Analysis of experimental error sources in a linear-optics quantum gate

Possible error sources in an experimentally realized linear-optics controlled-Z gate (Okamoto et al 2005 Phys. Rev. Lett. 95 210506) are analyzed by considering the deviations of the beam splitting ratios from the ideal values (δRH, δRV), the polarization-dependent phase shift (birefringence) of the optical components (δϕ) and the mode mismatch of input photons (δξ). It is found that the error rate is linearly dependent on δRV and δξ, while the dependence on δRH and δϕ is approximately quadratic. As a practical result, the gate is much more sensitive to small errors in RV than in RH. Specifically, the reflectivity error for vertical polarization must be less than 0.1% to realize a gate with an error of less than 0.1%, whereas the reflectivity error for horizontal polarization can be up to 1%. It is also shown that the effects of different error sources are not independent of each other (linear error model). Under certain conditions, the deviation from the linear error model exceeds 10% of the total error. The method of analysis used illustrates the basic features of errors in general linear optics quantum gates and circuits, and can easily be adapted to any other device of this type.


Introduction
Quantum information science enables us to improve the performance of certain tasks by exploiting quantum mechanical effects. Photons have been widely used for quantum communication [1], quantum metrology [2,3] and quantum lithography [4,5]. Using the dualrail scheme [6], qubits can be encoded in one photon in two modes, e.g. the polarization degrees of freedom. The main advantages of photonic qubits are their robustness against decoherence and their suitability for single-qubit operations. However, two-qubit gates require strong interaction between single photons, which may necessitate extremely high optical nonlinearity. In 2001, Knill et al discovered a way to overcome this problem by proposing a controlled-Z (CZ) gate where the correct gate operation is heralded by a success signal, resulting in a possible implementation of scalable quantum computation where only the heralded operations are selected [7]. This method uses linear optics, photon-number-resolving detectors [8,9] and single-photon sources [10]. Motivated by this discovery, a closely related proposal of a postselected CZ gate, where the successful gate operation is identified by post-selecting the correct photon distribution in the output modes, was introduced [11,12] and eventually realized by a very compact optical circuit using partially polarizing beam splitters (PPBSs) [13]- [15]. Similar linear optics quantum circuits have also been demonstrated, both for heralded [16] and post-selected [17,18] schemes.
In such quantum circuits, errors in one gate generally propagate to the next gate and thus the errors tend to accumulate. Consequently, it is critical to analyze the errors in each gate to achieve high-fidelity operation for quantum circuits. In an early study, Ralph et al [12] analyzed the output states of the linear-optics CZ gate for a typical set of input states. They considered two error sources, incorrect beam-splitting ratios and spatio-temporal mode mismatching at the two-photon interference, and calculated the fidelity of the output states to the ideal ones (Bell states). Kiesel et al [15] derived a full process matrix of a linear-optics CZ gate taking the gate 3 errors into account. Recently, Weinhold et al [19] considered the impact of errors observed in reconstructed process matrices on the fault tolerance of quantum computation. In particular, they pointed out the importance of controlling the photon numbers in the input modes.
In this paper, we consider the joint effect of errors in the optical elements and mode matching in the multi-photon interference by analyzing the dependence of the quantum coherent operation in the Hilbert space of photon polarization on the experimental parameters. Specifically, we include errors in the reflectivities and a possible polarization-dependent phase shift (birefringence), which has not been considered in the previous analysis [12,15,19] of the beam splitter. Another important source of errors may be the combination of excess input photons and photon loss [19]. However, this error mainly depends on the type of photon source used. Since this paper focuses on errors in the operation of the device itself, we chose not to include such errors.
We show that the effects of the reflectivity error in the vertical polarization are far more significant than the effects of errors in the horizontal polarization or the birefringence. We also investigate how the various causes of errors interact with each other. One of the problems in the comparison of different error sources is the fact that the errors are not independent of each other, so that a linear error model where the total error is given by a simple sum of all errors is not valid. We evaluate the extent to which the errors are influenced by the presence of other error sources and show that the deviation from the linear error model can exceed 10% of the total error under certain conditions. Furthermore, we discuss the requirements for achieving a small total error (e.g. 0.001), for future scalable optical quantum computations [20]. This paper is organized as follows. In section 2, we describe the method used to analyze the gate and the causes of errors included in our model. First, we briefly introduce a PPBS-based CZ gate and derive the operator describing the gate operation in the post-selected Hilbert space of two-photon polarization. Then, we introduce the experimental error sources and describe the modified gate operation in the post-selected Hilbert space. Finally, we show how the process fidelity and the associated error rate can be derived from this formulation of the imperfect gate operation. In section 3, we present and discuss the results of the analysis. We show how each error source contributes to the total error rate and identify the mutual dependence of error sources in terms of the deviation from the linear error model. Finally, the conditions for achieving a low error rate are discussed. Section 4 concludes the paper.

A compact CZ gate using PPBSs
The construction of CZ gates using PPBSs was reported in 2005 [13]- [15]. In this scheme, the separate path interferences of the original theoretical proposal [11,12] are replaced with a single polarization-dependent interference at a PPBS, making the gates more compact and robust against vibrations. Figure 1 shows a schematic diagram of the CZ gate. Two photons in modes C in and T in generated by single photon sources are simultaneously input to PPBS-A. The reflectivities of PPBS-A are 1/3 for horizontally polarized light and 1 for vertically polarized light. These reflectivities therefore generate two-photon quantum interference [21] for only the horizontally polarized photons. At the gate output, we select the cases in which single photons are emitted to both output modes, C out and T out . This combination of two-photon interference and post-selection flips the phase of the input state only when both of the input photons  are horizontally polarized. However, the operation also attenuates the probability amplitude of horizontally polarized photons while preserving that of vertically polarized photons. To compensate for this difference in the probability amplitudes of horizontally and vertically polarized photons, two PPBS-Bs are inserted at both output modes of PPBS-A. The gate then functions as a CZ gate for horizontal and vertical basis with a success probability of 1/9.
As described above, PPBS-A is the key component, while PPBS-B is a supplemental component. Indeed, the errors caused by PPBS-Bs can be easily compensated since the PPBS-B acts only on single-qubit inputs. Consequently, we consider only the errors caused by PPBS-A in the following analysis.

A unitary operator for a PPBS
A beam splitter is an optical element that has two spatial modes each as its input (a, b) and output (a , b ). We use the annihilation operatorsâ,b,â andb for photons in the corresponding modes (figure 2(A)). When there are no losses at the beam splitter, the relationship between the operators can be expressed using the following matrix representation for the beam splitter [22]: where R is the reflectivity of the beam splitter. In contrast, a PPBS has four modes for input and output by including the polarization degree of freedom. We use the annihilation operatorsâ H ,b H ,â V ,b V ,â H ,b H ,â V andb V for the corresponding modes (figure 2(B)). A horizontally (vertically) polarized photon at the input preserves its polarization at the output, but the reflectivities are different for the two orthogonal polarizations. The relationship between input and output operators can then be expressed as where φ is the phase difference between horizontally polarized light and vertically polarized light and R H and R V are the reflectivities of horizontally and vertically polarized light, respectively.

Operation formalisms for an ideal CZ gate
In the ideal case, the parameters of PPBS-A are R H = 1/3, R V = 1 and φ = 0, so that the matrix for PPBS-A can be expressed as When the input state of PPBS-A is | H, H a,b =â † Hb † H |0 corresponding to two horizontally polarized photons input from opposite signs, the output state can be derived using the transformation of the creation operators defined by equations (2) and (4) as follows: The complete transformation for an arbitrary state can be obtained by applying the transformation U PPBSA to the creation operators of the remaining three basis states, For the CZ operation, we post-select the cases where single photons are in both output modes, a and b . With ideal PPBS-Bs (R H = 0, R V = 2/3) in the output modes of the PPBS, the output state of our CZ gate will bê whereL = 1/3Û CZ describes the ideal gate process including the post-selection withÛ CZ = − |H, H H, H| + |H, V H, V| + |V, H V, H| + |V, V V, V|. This operation is essentially a CZ gateÛ CZ with a success probability of 1/9. By using the logical basis states |0 T ≡ (|V T + |H T ) and |1 T ≡ (|V T − |H T ) for the target qubit and |0 C ≡|V C and |1 C ≡|H C for the control qubit, this operation corresponds to a controlled-NOT gate.

Operation formalism for a CZ gate including error sources
In this paper, we consider the birefringence error and the non-ideal reflectivities introduced by imperfections in PPBS-A, along with the spatio-temporal mode mismatch at the two-photon interference. When using parametric down-conversion, another important source of errors may be the combination of excess input photons and photon loss [19]. However, this error mainly depends on the type of photon source used. Since this paper focuses on errors in the operation of the device itself, we chose not to include such errors. (3) shows how errors in the reflectivity and the phase shift between horizontal and vertical polarizations can be included in the operation of the gate. In the following, δ R H and δ R V will be used to describe the deviations of the reflectivities R H and R V from the ideal reflectivities (i.e. the reflectance errors):

Birefringence and non-ideal reflectivities of the PPBS. Equation
Likewise, φ in equation (3) becomes an error source (birefringence error). Here the 'birefringence error' originates from a polarization-dependent optical phase that is unintentionally added when light is reflected by the PPBS. We define the birefringence error as the deviation of φ from its ideal value of zero,

Spatio-temporal mode mismatch.
For the ideal operation, equation (9) was derived by assuming that the input photons are in single modes and are perfectly mode-matched (i.e. they are indistinguishable) at PPBS-A. In practice, however, the modes of these photons are partially mismatched in space and time or frequency due to factors such as misalignment or intrinsic jitter of the single-photon arrival time. These types of errors can be analyzed using ancillary modes (â H1 ,b H1 ,â V1 ,b V1 ) (figure 3) and the mode-matching parameter ξ . Here, we define new operators for input port A to include the effect of the mode mismatch aŝ whereâ H (â V ) interferes withb H (b V ) perfectly, whileâ H1 (â V1 ) does not interfere withb H (b V ) at all. In the following, the photon in input port A is considered to be a superposition of these two modes (â H0 (â V0 )), while the photon in input port B is in modeb H (b V ). Note that the use of a mixture of operators instead of a superposition in equations (13) and (14) results in the same result. The parameter ξ is equivalent to the visibility of the two-photon Hong-Ou-Mandel interference at a 50/50 beam splitter (appendix). Just as for the other error sources, we define the deviation of δξ from the ideal value of 1 as Since the modesâ H andâ H1 are orthogonal, the output modes ofâ H andâ H1 are also orthogonal. We can therefore assume that the ancillary modes form an additional set of modes transformed by the PPBS-A according to whereâ H1 ,b H1 (â V1 andb V1 ) are the output modes shown in figure 3.

Process operators and process fidelity
To analyze the process fidelity in the presence of errors, we need a complete description of the post-selected gate process with errors such as the one given in equation (9). To obtain such a description, we follow a procedure similar to that given by equations (5)- (8), substituting the modified values of R H , R V and φ. To include the mode matching effect, we first consider a complete mode mismatch. In that case, photons in the input modesâ i are distinguishable from photons in the input modesb i , and the reflection process of both photons at the PPBS does not interfere with the transmission process of both photons. Therefore, the effects of reflection and transmission processes on the polarization state of the two photons can be described by two separate operators,Ŝ R andŜ T . Assuming that the additional PPBS-Bs in the output are ideal (R H = 0, R V = 2/3), In the case of perfect mode matching (ξ = 1), two-photon reflection and two-photon transmission are indistinguishable and the amplitudes of the output states interfere. Taking into account the correct phase relation given by the field operators, the effects of this quantum coherent operation on the two-photon polarization state are then given bŷ In the general case described by equations (13) and (14), the coherent operation has an amplitude of √ ξ and the mismatched output components have an amplitude of √ 1 − ξ each. After tracing out the ancillary degrees of freedom, this results in a statistical mixture of the coherent operation S QC with a statistical weight of ξ = 1 − δξ and the mismatched componentsŜ R andŜ T with a statistical weight of 1 − ξ = δξ . The process can then be represented byρ out = E(ρ in ) [23] with This equation was also used in the error analysis of Kiesel et al [15] for the estimation of a mode matching error in their experiment. The process E describes a linear map between the elements of the input density matrix and the elements of the output density matrix within the Hilbert space of two-photon polarization. This linear map can be expressed as a process matrix obtained by expanding the input density matrix and the output density matrix in terms of the operators {|i j|} associated with their matrix elements ρ i j . Each column of the process matrix then corresponds to one of the 16 matrix elements of the input density matrix, and each line 9 corresponds to one of the 16 matrix elements of the output matrix. For example, the column of the process matrix for the input matrix element with i = j = HH is given by All the other 15 columns can be obtained in the same manner.
To compare the noisy process E with the quantum coherent operationÛ CZ given by equation (9), we use the process fidelity F proc. . In terms of the process matrix expansion given above, the process fidelity is [24] whereÎ is the identity operator in the four-dimensional Hilbert space of two photons each having two polarization degrees of freedom. Equations (20) and (22) define the process fidelity for any combination of experimental error sources. In the following, we will discuss the results for several interesting cases with the intention of defining the conditions for achieving fidelities close to one. We will therefore evaluate the effects of the errors in terms of the error rate , defined as the difference between the process fidelity and a perfect process fidelity of one, Since the error rate is zero when all the errors are zero, the dependence of the error rate on the experimental error sources δ R H , δ R V , δφ and δξ is a direct measure of the impact of these errors on the performance of the linear optics quantum gate.

Results and discussion
In this section, we discuss the effect of error sources on the total error rate of the CZ gate. The error sources considered are the reflectance errors due to the deviations (δ R H = R H − 1/3 and δ R V = 1 − R V ) of the actual reflectivities of PPBS-A (R H , R V ) from the ideal ones (R H = 1/3, R V = 1), the birefringence error (δφ) due to the phase difference (0 in the ideal case) between the horizontally polarized photons and the vertically polarized photons after passing through PPBS-A, and the mode mismatch error (δξ = 1 − δξ ) due to the spatio-temporal mode mismatch between the two input photons.

The reflectance error
The error rate for reflectance errors only (δφ = 0, δξ = 0) is shown in figures 4(A)-(C). Figure 4(A) shows the error rate as a function of δ R H with δ R V = 0 (red solid curve) and δ R V = 0.015 (blue dashed curve). For small δ R H , the dependence of on the reflectance errors is approximately quadratic. This means that the error due to δ R H rapidly diminishes for small δ R H . Note that the precise dependences of on δ R H are asymmetric to the vertical line δ R H = 0; this is discussed in detail below. Figure 4  diverge as δ R V increases. This indicates that the errors due to δ R H and δ R V interact with each other.
To analyze this interaction of the error sources, figure 4(C) shows a contour plot of . The horizontal axis represents δ R H and the vertical axis represents δ R V . The red solid line indicates the optimum value of δ R H that minimizes for a certain value of δ R V . The line leans to the right, showing that errors in R H can partially compensate for errors in R V . This is a significant practical difference from the predictions of a linear error model, where errors can only add up.
To evaluate the interaction between different error sources, we define the deviation from the linear error model δ as   Figure 5 shows the error rate caused by a birefringence error. Again, the dependence of on δφ is almost quadratic in all three cases, indicating that the effects of the birefringence error decrease rapidly as the error gets smaller. Note that, in the absence of other errors, the effect of the birefringence error (δφ) can be given in a particularly simple analytical form, by substituting δ R H = δ R V = δξ = 0 into equation (23): Figures 5(B) and (C) show contour plots of for combinations of δφ with δ R H and δ R V , respectively. In both cases, is symmetric about a vertical line at φ = 0. However, figure 5(B) shows a slight asymmetry with respect to the horizontal line at δ R H = 0. As in figure 4(C), the value of δ R H that minimizes for a certain value of δφ is not zero. This indicates that the errors of R H and φ can partially compensate each other, just like the errors of R H and R V . On the other hand, figure 5(C) does not show this kind of dependence. For any given δφ, is always minimized at δ R V = 0. However, this does not imply that these two factors are completely independent, since symmetric deviations from the linear error model are still possible.  Figure 6(A) shows that is asymmetric with respect to δ R H , similar to figure 4(A). In figure 6(C), this asymmetry results in compensation of errors (negative δ ) for positive δ R H , while the errors increase (positive δ ) for negative δ R H . At δ R H = 0.04 and δφ = 20 (deg), δ is −0.0067, which is 9.7% of the total error at this point ( = 0.0686). Thus, almost 10% of the error is compensated by the interaction between the two error sources. In contrast, δ between δ R V and δφ is relatively small. At δ R V = 0.015 and δφ = 20 (deg), δ is −0.0011, which is only 1.5% of the total error ( = 0.0735). Note that the comparison of these two cases is motivated by the fact that these combinations of δ R H and δ R V with δφ = 20 (deg) have almost the same value of in the linear error model. Figure 7 shows a plot of as a function of δξ . Figure 7(A) shows the region 0 < δξ < 1 and figure 7(B) shows the region 0 < δξ < 0.2. These plots are calculated assuming that there are no device errors (δ R H = 0, δ R V = 0, δφ = 0). In the region where δξ is sufficiently small, is almost linear in δξ . This means that the mode mismatch error remains a significant source of error even if δξ is very small. In the region shown in figure 7(B), an increase of δξ by about

Conditions for a low-error gate
In this section, we consider the quality of the PPBS-A and the mode matching required for high-fidelity low-error gates. We estimated the values of parameters required to suppress to under 0.05, 0.01 and 0.001 for several conditions (table 1). About the error probability per gate for scalable quantum computation, 10 −4 has been often quoted but 10 −3 was suggested recently by Knill [20]. Unfortunately, the gate we analyzed is probabilistic (1/9) even when we assume the QND photon number detector in the output modes. Thus the 10 −3 error threshold cannot be applicable to our case directly. However, we think that the analysis of error parameters for this threshold is informative. Firstly, we evaluated the cases where only one error source exists and the other parameters are ideal. δ R H has positive and negative values, but here we show only the positive ones (δ R H > 0). To satisfy = 0.001, δ R V must be ten times smaller than δ R H . At = 0.05, the ratio is only about 1.5. This difference illustrates the difference between the quadratic dependence of on δ R H and the linear dependence of on δ R H . Likewise, the condition for mode matching becomes very severe at low errors. Table 1 thus shows that, for small errors, the most significant error sources are δ R V and δξ .
Condition (ii) shows the mode matching requirement for an actual PPBS-A tested in our laboratory. The properties of this PPBS-A were δ R H = 0.022, δ R V = 0.020 and δφ = 3.5 (deg). The PPBS-A is a cubic beam splitter made of an optical glass prism (BK7) with a specially designed dielectric multi-layer coating. This PPBS-A was designed to reduce the birefringence error; a conventional PPBS is therefore likely to have an even larger value of δφ. As indicated in the table, we cannot suppress to 0.01 or below in this case, because the value of δ R V = 0.020 is larger than the limit of 0.010 required to achieve such error rates.
Finally, we calculated the case when the errors from all parameters are equal (condition (iii)). We searched for values of the parameters (δ R H , δ R V , δφ, δξ ) that satisfy (δ R H ) = (δ R V ) = (δφ) = (δξ ) and (δ R H , δ R V , δφ, δξ ) = 0.05 or 0.01, or 0.001. As a result, the errors shown in table 1 decrease linearly with the error requirement for δ R V and δξ , while the corresponding errors of δ R H and δφ decrease only with the square root. Again, the result shows that δ R V and δξ are the main sources of error for low-error gates.

Conclusions
We analyzed the errors in a linear-optics CZ gate caused by combinations of imperfect reflectivities and birefringence in the main optical component and a mode mismatch error between the separate input photons. We found that error rate depends linearly on the reflectivity error δ R V for the vertically polarized photons and the mode matching error δξ , while its dependence on the reflectivity error δ R H for the horizontally polarized photons and the birefringence error δφ is only quadratic. We also evaluated the interaction between the error sources. In particular, we found that the effect of δ R H is strongly correlated with the effects of δφ and δξ . In the case of a phase error of 20 degrees shown in figure 6, the deviation from the linear error model was found to be around 10% of the total error, indicating that the deviation from the linear error model can be quite significant under certain conditions. The requirements for achieving high-fidelity gates were investigated for some specific cases, illustrating the significance of limiting the errors given by δ R V and δξ for the construction of low-error gates. Even though high-reflectivity mirrors with the sum of the transmittance and the loss (∼1 − R) less than 10 −5 has been realized [25], accurate control of the reflectivity and the phase may need future technical development. For minimizing spatial mode mismatch, gates in fibers [26] or waveguides [27] would be a suitable solution. Single-photon sources with small temporal jitter [28] are very important to minimize temporal mode mismatch. The analysis in this paper can easily be adapted to other linear optics quantum gates and may thus provide the foundation for a better understanding of experimental errors in a wide range of optical quantum information circuits. The cause of the different behavior of errors for the gate fidelity will be an interesting study.
Therefore, the probability of coincidence detection at both outputs is On the other hand, the coincidence detection probability when two photons arrive at the beam splitter with a sufficiently large time difference is P N Q = 1/2. Therefore, the visibility of the Hong-Ou-Mandel interference in this case is