NOT Photonic Quantum Circuit Combining Effective Optical Nonlinearities

Realization of a Knill-Laflamme-Milburn controlled- NOT photonic quantum circuit combining effective optical nonlinearities

Ryo Okamotoa,b, Jeremy L. O’Brienc, Holger F. Hofmannd, and Shigeki Takeuchia,b,1

aResearch Institute for Electronic Science, Hokkaido University, Sapporo 060-0812, Japan; bThe Institute of Scientific and Industrial Research, Osaka University, 8-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan; cCenter for Quantum Photonics, H. H. Wills Physics Laboratory and Department of Electrical and Electronic Engineering, University of Bristol, Merchant Venturers Building, Woodland Road, Bristol BS8 1UB, United Kingdom; and dGraduate School of Advanced Sciences of Matter, Hiroshima University, Hiroshima 739-8530, Japan

Edited by Alain Aspect, Institut d'Optique, Orsay, France, and approved May 5, 2011 (received for review December 21, 2010)

Quantum information science addresses how uniquely quantum In 2001, KLM made the surprising discovery that a scalable mechanical phenomena such as superposition and entanglement quantum computer could be built from only linear optical can enhance communication, information processing, and precision networks, and single photon sources and detectors (11). In fact, measurement. Photons are appealing for their low-noise, light- it was even surprising to KLM themselves, as they had initially speed transmission and ease of manipulation using conventional intended to prove the opposite. The KLM recipe consists of optical components. However, the lack of highly efficient optical two parts: an optical circuit for a CNOT gate using linear optics, Kerr nonlinearities at the single photon level was a major obstacle. single photon sources (12), and photon number-resolving detec- In a breakthrough, Knill, Laflamme, and Milburn (KLM) showed tors (13); and a scheme (14, 15) for increasing the success prob- P 1∕16 that such an efficient nonlinearity can be achieved using only linear ability of this CNOT gate ( ¼ ) arbitrarily close to unity, optical elements, auxiliary photons, and measurement [Knill E, where the probabilistic CNOT gates generate the entangled Laflamme R, Milburn GJ (2001) Nature 409:46–52]. KLM proposed states used as a resource for the implementation of controlled a heralded controlled-NOT (CNOT) gate for scalable quantum com- unitary operation based on quantum teleportation (16, 17). This putation using a photonic quantum circuit to combine two such discovery opened the door to linear optics quantum computation nonlinear elements. Here we experimentally demonstrate a KLM and has spurred world-wide theoretical and experimental efforts CNOT gate. We developed a stable architecture to realize the to realize such devices (18), as well as new quantum communication schemes (2) and optical quantum metrology (5). Inspired by required four-photon network of nested multiple interferometers the KLM approach, a number of quantum logic gates using her- based on a displaced-Sagnac interferometer and several partially alded photons and event postselection have been proposed and polarizing beamsplitters. This result confirms the first step in the – “ ” demonstrated (19 28). Furthermore, optical quantum circuits original KLM recipe for all-optical quantum computation, and combining these gates have been demonstrated (29–33). In this should be useful for on-demand entanglement generation and pur- context, photonic quantum information processing using linear ification. Optical quantum circuits combining giant optical nonli- optics and postselection is one of the promising candidates in nearities may find wide applications in quantum information the quest for practical quantum information processing (18). processing, communication, and sensing. Knill-Laflamme-Milburn C-NOT Gate nonlinear optics ∣ quantum optics ∣ linear optics ∣ quantum gates Interestingly, none of these gates realized so far (19–28) actually used the original KLM proposal of a simple measurement- everal physical systems are being pursued for quantum com- induced nonlinearity: either the gates are not heralded (the Sputing (1)—promising candidates include trapped ions, resultant output photons themselves have to be measured and neutral atoms, nuclear spins, quantum dots, superconducting destroyed) or rely on additional entanglement effects; as we systems, and photons—while photons are indispensable for quan- explain below, the KLM scheme is based on a direct implementa- tum communication (2, 3) and are particularly promising for tion of the nonlinear sign-shift (NS) gate that relies on the inter- quantum metrology (4, 5). In addition to low-noise quantum action with a single auxiliary photon at a beam splitter (BS). The systems (typically two-level “qubits”) quantum information pro- NS gate is thus based on the efficient optical nonlinearity induced tocols require a means to interact qubits to generate entangle- by single photon sources and detectors. While a measurement-in- ment. The canonical example is the controlled-NOT (CNOT) duced nonlinearity has been verified by a conditional phase shift gate, which flips the state of the polarization of the “target” for one specific input (21), the complete function of a NS gate for COMPUTER SCIENCES photon conditional on the “control” photon being horizontally arbitrary inputs has not been demonstrated. Moreover, it is an polarized (the logical “1” state). The gate is capable of generating important remaining challenge to combine the nonlinearities into maximally entangled two-qubit states, which together with one- a network such as the KLM-CNOT gate, because this requires a qubit rotations provide a universal set of logic gates for quantum more reliable control of optical coherence than a nonlinearity acting on a single beam, especially because nonlinearities tend computation. The low-noise properties of single photon qubits are a result of to couple modes to produce additional and often unexpected noise patterns. Specifically, it is a difficult task to implement the their negligible interaction with the environment, however, the nested interferometers needed to perform the multiple classical fact that they do not readily interact with one-another is proble- matic for the realization of a CNOT or other entangling interaction. Consequently it was widely believed that matter systems, Author contributions: R.O., J.L.O’B., H.F.H., and S.T. designed research; performed such as an atom or atom-like system (6), or an ensemble of such research; contributed new reagents/analytic tools; analyzed data; and wrote the paper. systems (7), would be required to realize such efficient optical The authors declare no conflict of interest. nonlinearities. Indeed the first proposals for using linear optics This article is a PNAS Direct Submission. to benchmark quantum algorithms require exponentially large Freely available online through the PNAS open access option. physical resources (8–10). 1To whom correspondence should be addressed. E-mail: [email protected].

www.pnas.org/cgi/doi/10.1073/pnas.1018839108 PNAS ∣ June 21, 2011 ∣ vol. 108 ∣ no. 25 ∣ 10067–10071 Downloaded by guest on September 25, 2021 1 1 → 2 0 − 0 2 and quantum interferences that form the elements of the quan- photons occurs at BS1: j iC1 j iT0 j iC1 j iT0 j iC1 j iT0 .In tum gate operation, which has prevented the realization of the this case the NS gates each impart a π phase shift to these KLM-CNOT gate. 2 → − 2 two-photon components: j iC1∕T0 j iC1∕T0 . At BS2 the re- The key element in the KLM CNOT gate is the nondetermi- verse quantum interference process occurs, separating the A nistic NS gate (Fig. 1 ), which operates as follows: When a super- photons into the C1 and T0 modes, while preserving the phase 0 1 position of the vacuum state j i, one photon state j i and shift that was implemented by the NS gates. In this way the re- 2 two-photon state j i is input into the NS gate, the gate flips quired π phase shift is applied to the upper path of the target MZ, the sign (or phase) of the probability amplitude of the 2 j i and so CNOT operation is realized. component: jψi¼αj0iþβj1iþγj2i → jψ 0i¼αj0iþβj1i − γj2i. An NS gate can be realized using an optical circuit consisting of Note that this operation is nondeterministic—it succeeds with P 1∕4— three beam splitters, one auxiliary single photon, and two-photon probability of ¼ however, the gate always gives a signal B (photon detection) when the operation is successful. number-resolving detectors (Fig. 1 ) (11). The NS gate is successful, i.e., jψi → jψ 0i, when one photon is detected at the upper The Nonlinear Sign-Shift Gate detector and no photons at the lower detector. This outcome A CNOT gate can be constructed from two NS gates as shown schematically in Fig. 2A (11). Here the control and target qubits are encoded in optical mode or path (“dual-rail encoding”), with a photon in the top mode representing a logical 0 and in the bot- tom a logical 1. The target modes are combined at a 1∕2 reflectivity BS (BS3), interact with the control 1 mode via the central Mach-Zehnder interferometer (MZ), and are combined again at a 1∕2 reflectivity BS (BS4) to form another MZ with the two target modes, whose relative phase is balanced such that, in the ab- sence of a control photon, the output state of the target photon is the same as the input state. The goal is to impart a π phase shift in the upper path of the target MZ, conditional on the control photon being in the 1 state such that the NOT operation will be implemented on the target qubit. When the control input is 1, quantum interference (34) between the control and target

Fig. 2. The KLM CNOT gate. (A) The gate is constructed of two NS gates; the output is accepted only if the correct heralding signal is observed for each NS gate. Gray indicates the surface of the BS from which a sign change occurs upon reflection. (B) The KLM CNOT gate with simplified NS gate. (C) The same circuit as (B) but using polarization encoding and PPBSs. (D) The stable optical quantum circuit used here to implement the KLM CNOT gate using PPBSs and Fig. 1. The KLM NS gate. (A) If the NS gate succeeds it is heralded; indicated a displaced-Sagnac architecture. The target MZ, formed by BS11 and BS12 in conceptually by the light globe. (B) The original KLM NS gate is heralded by Fig. 2B, can be conveniently incorporated into the state preparation and detection of a photon at the upper detector and no photon at the lower de- measurement, corresponding to a change of basis, as described in the caption tector. Gray indicates the surface of the BS from which a sign change occurs to Fig. 3. The blue line indicates optical paths for vertically polarized compo- upon reflection. (C) A simplified KLM NS gate for which the heralding signal is nents, and the red line indicates optical paths for horizontally polarized com- detection of one photon. ponents.

10068 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1018839108 Okamoto et al. Downloaded by guest on September 25, 2021 occurs with probability 1∕4 and so the success probability of the photons C and A1, respectively, and measured the simultaneous CNOT gate is ð1∕4Þ2 ¼ 1∕16. single photon detection counts between detectors DC and DA1 The key to NS gate operation is multiphoton quantum inter- while scanning the arrival time of the C photon. Note that the ference, which can be understood by considering the simplified reflectivity of PPBS2 for horizontal polarization is 23% and thus V 54% NS gate shown in Fig. 1C (35). The probability amplitude for the visibility for perfect interference is th ¼ , rather than V one photon to be detected at the output detector (which is the 100% in the case of a 50% reflectivity BS. The visibility exp success signal) can be calculated by summing up the amplitudes of the observed dips are 48 4% and 49 3% (with bandpass of the indistinguishable processes leading to this result: For the filters of center wavelength 780 nm and FWHM 2 nm), corre- 0 V ≡ V ∕V 89% j i input only reflection of the auxiliarypffiffiffiffi photon contributes and sponding to relative visibilities of r exp th ¼ and the amplitude is simply given by R, where R is the reflectivity of 91%. To test the performance of our CNOT gate circuit, we used the beamsplitter. For the j1i input the total probability amplitude coincidence measurements between the four threshold detectors 1 − 2R is given by the sum of the probability amplitudes for two at DA1, DA2, DC, and DTrather than using photon number dis- photons to be reflected (−R) and two photons to be transmitted criminating detectors for DA1 and DA2 and loss detection at 1 − R 2 (pffiffiffiffi ). Finally for the j i input the probability amplitude is PPBS3s because we needed to analyze the polarization state Rð3R − 2Þ. This probability amplitude shows that nonlinear sign of the output to confirm correct operation. We performed this flip of the j2i term, required for NS gate operation, is possible for polarization analysis using a half-wave plate (HWP in Fig. 2D) any R < 2∕3, however, the amplitudes of the j0i, j1i and j2i com- or quarter-wave plate (QWP in Fig. 2D) together with a polariz- ponents are also modified by the operation, which is not desired. ing beam splitter (PBS). In the original NS gate (Fig. 1B), the path interferometer is used to balance these amplitudes. To preserve these amplitudes in the Experimental Results “ ” case where the simplified NS gates are used small losses (0.24 for We first checked the logical basis operation of the CNOT gate C T 0 1 R ¼ 0.23 in Fig. 1C) should be deliberately introduced in the out- by preparing and in the four combinations of j i and j i (the ZZ put using BS9 and BS10 in Fig. 2B (35), at the cost of reducing the basis states) and measured the probability of detecting these ZZ “ success probability slightly (from 0.25 to 0.23), but with the ben- states in the output for each input state, to generate the truth ” A efit of removing the need for the interferometer in the NS gates. table shown in Fig. 3 . The experimental data show the ex- T ’ Even with this simplification significant technical difficulties re- pected CNOT operation, i.e., the photon s state is flipped only main: nested interferometers, two auxiliary photons, and several classical and quantum interference conditions. Experimental Implementation of the KLM C-NOT Gate We designed the inherently stable architecture shown in Fig. 2D to implement the KLM CNOT gate of Fig. 2B , using polarization to encode photonic qubits. This design takes advantage of two recent photonic quantum circuit techniques: partially polarizing beam splitters (24–26, 29) (PPBSs), which results in the circuit shown in Fig. 2C, and the displaced-Sagnac architecture (5, 29), which results in the circuit shown in Fig. 2D. The PPBSs have a different reflectivity R and transmissivity T for horizontal H and vertical V polarizations. We used three kinds of PPBSs: PPBS1 (RH ¼ 50%, RV ¼ 100%), PPBS2 (RH ¼ 23%, RV ¼ 100%), and PPBS3 (TH ¼ 76%, TV ¼ 100%). The control (C) and target (T) photons are first incident on PPBS1 (first PPBS1 in Fig. 2C) where two-photon quantum interference transfers pairs of H- polarized photons to the same output port by photon bunching. The outputs are then routed to PPBS2 (two PPBS2s in Fig. 2C), where quantum interferences of the H components with two auxiliary horizontally polarized photons induce the effective nonlinearity. The photons then return to PPBS1 (second PPBS1 in Fig. 2C) where a final quantum interference reverses the initial operation of PPBS1, separating pairs of H-polarized photons into separate outputs. The PPBS3 at each of the outputs (PPBS3s in Fig. 2C) balance the output polarization components. To charac- C COMPUTER SCIENCES terize the operation of the gate, the output modes out and T out were detected by the photon counters (DC and DT) with polarization analyzers. Note that all the four polarization modes of the control and target photons pass through all the optical components inside the interferometer so that the path difference between those four polarization modes are robust to drifts or vibrations of these optical components. Fig. 3. Experimental demonstration of a KLM CNOT gate. Left: ideal opera- We used four photons generated via type-I spontaneous para- tion. Right: fourfold coincidence count rates (per 5,000 s) detected at DC, DT, 0 V 1 H metric down-conversion. The pump laser pulses (76 MHz at DA1, andp DA2.ffiffiffi (A) For control qubit,pffiffiffi j Z i¼j i, j Z i¼j i; for target qubit, 0 1∕ 2 V H 1 1∕ 2 V − H “ ” C 1 390 nm, 200 mW) pass through a beta-barium borate crystal j Z i¼ ðj iþj iÞ, j Z i¼ ðj piffiffiffi j iÞ. 10 indicatespffiffiffi ¼ and (1.5 mm) twice to generate two pairs of photons. One pair was T ¼ 0.(B) For control qubit, j0X i¼1∕ 2ðjViþjHiÞ, j1X i¼1∕ 2ðjVi − jHiÞ; 0 V 1 H 0 C T forpffiffiffi target qubit, j X i¼jpffiffiffii, j X i¼j i.(C) For control qubit,pffiffiffij Y i¼ used as the and qubits, and the other as the auxiliary photons 1∕ 2 V i H 1 1∕ 2 V − i H 0 1∕ 2 V − ðj iþ j iÞp,ffiffiffij Y i¼ ðj i j iÞ; for target qubit, j Y i¼ ðj i A1 and A2. We first checked the quality of quantum interference i H 1 1∕ 2 V i H C∕T j iÞ, j Y i¼ ðj iþ j iÞ. The events in which two pairs of photons are (34) between a photon and an auxiliary photon at PPBS2. simultaneously incident to the ancillary inputs and no photons are incident to For example, to test the interference between C and A1, we de- the signal inputs are subtracted, as confirmed by a reference experiment tected photons T and A2 just after the photon source to herald without input photons.

Okamoto et al. PNAS ∣ June 21, 2011 ∣ vol. 108 ∣ no. 25 ∣ 10069 Downloaded by guest on September 25, 2021 when the C qubit is 1. The (classical) fidelity of this process Note that alternative approaches that do not follow the KLM FZZ→ZZ, defined as the ratio of transmitted photon pairs in recipe as closely can be useful for scalable linear optics quantum the correct output state to the total number of transmitted information processing (18). For all these approaches, further photon pairs, is 0.87 0.01. progress in on-demand single photon sources and practical Because almost all the errors conserve horizontal/vertical photon resolving detectors will be crucial to ensure reliable polarization, the process fidelity FP of the quantum coherent operation. gate operation can be determined from the fidelities obtained from only three sets of orthogonal input- and output states Appendix: Derivation of the Process Fidelity (see Appendix: Derivation of the Process Fidelity), The PPBSs used to realize the KLM CNOT gate preserve the horizontal/vertical polarization with high fidelity. In the quantum FP ¼ðFZZ→ZZ þ FXX→XX þ FXZ→YY − 1Þ∕2. [1] CNOToperation, these polarizations correspond to the ZX-basis of the qubits. In the data shown in Fig. 3, this means that the The measurement result of the input-output probabilities in the number of flips observed for the control qubit in Fig. 3A and XX B basisp areffiffiffi shown in Fig. 3 p,ffiffiffi where the basis states are for the target qubit in Fig. 3B are negligibly small, i.e., 0 error fj0X i ≡ 1∕ 2ðj0iþj1iÞ;j1X i ≡ 1∕ 2ðj0i − j1iÞg; the fidelity is event and only 1 error event respectively over 943 total events. FXX→XX ¼ 0.88 0.02. To obtain FXZ→YY, we detected the We can therefore describe the errors of the quantum gate in YY XZ C basis output from basispffiffiffi inputs, as shown in Fig.pffiffiffi 3 . terms of dephasing between the ZX-eigenstates. In terms of The Y basis states are fj0Y i ≡ 1∕ 2ðj0iþij1iÞ;j1Y i ≡ 1∕ 2ðj0i− the operator expansion of errors, we can define the correct i 1 F 0 81 0 02 1 U^ j iÞg. The fidelity is XZ→YY ¼ . . . Using Eq. , we find operation gate and three possible phase flip errors as a process fidelity of FP ¼ 0.78. A more intuitive measure of how all other possible gate opera- U^ VV VV VH VH HV HV − HH HH ; tions (input and output states) perform is given by the average gate ¼j ih jþj ih jþj ih j j ih j ¯ gate fidelity F, which is defined as the fidelity of the output state ^ UT ¼jVVihVVj − jVHihVHjþjHVihHVjþjHHihHHj; averaged over all possible input states. This measure of the gate ^ performance is related to the process fidelity by (36, 37) UC ¼jVVihVVjþjVHihVHj − jHVihHVjþjHHihHHj; ¯ U^ VV VV − VH VH − HV HV − HH HH : [3] F ¼ðdFp þ 1Þ∕ðd þ 1Þ; [2] CT ¼j ih j j ih j j ih j j ih j

d d 4 where is the dimension of the Hilbert space ( ¼ for a 2 qubit The operation of the gate can then be written as gate). Based on Eqs. 1 and 2, our results show that the average gate fidelity of our experimental quantum CNOT gate is ^ ^ ¯ Eðρ Þ¼ χnmUnρ Um; [4] F ¼ 0.82 0.01. in ∑ in n;m Discussion The data presented above confirm the realization of the CNOT where n, m ∈ fgate;T;C;CTg, and χnm define the process matrix gate proposed by KLM, which is an optical circuit combining a of the noisy quantum process. pair of efficient nonlinear elements induced by measurement. Each of our experimentally observed truth table operations i → j U^ This result confirms the first step in the KLM recipe for all- is correctly performed by gate and one other operation optical quantum computation and illustrates how efficient non- ^ Un. Therefore, the fidelities Fi→j can be given by the sums of linearities induced by measurement can be utilized for quantum F χ U^ information science; such measurement-induced optical nonli- the probability p ¼ gate;gate for the correct operation gate ^ nearities could also be an alternative to nonlinear media used and the probabilities ηn ¼ χnn for the errors Un as follows. for quantum nondemolition detectors (39) or photonic pulse shaping (40). By emulating fundamental nonlinear processes, FZZ→ZZ ¼ Fp þ ηT FXX→XX ¼ Fp þ ηC such measurement-induced optical nonlinearities can also im- prove our understanding of the quantum dynamics in nonlinear FXZ→YY ¼ Fp þ ηCT : [5] media. Conversely, future technical progress may permit the re- placement of these effective optical nonlinearities in the network Note that these relations between the diagonal elements of the by approaches based on nonlinearities in material systems such as process matrix and the experimentally observed fidelities can also atoms (6), solid state devices (41), hybrid systems (42), or optical be derived from Eq. 4 using the formal definition of the experi- fiber Kerr nonlinearities (43). In this context, our demonstration mental fidelities. In this case the fidelities are determined by the provides an experimental test for quantum networks based on sums over the correct outcomes jðjÞli in EðjðiÞkihðiÞkjÞ, averaged nonlinear optical elements and may serve as a reference point over all inputs jðiÞki, for comparisons with future networks using other optical nonlinearities. In particular, the present results may be useful as a F j E i i j ∕4 starting point for a more general analysis of quantum error pro- i→j ¼ ∑hð Þlj ðjð Þkihð ÞkjÞjð Þli Þ pagation in nonlinear optical networks. Our device will be useful l;k for conventional and cluster state approaches to quantum computing (38), as well as quantum communication (2), and optical χ j U^ † i i U^ j ∕4 : [6] ¼ ∑ nm ∑hð Þlj njð Þkihð Þkj mjð Þli quantum metrology (5). This circuit could be implemented using n;m l;k an integrated waveguide architecture (28), in which case a dual- rail encoding could conveniently be used. k l ∈ 1;2;3;4 i k i In the present tests of the performance of CNOT gate opera- Here , f g, and ð Þk denote the th state of the basis tion, we used threshold detectors to monitor the output state. For states. For example, jðiÞ1i¼jVVi, jðiÞ2i¼jVHi, jðiÞ3i¼jHVi, applications in which the output state cannot be monitored, high- jðiÞ4i¼jHHi for i ¼ ZX. The sums over initial states k and efficiency number-resolving photon detectors (13) could be used final states l are one for n ¼ m ¼ 0 and for a single other error, at DA1 and DA2 to generate the heralding signals. We also used n ¼ m ¼ nðijÞ. All remaining sums are zero, confirming the spontaneous parametric fluorescence as single photon sources. results in Eq. 5).

10070 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1018839108 Okamoto et al. Downloaded by guest on September 25, 2021 ¯ Because the diagonal elements of the process matrix corre- F ¼ðdFp þ 1Þ∕ðd þ 1Þ¼0.82; [8] spond to the probabilities of the orthogonal basis operations, their sum is normalized to one, so that ∑ χnn ¼ Fp þ ηT þ n where d is the dimension of the Hilbert space (d ¼ 4 for a ηC þ ηCT ¼ 1. It follows that the sum of all three experimentally two-qubit gate). determined fidelities is FZZ→ZZ þ FXX→XX þ FXZ→YY ¼ 2Fp þ 1. Therefore, the process fidelity of our KLM CNOT gate is given by ACKNOWLEDGMENTS. We thank T. Nagata and M. Tanida for help and discussions. This work was supported in part by Quantum Cybernetics project, Fp ¼ðFZZ→ZZ þ FXX→XX þ FXZ→YY − 1Þ∕2 ¼ 0.78: [7] the Japan Science and Technology Agency (JST), Ministry of Internal Affairs and Communication (MIC), Japan Society for the Promotion of Science (JSPS), 21st Century Center of Excellence (COE) Program, Special Coordination Funds This number clearly exceeds the threshold Fp ≥ 0.5 for the gate to — for Promoting Science and Technology, Daiwa Anglo-Japanese Foundation, produce entanglement a key quantum operation of the gate. European Research Council (ERC), Engineering and Physical Sciences Research The fidelity of the output states of the gate, averaged over all Council (EPSRC), and Leverhulme Trust. J.L.O’B. acknowledges a Royal Society input states is related to the process fidelity Wolfson Merit Award.

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Okamoto et al. PNAS ∣ June 21, 2011 ∣ vol. 108 ∣ no. 25 ∣ 10071 Downloaded by guest on September 25, 2021