Title Demonstration of an Otical Quantum Controlled-NOT G Author(s)Okamoto, Ryo; Hofmann, Holger F.; Takeuchi, Shigeki; CitationPhysical Review Letters, 95: 210506 Issue Date 2005-11-18 Doc URL htt://hdl.handle.net/2115/5550 Rights Coyright 2005 American Physical Society Tye article File Information PRL95-21.df Instructions for use Hokkaido University Collection of Scholarly and Aca
Demonstration of an Otical Quantum Controlled-NOT Gate without Path Interference Ryo Okamoto, 1 Holger F. Hofmann, 2 Shigeki Takeuchi, 1 and Keiji Sasaki 1 1 Research Institute for Electronic Science, Hokkaido University, Saoro 060-0812, Jaan 2 Graduate School of Advanced Sciences of Matter, Hiroshima University, Hiroshima 739-8530, Jaan (Received 30 June 2005; ublished 18 November 2005) We reort the first exerimental demonstration of an otical quantum controlled-not gate without any ath interference, where the two interacting ath interferometers of the original roosals [Phys. Rev. A 66, 024308 (2002); 65, 062324 (2002)] have been relaced by three artially olarizing beam slitters with suitable olarization deendent transmittances and reflectances. The erformance of the device is evaluated using a recently roosed method [Phys. Rev. Lett. 94, 160504 (2005)], by which the quantum rocess fidelity and the entanglement caability can be estimated from the 32 measurement results of two classical truth tables, significantly less than the 256 measurement results required for full quantum tomograhy. DOI: 10.1103/PhysRevLett.95.210506 PACS numbers: 03.67.Lx, 03.65.Yz, 03.67.Mn, 42.50.Ar Quantum comuting romises to solve roblems such as factoring large integers [1] and searching over a large database [2] efficiently. One of the greatest challenges is to imlement the basic elements of quantum comutation in a reliable hysical system and to evaluate the erformance of the oeration in a sufficient manner. In one of the earliest roosals for imlementing quantum comutation [3], each qubit was encoded in a single hoton existing in two otical modes. The main advantage of the hotonic imlementation of qubits is the robustness against decoherence and the availability of one-qubit oerations. However, the difficulty of realizing the nonlinear interactions between hotons that are needed for the imlementation of two-qubit oerations has been a major obstacle. In recent work, Knill, Laflamme, and Milburn [4] have shown that this obstacle can be overcome by using linear otics, single hoton sources, and hoton number detectors. By now, various controlled-not (CNOT) gates for hotonic qubits using linear otics have been roosed [5 10] and demonstrated [11 16]. In articular, it has been shown in Refs. [7,8] that a comact CNOT gate can be realized by interaction at a single beam slitter and ostselection of the outut. Since this gate requires no ancillary hoton inuts or additional detectors, it should be esecially useful for exerimental realizations of otical quantum circuits [17]. However, there have been two crucial difficulties. In the original scheme [7,8], the olarization sensitivity of the oeration was achieved by searating the aths of the orthogonal olarizations, essentially creating two interacting two-ath interferometers. Therefore, the initial exerimental realizations based the original roosal [14,15] are very sensitive to the noisy environment (thermal drifts and vibrations), making it necessary to control and to stabilize nanometer order ath-length differences. In addition to these roblems, erfect mode matching is required in each outut of the interferometer. Thus, it is very difficult to construct quantum circuits using devices based on those exerimental setus. Another difficulty is the evaluation of exerimental errors in multiqubit gates. In order to obtain the most comlete evaluation of gate erformance ossible, quantum rocess tomograhy has been used in the revious exeriment [15]. However, 256 different measurement setus are required to evaluate only one CNOT device. When we have to evaluate even more comlicated quantum devices realized by a combination of gates, the number of measurements required for tomograhy raidly increases as the number of inut and outut qubits increases. In this Letter, we resent an exerimental realization of the comact otical CNOT gate [7,8] without any ath interference. We show that the CNOT gate can be imlemented using three artially olarizing beam slitters (PPBSs) with suitable olarization deendent transmittances and reflectances, where the essential interaction is realized by a single intrinsic PPBS, while the other two sulemental PPBSs act as local olarization comensators. The gate oeration can then be obtained directly from the olarization deendence of the reflectances of the PPBSs, removing the need for interference between different aths for orthogonal olarizations. We have evaluated the device oeration using a recently roosed method [18], by which we can determine the lower and uer bounds of the rocess fidelity from measurements of only 32 inut-outut combinations. We can thus characterize the gate oeration with 1=8 the number of inut-outut measurements required for comlete quantum rocess tomograhy. We hoe that these results will oen a door to the realization of more comlex quantum circuits for quantum comuting. Figure 1(a) shows the reviously roosed otical circuit for the CNOT gate [7,8]. The beam slitter sitting in the center of the circuit is the essential one which realizes the quantum hase gate oeration by fliing the hase of the state jh; Hi, where both hotons are horizontally olarized, to jh; Hi due to two-hoton interference. Since this oeration attenuates the amlitudes of horizontally olarized comonents by a factor of 1= 3, the other two beam slitters with reflectivity 1=3 are inserted in each of the 0031-9007=05=95(21)=210506(4)$23.00 210506-1 2005 The American Physical Society
FIG. 1 (color online). Schematics of the comact CNOT gate. (a) The otical circuit in the original roosals [7,8]. Here the olarizing beam slitters (PBSs) reflect (transmit) hotons with vertical (horizontal) olarization. (b) The otical circuit without any ath interference using artially olarizing beam slitters. Note that PPBS-Bs can be laced either before or after the PPBS-A. interferometer aths in order to also attenuate the amlitudes of vertically olarized comonents jvi, so that the total amlitude of any two-hoton inut is uniformly attenuated to 1=3. If we now define the comutational basis of the gate as j0 z i C jvi C, j1 z i C jhi C for the control qubit and j0 z i T 1= 2 jvit jhi T, j1 z i T 1= 2 jvit jhi T for the target qubit, the gate erforms the unitary oeration ^U CNOT of the quantum CNOT on the inut qubits. The difficulty in the original roosal is that we have to stabilize the two interferometers of the horizontally and vertically olarized aths by controlling the length of four otical aths with an accuracy on the order of nanometers in order to achieve reliable oeration of the device. In addition, the modes in each ath have to be aligned recisely at the outut orts of the olarizing beam slitters. Such difficulties have been crucial obstacles for the future realization of otical quantum circuits consisting of several CNOT gates. Figure 1(b) shows our solution to this roblem. We use one intrinsic PPBS (PPBS-A) and two sulemental PPBSs (PPBS-B) in the otical circuit. The intrinsic PPBS-A, which corresonds to the central beam slitter in Fig. 1(a), imlements the quantum hase gate oeration by reflecting vertically olarized light erfectly and reflecting (transmitting) 1=3 2=3 of horizontally olarized light. The two sulemental PPBS-Bs are inserted to adjust the amlitudes of the local horizontal and vertical comonents of the hotonic qubits by transmitting (reflecting) 1=3 2=3 of vertically olarized light and transmitting horizontally olarized light erfectly. As Fig. 1(b) shows, the use of PPBSs allows us to reduce the four otical aths in Fig. 1(a) to only two otical aths, and ath interferometers are no longer required for the imlementation of the comact quantum CNOT gate. In the following exerimental demonstration, we used a simle olarization comensation instead of the two sulemental PPBS-Bs. The only urose of the PPBS-Bsisto reduce the amlitude of the vertical comonent in the inut to 1= 3 of the original inut value, while leaving the 210506-2 horizontal comonent unchanged. Therefore, we can easily simulate the function of the sulemental PPBS-Bs by using comensated inut states whose vertical comonent is reduced to 1= 3. To simulate a general inut state j effective i c H jhi c V jvi, we thus use a comensated inut state of j com: i c H jhi c V = 3 jvi. For examle, the inut for the target qubit state j0 z i T becomes j0 0 zi T 3 jhi jvi = 6, which can be easily reared just by changing the angle of linear olarization by rotating the half-wave late (HWP) in the target inut. The schematic of our exerimental setu is shown in Fig. 2. We used a air of hotons generated through sontaneous arametric down-conversion for our inut. A beta barium borate (BBO) crystal cut for the tye II twin-beam condition [19] was umed by an argon ion laser at a wavelength of 351.1 nm. The um beam was focused in the BBO crystal using a convex lens to increase the hoton flux [20]. Pairs of twin hotons are emitted at 702.2 nm with orthogonal olarizations. Glan-Thomson olarizers are used to increase the extinction ratio. After removing the scattered um light using bandass filters (IF, center wavelength 702.2 nm, FWHM 0.3 nm), each of the hotons was guided to olarization maintaining single-mode fibers (PMFs) via objective lens and then delivered to the CNOT verification setu. After the collimation lens for the outut of the PMFs, the olarization of hotons are controlled by HWPs. The timing of the two hotons injected to the PPBS-A was controlled using an otical delay. A quarterwave late was inserted to comensate the hase change between horizontal and vertical olarization in the otical delay. After the quantum interference which occurs at PPBS-A, a dielectric mirror secially made to order, the olarization of outut hotons was analyzed using HWPs and PBSs. Finally, those hotons are couled into singlemode fibers and counted by the single hoton counters (SPCM-AQ-FC, Perkin Elmer). Our setu ermits us to select various linear olarizations for the inut and to detect the corresonding linear olarizations in the outut. As has been shown in Ref. [18], it is ossible to characterize the essential quantum roerties of the gate oeration by using the comutational ZZ basis given above and the comlementary linearly olarized XX basis given by j0 x i C 1= 2 jvic jhi C, j1 x i C 1= 2 jvic jhi C for the control qubit and FIG. 2 (color online). Exerimental setu for the demonstration of the CNOT gate without any ath interference. M is for reflecting mirrors.
TABLE I. Measurement results for the inut-outut robabilities of the CNOT oeration in the ZZ basis (a) and the reverse CNOT oeration in the XX basis (b). (a) h0 z 0 z j h0 z 1 z j h1 z 0 z j h1 z 1 z j j0 z 0 z i 0.898 0.031 0.061 0.011 j0 z 1 z i 0.021 0.885 0.006 0.088 j1 z 0 z i 0.064 0.027 0.099 0.810 j1 z 1 z i 0.031 0.096 0.819 0.054 (b) h0 x 0 x j h0 x 1 x j h1 x 0 x j h1 x 1 x j j0 x 0 x i 0.854 0.044 0.063 0.039 j0 x 1 x i 0.013 0.099 0.013 0.874 j1 x 0 x i 0.050 0.021 0.871 0.058 j1 x 1 x i 0.019 0.870 0.040 0.071 j0 x i T jvi T, j1 x i T jhi T for the target qubit. The oeration of the gate on this inut basis also corresonds to a CNOT, with the target qubit acting on the control qubit (reverse CNOT). The measurement result of the inut-outut robabilities of our CNOT gate in the ZZ basis and in the XX basis are shown in Table I(a) and (b), resectively. We measured the coincidence counts between two SPCMs by aroriately setting the HWPs for 16 different combinations of inut and outut states. In order to convert the coincidence rates to robabilities, we normalize them with the sum of coincidence counts obtained for the resective inut state. The three dimensional bar grahs of Table I are shown in Fig. 3. The fidelity F zz of the CNOT oeration in the ZZ basis, defined as the robability of obtaining the correct outut averaged over all four ossible inuts, is 0.85. Similarly, the fidelity F xx of the reverse CNOT oeration in the XX basis is 0.87. As discussed in detail in Ref. [18], the two comlementary fidelities F zz and F xx define an uer and a lower bound for the quantum rocess fidelity F rocess of the gate with F zz F xx 1 F rocess minff zz ;F xx g: (1) Thus, our exerimental results show that the rocess fidelity of our exerimental quantum CNOT gate is 0:72 F rocess 0:85: (2) The lower bound of the rocess fidelity also defines a lower bound of the entanglement caability of the gate, since the fidelity of entanglement generation is at least equal to the rocess fidelity. In terms of the concurrence C that the gate can generate from roduct state inuts, the minimal entanglement caability is, therefore, given by C 2F rocess 1 [18]. Since our exerimental results show that the minimal rocess fidelity of the gate is 0.72, the lower bound of the entanglement caability is C 0:44: (3) The exerimental results shown in Table I are therefore sufficient to confirm the entanglement caability of our gate. In order to gain a better understanding of the noise effects in our exerimental quantum gate, we can analyze the errors in the classical oerations shown in Table I and associate them with quantum errors reresented by elements of the rocess matrix. For this urose, it is useful to classify the errors according to the bit fli errors in the outut of the oerations in the ZZ and the XX basis, using 0 for the correct gate oeration, C for a fli of the control bit outut, T for a fli of the target bit outut, and B for a fli of both oututs. It is then ossible to exand the rocess matrix in terms of 16 orthogonal unitary oerations ^U i, where the index i f00; C0; T0; B0; 0C; CC; TC; BC; 0T; CT; TT; BT; 0B; CB; TB; BBg defines the air of error syndromes in the comlementary oerations in the ZZ and the XX basis, e.g., ^U TC for a target fli error in the ZZ oeration and a control fli error in the XX oeration and ^U 00 ^U CNOT for the ideal gate oeration. The rocess matrix describing the noisy gate oeration is given by the oerator sum reresentation of the relation between an arbitrary inut density matrix ^ in and its outut density matrix ^ out, ^ out X i;j i;j ^U i ^ in ^U y j : (4) Since each oeration ^U i describes a well defined combination of errors in the ZZ and XX oerations, it is now ossible to relate the error robabilities observed in Table I to sums over the diagonal elements i;i of the rocess matrix, as shown in Table II. Since the correlations be- FIG. 3 (color online). (a) Bar grah of the exerimental results for the CNOT oeration in the ZZ basis. (b) Bar grah of the exerimental results for the reverse CNOT oeration in the XX basis. 210506-3 TABLE II. Relation between exerimentally observed errors and rocess matrix elements. (X 0; C; T; B). i;i X0 XC XT XB Sum 0X 00;00 0C;0C 0T;0T 0B;0B 0.853 CX C0;C0 CC;CC CT;CT CB;CB 0.052 TX T0;T0 TC;TC TT;TT TB;TB 0.051 BX B0;B0 BC;BC BT;BT BB;BB 0.044
TABLE III. Diagonal elements of the rocess matrix for (a) the worst case of rocess fidelity 0.72 and (b) the otimal case of rocess fidelity 0.85. (a) X0 XC XT XB Sum 0X 0.720 0.071 0.034 0.028 0.853 CX 0.052 0.000 0.000 0.000 0.052 TX 0.051 0.000 0.000 0.000 0.051 BX 0.044 0.000 0.000 0.000 0.044 (b) X0 XC XT XB Sum 0X 0.853 0.000 0.000 0.000 0.853 CX 0.005 0.025 0.012 0.010 0.052 TX 0.005 0.024 0.012 0.010 0.051 BX 0.004 0.022 0.010 0.008 0.044 tween errors in ZZ and errors in XX are unknown, a range of different distributions of diagonal elements i;i is consistent with the exerimental data. It is ossible to illustrate this range of ossibilities by considering the scenarios of lowest and highest rocess fidelity. The matrix elements of these cases are shown in Table III(a) and (b). In the worst case scenario of a rocess fidelity of 0.72 shown in Table III(a), each error syndrome is observed only in either the ZZ or the XX basis. Therefore, the errors observed in the ZZ oeration can be identified directly with i C0; T0; B0, and the errors observed in the XX oeration can be identified with i 0C; 0T; 0B. In the otimal case of a rocess fidelity of 0.85 shown in Table III(b), the diagonal elements of i C0; T0; B0 for errors only in ZZ are all zero. The remaining errors have been distributed over the other diagonal matrix elements within the constraints given by Table II. Note that the distribution of matrix elements in the otimal case is far more homogeneous than the worst case scenario. The actual rocess fidelity is, therefore, likely to be closer to the uer limit of 0.85 than to the lower limit of 0.72. In conclusion, we have demonstrated the first exerimental realization of an otical quantum CNOT gate without any ath interference by measuring the fidelities of the classical CNOT oerations in the comutational ZZ basis and in the comlementary XX basis. The erformance of both oerations by the same quantum gate at fidelities of 0.85 and 0.87 indicates that our device has a quantum rocess fidelity of 0:72 F rocess 0:85 and an entanglement caability of C 0:44. Since the gate resented in this Letter requires no ath-length adjustments, it should be ideal for the construction of quantum circuits using multile gates. The resent work may, therefore, rovide an imortant first ste towards the realization of otical quantum comutation with larger numbers of qubits. We thank H. Fujiwara, K. Tsujino, and D. Kawase for technical suggestions. This work was suorted in art by Core Research for Evolutional Science and Technology, Jaan Science and Technology Agency, Grant-in-Aid of Jaan Science Promotion Society and 21st century COE rogram. [1] P. W. Shor, SIAM J. Comut. 26, 1484 (1997). [2] L. K. Grover, Phys. Rev. Lett. 79, 325 (1997). [3] G. J. Milburn, Phys. Rev. Lett. 62, 2124 (1989). [4] E. Knill, R. Laflamme, and G. J. Milburn, Nature (London) 409, 46 (2001). [5] M. Koashi, T. Yamamoto, and N. Imoto, Phys. Rev. A 63, 030301(R) (2001). [6] T. B. Pittman, B. C. Jacobs, and J. D. Franson, Phys. Rev. A 64, 062311 (2001). [7] H. F. Hofmann and S. Takeuchi, Phys. Rev. A 66, 024308 (2002). [8] T. C. Ralh, N. K. Langford, T. B. Bell, and A. G. White, Phys. Rev. A 65, 062324 (2002). [9] T. C. Ralh, A. G. White, W. J. Munro, and G. J. Milburn, Phys. Rev. A 65, 012314 (2002). [10] M. A. Nielsen, Phys. Rev. Lett. 93, 040503 (2004). [11] T. B. Pittman, B. C. Jacobs, and J. D. Franson, Phys. Rev. Lett. 88, 257902 (2002). [12] K. Sanaka, K. Kawahara, and T. Kuga, Phys. Rev. A 66, 040301(R) (2002). [13] T. B. Pittman, M. J. Fitch, B. C. Jacobs, and J. D. Franson, Phys. Rev. A 68, 032316 (2003). [14] J. L. O Brien, G. J. Pryde, A. G. White, T. C. Ralh, and D. Branning, Nature (London) 426, 264 (2003). [15] J. L. O Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralh, and A. G. White, Phys. Rev. Lett. 93, 080502 (2004). [16] Z. Zhao, A. N. Zhang, Y. A. Chen, H. Zhang, J. F. Du, T. Yang, and J. W. Pan, Phys. Rev. Lett. 94, 030501 (2005). [17] It should be noted, however, that the architecture of such circuits will be restricted by the ostselection requirement that the two hotons are outut to different orts [7]. To overcome this roblem, it would be useful to develo a new device to filter out the two-hoton comonents in the outut. [18] H. F. Hofmann, Phys. Rev. Lett. 94, 160504 (2005). [19] S. Takeuchi, Ot. Lett. 26, 843 (2001). [20] C. Kurtsiefer, M. Oberarleiter, and H. Weinfurter, Phys. Rev. A 64, 023802 (2001). 210506-4