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linear - optical architectures belong to the most prominent platforms for realizing protocols of quantum information processing @xcite . they are experimentally feasible and they work directly with photons without the necessity to transfer the quantum state of a photonic qubit into another quantum system like an ion etc . the latter feature is quite convenient because photons are good carriers of information for communication purposes . linear - optical quantum gates achieve the non - linearity necessary for the interaction between qubits by means of the non - linearity of quantum measurement . unfortunately , quantum measurement is not only non - linear but also probabilistic . therefore linear - optical implementations of quantum gates are mostly probabilistic too  their operation sometimes fails . partly , this is a fundamental limitation . but in many cases when data qubits appear in an improper state after the measurement on an ancillary system they can still be corrected by applying a proper unitary transformation which depends only on the measurement result . in these situations , implementation of the feed forward can increase the probability of success of the gate @xcite . in the present paper , we apply this approach to a linear - optical programmable quantum gate . the most of conventional computers use fixed hardware and different tasks are performed using different software . this concept can be , in principle , applied to quantum computers as well : the executed unitary operation can be determined by some kind of a program . however , in 1997 nielsen and chuang @xcite showed that an @xmath1-qubit quantum register can perfectly encode at most @xmath2 distinct quantum operations . although this bound rules out perfect universally - programmable quantum gates ( even unitary transformations on only one qubit form a group with uncountably many elements ) , it is still possible to construct approximate or probabilistic programmable quantum gates and optimize their performance for a given size of the program register . such gates can either operate deterministically , but with some noise added to the output state @xcite , or they can operate probabilistically , but error free @xcite . combination of these regimes is also possible . a probabilistic programmable phase gate was proposed by vidal , masanes , and cirac @xcite . it carries out rotation of a single - qubit state along the @xmath0-axis of the bloch sphere . the angle of rotation ( or the phase shift ) is programmed into a state of a single - qubit program register . it is worth noting that an exact specification of an angle of rotation would require infinitely many classical bits . but here the information is encoded into a single qubit only . the price to pay is that the success probability of such a gate is limited by 50% @xcite . the programmable phase gate was experimentally implemented for the first time in 2008 @xcite . however , the success probability of that linear - optical implementation reached only 25% . in the present paper we will show how to increase the success probability of this scheme to its quantum mechanical limit of 50% by means of electronic feed forward . the programmable phase gate works with a data and program qubit . the program qubit is supposed to contain information about the phase shift @xmath3 encoded in the following way : @xmath4 the gate performs a unitary evolution of the data qubit which depends on the state of the program qubit : @xmath5 without loss of generality we can consider only pure input states of the data qubit : @xmath6 so the output state of the data qubit reads : @xmath7    experimentally the programmable phase gate can be implemented by an optical setup shown in fig . [ fig - scheme ] . each qubit is represented by a single photon which may propagate in two optical fibers . the basis states @xmath8 and @xmath9 correspond to the presence of the photon in the first or second fiber , respectively . when restricted only to the cases where a single photon emerges in each output port , the conditional two - photon output state reads ( the normalization reflects the fact that the probability of this situation is @xmath10 ) : @xmath11 , \ ] ] where @xmath12 . if we make a measurement on the program qubit in the basis @xmath13 then also the output state of the data qubit collapses into one of the two following states according to the result of the measurement : @xmath14 . if the measurement outcome is @xmath15 then the unitary transformation @xmath16 has been applied to the data qubit . if the outcome is @xmath17 then the state acquires an extra @xmath18 phase shift , i.e. , @xmath19 has been executed . this is compensated by a fast electro - optical modulator which applies a corrective phase shift @xmath20 ( in practice we apply phase shift @xmath18 what is equivalent ) . the scheme of the setup is shown in fig . [ fig - scheme ] . pairs of photons are created by type - ii collinear frequency - degenerate spontaneous parametric down conversion ( spdc ) in a two - millimeter long bbo crystal pumped by a diode laser ( coherent cube ) at 405 nm . the photons are separated by a polarizing beam splitter and coupled into single - mode fibers . using fiber polarization controllers the same polarizations are set on the both photons . by means of fiber couplers and electro - optical phase modulators the required input states of the program and data qubits are prepared . to prepare state ( [ eq - prog_qubit ] ) of the program qubit a fiber coupler ( fc ) with fixed splitting ratio 50:50 is used . an arbitrary state of the data qubit ( [ eq - dat_qubit_in ] ) is prepared using electronically controlled variable ratio coupler ( vrc ) . all employed phase modulators ( eo space ) are based on the linear electro - optic effect in lithium niobate . their half - wave voltages are about 1.5v . these phase modulators ( pm ) exhibit relatively high dispersion . therefore one pm is placed in each interferometer arm in order to compensate dispersion effects . because the overall phase of a quantum state is irrelevant it is equivalent to apply either a phase shift @xmath21 to @xmath9 or @xmath22 to @xmath8 . the gate itself consists of the exchange of two rails of input qubits and of the measurement on the data qubit ( see fig . [ fig - scheme ] ) . the measurement in basis @xmath23 is accomplished by a fiber coupler with fixed splitting ratio 50:50 and two single photon detectors . in this experiment we use single photon counting modules ( perkin - elmer ) based on actively quenched silicon avalanche photodiodes . detectors d@xmath24 , d@xmath25 , and d@xmath26 belongs to a quad module spcm - aq4c ( total efficiencies 5060% , dark counts 370440s@xmath27 , response time 3340ns ) . as detector d@xmath28 , serving for the feed forward , a single module spcm aqr-14fc is used because of its faster response ( total efficiency about 50% , dark counts 180s@xmath27 , response time 17ns ) . the output of the detector is a 31ns long ttl ( 5v ) pulse .    to implement the feed forward the signal from detector d@xmath28 is led to a passive voltage divider in order to adapt the 5v voltage level to about 1.5v ( necessary for the phase shift of @xmath18 ) and then it is led directly to the phase modulator . the coaxial jumpers are as short as possible . the total delay including the time response of the detector is 20ns . to compensate this delay , photon wave - packets representing data qubits are retarded by fiber delay lines ( one coil of fiber of the length circa 8 m in each interferometer arm ) . timing of the feed - forward pulse and the photon arrival was precisely tuned . coherence time of photons created by our spdc source is only several hundreds of femtoseconds . the right - most block in fig .  [ fig - scheme ] enables us to measure the data qubit at the output of the gate in an arbitrary basis . these measurements are necessary to evaluate performance of the gate . the whole experimental setup is formed by two mach - zehnder interferometers ( mzi ) . the length of the arms of the shorter mzi is about 10.5 m ( the upper interferometer in fig . [ fig - scheme ] ) . the length of the arms of the longer one is about 21.5 m ( the lower interferometer one in fig . [ fig - scheme ] ) . to balance the arm lengths we use motorized air gaps with adjustable lengths . inside the air gaps , polarizers and wave plates are also mounted . they serve for accurate setting of polarizations of the photons ( to obtain high visibilities the polarizations in the both arms of each mzi must be the same ) .    to reduce the effect of the phase drift caused by fluctuations of temperature and temperature gradients we apply both passive and active stabilization . the experimental setup is covered by a shield minimizing air flux around the components and the both delay fiber loops are winded on an aluminium cylinder which is thermally isolated . besides , after each three seconds of measurement an active stabilization is performed . it measures intensities for phase shifts 0 and @xmath29 and if necessary it calculates phase compensation and applies corresponding additional corrective voltage to the phase modulator . this results in the precision of the phase setting during the measurement period better than @xmath30 . for the stabilization purposes we use a laser diode at 810 nm . to ensure the same spectral range , both the laser beam and spdc generated photons pass through the same band - pass interference filter ( spectral fwhm 2 nm , andover ) . during the active stabilization the source is automatically switched from spdc to a laser diode . any quantum operation can be fully described by a completely positive ( cp ) map . according to the jamiolkowski - choi isomorphism any cp map can be represented by a positive - semidefinite operator @xmath31 on the tensor product of input and output hilbert spaces @xmath32 and @xmath33 @xcite . the input state @xmath34 transforms according to @xmath35.\ ] ] combinations of different input states with measurements on the output quantum system represent effective measurements performed on @xmath36 . a proper selection of the input states and final measurements allows us to reconstruct matrix @xmath31 from measured data using maximum likelihood ( ml ) estimation technique @xcite .    for each phase shift , i.e , for a fixed state of the program qubit , we used six different input states of the data qubit , namely @xmath37 and @xmath38 . for each of these input states the state of the data qubit at the output of the gate was measured in three different measurement basis , @xmath39 and @xmath40 . each time we simultaneously measured two - photon coincidence counts between detectors d@xmath24 & d@xmath25 , d@xmath24 & d@xmath26 , d@xmath28 & d@xmath25 , d@xmath28 & d@xmath26 in 12 three - second intervals . the unequal detector efficiencies were compensated by proper rescaling of the measured coincidence counts . from these data we have reconstructed choi matrices describing the functioning of the gate for several different phase shifts . in figs . [ fig - choi - halfpi ] and [ fig - choi - pi ] there are examples of the choi matrices of the gate for @xmath41 and @xmath42 , respectively .    reconstructed : ideal : +   +    reconstructed : ideal : +   +    to quantify the quality of gate operation we have calculated the process fidelity . if @xmath43 is a one - dimensional projector then the common definition of process fidelity is @xmath44 / ( \mathrm{tr}[\chi]\mathrm{tr}[\chi_{\mathrm{id}}]).\ ] ] here @xmath43 represents the ideal transformation corresponding to our gate . in particular , @xmath45 where @xmath46 stays for the unitary operation ( [ eq - uphi ] ) applied by the gate . we have also reconstructed density matrices of the output states of the data qubit corresponding to all input states and calculated fidelities and purities of the output states . the fidelity of output state @xmath47 is defined as @xmath48 with @xmath49 being the ( pure ) input state . the purity of the output state is defined as @xmath50 $ ] . if the input state is pure the output state is expected to be pure as well . table  [ tab - withff ] shows process fidelities for seven different phase shifts . it also shows the average and minimal values of output state fidelities and average and minimal purities of output states . fidelities and purities are averaged over six output states corresponding to six input states described above . also the minimum values are related to these sets of states . .process fidelities ( @xmath51 ) , average ( @xmath52 ) and minimal ( @xmath53 ) output - state fidelities , average ( @xmath54 ) and minimal ( @xmath55 ) output - state purities for different phases ( @xmath3 ) _ with _ feed forward ( @xmath56 ) . [ cols="^,^,^,^,^,^",options="header " , ] we have implemented a reliable and relatively simple electronic feed - forward system which is fast and which does not require high voltage . we employed this technique to double the success probability of a programmable linear - optical quantum phase gate . we showed that the application of the feed forward does not affect substantially either the process fidelity or the output - state fidelities . beside the improvement of efficiency of linear - optical quantum gates , this feed forward technique can be used for other tasks , such as quantum teleportation experiments or minimal disturbance measurement . the authors thank lucie elechovsk for her advice and for her help in the preparatory phase of the experiment . this work was supported by the czech science foundation ( 202/09/0747 ) , palacky university ( prf-2011 - 015 ) , and the czech ministry of education ( msm6198959213 , lc06007 ) . increasing the number of program bits in semi - classical deterministic case one increases the precision , increasing the number of program qubits in quantum probabilistic case one increases the probability of success ( see also ref .
we have successfully used a fast electronic feed forward to increase the success probability of a linear optical implementation of a programmable phase gate from 25% to its theoretical limit of 50% . 
 the feed forward applies a conditional unitary operation which changes the incorrect output states of the data qubit to the correct ones . 
 the gate itself rotates an arbitrary quantum state of the data qubit around the @xmath0-axis of the bloch sphere with the angle of rotation being fully determined by the state of the program qubit . 
 the gate implementation is based on fiber optics components . 
 qubits are encoded into spatial modes of single photons . 
 the signal from the feed - forward detector is led directly to a phase modulator using only a passive voltage divider . 
 we have verified the increase of the success probability and characterized the gate operation by means of quantum process tomography . 
 we have demonstrated that the use of the feed forward does not affect either the process fidelity or the output - state fidelities .