?tout, where tout = L/vg is the interaction time, with vg being the light velocity inside the medium. For ?c = ? we then obtain |?out) = ? |?in). Attaining large conditional phase- shift within a reasonable interaction length L (of a few centimeters) requires, however, a very large value of the coe?cient ? and long interaction time (small group velocity vg ), which conventional media can not provide. Developing schemes for achieving giant cross-phase modulation is therefore crucial for the implementation of deterministic optical quantum logic gates, which is an active topic of current research. Recently, several promising proposals towards this goal, based on electromagnetically induced transparency (EIT) in atomic media, have been reported. After leaving the medium, the | ?) component of each photon is recombined with its | ‡) component on another polarizing beam-splitter. Complete temporal overlap of the two polarization components of each photon can be achieved by delaying the | ‡) component in a ?ber loop or sending it though an EIT medium, in which the light propagates with a small group velocity vg c (see Sect. 6.3). At the output, we then have the transformation

A B
| ‡ )| ‡
A B

A
)? | ‡
A

B
)| ‡ )
B

| ‡ )| ?
A B

)? | ‡

)| ? )
A B

|? )| ‡
A B
|? )| ?

)? | ?
) ? ? | ?

)| ‡ )
A B
)| ? )


which corresponds to the truth-table of the cz logic gate between a pair of qubits represented by photons A and B.
Another important prerequisite for the optical quantum computer is the availability of single-photon sources. Currently, the most accessible scheme


for generating single photons relies on the process of spontaneous parametric down-conversion, in which a single pump photon of frequency ?p and wave vector kp is converted to two polarization- and momentum-correlated photons (signal and idler) of frequencies ?s and ?i and wave vectors ks and ki, such that ?p = ?s + ?i and kp = ks + ki. The crystal thus produces pairs of entangled photons in state
s i s i
|?2) = 1 ( |‡ )| ? ) + |? )| ‡ )) . (10.37)
Then, if we detect one photon of the pair, say the idler, along a particular direction ki and polarization i), we know that there is one signal pho- ton in the mode with ks = kf ? ki and in state s). This, however, is not a deterministic source of single-photons, as it relies on the spontaneous generation of entangled photon pairs and conditional detection of one photon, which projects the other photon onto the desired polarization and momentum state. Hence, we can not control the timing and the temporal characteristics of the single-photon pulses. If we need to initialize the quantum register with a certain number of qubits represented by the signal photons, we will need a coincidence detection of the corresponding number of idler photons, which for a large register is a very unlikely event. Thus, for an e?cient initializa- tion of the register, deterministic sources of single photons are required. In Sect. 5.3 we have presented one such deterministic and e?cient source of tai- lored single-photon pulses based on intracavity STIRAP with a single atom.


PBS
D0
?1




Fig. 10.7. Photonic qubit measurement in the computational basis { |0), |1)} is implemented with a polarizing beam-splitter (PBS) and two photodetectors D0 and D1.


To complete the description of the optical quantum computer, we need to consider a measurement scheme capable of reliably detecting the polarization states of single photons. When the photonic qubit |?1) = ? | ‡) + ? | ?) goes though a polarizing beam-splitter, its vertically and horizontally polarized components are sent to two di?erent spatial modes—photonic channels. Plac- ing e?cient photon detectors at each channel would therefore accomplish the projective measurement of the qubits in the computational basis { |0), |1)}, as shown in Fig. 10.7. Avalanche photodetectors, mentioned in the beginning of Sect. 2.3, have very high quantum e?ciencies of ? ” 70% and can therefore realize reliable qubit measurements. Finally, all of the constituent parts of the


optical quantum computer described above can be interconnected by optical ?bers, according to the algorithm or program under execution.