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Abstract
Linear optical multiports are widely used in photonic quantum information processing. Naturally, these devices are directionally-biased since photons always propagate from the input ports toward the output ports. Recently, the concept of directionally-unbiased linear optical multiports was proposed. These directionally-unbiased multiports allow photons to propagate along a reverse direction, which can greatly reduce the number of required linear optical elements for complicated linear optical quantum networks. Here, we report an experimental demonstration of a 3 × 3 directionally-unbiased linear optical fiber multiport using an optical tritter and mirrors. Compared to the previous demonstration using bulk optical elements which works only with light sources with a long coherence length, our experimental directionally-unbiased 3 × 3 optical multiport does not require a long coherence length since it provides negligible optical path length differences among all possible optical trajectories. It can be a useful building block for implementing large-scale quantum walks on complex graph networks.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Quantum walks (QW), the quantum analogue of classical random walks, provide promising methodologies to execute complex quantum algorithms for quantum computing [1,2]. As random walks are used for algorithm design in classical computer, many quantum algorithms are developed based on QW which provide speed-up over the best known classical algorithms for the same problem [3]. Furthermore, this approach has capability of performing algorithms for universal quantum computation [4–8]. Linear optics has been considered as one of the most promising candidates for implementing QW, because of its robustness against decoherence, ease of preparation of quantum states and its scalability through on-chip integration [9–12].
QW are movements of walkers on a graph, which consist of connected vertices with N edges [13–16]. A walker moves from a vertex to one of the neighboring vertices at each step. In linear optics, each vertex can be implemented by an $\textit {N} \times \textit {N}$ multiport which is a linear optical element having N input ports and $N$ output ports. A set of phase shifters and $2 \times 2$ beam splitters (BSs) can construct an $N \times N$ multiport with $N\geq 3$ [17,18]. Since directionality of a BS forbids input photons to move back to the same port where the input photons entered, an implementation of an $N \times N$ multiport forms a unidirectional expanding tree-like structure to cover all possible optical paths. From the inability of reverse propagation, this unidirectional implementation is called a directionally-biased multiport.
Recently, a directionally-unbiased multiport has been proposed [19]. Unlike a directionally-biased multiport, it allows an input photon to leave the same port where the input photon entered. Hence, it is suitable for implementing undirected graphs for QW.
It has been reported only an experimental implementation of an $N \times N$ directionally-unbiased multiport for $N\geq 3$ [20]. Osawa et al. implemented a $3 \times 3$ directionally-unbiased multiport based on an optical cavity in free space using bulk optical elements [20]. However, this scheme requires input photons having a long coherence length due to large path length differences among the different trajectories in the cavity. Indeed, it was experimentally tested with a HeNe laser beam whose coherence length is longer than 1 km. It does not show the quantum optical phenemoena in directionally-unbiased multiports, and yet there has been no experimental demonstration of directionally-unbiased multiport using quantum linght sources.
In this paper, we experimentally demonstrate a $3 \times 3$ directionally-unbiased linear optical multiport using a $3 \times 3$ optical fiber BS, or a tritter. By measuring the two-photon Hong-Ou-Mandel (HOM) interference [21] and amplitude distribution ratios of the $3 \times 3$ directionally-unbiased multiport, we reconstructed a transfer matrix of the implemented multiport. All results are obtained using quantum light sources. We believe that our directionally-unbiased multiport device can be useful building blocks for simulating QWs on complex graph networks.
2. Theory
2.1 Directionally-biased/unbiased multiports using fiber optical multiports
Let us consider a $3 \times 3$ linear optical multiport as shown in Fig. 1(a) and (b) where a tritter is a $3 \times 3$ BS based on a fused optical fiber [22]. These multiports form an interferometer to control an amplitude distribution among input and output modes using phase shifters. Figure 1(a) describes a directionally-biased $3 \times 3$ linear optical multiport. This model, an expanded Mach-Zehnder interferometer, consists of two tritters and two phase shifters. Then, the transfer matrix of the $3 \times 3$ directionally-biased linear optical multiport is given by
where $U_{F}$ is a transfer matrix of a forward directional tritter and $\varPhi$ is a phase shift operation. For example, the transfer matrix of an ideal tritter is given byThe phase shift operation $\varPhi$ can be represented by the following unitary phase matrix
Fig. 1. Schematic designs of (a) a directionally-biased $3 \times 3$ optical multiport (b) a directionally-unbiased $3 \times 3$ optical multiport, and (c) a directionally-unbiased $N \times N$ optical multiport. The relative phase of the mode 0, $\phi _{0}$, is set to zero as a reference. A directionally-biased mulitport consists of two tritters and two phase shifters $\phi _{1}$ and $\phi _{2}$, which can control the amplitude distribution of the multiport. Photons are inserted to the input ports $a_0$, $a_1$ and $a_2$, and then come out from the output ports $b_0$, $b_1$ and $b_2$. On the other hand, a directionally-unbiased multiport consists of one tritter and three mirror units. In this design, the input photons can move back to the same input ports $a_0 \& b_0$, $a_1 \& b_1$, and $a_2 \& b_2$, which means that the input and output ports are not separated. A directionally-unbiased $N\times N$ optimal multiport can be implemented by replacing a tritter with an $N\times N$ multiport and adding a mirror in each mode. The red and blue arrows correspond to the directions of the input and the output modes, respectively.
On the other hand, a directionally-unbiased $3 \times 3$ linear optical multiport is shown in Fig. 1(b). This scheme can be considered as an expanded Michelson interferometer, where the input and the output ports are the same. It can be realized by placing a mirror at each output mode of a tritter, and then the photons coming out from the output modes of a tritter is reflected back and enter the output modes of a tritter. The relative phase shift at each mode can be controlled by adjusting the position of mirrors. Note that the relative phase shift $\phi _{1}$ and $\phi _{2}$ provide the tunability on the transfer matrix of a directionally-unbiased multiport.
The transfer matrix of a directionally-unbiased $3 \times 3$ multiport $U_{\textrm {unbiased}}$ is similar to a directionally-biased one $U_{\textrm {biased}}$ except the fact that an operation of the second tritter corresponds to the backward propagation of the first one. Therefore, the overall transfer matrix is given by
where $U_{B}$ is a transfer matrix of a tritter in a backward direction, and it is equivalent to a transpose of a tritter operation in a forward direction (i.e. $U_{B}=U_{F}^{T}$) [22]. As shown in Fig. 1(c), a directionally-unbiased multiport can be extended to an $N \times N$ multiport.Although a directionally-biased device can implement any unitary matrices, reversibility of the directionally-unbiased linear-optical device imposes symmetry of the self-transpose property $U_{i,k}=U_{k,i}$ where $i,k$ are matrix indices. Therefore only a subset of symmetric unitary matrices can be implemented by a directionally-unbiased device [22]. However, note that a directionally-unbiased design can still produce the Grover coin and the Fourier coin, which are considered important quantum algorithms for quantum simulation [13,22].
2.2 Reconstruction of a transfer matrix
Here we discuss the experimental reconstruction method of a transfer matrix of a general $3 \times 3$ linear optical multiport, which has three input ports {$a_{0}$, $a_{1}$, $a_{2}$} and three output ports {$b_{0}$, $b_{1}$, $b_{2}$}. It is known that an unknown $3 \times 3$ linear optical multiport can be reconstructed by measuring the classical amplitude distribution of the multiport and the two-photon HOM quantum interference [23,24].
An unknown $3 \times 3$ unitary matrix $U$ can be represented by
By varying the relative time delay between two input photons, we can observe the HOM interference that leads to peaks or dips in the output coincidence events when the relative time delay is zero. The HOM visibility between two input photons is determined by the phase values in the matrix $U$ and is given by
Note that a positive (negative) value of $\mathcal {V}_{ij}^{kl}$ corresponds to a dip (peak).
From the amplitude distribution and the HOM visibility results, an unknown transfer matrix $U$ can be numerically reconstructed by minimizing the root mean square distance between two visibility matrices, which is given by
where $\mathcal {V}_{M}$ is an experimentally measured visibility matrix and $\mathcal {V}_{U}$ is the visibility matrix calculated from the reconstructed transition matrix $U$. Numerical optimization consists of two steps. At first, relative phases $\phi _\alpha$, $\phi _\beta$, $\phi _\gamma$, and $\phi _\delta$ vary while the probability distribution ratio is fixed. The fixed distribution ratio is obtained from the measured amplitude distribution ratio $|u_{ki}|^{2}$. This step is for determining initial relative phase values. Secondly, both phases and probability distribution ratios vary for optimization.We quantified the quality of reconstruction using the similarity $S$, which is defined as [24]
where the similarity can be $0\le S\le 1$. Note that $S=0$ and $1$ correspond to a completely anti-correlated and perfect overlap, respectively.3. Experiment
The schematic of our experimental setup to implement and analyze a $3 \times 3$ directionally-unbiased optical fiber multiport is shown in Fig. 2. We generate a pair of identical single photons with the center wavelength of 1556 nm via type-2 spontaneous parametric down conversion (SPDC) process using a 10 mm-thick PPKTP crystal (poling period of 46.15 $\mu$m) pumped by femtosecond laser pulses (not shown in Fig. 2). The power and repetition rate of the pump are 20 mW and 125 MHz, respectively. The indistinguishability between photons is verified by the HOM interference visibility of $\mathcal {V} \sim 0.99$ [25]. In our experiment, the probability that two photons enter the same input mode is highly suppressed since multiple photon pair generation rate is negligible. Note that the coherence length of our down-converted photons is $\sim 440 \mu$m, which is much narrower than the coherence length of a He-Ne laser.
Fig. 2. Experimental setup for implementing a directionally-unbiased $3 \times 3$ multiport. Whole system is enclosed by an outer cage for minimizing detrimental noise from external environment so that the relative phases $\varPhi$ is maintained. In our experiment, $\varPhi$ is maintained for tens of minutes. Since it takes 4 minutes to obtain the whole data set for the transfer matrix reconstruction, it is enough to obtain the experimental result without phase fluctuation. In order to confirm that the $\varPhi$ is maintained during the experiment, we have checked that the amplitude distribution ratios are maintained during the experiment. HWP: half-waveplate, PBS: polarizing beam splitter, SNSPD: superconducting nanowire single photon detectors.
By using a set of waveplates and polarizing beam splitters (PBSs), we can prepare the input state $\hat {a}^{\dagger }_{i}\hat {a}^{\dagger }_{j}|0\rangle$ and $\hat {a}^{\dagger }_{i}|0\rangle$ with horizontal polarization where $i,j=0,1,2~(i\neq j)$ for measuring visibility of the two-photon HOM interference and amplitude distributions, respectively. In order to choose the combination of the input ports, we adjust the polarization of the input photons using a set of half waveplates (Hs) as shown in the optical switch of Fig. 2. For example, $\hat {a}^{\dagger }_{0}\hat {a}^{\dagger }_{1}|0\rangle$ input state can be prepared by setting the angles of (H1, H2, H3, H4, H5) as (0$^{\circ }$, 0$^{\circ }$, 0$^{\circ }$, 0$^{\circ }$, 45$^{\circ }$). Additionally, we can obtain either $\hat {a}^{\dagger }_{0}|0\rangle$ or $\hat {a}^{\dagger }_{1}|0\rangle$ input state by setting the angles of (H3, H4) as either (0$^{\circ }$, 45$^{\circ }$) or (45$^{\circ }$, 0$^{\circ }$), respectively. Using this method, we can send all combinations of single and two-photon input states into our $3 \times 3$ multiport.
Note that photons at each input mode should have the same polarization for indistinguishability between two photons. We can adjust the polarization of the photons inside a tritter using fiber polarization controllers. Moreover, Faraday rotators located between the output mode of a tritter and a mirror guarantee that photons propagating in the opposite directions do not interfere with each other. We can isolate the photons coming out from the input ports using optical fiber circulators, and send them toward superconducting nanowire single photon detectors (SNSPDs) for single photon detection. Single photon count rates for $\hat {b}^{\dagger }_{k}|0\rangle$ with input states $\hat {a}^{\dagger }_{i}|0\rangle$ are obtained for measuring amplitude distribution ratios, and all possible combinations of the two-photon coincidence detection events $\hat {b}^{\dagger }_{k}\hat {b}^{\dagger }_{l}|0\rangle$ where $k,l=0,1,2~(k\neq l)$ with input states $\hat {a}^{\dagger }_{i}\hat {a}^{\dagger }_{j}|0\rangle ~(i\neq j)$ for measuring the two-photon HOM interference are obtained using a home-made coincidence counting unit (CCU) [26,27].
For measuring visibilities of the two-photon HOM interference, we introduced an optical delay $\Delta$ between two input photons by scanning the position of one collimator using a motorized linear stage. We measure all possible combinations of the two-photon HOM interference fringe for each input state $\hat {a}^{\dagger }_{i}\hat {a}^{\dagger }_{j}|0\rangle$. In order to obtain amplitude distribution ratios of the multiport, all possible single photon count rates $n_{ki}$ are measured for each input state $\hat {a}^{\dagger }_{i}|0\rangle$. The position of the motorized linear stage is rearranged whenever the input state is changed, to set an optical delay $\Delta =0$.
4. Experimental results
Here, we experimentally reconstruct a transfer matrix of our implemented directionally-unbiased multiport devices as shown in Fig. 2 using two independent reconstruction methods. First, we directly reconstruct a transfer matrix $V_{1}$ as Eq. (5) by minimizing Eq. (10) from data set 1: $\mathcal {V}_{M_{1}}$ and $|u_{M_{1}}|^{2}$ where measured HOM visibility and the amplitude distribution. Second, we reconstruct the transfer matrix $W_{1}=U_{B} \varPhi_1 U_{F}$ as Eq. (4), and by independently reconstructing $U_{F}$ and $U_{B}$, and $\varPhi _{1}$. Then, we compare two reconstructed transfer matrices $V_{1}$ and $W_{1}$. We also obtained another set of transfer matrices $V_{2}$ and $W_{2}$ with phase value $\varPhi _{2}$ to show the controllability of phase $\varPhi$ in our implementation from data set 2: $\mathcal {V}_{M_{2}}$ and $|u_{M_{2}}|^{2}$. We provide these results in the Appendix C.
We first experimentally reconstruct the optimal transfer matrix $V_{1}$ of our $3 \times 3$ directionally-unbiased multiport using the measured HOM visibility matrix $\mathcal {V}_{M_{1}}$ and the amplitude distribution $|u_{M_{1}}|^{2}$. All possible combinations of the HOM interferences are measured, and experimental results and corresponding fitting functions are shown in Fig. 3 (a), (b), and (c). From the experimental results, $\mathcal {V}_{M_{1}}$ is obtained to be
Fig. 3. Two-photon HOM interference results of data set 1 and data set 2 for input states (a) (d) $\hat {a}^{\dagger }_{0}\hat {a}^{\dagger }_{1}|0\rangle$, (b) (e) $\hat {a}^{\dagger }_{0}\hat {a}^{\dagger }_{2}|0\rangle$, and (c) (f) $\hat {a}^{\dagger }_{1}\hat {a}^{\dagger }_{2}|0\rangle$. Each data set is measured to obtain a transfer matrix $V_{1}$ and $V_{2}$. Green, red, and blue dots correspond to the experimental two-photon coincidence counts with detection basis $\hat {b}^{\dagger }_{0}\hat {b}^{\dagger }_{1}|0\rangle$, $\hat {b}^{\dagger }_{0}\hat {b}^{\dagger }_{2}|0\rangle$, and $\hat {b}^{\dagger }_{1}\hat {b}^{\dagger }_{2}|0\rangle$, respectively. Solid lines are fitting functions and we calculate the visibility of the HOM interference from the fitting results. Error bars correspond to one standard deviation.
From the measured count rate, the amplitude distribution $|u_{M_{1}}|^{2}$ is obtained by using Eq. (6). The matrix of the amplitude distribution $|u_{M_{1}}|^{2}$ is obtained to be
From the measurement results $\mathcal {V}_{M_{1}}$ and $|u_{M_{1}}|^{2}$, we numerically found a transfer matrix $V_{1}$ of our implemented directionally-unbiased $3 \times 3$ linear optical multiport. The experimentally reconstructed matrix $V_{1}$ and the corresponding standard deviation $\sigma _{V_{1}}$ are given by
Fig. 4. Real and imaginary parts of the experimentally reconstructed transfer matrices of (a) $V_{1}$ and (c) $V_{2}$ obtained by optimizing Eq. (10) and (b) $W_{1}$ and (d) $W_{2}$ obtained by reconstructing each operation in Eq. (4). The fidelity between (a) and (b) [(c) and (d)] is calculated to be $F(V_{1}, W_{1})=0.971\pm 0.005$ [$F(V_{2},W_{2})=0.966\pm 0.006$].
In order to confirm that our implemented multiport is a directionally-unbiased $3 \times 3$, we independently reconstruct a transfer matrix $W=U_{B}\varPhi U_{F}$. To obtain $W_{1}$, we independently reconstruct $U_{F}$ and $U_{B}$ by measuring the HOM visibility matrix and the amplitude distribution, respectively. We performed this measurement by removing mirror units and connecting the outputs of the tritter into the SNSPDs. The obtained HOM visibility matrix and the amplitude distribution data are presented in Appendix A. With these data, the experimentally reconstructed $U_{F}$ and the corresponding standard deviation $\sigma _{F}$ are
By substituting reconstructed $U_F$ and $U_{B}$ to Eq. (4), we numerically obtain the $\varPhi _{1}$ by minimizing the $RMS2=\Sigma (\mathcal {V}_{W_{1}}-\mathcal {V}_{M_{1}})^{2}$ by varying the $\phi _1$ and $\phi _2$ of $\varPhi _{1}$. Here $\mathcal {V}_{W_{1}}$ denotes the calculated visibility matrix from $W_{1}$. By combining the obtained phase parameters $[\phi _{1}, \phi _{2}]=[(0.383\pm 0.014)\pi , (-0.596\pm 0.010)\pi ]$ and the transfer matrices $U_{B}$ and $U_{F}$, we obtain the transfer matrix of the implemented $3 \times 3$ directionally-unbiased multiport $W_{10}$. Then, we can obtain the corresponding equivalent real-bordered matrix $W_{1}$ and standard deviation $\sigma _{W_{1}}$, which are given by
In order to compare the reconstructed transfer matrices $V_{1}$ and $W_{1}$, we use fidelity $F$ between two matrices, which is defined by [28]
where $0\le F\le 1$. High fidelity means matrices are similar with each other. The obtained fidelity is $F(V_{1}, W_{1})=0.971\pm 0.005$, meaning that these two transfer matrices are highly similar to each other and verifies Eq. (4).5. Conclusion
We experimentally demonstrated a $3 \times 3$ directionally-unbiased linear optical multiport consisting of a tritter and mirrors. We reconstructed the transfer matrix from the experimental two-photon HOM visibility and amplitude distributions via two independent reconstruction methods, and confirmed that two methods give the similar results in terms of the fidelity. Comparing to the previous demonstration of a $3 \times 3$ directionally-unbiased multiport based on bulk optics elements [20], our $3 \times 3$ directionally-unbiased linear optical multiport does not require a long coherence length of the light source. We also note that our experiment is the first demonstration of quantum optical phenomenon using $3 \times 3$ directionally-unbiased multiport. Although the total transmittance in our setup is relatively low value of 0.39, it does not influence on the reconstruction. We carefully adjusted the in-and-out fiber coupling efficiency so that the amount of loss in each port is identical. In addition, we used a post-selection method to obtain normalized data in our experiment. Thus, the reconstructed transfer matrix is still a unitary matrix. We expect that this amount of loss can be strongly reduced by using optical fiber based Faraday mirrors instead of the free-space collimator and mirrors. We emphasize that our scheme can be easily extended to implement a $N \times N$ directionally-unbiased linear optical multiport by replacing a tritter into a $N \times N$ multiport BS.
Since a directionally-unbiased multiport can be a single vertex having multiple edges in a graph, we can perform QW simulation on a complex graph networks by combining a number of directionally-unbiased multiports. For example, our $3 \times 3$ directionally-unbiased multiport can construct a hexagonal lattice. Complex graph networks composed of directionally-unbiased multiports can also be used for simulating various Hamiltonians and topological physics [5,29–31]. Moreover, our directionally-unbiased multiport can be imprinted on an integrated photonic circuits [23] so that the phases of the multiport can be even more stabilized and the relative phases can be tunable electrically. We believe that our results can provide a practical platform to perform quantum simulations on complex graph networks.
Appendix A: experimental results of a forward and backward propagating tritter
For direct reconstruction of a single tritter transfer matrix $U_{F}$ and $U_{B}$, we experimentally obtained the HOM visibility and amplitude distributions of our tritter in both forward and reverse directions. Reverse propagation of the tritter is realized by exchanging the input and output ports of the tritter device. Experimental results are
Appendix B: phase parameters to obtain the equivalent real-bordered unitary matrix
The relationship between two equivalent matrices $W_{1}$ and $W_{10}$ is presented as
Since both $W_{1}$ and $W_{10}$ give the same HOM visibility and amplitude distributions, $W_{1}$ is also a transfer matrix of our $3 \times 3$ directionally-unbiased multiport. The phase factor $\phi _{m}$ with $m=a, b, c, d, e$ correspond to the relative phase on the input and output ports of the multiport. Each $\phi _{m}$ is determined by solving first order equations, which satisfy the real-border condition that $\left \lbrace W_{1}\right \rbrace _{k1}$ and $\left \lbrace W_{1}\right \rbrace _{1i}$ are real numbers for all $k$ and $i$.
Appendix C: another reconstructed matrix $V_{2}$ and $W_{2}$
To obtain a transfer matrix of the directionally-unbiased multiport with phase $\varPhi _{2}$, we followed the same matrix reconstruction process as described in Section 4. From the experimental results shown in Fig. 3(d), (e) and (f), measured visibility $\mathcal {V}_{M_{2}}$ is obtained to be
The amplitude distribution matrix $|u_{M_{2}}|^{2}$ is also obtained to be
From Eq. (29), the numerically found transfer matrix $V_2$ and the corresponding standard deviation $\sigma _{V_{2}}$ are given by
Then we numerically obtain the $\varPhi _{2}$ by minimizing the $RMS2=\Sigma (\mathcal {V}_{W_{2}}-\mathcal {V}_{M_{2}})^{2}$, where $\mathcal {V}_{W_{2}}$ is the calculated visibility matrix from $W_{2}$. The obtained phase parameters are $[\phi _{1},\phi _{2}]=[(0.061\pm 0.011)\pi , -(0.291\pm 0.017)\pi ]$, and the corresponding real-bordered matrix $W_{2}$ and the standard deviation $\sigma _{W_{2}}$ are given by
Funding
Korea Institute of Science and Technology (2E31021); Institute for Information and Communications Technology Promotion (2020-0-00947, 2020-0-00972); National Research Foundation of Korea (2019M3E4A1078660, 2019M3E4A1079777, 2021R1C1C1003625).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. J. Kempe, “Quantum random walks: an introductory overview,” Contemp. Phys. 44(4), 307–327 (2003). [CrossRef]
2. S. E. Venegas-Andraca, “Quantum walks: a comprehensive review,” Quantum Inf. Process. 11(5), 1015–1106 (2012). [CrossRef]
3. A. M. Childs, E. Farhi, and S. Gutmann, “An example of the difference between quantum and classical random walks,” Quantum Inf. Process. 1(1/2), 35–43 (2002). [CrossRef]
4. A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F. Galvão, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, “Integrated multimode interferometers with arbitrary designs for photonic boson sampling,” Nat. Photonics 7(7), 545–549 (2013). [CrossRef]
5. D. S. Simon, S. Osawa, and A. V. Sergienko, “Joint entanglement of topology and polarization enables error protected quantum registers,” New J. Phys. 20(9), 093032 (2018). [CrossRef]
6. A. M. Childs, “Universal computation by quantum walk,” Phys. Rev. Lett. 102(18), 180501 (2009). [CrossRef]
7. Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48(2), 1687–1690 (1993). [CrossRef]
8. D. Aharonov, A. Ambainis, J. Kempe, and U. Vazirani, “Quantum walks on graphs,” in Proceedings of the 33rd annual ACM symposium on theory of computing (ACM, 2001), pp. 50–59. [CrossRef]
9. J. Carolan, C. Harrold, C. Sparrow, E. Martin-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Mattews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349(6249), 711–716 (2015). [CrossRef]
10. J. Carolan, J. D. A. Meinecke, P. J. Shadbolt, N. J. Russell, N. Ismail, K. Wörhoff, T. Rudolph, M. G. Thompson, J. L. O’Brien, J. C. F. Matthews, and A. Laing, “On the experimental verification of quantum complexity in linear optics,” Nat. Photonics 8(8), 621–626 (2014). [CrossRef]
11. M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” Nat. Photonics 7(7), 540–544 (2013). [CrossRef]
12. H. Tang, X. F. Lin, Z. Feng, J. Y. Chen, J. Gao, K. Sun, C. Y. Wang, P. C. Lai, X. Y. Xu, Y. Wang, L. F. Qiao, A. -L. Yang, and X. M. Jin, “Experimental two-dimensional quantum walk on a photonic chip,” Sci. Adv. 4(5), eaat3174 (2018). [CrossRef]
13. H. Tang, C. D. Franco, Z. Y. Shi, T. S. He, Z. Feng, J. Gao, K. Sum, Z. M. Li, Z. Q. Jiao, T. Y. Wang, M. S. Kim, and X. M. Jin, “Experimental quantum fast hitting on hexagonal graphs,” Nat. Photonics 12(12), 754–758 (2018). [CrossRef]
14. M. Faccin, T. Johnson, J. Biamonte, S. Kais, and P. Migdal, “Degree Distribution in Quantum Walks on Complex Networks,” Phys. Rev. X 3(4), 041007 (2013). [CrossRef]
15. B. Tregenna, W. Flanagan, R. Maile, and V. Kendon, “Controlling discrete quantum walks: coins and initial states,” New J. Phys. 5, 83 (2003). [CrossRef]
16. Z. H. Bian, J. Li, X. Zhan, J. Twamley, and P. Xue, “Experimental implementation of a quantum walk on a circle with single photons,” Phys. Rev. A 95(5), 052338 (2017). [CrossRef]
17. M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73(1), 58–61 (1994). [CrossRef]
18. W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walmsley, “Optimal design for universal multiport interferometers,” Optica 3(12), 1460–1465 (2016). [CrossRef]
19. D. S. Simon, C. A. Fitzpatrick, and A. V. Sergienko, “Group transformations and entangled-sate quantum gates with directionally unbiased linear-optical multiports,” Phys. Rev. A 93(4), 043845 (2016). [CrossRef]
20. S. Osawa, D. S. Simon, and A. V. Sergienko, “Experimental demonstration of a directionally-unbiased linear optical multiport,” Opt. Express 26(21), 27201–27211 (2018). [CrossRef]
21. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59(18), 2044–2046 (1987). [CrossRef]
22. S. Osawa, D. S. Simon, and A. V. Sergienko, “Directionally-unbiased unitary optical devices in discrete-time quantum walks,” Entropy 21(9), 853 (2019). [CrossRef]
23. E. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’Brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun. 2(1), 224 (2011). [CrossRef]
24. N. Spagnolo, C. Vitelli, L. Aparo, P. Mataloni, F. Sciarrino, A. Crespi, R. Ramponi, and R. Osellame, “Three-photon bosonic coalescence in an integrated tritter,” Nat. Commun. 4(1), 1606 (2013). [CrossRef]
25. D. Lee, Y. W. Cho, H. T. Lim, S. W. Han, H. Jung, S. Moon, and Y. S. Kim, “Characterizing Bell state analyzer using weak coherent pulses,” Quantum Inf. Process. 20(4), 149 (2021). [CrossRef]
26. B. K. Park, Y. S. Kim, O. Kwon, S. W. Han, and S. Moon, “High-performance reconfigurable coincidence counting unit based on a field programmable gate array,” Appl. Opt. 54(15), 4727–4731 (2015). [CrossRef]
27. B. K. Park, Y. S. Kim, Y. W. Cho, S. Moon, and S. W. Han, “Arbitrary configurable 20-channel coincidence counting unit for multi-qubit quantum experiment,” Electronics 10(5), 569 (2021). [CrossRef]
28. E. Polino, M. Riva, M. Valeri, R. Silvestri, G. Corrielli, A. Crespi, N. Spagnolo, R. Osellame, and F. Sciarrino, “Experimental multiphase estimation on a chip,” Optica 6(3), 288–295 (2019). [CrossRef]
29. D. S. Simon, C. A. Fitzpatrick, S. Osawa, and A. V. Sergienko, “Quantum simulation of discrete-time hamiltonians using directionally unbiased linear optical multiports,” Phys. Rev. A 95(4), 042109 (2017). [CrossRef]
30. D. S. Simon, C. A. Fitzpatrick, S. Osawa, and A. V. Sergienko, “Quantum simulation of topologically protected states using directionally unbiased linear-optical multiports,” Phys. Rev. A 96(1), 013858 (2017). [CrossRef]
31. D. S. Simon, S. Osawa, and A. V. Sergienko, “Quantum clustered two-photon walks,” Phys. Rev. A 101(3), 032118 (2020). [CrossRef]
