Scheme for a linear-optical controlled-phase gate with programmable phase shift
SScheme for a linear-optical controlled-phase gate with programmable phase shift
Karel Lemr, ∗ Karol Bartkiewicz,
2, 1 and Anton´ın ˇCernoch RCPTM, Joint Laboratory of Optics of Palack´y University andInstitute of Physics of Academy of Sciences of the Czech Republic,17. listopadu 12, 771 46 Olomouc, Czech Republic Faculty of Physics, Adam Mickiewicz University, PL-61-614 Pozna´n, Poland (Dated: November 7, 2018)We present a linear-optical scheme for a controlled-phase gate with tunable phase shift pro-grammed by a qubit state. In contrast to all previous tunable controlled-phase gates, the phaseshift is not hard-coded into the optical setup, but can be tuned to any value from 0 to π by thestate of a so-called program qubit. Our setup is feasible with current level of technology usingonly linear-optical components. We provide an experimental feasibility study to assess the gate’simplementability. We also discuss options for increasing the success probability up to 1/12 whichapproaches the success probability of a optimal non-programmable tunable controlled-phase gate. PACS numbers: 42.50.-p,42.79.Sz
I. INTRODUCTION
Quantum computing is a promising approach allow-ing, in principle, considerably increasing computing effi-ciency [1, 2]. It has been demonstrated that any quan-tum circuit can be decomposed into a set of standardsingle and two-qubit gates [3]. While the single-qubitgates represent just single qubit rotations, the two-qubitgates make the qubits interact and thus process the in-formation. A prominent example of such a two-qubitgate is the controlled-phase gate (or its close relative thecontrolled-NOT gate) [4].The controlled-phase gate performs the followingtransformation on the target and control qubit states: | (cid:105) → | (cid:105) , | (cid:105) → | (cid:105) , | (cid:105) → | (cid:105) , | (cid:105) → e iϕ | (cid:105) , (1)where 0 and 1 in the brackets stand for logical states ofthe target and control qubits respectively. The parameter ϕ then denotes the introduced phase shift. There havebeen a number of experimental implementations of thecontrolled-phase gate achieved on various physical plat-forms including nuclear magnetic resonance [5], trappedions [6] or superconducting qubits [7]. On the platform oflinear optics, this gate has been implemented using var-ious schemes [8–10] (for review papers see also [11, 12]).All these implementations however only considered phaseshift ϕ = π also known as the controlled-sign transfor-mation.Operating the controlled-phase gate at phase shiftsother then π has been investigated for the first time ina seminal paper by Lanyon et al. from 2009 [13]. In or-der to achieve divers phase shifts, the authors increased ∗ Electronic address: [email protected]
FIG. 1: Conceptual scheme of a programmable c-phase gate.“T”, “C” and “P” denote target, control and program portsrespectively. The phase shift ϕ encoded into the state of theprogram qubit translates into the phase shift introduced bythe gate according to the Eq. (1). the Hilbert space by introducing axillary modes. Theirimplementation however does not have optimal successprobability. In 2010, Konrad Kieling and his colleaguesproposed a scheme for optimal linear-optical c-phase gatewith tunable phase shift [14]. In 2011, this scheme hasbeen experimentally implemented and tested in our lab-oratory [15]. Both our [15] and the Lanyon et al. [13]scheme have the phase-shift hard-coded by the specificsetting of various optical elements. This fact limits thegates in their adaptability and use in multi-purpose quan-tum circuits.In order to make quantum circuits more versatile,researchers have proposed the so-called programmablegates [16]. Instead of hard-coding the transformationsinto the experimental setup, these gates have their prop-erties programmed by quantum state of the so-called pro-gram qubit. While it would be necessary to use infiniteamount of classical information, to precisely set a real-valued parameter of a quantum gate (or transformation),one qubit of quantum information suffices. Such qubitcan also be transmitted over a quantum channel thus al-lowing for remote programming of a quantum gate sim-ilar to classical software distribution over computer net-works. The quantum transformation in question is basi- a r X i v : . [ qu a n t - ph ] J un cally teleported to its user.As a proof-of-principle, Miˇcuda et al. have constructeda programmable phase gate [17]. This gate introduces aprogrammable phase shift between logical states | (cid:105) and | (cid:105) of a signal qubit and thus it achieves programmablesingle-qubit rotation along one axis. The success proba-bility of this scheme has been recently improved to thetheoretical limit of 1 / | ψ T (cid:105) , | ψ C (cid:105) and | ψ P (cid:105) denote the target, control and program qubitsrespectively. The program qubit takes the form of | ψ P (cid:105) = 1 √ (cid:0) | (cid:105) − e iϕ | (cid:105) (cid:1) , (2)where ϕ is the phase shift to be introduced by the gateif both the target and control qubits are in the logicalstate | (cid:105) as requested by the gate’s definition [see Eq.(1)]. The program qubit is destroyed by detection in theprocess while the target and control qubits leave the gateand can be used for further processing.The paper is organized as follows: in Sec. II we de-rive the basic functioning of the gate and in Sec. III weshow what techniques can be used to increase the successprobability of the scheme. Subsequently in Sec. IV wediscuss the scheme’s experimental feasibility. II. LINEAR-OPTICAL SCHEME
Linear-optical scheme for a c-phase gate with pro-grammable phase shift is depicted in Fig. 2. In thisscheme we consider encoding logical states | (cid:105) and | (cid:105) into horizontal (H) and vertical (V) polarization states ofindividual photons. Similarly to the Lanyon et al. gate[13], we also introduce an auxiliary mode. In our casehowever, we use this mode for interaction between thetarget and program qubits. In this section, we derive theprinciple of operation by showing what transformationthe gate implements on all four basis states as defined inEq. (1) having simultaneously the phase shift ϕ encodedin the state of the program qubit (2). The gate is nec-essarily probabilistic (all linear-optical c-phase gates are[14]) and its successful operation is heralded by observ-ing one photon at each of the target and control outputport and also by detecting a photon on detector D. Let us FIG. 2: Linear-optical setup for the c-phase gate with pro-grammable phase shift. The target, control and programqubit enter the setup at T IN , C IN and P IN respectively whilethe target and control output are denoted T OUT and C
OUT .The program qubit is detected by polarization sensitive de-tector D projecting it onto diagonally polarized state. Polar-izing beam splitters PBS x ( x = 1 , ,
3) transmit horizontallypolarized photons while reflecting vertical polarization. Thepartially polarizing beam splitter PPBS has unit transmis-sivity for horizontal polarization and t V = 1 / √ r V = (cid:112) / is a neutral density filter withamplitude transmissivity t F1 = while the filter F only fil-ters horizontal polarization with transmissivity t F2H = 1 / √ @ − . and HWP @22.5 deg. The gate succeeds if one photon leaves by the targetoutput port, one photon by the control output port and onephoton is detected by detector D. start with the evaluation of the state | ψ P (cid:105) (we maintainthe order of qubits: target, control and program). Thetarget photon impinges on the polarizing beam splitterPBS that sends it to the upper path. There the targetphoton is subjected to a neutral density filter F withamplitude transmissivity t F = and subsequently con-tinues to the target output port by passing through thesecond polarizing beam splitter PBS . So far, one canwrite down the transformation of the state as | ψ P (cid:105) → | ψ P (cid:105) . Meanwhile the control photon is transmitted by the par-tially polarizing beam splitter PPBS (with transmissivity t H = 1 for horizontal polarization and t V = (cid:112) − r V = √ for vertical polarization) and after being subjected topolarization filtering by the filter F (filtering horizontalpolarization with transmissivity t F H = √ and lettingthe vertical polarization unfiltered t F V = 1) it leavesthe setup by the control output port. At this point thetransformation by the gate reads | ψ P (cid:105) → √ | ψ P (cid:105) . Finally the program photon impinges on the polarizingbeam splitter PBS , where it gets transmitted and re-flected with equal amplitude 1 / √
2. Since the gate suc-ceeds only if a photon is detected on detector D, only thetransmission of the program photon through PBS has tobe taken into account. Considering the program photonis in the state (2), the overall state gets transformed into | ψ P (cid:105) → √ | (cid:105) . Once the program photon leaves PBS , we project it ontodiagonal polarization | D (cid:105) = √ ( | (cid:105) + | (cid:105) ) resulting inthe final form of the transformation | ψ P (cid:105) → √ | D (cid:105) . This projection is needed to erase the which-path infor-mation about the photon detected on detector D.In the same way, we now evaluate the transformationof the second state | ψ P (cid:105) . The only difference this timeis in the control qubit. It impinges on the PPBS havingvertical polarization and is therefore transmitted withamplitude √ and reflected with amplitude (cid:113) . Onlythe transmission of the control photon by the PPBS con-tributes to the successful operation of the gate. Furtherto that, no attenuation of the vertically polarized controlphoton takes place on F . Having the same transforma-tion for the target and program qubits as in the previousparagraph, one can identify the overall action of the gate | ψ P (cid:105) → √ | D (cid:105) . A different situation happens for the third state | ψ P (cid:105) . The target photon is reflected by PBS enter-ing the lower path, where it is subjected to a half-waveplate HWP oriented by − . | (cid:105) → | (cid:105) + √ | (cid:105) (3)and thus causes the overall state to get transformed into | ψ P (cid:105) → | ψ P (cid:105) + √ | ψ P (cid:105) . At this point the target and control photons interactson the PPBS. Since the control photon is transmittedthrough the PPBS (having horizontal polarization in thiscase), we only take into account the transmission of thetarget photon to assure successful outcome of the gate | ψ P (cid:105) → √ | ψ P (cid:105) + 12 √ | ψ P (cid:105) , where we have already incorporated the action of the po-larization sensitive filter F . Now the target state entersa Hadamard transform implemented by a half-wave plate HWP rotated by 22.5 degrees with respect to horizon-tal polarization providing target photon transformationof the form of | (cid:105) → √ | (cid:105) + | (cid:105) ) (4) | (cid:105) → √ | (cid:105) − | (cid:105) ) , which then translates into the overall state evolution | ψ P (cid:105) → √ | ψ P (cid:105) . The target photon passes through PBS having horizon-tal polarization. Therefore identically to the cases de-rived above, the gate can only succeed if the programphoton passes through the PBS and then gets projectedonto diagonal polarization. Thus we obtain the transfor-mation in the form of | ψ P (cid:105) → √ | D (cid:105) . Finally, the target photon is again subjected to aHadamard transform (HWP ) resulting in | ψ P (cid:105) → √ | D (cid:105) + | D (cid:105) ) . Only the target photon reflected by the PBS leaves thegate by designated output port and thus we obtain thefinal form of the transformation | ψ P (cid:105) → √ | D (cid:105) . To complete our analysis, we now evaluate the trans-formation of the last basis state | ψ P (cid:105) . Similarly to theprevious case, the target photon gets reflected on PBS and transformed by the half-wave plate HWP accordingto prescription (3). The overall state thus takes the formof | ψ P (cid:105) → | ψ P (cid:105) + √ | ψ P (cid:105) . At this point two-photon interference on the PPBS takesplace resulting in | ψ P (cid:105) → √ | ψ P (cid:105) + | ψ P (cid:105) − | ψ P (cid:105) )= 12 √ | ψ P (cid:105) − | ψ P (cid:105) ) , (5)where only the terms contributing to success of the gateare shown. Note that when both the target and controlphotons enter the PPBS in vertical polarization state,the interference of both the photons being transmittedand both the photons being reflected (Hong-Ou-Mandelinterference) occurs introducing a phase shift π to theterm | ψ P (cid:105) [10]. By means of the subsequent Hadamardtransform in the target mode (HWP ), the state trans-forms into | ψ P (cid:105) → √ | ψ P (cid:105) . On PBS the target photon gets reflected (being verti-cally polarized) and so only the program photon reflec-tion can contribute to the gate’s successful operation.This means that just its vertical polarization term con-tributes yielding the overall state in the form of | ψ P (cid:105) → − e iϕ √ | (cid:105) which after projecting the photon in program mode ontodiagonal polarization gives | ψ P (cid:105) → − e iϕ √ | D (cid:105) . Action of the Hadamard gate in the target mode (HWP )and reflection of the target photon on PBS to its outputport results in the final transformation | ψ P (cid:105) → e iϕ √ | D (cid:105) . In contrast to the three previous cases, the state is nowphase-shifted by angle ϕ exactly as prescribed in (1).We have shown that the setup depicted in Fig. 2 imple-ments the tunable c-phase gate with the phase shift pro-grammed by the program qubit. This has been demon-strated on all four basis states of the control and tar-get qubits together with an arbitrary program state.Since all transformations are linear, the entire operationholds also for any superposition of the above mentionedfour basis states. The success probability of the gate is1 / (4 √ = and is state independent (in order notto deform superpositions of basis states). In the nextsection, we will consider potential improvements to thesetup in order to increase the success probability. III. INCREASING THE SUCCESSPROBABILITY
So far we have discussed the basic scheme for the pro-grammable c-phase gate which is experimentally also themost easily implementable. Its success probability ishowever more then four times lower then the successprobability of the optimal non-programmable c-phasegate. In this section we will discuss two optimizationapproaches allowing for considerable improvement in suc-cess probability (see optimized scheme in Fig. 3).Firstly, one can increase the success probability of theprogram photon projection implemented before its de-tection. As derived in the previous section, the gatesucceeds if the program photon is projected onto di-agonal polarization and detected by D. This way, we
FIG. 3: Optimized setup for the programmable c-phase gate.The components are designated as in Fig. 2 with the newlyadded half-wave plates HWP (rotated by 22.5 deg.) andHWP (rotated by 45 deg.) and a phase modulator PLMimplementing a feed forward operation. neglect half of the cases corresponding to the programphoton being projected onto anti-diagonal polarization[ | A (cid:105) = √ ( | (cid:105) − | (cid:105) )]. As experimentally demonstratedon a simpler unconditional programmable gate [18], onecan increase the success probability by a factor of two ifthe anti-diagonal projections of the program photon areincluded. In such cases a feed-forward transformation | (cid:105) → −| (cid:105) has to by applied to the target photon im-mediately as it exits PBS [24]. Such transformation canbe achieved using for instance a phase modulator (PLM)[18].The second possibility to increase the overall successprobability again by a factor of two is to use both theoutput ports of PBS (designated T OUT1 and T
OUT2 inFig. 3). In this case, the amplitude transmissivity of thefilter F shall be reduced to t F1 = √ and a half-waveplate HWP inserted behind it. This newly added wave-plate is rotated by 22.5 degrees with respect to horizontalpolarization to implement the Hadamard transform (4).The target state at T OUT1 is thus kept unchanged, butit allows for the target photon to leave also by the out-put port T
OUT2 . The target photon exiting PBS byits second output is however polarization-swapped withrespect to the target photon in the first output. A half-wave plate HWP (rotated by 45 degrees) is thereforeinserted to the output port T OUT2 to perform the swapoperation ( | (cid:105) → | (cid:105) , | (cid:105) → | (cid:105) ). The second output portcan be used only if one does not require the target photonto leave by a specified output. Such situation occurs forinstance if the target qubit is immediately measured afterbeing processed by the gate (the gate is the last elementin a quantum circuit). The measurement apparatus canthen be installed to both output ports.If only one of the mentioned optimization strategies isused, the success probability of the gate increases to 1/24and if both of them are used it reaches the value of 1/12(for all phases ϕ ). Note that optimal non-programmablec-phase gate has the minimum success probability beingabout 1/11 for ϕ close to π [15]. If completely opti-mized, the programmable gate thus performs with almostthe same probability as the optimal non-programmablegate at its success probability minimum. IV. EXPERIMENTAL FEASIBILITY
In this section, we discuss the feasibility of the pro-posed scheme based on the current level of technologicaldevelopment in linear-optical quantum information pro-cessing with discrete photons. Firstly, in order to achieveany linear-optical quantum gate, one needs to generateadequate input photon state. Our gate requires threephotons each bearing one polarization-encoded qubit.Generation of three separate photons has already beenachieved in various experiments. Either one photon pairfrom spontaneous parametric down-conversion (SPDC)is combined with one additional photon from attenuatedfundamental laser beam [25] or two photon pairs are gen-erated via SPDC with one photon serving just as a trig-ger [20]. With either of these techniques one can generateinput states with sufficient fidelity, typically more than95%, with repetition rate at about 1 per 5 s [26].In the next step, we asses the feasibility of stabilizingthe proposed scheme. There are two types of stabiliza-tions required: two-photon temporal and spatial over-lap stabilization (Hong-Ou Madel interference [27]) andsingle-photon interference stabilization. The first men-tioned has to be achieved with precision of about 1/50of the photons wave packet length. Such wave packet istypically 100 µ m long (FWHM) in space and thereforestability with precision in units of µ m suffices. One canuse motorized translation to achieve this task. From ourexperience, the two-photon overlap is stable for aboutone hour [15]. After that, one needs to perform two-photon interference measurement and observe the Hong-Ou-Mandel dip to reset the actual position of maximaltwo-photon overlap. Since two-fold coincidences occurmuch more frequently then three-fold or four-fold ones,this restabilization measurement, when optimized, cantake no more that a minute. In our scheme, we requireto stabilize the two-photon overlap on the PPBS and onPBS . Note that the stabilization procedures can be per-formed separately and consecutively and hence the needfor two of them does not pose significant technologicaldifficulties. There is also one instance of single-photoninterference occurring in the setup between the polariz-ing beam splitters PBS and PBS . In contrast to thetwo-photon overlap, the single-photon interference needsto be stabilized typically to at least λ/
50, where λ is thephoton’s wavelength. Such precision requires combiningboth motorized translation for larger steps with piezotranslation for fine adjustments. In a typical bulk inter-ferometer on decimeter scale, the single-photon stability only lasts for less than a minute [15]. One can howeversignificantly increase such time by replacing classical in-terferometer by a compact design using beam dividers[28]. In general, there are two ways of how to perform sta-bilization of this kind. One option is to use the individualphotons themselves and perform a set of measurementsfor various settings of the piezo shift. Note that single-photon detections occur even with much higher rate thattwo-fold coincident detections and therefore there is usu-ally fair amount of signal to work on [15, 29]. Otherexperimentalists prefer using a strong laser beam propa-gating along the weak quantum signal. This strong sig-nal is at sufficiently distant wavelength to allow mixingand subsequent decoupling from the quantum signal us-ing dichroic filters. In this case the stabilization can beperformed “on-line” during the entire experiment [30].Using either of the above described techniques, it is quitefeasible to stabilize the single-photon interference in theproposed setup.Finally, the feasibility considerations have to be ded-icated to the final detection procedure. In order to beexperimentally implementable, the detection has to berobust against non-unit quantum detection efficiency oftypical detectors and technological losses (e.g. back-reflection, coupling efficiency). This requirement rulesout vacuum detection-based schemes (schemes where suc-cess is heralded by vacuum detection or no-detection)[31]. Similarly, photon-number resolving detection is notcompletely reliable because of detection (in)efficiency [32]and technological losses. In our case however, the schemeonly requires post-selection on three-fold coincidence de-tections and thus the quantum efficiency of the detectorsonly affects the detection rate and not the detected quan-tum state. This is a key feature of the proposed schemewith respect to its feasibility. V. CONCLUSIONS
In this paper, we have provided a linear-optical schemefor a programmable c-phase gate. The phase shift in-troduced by this gate is set by the state of a programqubit which makes the gate more versatile than previ-ously implemented tunable c-phase gates with phase shifthard-coded to the setup setting. The setup is designedwith experimental feasibility in mind. It does not requirephoton-number resolving detectors nor post-selection onvacuum detection and is therefore implementable withcurrent level of know-how in experimental linear-opticalquantum information processing.We have also presented two optimization options. Eachof them allows doubling the overall success probability ofthe gate. Using both these optimizations, the gate suc-ceeds with probability of 1/12 which is close to the suc-cess probability of a non-programmable tunable c-phasegate [15]. Further the two optimization steps can be usedindependently allowing to obtain success probability of1/24 if only one of them is used. The first optimiza-tion method consists of applying an experimentally fea-sible feed-forward operation. The second optimizationinvolves using both output ports of the final polarizingbeam splitter in the target mode.
Acknowledgment
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