IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 15, NO. 6, NOVEMBER/DECEMBER 2009 1685 Experimental Characterization of a Telecommunications-Band Quantum Controlled-NOT Gate Monika Patel, Joseph B. Altepeter, Matthew A. Hall, Milja Medic, and Prem Kumar, Fellow, IEEE Abstract—The quantum controlled-NOT gate is an example of the maximally entangling gate, which is a broad class of operations that are necessary for scalable linear optics quantum computation. Here, we characterize a telecommunications-wavelength (1550 nm) quantum controlled-NOT gate, and for the first time, experimentally bound its process fidelity by measuring its operation in two complementary polarization bases. The gate’s final process fidelity F is given by 91% ≤ F ≤ 95%. Index Terms—Controlled-NOT (CNOT), entangling gate, process tomography. I. INTRODUCTION UANTUM information processing holds the promise of exponentially faster computation [1]–[3] and fundamentally nonclassical communication [4]. Although implementing a full-scale quantum computer is prohibitively difficult using existing technology, it is feasible that near-term quantum communications applications, which require relatively few qubits and gates, might be realized. By designing quantum states and quantum gates that operate at telecommunications wavelengths (1.3–1.6 µm), it is possible to leverage the existing telecommunications infrastructure for communications-based quantum information processing (for example, to perform symmetrically private queries on small databases [5]). Here, we report on the experimental characterization of a telecommunicationswavelength linear optics quantum controlled-NOT (CNOT) gate. To date, there have been several experimental implementations of linear optics CNOT gates that use visible or nearinfrared wavelength (<800 nm) photons [6]– [18]. Our group Q has previously reported on the first telecommunications-band CNOT gate [19]; although this paper verified its entangling operation, it placed no quantitative bounds on its performance. Here, we quantitatively bound the process fidelity of the same polarization-encoded CNOT gate, after significantly improving the measurement source, the detection system, and the CNOT gate itself. This report is organized as follows. We first provide a review of the basic operation of the CNOT gate, followed by a detailed analysis of its implementation using only linear optical components—an analysis that applies to either spatially encoded or polarization-encoded CNOT gates. After this introduction, we describe in detail the experimental setup (source, gate, and detection) for a polarization-encoded, telecommunications-band CNOT gate. Finally, the experimental results are presented. A. CNOT Operator Entangling gates are a fundamental primitive for scalable quantum information processing [3]. The CNOT gate is an example of a maximally entangling gate that allows the state of one qubit (the “control” qubit) to conditionally flip the state of another qubit (the “target” qubit). In the two-qubit basis {00, 01, 10, 11} (the first digit denotes the value of the control and the second the value of the target), the CNOT gate is defined by the unitary matrix |00 UCNOT = Manuscript received February 20, 2009; revised May 18, 2009. First published November 10, 2009; current version published December 3, 2009. This work was supported in part by Defense Advanced Research Projects Agency (DARPA) under Grant W31P4Q-08-1-0006. The work of M. A. Hall was supported by the National Science Foundation (NSF) Integrative Graduate Education and Research Traineeship (IGERT) Fellowship under Grant DGE-0801685. M. Patel and M. Medic are with the Department of Physics, Northwestern University, Evanston, IL 60208 USA (e-mail: monika@u.northwestern.edu; milja@u.northwestern.edu). J. B. Altepeter and M. A. Hall are with the Center for Photonic Communication and Computing, Northwestern University, Evanston, IL 60208 USA (e-mail: joe.altepeter@gmail.com; matt.a.hall@gmail.com). P. Kumar is with the Department of Physics and the Center for Photonic Communication and Computing, Northwestern University, Evanston, IL 60208 USA (e-mail: kumarp@northwestern.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JSTQE.2009.2025704 |01 |10 |11
00|  1 0   0 0
01| 0 1 0 0
10| 0 0 0 1
11| 0  0  . 1 0 (1) In this canonical basis, the CNOT gate appears to perform a very classical function. Its entangling character is not revealed until it operates on superposed input states. √ When operating on the completely separable input state (1/ 2) (|0 + |1) ⊗ |0, the √CNOT gate outputs the maximally entangled Bell state (1/ 2) (|00 + |11). This entangling operation can furthermore be utilized in reverse, transforming a CNOT gate into a Bell measurement 1 1 CNOT √ (|00 + |11) ←→ √ (|0 + |1) ⊗ |0 2 2 1 1 CNOT √ (|00 − |11) ←→ √ (|0 − |1) ⊗ |0 2 2 1077-260X/$26.00 © 2009 IEEE 1686 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 15, NO. 6, NOVEMBER/DECEMBER 2009 Fig. 1. Pictorial diagrams of two physical implementations for a linear optics CNOT gate. (a) Spatially encoded CNOT gate. C 0 and C 1 label the 0 and 1 modes of the control qubit, whereas T 0 and T 1 denote the modes of the target qubit. Beamsplitters are colored to indicate their reflectivity: green for R = 1/3 and blue for R = 1/2. In each case, the gray side of the beamsplitter provides an ei π phase on reflection. (b) Same gate implemented using two polarizationencoded qubits. Here, the horizontal and vertical polarization states define the logical qubit basis according to the rules |0 ≡ |H  and |1 ≡ |V . Swap and Hadamard gates can be implemented with half-wave plates at 45◦ and 22.5 ◦ , respectively. The PPBS perfectly reflects vertically polarized light (R V = 1) and partially reflects horizontally polarized light (R H = 1/3). The gray side of the PPBS provides an ei π phase to horizontally polarized light on reflection. 1 1 CNOT √ (|01 + |10) ←→ √ (|0 + |1) ⊗ |1 2 2 1 1 CNOT √ (|01 − |10) ←→ √ (|0 − |1) ⊗ |1. 2 2 ation using only single-photon transformations. When a single control photon and a single target photon are input to the gate, the CNOT operates on a vector space spanned by the four-element two-photon basis {C0 T0 , C0 T1 , C1 T0 , C1 T1 }. When a single photon in either the control or target is input, the gate operates on a vector space defined by a four-element single-photon basis: {C0 , C1 , T0 , T1 }, where Ci and Ti denote the control and target modes for the state |i. By breaking up the CNOT’s operation into four steps and tracking how these four single-photon inputs evolve at each stage, Fig. 2 plots the step-by-step evolution of each single-photon input for both the spatial and polarization encodings. Note that at every step, the spatial and polarization encodings are equivalent. The single-photon creation operators a†C i and a†T i acting on the total vacuum state |0 describe photons that populate the modes Ci and Ti . For readability, we will refer to these creation operators through the use of the “hatted” operators Ĉi† ≡ a†C i and T̂i† ≡ a†T i . Using this notation, the CNOT single-photon transformations are   √ † 1 Ĉ0† + 2ĈX 3   1 CNOT Ĉ1† −→ −Ĉ1† + T̂0† + T̂1† 3   1 CNOT T̂0† −→ Ĉ1† + T̂0† + T̂X† 3   1 † CNOT T̂1 −→ Ĉ1† + T̂1† − T̂X† 3 CNOT Ĉ0† −→ (2) Each of the four maximally entangled Bell states is rotated to or from one of four separable states (which can be more easily experimentally measured or created). This operation, while clearly very useful for quantum information processing, requires there to be a direct interaction between the two qubits. For photons, where there is no appreciable coupling between two single photons, this is a daunting requirement. In order to overcome this obstacle, Knill et al. instead proposed using the quantum mechanical measurement process to provide the massive nonlinearity necessary to couple two single photons [20], [21]. This computational paradigm allows nondeterministic but scalable two-photon gates to be created using only linear optics. B. Implementing the CNOT Using Linear Optics Linear optics quantum computing (LOQC) [20], [21] is a quantum information processing paradigm that relies solely on linear optical elements and single-photon counters to achieve scalable computation. The LOQC CNOT gate described here, for example, acts exactly as a standard quantum CNOT gate, except that it is only successful one-ninth of the time, where success is defined by exactly one photon being measured in each of the control and target outputs. This general LOQC gate can be encoded using either spatial or polarization qubits, and physical implementations for both encodings are shown in Fig. 1. The spatial and polarization encodings are equivalent, and each performs the CNOT operation in (1) with a success probability of 1/9. Because these gates are constructed using only linear optical elements, it is possible to describe their complete oper- (3) (4) (5) (6) † where ĈX and T̂X† represent creation operators for two ancillary dump modes into which input photons are probabilistically lost. In order to derive the two-photon operation for the same gate, we only have to apply the aforementioned transformations to the standard basis of two-photon inputs 1 † √ †  † Ĉ1 + T̂0† + T̂X† Ĉ0 + 2ĈX 3 1 † † Ĉ Ĉ + Ĉ0† T̂0† + Ĉ0† T̂X† = 3 0 1 √  † † † † Ĉ1 + ĈX T̂0† + ĈX T̂X† + 2 ĈX Ĉ0† T̂0† −→ CNOT 1 † √ †  † Ĉ0 + 2ĈX Ĉ1 + T̂1† − T̂X† 3 1 † † Ĉ Ĉ + Ĉ0† T̂1† − Ĉ0† T̂X† = 3 0 1 √  † † † † Ĉ1 + ĈX T̂1† − ĈX T̂X† + 2 ĈX (7) Ĉ0† T̂1† −→ CNOT (8)  1 −Ĉ1† + T̂0† + T̂1† Ĉ1† + T̂0† + T̂X† 3 1 − Ĉ1† Ĉ1† − Ĉ1† T̂X† + T̂0† T̂0† + T̂0† T̂X† = 3 Ĉ1† T̂0† −→ CNOT + Ĉ1† T̂1† + T̂1† T̂0† + T̂1† T̂X† (9) PATEL et al.: EXPERIMENTAL CHARACTERIZATION OF A TELECOMMUNICATIONS-BAND QUANTUM CONTROLLED-NOT GATE 1687 Fig. 2. Graphical representation of the single-photon transformations performed by both the spatially encoded and polarization-encoded linear optics CNOT gates. Optical elements are labeled as in Fig. 1. As it travels through successive components of the gate, each photon evolves into a superposition of different spatial/polarization modes. These superposed modes are graphically depicted after each major CNOT component, with each vertical box depicting a single term of the superposition. Each box is faded in inverse proportion to its term’s amplitude (the amplitude is explicitly noted below the box). (a) Evolution of the state Ĉ 0† |0 √ † √ † † † into the single-photon superposition |0 into the single-photon superposition )|0, and the state Ĉ H + 2 Ĉ X )|0. Note (1/3)(Ĉ 0† + 2 Ĉ X (1/3)(Ĉ H that these two processes are identical. (b) Evolution of C 1† / V |0 into T̂ 0†/ H + T̂ X† )|0. (d) Evolution of T 1†/ V |0 into (1/3)(−Ĉ 1† / V + T̂ 0†/ H + T̂ 1†/ V )|0. (c) Evolution of T 0†/ H |0 into (1/3)(Ĉ 1† / V + T̂ 1†/ V − T̂ X† )|0. (1/3)(Ĉ 1† / V + 1688 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 15, NO. 6, NOVEMBER/DECEMBER 2009 Fig. 3. Operation of spatially encoded and polarization-encoded linear optical CNOT gates on the input state |10, i.e., the two-photon evolution of the state Ĉ 1† T̂ 0† |0 into the superposition state (1/3)(−Ĉ 1† Ĉ 1† − Ĉ 1† T̂ X† + T̂ 0† T̂ 0† + T̂ 0† T̂ X† + Ĉ 1† T̂ 1† + T̂ 1† T̂ 0† + T̂ 1† T̂ X† )|0. Note that this depiction uses the same style as Fig. 2, except that now each superposed term is represented by a box with two photons. At the conclusion of the gate the two-photon state is in a seven-term superposition, only one of which represents exactly one control photon and one target photon. This term, (1/3)(Ĉ 1† T̂ 1† )|0, corresponds to the correct CNOT output |11. The square of the amplitude of this term is 1/9, which is the success probability of the linear optical CNOT gate.  1 −Ĉ1† + T̂0† + T̂1† Ĉ1† + T̂1† − T̂X† 3 1 − Ĉ1† Ĉ1† + Ĉ1† T̂X† + Ĉ1† T̂0† + T̂0† T̂1† = 3 quantum measurement in an arbitrary two-qubit polarization basis. Ĉ1† T̂1† −→ CNOT − T̂0† T̂X† + T̂1† T̂1† − T̂1† T̂X† . A. Source (10) In each case, the superposed term corresponding to successful CNOT operation has been underlined; all other terms do not have a single control photon (in mode C0 or C1 ) and a single target photon (in mode T0 or T1 ). The derivation of (9) is graphically depicted in Fig. 3. II. EXPERIMENTAL CONFIGURATION The creation and characterization of any LOQC gate requires three distinct experimental apparatuses: a photon source that will act as the input to the LOQC gate, the LOQC gate itself, and a system for measurement and detection of the LOQC gate output. This section describes each of these experimental apparatuses in detail: a fiber-based source of degenerate photon pairs, a fiber-coupled and polarization-encoded LOQC CNOT gate, and an 8.3-MHz-rate four-detector setup coupled to the system for Fig. 5 shows the experimental schematic for the fiber-based source of degenerate photon pairs that act as the CNOT input [22], [23]. This source utilizes the Kerr nonlinearity of dispersionshifted optical fiber (DSF) to produce photon pairs through a process known as spontaneous four-wave mixing (FWM). In standard FWM, two degenerate pump photons at frequency ω0 scatter, thus producing signal and idler photons at frequencies ω1 and ω2 that satisfy the energy conservation relation 2ω0 = ω1 + ω2 . This experiment exploits the complementary FWM process, through which two pump photons at frequencies ω1 and ω2 scatter into two daughter photons at the degenerate frequency ω0 . This dual-frequency pump is created by passing the output of a 50-MHz-rate femtosecond fiber laser (IMRA femtolite BS-60) through two free-space transmission grating filters connected via an erbium-doped fiber amplifier. The photon pairs are generated in an optical fiber Sagnac loop [see Fig. 4(b)], consisting of the FWM fiber spool of 300 m of DSF, a fiber polarization PATEL et al.: EXPERIMENTAL CHARACTERIZATION OF A TELECOMMUNICATIONS-BAND QUANTUM CONTROLLED-NOT GATE Fig. 4. Quantum interference at a beamsplitter and the reverse Hong–Ou– Mandel effect. (a) a and b are the input ports, and c and d are the output ports of the beamsplitter. (b) Quantum interference in the Sagnac loop used in the experiment. By adjusting the phase δ using the FPC, we set the loop such that one of each pair of identical photons goes down each output arm of the 50/50 splitter. CW: clockwise, CCW: counterclockwise, FPC: fiber polarization controller, P: input power (this figure first appeared in [22]). 1689 d refer to the output ports. This phenomenon is known as the “reverse Hong–Ou–Mandel effect” [22], [24]. In this experiment, we set the Sagnac loop’s fiber polarization controller to a “50/50” or “quantum splitter” configuration [22] to exploit the reverse Hong–Ou–Mandel effect, which leads to these photon pairs being deterministically split, exactly one photon exiting from each of the two output single-mode fibers. When correctly aligned, a superposition of “bunched” photon pairs in the clockwise and counterclockwise modes interferes at the Sagnac’s output 50/50 beamsplitter to create a photon pair that is “split” between the two fiber modes. This interference occurs when two conditions are met: 1) there is no phase difference between the clockwise and counterclockwise amplitudes and 2) the polarizations of the clockwise and counterclockwise pairs are identical. The symmetry of the Sagnac loop automatically satisfies the first condition (although asymmetric pump losses coupled with self-phase modulation could break this symmetry in some systems). The consequences of not satisfying the second condition are explained in [25]. As detailed later in this paper, bunched photon outputs due to imperfect polarization matching are a significant source of error for our CNOT characterization. Later sections describe how these bunched contributions are directly measured, and then, incorporated into the CNOT characterization, thus eliminating the effect of bunching errors on the final measurement results. FWM pairs generated in the Sagnac loop—whether bunched or split—are separated from the dual-frequency pump via optical bandpass filters before being routed to the inputs of the CNOT gate. B. CNOT Gate Fig. 5. Experimental layout of the source connected to the CNOT gate. The two pumps, designated as pump1 and pump2, enter the Sagnac loop where they produce degenerate-frequency pairs via FWM. These pairs form the control and the target input to the CNOT gate, which meet at the first partially polarizing beamsplitter (PPBS1). In the CNOT gate, PPBS1 is followed by swap gates in the two paths, which are followed by two more PPBSs, PPBS2 and PPBS3, one in each arm. The measurement apparatus consists of a half-wave plate, a quarter-wave plate, and a polarizing beamsplitter in each arm followed by the single-photon detectors D1, D2, D3, and D4. PBS: polarizing beamsplitter, HWP: half-wave plate, QWP: quarter-wave plate, OBPF: optical bandpass filter, QS: quantum splitter, BS: beamsplitter. controller, and a 50/50 fiber beamsplitter. The spool of DSF is liquid nitrogen cooled to 77 K in order to suppress spontaneous Raman scattering. Identical photon pairs that occupy the same spatial mode are created via spontaneous dual-pump FWM inside the DSF. CNOT inputs are required to be in distinct spatial modes; to deterministically separate these photons into separate spatial modes, we note that a superposition of pairs of photons incident on the inputs of a beamsplitter (|ψin  = (1/2) [|0a |2b + |2a |0b ]) results in exactly one photon in each output arm (|ψout  = |1c |1d ). As shown in Fig. 4(a), a and b denote the input ports of the beamsplitter while c and Once created, the degenerate input pairs are routed through single-mode fibers to the CNOT gate inputs. This telecom-band gate, although fiber-coupled, is constructed from free-space linear optical components, and operates on spatially distinct, polarization-encoded photonic qubits (|H ≡ |0 , |V  ≡ |1). The gate’s central components are three custom-made partially polarizing beamsplitters (PPBSs), following the approach first outlined in [15] and [16]. Each PPBS perfectly reflects incident vertically polarized light while reflecting one-third and transmitting two-thirds of incident horizontally polarized light. Two swap gates (half-wave plates at 45◦ ) and two Hadamard gates (half-wave plates at 22.5◦ ) complete the CNOT architecture, as shown in Fig. 5. Note that this architecture uses two fewer swap gates than the polarization-encoded CNOT gate shown in Figs. 1–3. The elements shown in the “CNOT gate” shaded block of Fig. 5, therefore, perform a different—yet still maximally entangling—two-qubit operation. Because the missing wave plates are at the inputs and outputs of the device, it is possible to use adjacent input wave plates or measurement wave plates to compensate, in effect achieving perfect CNOT operation while relying on fewer total components. It is these same input and output wave plate/polarizer combinations that allow the creation and measurement of arbitrarily polarized input and output states. Vibrations on one of the input or output steering mirrors (not pictured) can cause a global phase on either the control or target 1690 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 15, NO. 6, NOVEMBER/DECEMBER 2009 qubit, a phase that will naturally fluctuate with time. Because all CNOT inputs and outputs are joint two-photon control-target states, the CNOT operation is immune to this noise. However, noise from the stray pump—which passes through what is effectively a huge Mach–Zehnder fiber interferometer bounded by the Sagnac loop and PPBS1—is affected by this time-varying phase. In order to ensure phase averaging over any stray pump light that reaches the gate, a slowly varying (at 8 Hz) piezoelectric transducer was affixed to a steering mirror in the path of the input target photon. C. Detection Fast, efficient detection is crucial for any LOQC gate characterization. LOQC gates are inherently nondeterministic (this CNOT gate has a probability of success of 1/9); coupled with the general lack of high-efficiency telecom-band single-photon counters, this dictates that adequate detection systems are a key experimental concern. Previous experiments [19] addressed this problem by using a single superconducting single-photon detector to herald an InGaAs/InP photodiode with a gate rate of 0.8 MHz. Here, we have greatly improved the total detection system by installing a set of four single-photon detectors (NuCrypt LLC, Model CPDS-4) to characterize the CNOT gate. Of the four detector chips in our system, two were made by Princeton Lightwave, Inc., and two by Goodrich Sensors Unlimited. We recorded coincidence counts on all pairwise combinations of the four detectors. Each detector was adjusted for optimal operation by synchronizing its gate-pulse timing with a clock provided by the laser source. In addition, the bias voltage on each detector was adjusted for minimal afterpulsing, and the operating rate was increased to 8.3 MHz. The average dark-count rate for these detectors was 3 × 10−4 per pulse. The detector quantum efficiencies were approximately 20%. By using a set of four detectors, we can simultaneously measure the complete four-element coincidence basis, increasing our effective data collection rate by a factor of 4, as well as automatically compensating for any pump intensity fluctuations. Because our coincidence counts were measured using four different pairs of detectors, we used direct measurements and an in situ maximum-likelihood program to compensate for both differences in individual detector efficiencies and output polarizing beamsplitter crosstalk. The ten-fold increase in detection rate and fourfold increase in collection rate resulted in a total data rate that was 40 times faster than in previous experiments. III. CNOT GATE CHARACTERIZATION In order to completely characterize a two-qubit quantum process, it is necessary to record the results of at least 256 separate measurements. In practice, these many measurements can be inconvenient, or in some cases, prohibitively difficult. Luckily, it is possible to use only 32 polarization measurements to bound the total process fidelity of any two-qubit gate [26]. This bound is given by F B 1 + F B 2 − 1 ≤ Fχ ≤ min F B 1 , F B 2 (11) Fig. 6. Experimentally measured truth tables characterizing the output of the CNOT gate. (a) Truth tables without correction for bunching in the H/V and D/A bases (|D ≡ (1/2) (|H  + |V ), |A ≡ (1/2) (|H  − |V )). The average truth table fidelity was 88.6 ± 0.3% and 89.1 ± 0.3% for the H/V and D/A bases, respectively. Each of the large data peaks corresponds to approximately 30 000 coincidence counts and 10 000 accidentals. (b) Same data after correction for bunching. After subtracting bunched coincidence counts, the average H/V fidelity is 94.8 ± 0.4% and the average D/A fidelity is 95.9 ± 0.4%. Imperfect alignment causes the degenerate photonpair source—with small probability—to produce two photons in the control or two photons in the target, i.e., bunching. The side peaks due to bunching are in the H H In /V H O u t , H V In /V V O u t , V H In /V H O u t , V V In /V V O u t , DD In /AD O u t , DA In /AD O u t , AD In /DD O u t , and AA In /DD O u t elements of the truth table. Systematic errors in coincidence rates due to this bunching effect can be directly measured by blocking either the control or the target port of the CNOT gate, and then, subtracted in order to reconstruct the true CNOT gate performance. These “bunching peaks,” on average, are characterized by approximately 9000 coincidences counts, 7000 accidental coincidences, and 2000 directly measured bunching coincidences. The remaining nonnegligible side peaks, H H In /H V O u t , H V In /H H O u t , DA In /DA O u t , and AA In /AA O u t , result from imperfections in the CNOT gate’s optical components (see text). where F B i is the average fidelity of the experimental results with theoretical expectations when using the basis Bi for both the inputs and the outputs of the CNOT gate. (In the experimental characterization which follows, B1 ≡ BH /V = {HH, HV, V H, V V } and B2 ≡ BD /A = {DD, DA, AD, AA}, where D and A denote diagonally and antidiagonally polarized light, respectively.) In other words, the bounds of the fidelity of an experimental process can be obtained by measuring two complementary 16-element datasets. After measuring the CNOT gate’s performance in these two canonical bases, we obtain the average fidelities F H /V = 88.6 ± 0.3% and F D /A = 89.1 ± 0.3%. The measured truth tables for these two bases are shown in Fig. 6(a). These results bound the process fidelity between 77.7% and 88.6%. These results, while confirming the measured gate’s entangling character, are clearly being limited by systematic rather than statistical errors. Studying Fig. 6(a), one observes side peaks of up to 10% of the height of the main peaks. PATEL et al.: EXPERIMENTAL CHARACTERIZATION OF A TELECOMMUNICATIONS-BAND QUANTUM CONTROLLED-NOT GATE We have examined several potential sources of error, including multipair production from the source and imperfect optics in the setup. Although our previous CNOT characterization was limited by four-photon production from the source, this cause of systematic error has been largely eliminated in the present experiment. This is accomplished by lowering the pump power until four-photon events are directly measured to account for less than 5% of FWM events, thus limiting the probability of any multipair-induced errors to <2.5% [19]. Inaccurately aligned and imperfect optical elements (such as wave plates and beamsplitters) are another potential source of error. To estimate the contribution from imperfections in the optical components to the total error, we inject classical 1550-nm light (Santec TSL-210) into the horizontalcontrol, vertical-control, horizontal-target, and vertical-target input modes, measuring the intensity at each of the four output modes for each of the four inputs. This provides a direct measurement of the absolute squares of the amplitude terms in (3)–(6). Using these directly measured values in conjunction with (3)–(6) for single-photon evolution, we are able to predict exactly which side peaks in the truth tables could have been caused by imperfections in the optical elements. From these calculations, we determine that optical imperfections of the type we measure account for nonzero probabilities in the HH In /HV O ut , HV In /HH O ut , DAIn /DAO ut , and AAIn /AAO ut elements of the truth table. Even after accounting for these, each truth table contains four unexplained yet nonnegligible side peaks. To account for these systematic errors, we are led to investigate the effects of photon bunching at the source’s output. Turning again to the single-photon-transformation picture of the CNOT gate, we note that the remaining errors are consistent with bunched photon inputs, i.e., when two photons instead of one are present in the control or target input. The CNOT operates on such inputs as follows (using the transformations given in (3)–(6), with coincidence terms underlined and 0 and 1 replaced with H and V , respectively): √ † √ †  † 1 † ĈH + 2ĈX ĈH + 2ĈX 3 √ † † 1 † † † † ĈX + 2ĈX ĈX ĈH ĈH + 2 2ĈH (12) = 3   CNOT 1 −ĈV† + T̂H† + T̂V† −ĈV† + T̂H† + T̂V† ĈV† ĈV† −→ 3 1 † † Ĉ Ĉ − 2ĈV† T̂H† − 2ĈV† T̂V† = 3 V V † † ĈH ĈH −→ CNOT + 2T̂V† T̂H† + T̂H† T̂H† + T̂V† T̂V† (13)  1 † ĈV† + T̂H† + T̂X† ĈV + T̂H† + T̂X† 3 1 † † Ĉ Ĉ + 2ĈV† T̂H† + 2ĈV† T̂X† = 3 V V T̂H† T̂H† −→ CNOT + 2T̂H† T̂X† + T̂H† T̂H† + T̂X† T̂X† (14) 1691  1 † ĈV† + T̂V† − T̂X† ĈV + T̂V† − T̂X† 3 1 † † = 2ĈV T̂V + ĈV† ĈV† − 2ĈV† T̂X† 3 T̂V† T̂V† −→ CNOT + T̂V† T̂V† − 2T̂V† T̂X† + T̂X† T̂X† . (15) Note that the underlined terms correspond to the error side peaks in Fig. 6(a) that remain unaccounted for. As mentioned in Section II-A, this bunching phenomenon is an artifact of an imperfectly aligned Sagnac loop in the degenerate photon source (even in the 50/50 configuration, if the quantum splitter’s counterclockwise and clockwise FWM components are not aligned to have identical polarizations, imperfect splitting will occur [25]). We measure the bunching probability by monitoring coincidences on a 50/50 fiber beamsplitter inserted in the signal arm. By adjusting the fiber polarization controller inside the Sagnac loop to minimize these coincidences, we are able to align the source for minimal emission of bunched photons. Unfortunately, imperfect initial alignment coupled with drifts in the source still lead to some bunched photons during the measurement time. These bunched photons can be directly measured during gate characterization, and the increase in measurement time necessary to completely correct for this effect is small (less than 5%–10% of normal measurement time). To directly measure the coincidence counts due to photon bunching, we measure the output coincidence counts when either the target or the control arm is blocked (in addition to the standard measurement with both unblocked). By subtracting these bunching-induced coincidences, we can directly measure our gate’s performance if it had been supplied with perfect input states. Fig. 6(b) shows the Hofmann characterization of the CNOT gate after subtraction of bunched coincidences. The compensation results in truth table fidelities of F H /V = 94.8 ± 0.4% and F D /A = 95.9 ± 0.4%, which bounds the process fidelity to between 90.7% and 94.8%. Although this compensation protocol completely corrects for errors due to bunching, it is instructive to estimate the uncompensated truth table fidelities F H /V and F D /A as a function of the bunching probability b. Using (7)–(10), we can construct a truth table Msplit corresponding to the probability that a given input state will be measured in a given output state Msplit = † CH TH† O ut O ut † CH TV† O ut CV† TH† O ut CV† TV† † CH TH† 1 9  0   0 0 In † CH TV† In CV† TH† 0 0 1 9 0 0 0 0 1 9 In In CV† TV†  0  0  (16) 1  9  0 We can construct similar truth tables for bunched photons at the control and target input ports using (12)–(15) (taking care to correctly normalize the states generated by repeated creation 1692 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 15, NO. 6, NOVEMBER/DECEMBER 2009 operators) A similar calculation can be performed in the D/A basis, leading to the following corrected truth table: C Mbunch = M D /A = † † In CH CH † CH TH† † CH TV† CV† TH† O ut O ut O ut O ut CV† TV† T Mbunch † † In CH CH In CV† CV† In CV† CV† 0 0 0 0  0   0 0 0 0 2 9  0  2  9   0  † CD TD† (17) O ut O ut O ut O ut CA† TA† = In † CD TA† In CA† TD† In CA† TA† In 1 9 0 2b 9 b 9   0    2b  9 0 0 1−b 9 b 9 1 9 1−b 9 0  0      0   0 (20) In TH† TH† O ut † CH TH† O ut † CH TV† O ut CV† TH† O ut CV† TV† † CD TA† CA† TD† 2 9 2 9 0 † CD TD† TV† In TV† In TH† TH† TV† In TV† The expected truth table fidelities as a function of b for M H /V and M D /A are identical, i.e., F H /V = F D /A . 0 0 0 0  0  2 9 0 0 0 2 9 2 9 0  0  (18)  0  0  2 9 Note that both of these matrices contain repeated columns in order to match the appropriate bunching error with the corresponding nonbunched input. For example, each of the nonIn In bunched inputs CV† TH† and CV† TV† suffers from bunching In errors of the CV† CV† type. This column must, therefore, be C repeated in the Mbunch matrix. Let us assume that bunched photons are equally likely to enter the gate via the control or the target port (because of the piezoelectric transducer on the target steering mirror, these inputs are incoherent). The complete truth table M as a function of bunching probability b can then be given by IV. CONCLUSION We have experimentally bounded the process fidelity of the first telecommunications-wavelength CNOT gate, and quantified the major sources of systematic error in its performance. In addition, we have used a convenient single-photontransformation representation for multiple-qubit LOQC processes to explicitly show the equivalence between spatially encoded and polarization-encoded CNOT operations. Because one of the primary advantages of an entangling gate, which operates in the 1550-nm C-band, is its potential for integration into the existing telecommunications infrastructure, the next steps for CNOT development will focus on implementations that do not require free-space components and that can be easily networked with other quantum computation sources and gates. REFERENCES M H /V = (1 − b)Msplit + † CH TH† † CH TH† † CH TV† CV† TH† CV† O ut O ut O ut O ut TV† In  b C T = Mbunch + Mbunch 2 † CH TV†  1−b 9   0    b  9 0 In CV† TH† In CV† TV† In 0 0 0 1−b 9 0 0 2b 9 b 9 1 9  0    1  9   2b 9 (19) The expected fidelity F H /V as a function of b (calculated using the truth table M H /V ) is therefore F H /V 1 = 2   1 +1−b . 1 + 2b For small b, this simplifies to F H /V = 1 − (3/2)b + 2b2 . [1] P. W. Shor, “Algorithms for quantum computation: Discrete logarithm and factoring,” in Proc. 35th Annu. Symp. Found. Comput. Sci., S. Goldwasser, Ed. 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Monika Patel received the B.S. degree in physics from the University of Mumbai, Mumbai, India, in 2003, and the M.S. degree from the Indian Institute of Technology, Mumbai, in 2005. She is currently working toward the Ph.D. degree in physics at the Center for Photonic Communication and Computing, Northwestern University, Evanston, IL. 1693 Joseph B. Altepeter received the B.Sc. degree in physics from Washington University in St. Louis, St. Louis, MO, in 2000, and the Ph.D. degree in physics from the University of Illinois at Urbana-Champaign, Urbana, in 2006. In 2001, he was a Fulbright Scholar at the University of Queensland, Australia. He was a National Science Foundation Graduate Research Fellow and a Postdoctoral Fellow in the Kwiat Quantum Information Group, University of Illinois at Urbana-Champaign. Since 2007, he has been an Intelligence Community Postdoctoral Fellow in the Kumar Group, Center for Photonic Communication and Computing, Northwestern University, Evanston, IL. Matthew A. Hall received the B.S. degree in electrical engineering in 2004 from Northwestern University, Evanston, IL, where he is currently working toward the Ph.D. degree in electrical engineering. Milja Medic received the B.S. degree in physics in 2004 from the University of Belgrade, Belgrade, Serbia, and the M.S. degree in physics and astronomy in 2006 from Northwestern University, Evanston, IL, where she is currently working toward the Ph.D. degree in physics and astronomy at the Center for Photonic Communication and Computing. Her current research interests include experimental quantum optics and photon pair entanglement. Prem Kumar (M’89–SM’90–F’03) received the Ph.D. degree in physics from the State University of New York at Buffalo, Buffalo, in 1980. He is currently the AT&T Professor of Information Technology in the Department of Electrical Engineering and Computer Science and the Director of the Center for Photonic Communication and Computing in the McCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL, where he is also a Professor of physics and astronomy in the Weinberg College of Arts and Sciences. His current research interests include the applications of nonlinear and quantum optics in several areas of optical technology: in optical communications, the focus is on the development of novel optical amplifiers and devices for ultrahigh-speed (terabit per second) communications; in imaging, the emphasis is on the use of novel quantum states of light such as squeezed and entangled states to achieve sub-Rayleigh resolution and subshot noise sensitivity; and in quantum fiber optics, the focus is on the generation and distribution of quantum entanglement over fiber channels and practical quantum cryptography over fiber lines. He was the Associate Editor of Optics Letters. Prof. Kumar is a Fellow of the Optical Society of America, the American Physical Society, and the Institute of Physics, U.K. In 2006, he received the Martin E. and Gertrude G. Walder Research Excellence Award from Northwestern University. In 2004, he was the recipient of the 5th International Quantum Communication Award sponsored by Tamagawa University in Tokyo, Japan. He was the Program Cochair of the Quantum Electronics and Laser Science Conference (QELS), 2006, and the General Cochair of the QELS 2008. He was a Principal Organizer of the 4th International Conference on Quantum Communication, Measurement, and Computing, 1998.