Abstract

Universal multiport interferometers, which can be programmed to implement any linear transformation between multiple channels, are emerging as a powerful tool for both classical and quantum photonics. These interferometers are typically composed of a regular mesh of beam splitters and phase shifters, allowing for straightforward fabrication using integrated photonic architectures and ready scalability. The current, standard design for universal multiport interferometers is based on work by Reck et al. [Phys. Rev. Lett. 73, 58 (1994) [CrossRef]  ]. We demonstrate a new design for universal multiport interferometers based on an alternative arrangement of beam splitters and phase shifters, which outperforms that by Reck et al. Our design requires half the optical depth of the Reck design and is significantly more robust to optical losses.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

1. INTRODUCTION

Reconfigurable universal multiport interferometers, which can implement any linear transformation between several optical channels, are emerging as a powerful tool for fields such as microwave photonics [1,2], optical networking [3,4], and quantum photonics [5,6]. Such devices are typically built using planar meshes of beam splitters, which are easy to fabricate and to individually control, as recent demonstrations of large, yet non-universal, interferometers have shown [7,8]. While it had been known for some time that useful operations could be performed by such meshes [9], the seminal work by Reck et al. [5] demonstrated that a specific triangular mesh of 2×2 beam splitters and phase shifters could be programmed, using a simple analytical method, to implement any unitary transformation between a set of optical channels. Continued interest in universal multiport interferometers for classical and quantum applications has led to new applications and programming procedures for the same interferometer design [10,11]. Recent demonstrations of universal multiport interferometers are based on this design and have been used to interfere up to six channels [6].

In this article, we propose a new design (Fig. 1), which outperforms the Reck design in two key respects. First, our new design achieves the minimal optical depth, requiring roughly half the depth of the Reck design, which is important for minimizing optical losses and reducing fabrication resources. Second, the natural symmetry of this new design makes it significantly more robust to fabrication errors caused by mismatched optical losses. Our finding is based on a new mathematical decomposition of a unitary matrix. We use this decomposition both to prove the universality of the design and to construct an efficient algorithm to program interferometers based on it. In the following, we first provide an overview of both the Reck design and of our new design and discuss some advantages of the latter. We then explain the general principles of our decomposition procedure using a 5×5 universal transformation as an example. Finally, we quantitatively compare the loss tolerance of our design to that of the Reck design and discuss their tolerance to error.

 

Fig. 1. Universal N-mode multiport interferometer (shown here for N=9) can be implemented using a mesh of N(N1)/2 beam splitters such as (a) the one proposed by Reck et al. or (b) the one we demonstrate in this paper. As shown in (c), a line corresponds to an optical mode, and crossings between two modes correspond to a variable beam splitter described by a Tm,n(θ,ϕ) matrix, which can be implemented by a Mach–Zehnder interferometer consisting of two 50:50 directional couplers, preceded by a phase shift at one input port. Although the total number of beam splitters in both interferometers is identical, our scheme clearly has a much shorter optical depth and therefore suffers less propagation loss. This reduction in optical depth stems from the fact that each mode crosses its nearest neighbor at the first possible occasion in contrast to the Reck scheme, where the top modes must propagate for some distance before interacting with other modes. Furthermore, the high symmetry inherent to our design improves the loss tolerance of the interferometer, as we show in the main text.

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2. BACKGROUND

An ideal, lossless multiport interferometer between N channels performs an optical transformation which can be described by an N×N unitary scattering matrix U acting on electric fields as Eout=UEin. Equivalently, in quantum optics, U describes the transformation of the creation or annihilation operators of the input modes to those of the output modes.

Within this framework, the following transformation between channels m and n (m=n1),

Tm,n(θ,ϕ)=[10001eiϕcosθsinθeiϕsinθcosθ10001],
corresponds to a lossless beam splitter between channels m and n with reflectivity cosθ (θ[0,π/2]) and a phase shift ϕ (ϕ[0,2π]) at input m. In the following, we will generally omit the explicit dependence of these Tm,n(θ,ϕ) matrices on θ and ϕ for notational simplicity.

Both our scheme and the scheme by Reck et al. [5] are based on analytical methods of decomposing the U matrix into a product of Tm,n matrices. Specifically, these schemes provide an explicit algorithm for writing any unitary matrix U as

U=D((m,n)STm,n),
where S defines a specific ordered sequence of two-mode transformations and where D is a diagonal matrix with complex elements with a modulus equal to one on the diagonal. A physical interferometer composed of beam splitters and phase shifters in the configuration defined by S, with values defined by the θ and ϕ in the Tm,n matrices, will therefore implement transformation U. We note that D is physically irrelevant for most applications but can be implemented in an interferometer nonetheless by phase shifts on all individual channels at the output of an interferometer.

The formalism developed here for unitary transformations describing lossless N×N interferometers can be extended to include any M×N linear (non-unitary) transformation. Indeed, it has been noted that any M×N linear transformation, with, for example, MN (resp. MN), can be either directly embedded within a 2N×2N (resp. 2M×2M) unitary transformation [12] to within a scaling factor, or, in a more compact way, implemented by 2 separate N×N (resp. M×M) interferometers connected to each other via phase and amplitude modulators [10]. Furthermore, realistic, lossy interferometers can also be included in our formalism simply by rescaling U by a loss factor, as we explain later. Therefore, our design for universal multiport interferometers, as well as that by Reck et al. [5], can be used to implement any linear transformation to within a scaling factor on any number of input and output channels. While our design is very general and can be used for any interferometric transformation, more efficient architectures may exist for specific values of N or M when NM (see [13] for N or M=1, for example) or for specific transformations (see [14] for implementing Fourier transforms, for example).

3. OVERVIEW OF THE TWO DESIGNS

Schematic views of the Reck design and of our design are presented in Fig. 1. Figure 1(a) presents the Reck design, in which the matrix decomposition method determines a sequence S that corresponds to a triangular mesh of beam splitters. Figure 1(b) presents our design, in which every mode crosses its nearest neighbor at the first possible occasion. Our design has a shorter optical depth and is more symmetrical than the Reck design. We note that both interferometers use the same minimal number N(N1)/2 of beam splitters to implement an N×N interferometer [5].

We define the depth of an interferometer as the longest path through the interferometer, enumerated by counting the number of beam splitters traversed by that path. It is important to minimize the optical depth of an interferometer because the resulting circuits can then be more compact, which is an important factor for the fabrication of planar waveguide circuits. Furthermore, propagation losses are reduced for an interferometer with a smaller depth. It is easy to see that our design has the minimal possible optical depth, since every channel crosses its nearest neighbor at the first possible occasion. Specifically, for an N×N interferometer, the Reck design has an optical depth of 2N3, whereas our design has an optical depth of N. To illustrate this, the longest path through the interferometer shown in Fig. 1(a) follows the edges of the triangle and crosses 2N3=15 beam splitters, whereas the longest paths through the interferometer in Fig. 1(b) cross N=9 beam splitters. The increased symmetry of our design also leads to significantly better loss tolerance, as discussed in a subsequent paragraph.

4. DECOMPOSITION METHOD

In the following, we present our decomposition method, which allows us to analytically calculate the values of the beam splitter elements Tm,n in our design. Beyond its use in proving that our design is capable of implementing universal interferometric transformations, this method directly provides a recipe for programming such interferometers. Our decomposition method relies on two important properties of the Tm,n matrices. Firstly, for any given unitary matrix U, there are specific values of θ and ϕ that make any target element in row m or n of matrix Tm,nU zero, as per Reck et al. [5]. We will refer to this process as nulling that element of U and will still refer to the modified matrix after this operation as U. Secondly, we note that any target element in column n or m of U can also be nulled by multiplying U from the right by a Tm,n1 matrix.

Our algorithm, shown in Fig. 2 for the 5×5 case, consists of nulling elements of U one by one in such a way that every Tm,n and Tm,n1 matrix used in the process completely determines both the reflectivity and phase shift of one beam splitter and phase shifter. The sequence of Tm,n and Tm,n1 matrices must both correspond to the desired order of beam splitters in the interferometer and guarantee that the nulled elements of U are not affected by subsequent operations. In the Reck decomposition, the entire matrix can be nulled using either only Tm,n matrices or only Tm,n1 matrices while still making sure nulled elements of U are not affected by subsequent operations. Both matrices are necessary to verify this condition in our decomposition. As illustrated in Fig. 2, we null successive diagonals of U by alternating between Tm,n and Tm,n1 matrices in such a way that every nulled diagonal in the matrix corresponds to one diagonal line of beam splitters through the interferometer.

 

Fig. 2. Illustration of the algorithm for programming a universal multiport interferometer for a 5×5 interferometer. The left-hand side presents our decomposition procedure, and the right-hand side shows how our decomposition corresponds to building up the corresponding interferometer. (1) We start with any random unitary matrix U, and a blank interferometer. (2) We first null the bottom-left element of U with a T1,21 matrix, which causes the first two columns of U to mix. This corresponds to adding the top-left beam splitter in the interferometer. (3) We then null the next two elements of U using a T3,4 matrix followed by a T4,5 matrix, which correspond to the two bottom-right beam splitters in the interferometer. T3,4 mixes rows 3 and 4, and T4,5 mixes rows 4 and 5. Since both the (4,1) and (5,1) elements of U had been nulled, they are not affected by T4,5. (4) and (5) At every step in the algorithm, we null a successive diagonal of the updated U matrix by alternating between Tm,n and Tm,n1 matrices, which corresponds to adding diagonal lines of beam splitters to the interferometer. Tm,n (resp. Tm,n1) matrices of a given color cause the rows (resp. columns) m and n, which are shown in the same color, to mix and null the corresponding element of that color in U. It is clear from this process that once a matrix element has been nulled, no subsequent operation can modify it. (6) After step 5, U is a lower triangular matrix, which by virtue of its unitarity must be diagonal. As explained in the main text, we can then write U in the way shown here, which by construction exactly corresponds to the desired interferometer.

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At the end of the decomposition process, we obtain the following expression for a 5×5 matrix:

T4,5T3,4T2,3T1,2T4,5T3,4UT1,21T3,41T2,31T1,21=D,
where D is a diagonal matrix, as in Eq. (2). This can be rewritten as
U=T3,41T4,51T1,21T2,31T3,41T4,51DT1,2T2,3T3,4T1,2.

It is easy to demonstrate that if D consists of single-mode phase shifts, then for any Tm,n1 matrix, one can find a matrix D of single-mode phases and a matrix Tm,n such that Tm,n1D=DTm,n. The previous equation can therefore be rewritten as

U=DT3,4T4,5T1,2T2,3T3,4T4,5T1,2T2,3T3,4T1,2,
which, mirroring Eq. (2), completes our decomposition.

By construction, Eq. (5) physically corresponds to the multiport interferometer shown in Fig. 2, and the values of the θ and ϕ of the Tm,n matrices in this equation determine the values of the beam splitters and phase shifts that must be programmed to implement U. This decomposition principle can be generalized to any N, and an explicit general algorithm is given in Supplement 1. We also note that this algorithm can be used to inform the design of fixed interferometric circuits, such as those demonstrated in [15,16], in which the same arrangement of beam splitters was used to provide specific instances of random interference. Our algorithm can also be used to implement any desired optical interference between any type of optical mode and in any optical platform beyond integrated photonics.

We note that whereas the decomposition method presented here can straightforwardly be applied to program a universal multiport interferometer built according to our design, other programming schemes may exist. For instance, for the Reck design, self-configuring methods have recently been proposed [10,11] and demonstrated [17,18]. These methods can be advantageous for some applications in that they do not require full characterization of the circuit elements corresponding to the Tm,n matrices. The existence of such schemes for our design is beyond the scope of this article.

5. LOSS TOLERANCE

Optical loss is unavoidable in realistic interferometers, and finding methods to mitigate its effects is an integral part of any photonic scheme. In the following, we study the tolerance of multiport interferometers built according to our decomposition to loss and compare their performance to interferometers built and programmed according to the Reck design.

We first distinguish between two types of loss. Balanced loss in a multiport interferometer, in which every path through the interferometer experiences the same loss, preserves the target interference to within an overall scaling factor. This is generally acceptable for applications in the classical domain, such as optical switching or microwave photonics. In the quantum domain, although loss severely affects the scalability of quantum experiments, post-selection can, in some situations, be used to recover the desired interference pattern. We note that propagation loss in an interferometer is expected to contribute to balanced loss, since every physical path length in an interferometer must be matched to within the coherence length of the input light to maintain high-fidelity interference. However, propagation loss must therefore be proportional to the longest path through the interferometer (i.e., the optical depth), so interferometers built according to our design will suffer from only about half the propagation loss of an interferometer built according to the Reck design.

Unbalanced loss, where different paths through the interferometer experience different losses, can be difficult to characterize and, critically, can result in poor fidelity to the intended operation [1922]. Unequal losses between paths in the interferometer are typically caused by beam splitters, which are unavoidably lossy due to additional bending losses and scattering. To compare the tolerance of multiport interferometers to the unbalanced losses caused by beam splitters, we adopt the following procedure. For a given N, we generate 500 random unitary matrices [23], implement our decomposition, add loss to both outputs of all the resulting beam splitters, and compare the fidelities in the overall transformations. We use a simple loss model that assumes equal insertion loss for every beam splitter, and we quantify the fidelity of the transformation implemented by a lossy N×N experimental interferometer, Uexp, to the intended transformation U using the following metric,

F(Uexp,U)=|tr(UUexp)Ntr(UexpUexp)|2,
which corresponds to a standard fidelity measure, normalized so we do not distinguish between matrices that differ by only a constant multiplicative factor. This allows us to focus on unbalanced loss instead of balanced loss in our simulations.

Figure 3 shows our simulation results for both a fixed loss and varying interferometer sizes and for a fixed interferometer size and varying loss. We conclude the interferometers that are implemented in our design are significantly more tolerant to unbalanced loss than those implemented in the Reck design. This is because in the Reck design, different paths through the interferometer go through different numbers of beam splitters, so they all experience different losses and the resulting interference is degraded. In our design, the path lengths are better matched, so equally distributed loss within the interferometer does not strongly affect the resulting interference. We note that whereas unbalanced loss can be compensated for in the Reck design by adding loss to shorter paths, for example, by adding dummy beam splitters to the shorter paths in the interferometer, as proposed by Miller [11], this is inefficient and it is better to start with a fundamentally loss-resistant interferometer.

 

Fig. 3. Left: average fidelity for an interferometer with a constant loss of 0.2 dB per beam splitter (as in the universal multiport interferometer in [6]) for interferometers built according to the Reck design (black) and our design (blue), for different interferometer sizes. Inset: close-up of the fidelity in our design. Right: fidelity as a function of loss for interferometers implementing 20×20 transformations. We see from our results that our design is much more loss-tolerant than the Reck design and maintains high fidelity with the target unitary even in the case of high loss. This is because the mismatched path lengths in the Reck design causes the loss to severely affect the resulting interference. See [24] for underlying data.

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6. ERROR TOLERANCE

In a realistic interferometer, there will always be some error when setting the values for the phases and beam splitter reflectivities. Furthermore, if the 2×2 elements are composed of single Mach–Zehnder interferometers, imperfections in the beam splitters will make it difficult to reach high values of transmission or reflection. These errors will affect the circuit fidelity in both our design and in the Reck design. Since the overall error in the interferometer caused by these individual errors depends on the total number of beam splitters, the layout of the interferometer mainly affects how that error is distributed among the output ports. Therefore, the average error caused by these imperfections is roughly equal in the Reck design and in our design.

However, we note that the problem of error can, if necessary, be mitigated by concatenating imperfect beam splitters to create one ideal reconfigurable beam splitter [11,25,26], at the expense of additional loss. Our design is particularly well suited for implementing such a solution, since a uniform beam splitter loss does not strongly affect the circuit fidelity.

7. CONCLUSION

In conclusion, we have demonstrated a design for universal multiport interferometers which outperforms the design proposed by Reck et al. [5] in several respects. Our design is programmed using a new method for decomposing unitary matrices into a sequence of beam splitters, requires half the optical depth, and, significantly, not only suffers less propagation loss but is more loss-tolerant than the previous design.

We expect that our compact and loss-tolerant design for fully programmable universal mulitport interferometers will play an important role in the development of optical processors for both classical and quantum applications. Furthermore, we anticipate that our matrix decomposition method will be of use in its own right for other systems that use mathematical structures analogous to beam splitters and phase shifters, such as ion traps [27] and some architectures for superconducting circuits [28,29].

Funding

H2020 European Research Council (ERC); Engineering and Physical Sciences Research Council (EPSRC) (EP/K034480/1, NQIT); European Commission (EC) (H2020-FETPROACT-2014 QUCHIP), Research Councils UK (RCUK).

Acknowledgment

W. R. C. and P. C. H. demonstrated the matrix decomposition method presented here. W. R. C. did the numerical simulations and wrote the manuscript with contributions from all authors. B. J. M. conceived the project, and W. S. K. and I. A. W. supervised it. This publication was supported by the Oxford RCUK Open Access Block Grant.

Competing Financial Interests

The authors have jointly applied for a patent for the work presented in this paper.

 

See Supplement 1 for supporting content.

REFERENCES

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3. L.-N. Chen, E. Hall, L. Theogarajan, and J. Bowers, “Photonic switching for data center applications,” IEEE Photon. J. 3, 834–844 (2011). [CrossRef]  

4. R. Stabile, A. Albores-Mejia, A. Rohit, and K. A. Williams, “Integrated optical switch matrices for packet data networks,” Microsyst. Nanoeng. 2, 15042 (2016). [CrossRef]  

5. M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994). [CrossRef]  

6. 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. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711 (2015). [CrossRef]  

7. K. Tanizawa, K. Suzuki, M. Toyama, M. Ohtsuka, N. Yokoyama, K. Matsumaro, M. Seki, K. Koshino, T. Sugaya, S. Suda, G. Cong, T. Kimura, K. Ikeda, S. Namiki, and H. Kawashima, “Ultra-compact 32 × 32 strictly-non-blocking Si-wire optical switch with fan-out LGA interposer,” Opt. Express 23, 17599–17606 (2015). [CrossRef]  

8. N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Bosonic transport simulations in a large-scale programmable nanophotonic processor,” arXiv preprint arXiv:1507.03406 (2015).

9. T. K. Gaylor, E. I. Verriest, and M. M. Mirsalehi, “Integrated optical givens rotation device,” U.S. patent 4950042 A (August 21, 1990).

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11. D. A. Miller, “Perfect optics with imperfect components,” Optica 2, 747–750 (2015). [CrossRef]  

12. G. Alber, T. Beth, M. Horodecki, P. Horodecki, R. Horodecki, M. Rötteler, H. Weinfurter, R. Werner, and A. Zeilinger, Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments (Springer, 2003), Vol. 173.

13. D. A. Miller, “Self-aligning universal beam coupler,” Opt. Express 21, 6360–6370 (2013). [CrossRef]  

14. A. Crespi, R. Osellame, R. Ramponi, M. Bentivegna, F. Flamini, N. Spagnolo, N. Viggianiello, L. Innocenti, P. Mataloni, and F. Sciarrino, “Suppression law of quantum states in a 3D photonic fast Fourier transform chip,” Nat. Commun. 7, 10469 (2016). [CrossRef]  

15. M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1, e1400255 (2015). [CrossRef]  

16. N. Spagnolo, C. Vitelli, M. Bentivegna, D. J. Brod, A. Crespi, F. Flamini, S. Giacomini, G. Milani, R. Ramponi, P. Mataloni, R. Osellame, E. F. Galvão, and F. Sciarrino, “Experimental validation of photonic boson sampling,” Nat. Photonics 8, 615–620 (2014). [CrossRef]  

17. F. Morichetti, A. Annoni, S. Grillanda, N. Peserico, M. Carminati, P. Ciccarella, G. Ferrari, E. Guglielmi, M. Sorel, and A. Melloni, “4-channel all-optical mimo demultiplexing on a silicon chip,” in Optical Fiber Communication Conference (2016), paper ThE3.7.

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23. These random unitary matrices were generated using the QR decomposition of random matrices, following the code developed by Toby Cubitt, available on www.dr-qubit.org/Matlab_code.html.

24. http://dx.doi.org/10.5287/bodleian:6gMmJYmab.

25. N. Thomas-Peter, N. K. Langford, A. Datta, L. Zhang, B. J. Smith, J. B. Spring, B. J. Metcalf, H. B. Coldenstrodt-Ronge, M. Hu, J. Nunn, and I. A. Walmsley, “Integrated photonic sensing,” New J. Phys. 13, 055024 (2011). [CrossRef]  

26. N. Thomas-Peter, “Quantum enhanced precision measurement and information processing with integrated photonics,” Ph.D. thesis (Balliol College, University of Oxford, 2012).

27. C. Shen, Z. Zhang, and L.-M. Duan, “Scalable implementation of boson sampling with trapped ions,” Phys. Rev. Lett. 112, 050504 (2014). [CrossRef]  

28. B. Peropadre, A. Aspuru-Guzik, and J. J. Garcia-Ripoll, “Spin models and boson sampling,” arXiv preprint arXiv:1509.02703 (2015).

29. B. Peropadre, G. G. Guerreschi, J. Huh, and A. Aspuru-Guzik, “Microwave boson sampling,” arXiv preprint arXiv:1510.08064 (2015).

References

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  1. “Birth of the programmable optical chip,” Nat. Photonics10, 1 (2016), Editorial.
  2. J. Capmany, I. Gasulla, and D. Perez, “Microwave photonics: the programmable processor,” Nat. Photonics 10, 6–8 (2016).
    [Crossref]
  3. L.-N. Chen, E. Hall, L. Theogarajan, and J. Bowers, “Photonic switching for data center applications,” IEEE Photon. J. 3, 834–844 (2011).
    [Crossref]
  4. R. Stabile, A. Albores-Mejia, A. Rohit, and K. A. Williams, “Integrated optical switch matrices for packet data networks,” Microsyst. Nanoeng. 2, 15042 (2016).
    [Crossref]
  5. M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
    [Crossref]
  6. 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. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711 (2015).
    [Crossref]
  7. K. Tanizawa, K. Suzuki, M. Toyama, M. Ohtsuka, N. Yokoyama, K. Matsumaro, M. Seki, K. Koshino, T. Sugaya, S. Suda, G. Cong, T. Kimura, K. Ikeda, S. Namiki, and H. Kawashima, “Ultra-compact 32 × 32 strictly-non-blocking Si-wire optical switch with fan-out LGA interposer,” Opt. Express 23, 17599–17606 (2015).
    [Crossref]
  8. N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Bosonic transport simulations in a large-scale programmable nanophotonic processor,” arXiv preprint arXiv:1507.03406 (2015).
  9. T. K. Gaylor, E. I. Verriest, and M. M. Mirsalehi, “Integrated optical givens rotation device,” U.S. patent4950042 A (August21, 1990).
  10. D. A. Miller, “Self-configuring universal linear optical component,” Photon. Res. 1, 1–15 (2013).
    [Crossref]
  11. D. A. Miller, “Perfect optics with imperfect components,” Optica 2, 747–750 (2015).
    [Crossref]
  12. G. Alber, T. Beth, M. Horodecki, P. Horodecki, R. Horodecki, M. Rötteler, H. Weinfurter, R. Werner, and A. Zeilinger, Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments (Springer, 2003), Vol. 173.
  13. D. A. Miller, “Self-aligning universal beam coupler,” Opt. Express 21, 6360–6370 (2013).
    [Crossref]
  14. A. Crespi, R. Osellame, R. Ramponi, M. Bentivegna, F. Flamini, N. Spagnolo, N. Viggianiello, L. Innocenti, P. Mataloni, and F. Sciarrino, “Suppression law of quantum states in a 3D photonic fast Fourier transform chip,” Nat. Commun. 7, 10469 (2016).
    [Crossref]
  15. M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1, e1400255 (2015).
    [Crossref]
  16. N. Spagnolo, C. Vitelli, M. Bentivegna, D. J. Brod, A. Crespi, F. Flamini, S. Giacomini, G. Milani, R. Ramponi, P. Mataloni, R. Osellame, E. F. Galvão, and F. Sciarrino, “Experimental validation of photonic boson sampling,” Nat. Photonics 8, 615–620 (2014).
    [Crossref]
  17. F. Morichetti, A. Annoni, S. Grillanda, N. Peserico, M. Carminati, P. Ciccarella, G. Ferrari, E. Guglielmi, M. Sorel, and A. Melloni, “4-channel all-optical mimo demultiplexing on a silicon chip,” in Optical Fiber Communication Conference (2016), paper ThE3.7.
  18. A. Ribeiro, A. Ruocco, L. Vanacker, and W. Bogaerts, “Demonstration of a 4 × 4-port universal coupler,” in Group IV Photonics Conference (2016), paper PD5.
  19. B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
    [Crossref]
  20. B. J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X.-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Quantum teleportation on a photonic chip,” Nat. Photonics 8, 770–774 (2014).
    [Crossref]
  21. D. Bonneau, E. Engin, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, J. L. O’Brien, and M. G. Thompson, “Quantum interference and manipulation of entanglement in silicon wire waveguide quantum circuits,” New J. Phys. 14, 045003 (2012).
    [Crossref]
  22. A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’Brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun. 2, 224 (2011).
    [Crossref]
  23. These random unitary matrices were generated using the QR decomposition of random matrices, following the code developed by Toby Cubitt, available on www.dr-qubit.org/Matlab_code.html .
  24. http://dx.doi.org/10.5287/bodleian:6gMmJYmab .
  25. N. Thomas-Peter, N. K. Langford, A. Datta, L. Zhang, B. J. Smith, J. B. Spring, B. J. Metcalf, H. B. Coldenstrodt-Ronge, M. Hu, J. Nunn, and I. A. Walmsley, “Integrated photonic sensing,” New J. Phys. 13, 055024 (2011).
    [Crossref]
  26. N. Thomas-Peter, “Quantum enhanced precision measurement and information processing with integrated photonics,” Ph.D. thesis (Balliol College, University of Oxford, 2012).
  27. C. Shen, Z. Zhang, and L.-M. Duan, “Scalable implementation of boson sampling with trapped ions,” Phys. Rev. Lett. 112, 050504 (2014).
    [Crossref]
  28. B. Peropadre, A. Aspuru-Guzik, and J. J. Garcia-Ripoll, “Spin models and boson sampling,” arXiv preprint arXiv:1509.02703 (2015).
  29. B. Peropadre, G. G. Guerreschi, J. Huh, and A. Aspuru-Guzik, “Microwave boson sampling,” arXiv preprint arXiv:1510.08064 (2015).

2016 (3)

J. Capmany, I. Gasulla, and D. Perez, “Microwave photonics: the programmable processor,” Nat. Photonics 10, 6–8 (2016).
[Crossref]

R. Stabile, A. Albores-Mejia, A. Rohit, and K. A. Williams, “Integrated optical switch matrices for packet data networks,” Microsyst. Nanoeng. 2, 15042 (2016).
[Crossref]

A. Crespi, R. Osellame, R. Ramponi, M. Bentivegna, F. Flamini, N. Spagnolo, N. Viggianiello, L. Innocenti, P. Mataloni, and F. Sciarrino, “Suppression law of quantum states in a 3D photonic fast Fourier transform chip,” Nat. Commun. 7, 10469 (2016).
[Crossref]

2015 (4)

M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1, e1400255 (2015).
[Crossref]

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. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711 (2015).
[Crossref]

K. Tanizawa, K. Suzuki, M. Toyama, M. Ohtsuka, N. Yokoyama, K. Matsumaro, M. Seki, K. Koshino, T. Sugaya, S. Suda, G. Cong, T. Kimura, K. Ikeda, S. Namiki, and H. Kawashima, “Ultra-compact 32 × 32 strictly-non-blocking Si-wire optical switch with fan-out LGA interposer,” Opt. Express 23, 17599–17606 (2015).
[Crossref]

D. A. Miller, “Perfect optics with imperfect components,” Optica 2, 747–750 (2015).
[Crossref]

2014 (3)

N. Spagnolo, C. Vitelli, M. Bentivegna, D. J. Brod, A. Crespi, F. Flamini, S. Giacomini, G. Milani, R. Ramponi, P. Mataloni, R. Osellame, E. F. Galvão, and F. Sciarrino, “Experimental validation of photonic boson sampling,” Nat. Photonics 8, 615–620 (2014).
[Crossref]

B. J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X.-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Quantum teleportation on a photonic chip,” Nat. Photonics 8, 770–774 (2014).
[Crossref]

C. Shen, Z. Zhang, and L.-M. Duan, “Scalable implementation of boson sampling with trapped ions,” Phys. Rev. Lett. 112, 050504 (2014).
[Crossref]

2013 (3)

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
[Crossref]

D. A. Miller, “Self-aligning universal beam coupler,” Opt. Express 21, 6360–6370 (2013).
[Crossref]

D. A. Miller, “Self-configuring universal linear optical component,” Photon. Res. 1, 1–15 (2013).
[Crossref]

2012 (1)

D. Bonneau, E. Engin, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, J. L. O’Brien, and M. G. Thompson, “Quantum interference and manipulation of entanglement in silicon wire waveguide quantum circuits,” New J. Phys. 14, 045003 (2012).
[Crossref]

2011 (3)

A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’Brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun. 2, 224 (2011).
[Crossref]

N. Thomas-Peter, N. K. Langford, A. Datta, L. Zhang, B. J. Smith, J. B. Spring, B. J. Metcalf, H. B. Coldenstrodt-Ronge, M. Hu, J. Nunn, and I. A. Walmsley, “Integrated photonic sensing,” New J. Phys. 13, 055024 (2011).
[Crossref]

L.-N. Chen, E. Hall, L. Theogarajan, and J. Bowers, “Photonic switching for data center applications,” IEEE Photon. J. 3, 834–844 (2011).
[Crossref]

1994 (1)

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref]

Alber, G.

G. Alber, T. Beth, M. Horodecki, P. Horodecki, R. Horodecki, M. Rötteler, H. Weinfurter, R. Werner, and A. Zeilinger, Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments (Springer, 2003), Vol. 173.

Albores-Mejia, A.

R. Stabile, A. Albores-Mejia, A. Rohit, and K. A. Williams, “Integrated optical switch matrices for packet data networks,” Microsyst. Nanoeng. 2, 15042 (2016).
[Crossref]

Annoni, A.

F. Morichetti, A. Annoni, S. Grillanda, N. Peserico, M. Carminati, P. Ciccarella, G. Ferrari, E. Guglielmi, M. Sorel, and A. Melloni, “4-channel all-optical mimo demultiplexing on a silicon chip,” in Optical Fiber Communication Conference (2016), paper ThE3.7.

Aspuru-Guzik, A.

B. Peropadre, A. Aspuru-Guzik, and J. J. Garcia-Ripoll, “Spin models and boson sampling,” arXiv preprint arXiv:1509.02703 (2015).

B. Peropadre, G. G. Guerreschi, J. Huh, and A. Aspuru-Guzik, “Microwave boson sampling,” arXiv preprint arXiv:1510.08064 (2015).

Baehr-Jones, T.

N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Bosonic transport simulations in a large-scale programmable nanophotonic processor,” arXiv preprint arXiv:1507.03406 (2015).

Barbieri, M.

B. J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X.-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Quantum teleportation on a photonic chip,” Nat. Photonics 8, 770–774 (2014).
[Crossref]

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
[Crossref]

Bentivegna, M.

A. Crespi, R. Osellame, R. Ramponi, M. Bentivegna, F. Flamini, N. Spagnolo, N. Viggianiello, L. Innocenti, P. Mataloni, and F. Sciarrino, “Suppression law of quantum states in a 3D photonic fast Fourier transform chip,” Nat. Commun. 7, 10469 (2016).
[Crossref]

M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1, e1400255 (2015).
[Crossref]

N. Spagnolo, C. Vitelli, M. Bentivegna, D. J. Brod, A. Crespi, F. Flamini, S. Giacomini, G. Milani, R. Ramponi, P. Mataloni, R. Osellame, E. F. Galvão, and F. Sciarrino, “Experimental validation of photonic boson sampling,” Nat. Photonics 8, 615–620 (2014).
[Crossref]

Bernstein, H. J.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref]

Bertani, P.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref]

Beth, T.

G. Alber, T. Beth, M. Horodecki, P. Horodecki, R. Horodecki, M. Rötteler, H. Weinfurter, R. Werner, and A. Zeilinger, Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments (Springer, 2003), Vol. 173.

Bogaerts, W.

A. Ribeiro, A. Ruocco, L. Vanacker, and W. Bogaerts, “Demonstration of a 4 × 4-port universal coupler,” in Group IV Photonics Conference (2016), paper PD5.

Bonneau, D.

D. Bonneau, E. Engin, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, J. L. O’Brien, and M. G. Thompson, “Quantum interference and manipulation of entanglement in silicon wire waveguide quantum circuits,” New J. Phys. 14, 045003 (2012).
[Crossref]

Bowers, J.

L.-N. Chen, E. Hall, L. Theogarajan, and J. Bowers, “Photonic switching for data center applications,” IEEE Photon. J. 3, 834–844 (2011).
[Crossref]

Brod, D. J.

M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1, e1400255 (2015).
[Crossref]

N. Spagnolo, C. Vitelli, M. Bentivegna, D. J. Brod, A. Crespi, F. Flamini, S. Giacomini, G. Milani, R. Ramponi, P. Mataloni, R. Osellame, E. F. Galvão, and F. Sciarrino, “Experimental validation of photonic boson sampling,” Nat. Photonics 8, 615–620 (2014).
[Crossref]

Broome, M. A.

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
[Crossref]

Capmany, J.

J. Capmany, I. Gasulla, and D. Perez, “Microwave photonics: the programmable processor,” Nat. Photonics 10, 6–8 (2016).
[Crossref]

Carminati, M.

F. Morichetti, A. Annoni, S. Grillanda, N. Peserico, M. Carminati, P. Ciccarella, G. Ferrari, E. Guglielmi, M. Sorel, and A. Melloni, “4-channel all-optical mimo demultiplexing on a silicon chip,” in Optical Fiber Communication Conference (2016), paper ThE3.7.

Carolan, J.

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. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711 (2015).
[Crossref]

Chen, L.-N.

L.-N. Chen, E. Hall, L. Theogarajan, and J. Bowers, “Photonic switching for data center applications,” IEEE Photon. J. 3, 834–844 (2011).
[Crossref]

Ciccarella, P.

F. Morichetti, A. Annoni, S. Grillanda, N. Peserico, M. Carminati, P. Ciccarella, G. Ferrari, E. Guglielmi, M. Sorel, and A. Melloni, “4-channel all-optical mimo demultiplexing on a silicon chip,” in Optical Fiber Communication Conference (2016), paper ThE3.7.

Coldenstrodt-Ronge, H. B.

N. Thomas-Peter, N. K. Langford, A. Datta, L. Zhang, B. J. Smith, J. B. Spring, B. J. Metcalf, H. B. Coldenstrodt-Ronge, M. Hu, J. Nunn, and I. A. Walmsley, “Integrated photonic sensing,” New J. Phys. 13, 055024 (2011).
[Crossref]

Cong, G.

Crespi, A.

A. Crespi, R. Osellame, R. Ramponi, M. Bentivegna, F. Flamini, N. Spagnolo, N. Viggianiello, L. Innocenti, P. Mataloni, and F. Sciarrino, “Suppression law of quantum states in a 3D photonic fast Fourier transform chip,” Nat. Commun. 7, 10469 (2016).
[Crossref]

M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1, e1400255 (2015).
[Crossref]

N. Spagnolo, C. Vitelli, M. Bentivegna, D. J. Brod, A. Crespi, F. Flamini, S. Giacomini, G. Milani, R. Ramponi, P. Mataloni, R. Osellame, E. F. Galvão, and F. Sciarrino, “Experimental validation of photonic boson sampling,” Nat. Photonics 8, 615–620 (2014).
[Crossref]

Datta, A.

N. Thomas-Peter, N. K. Langford, A. Datta, L. Zhang, B. J. Smith, J. B. Spring, B. J. Metcalf, H. B. Coldenstrodt-Ronge, M. Hu, J. Nunn, and I. A. Walmsley, “Integrated photonic sensing,” New J. Phys. 13, 055024 (2011).
[Crossref]

Dorenbos, S. N.

D. Bonneau, E. Engin, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, J. L. O’Brien, and M. G. Thompson, “Quantum interference and manipulation of entanglement in silicon wire waveguide quantum circuits,” New J. Phys. 14, 045003 (2012).
[Crossref]

Duan, L.-M.

C. Shen, Z. Zhang, and L.-M. Duan, “Scalable implementation of boson sampling with trapped ions,” Phys. Rev. Lett. 112, 050504 (2014).
[Crossref]

Engin, E.

D. Bonneau, E. Engin, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, J. L. O’Brien, and M. G. Thompson, “Quantum interference and manipulation of entanglement in silicon wire waveguide quantum circuits,” New J. Phys. 14, 045003 (2012).
[Crossref]

Englund, D.

N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Bosonic transport simulations in a large-scale programmable nanophotonic processor,” arXiv preprint arXiv:1507.03406 (2015).

Ezaki, M.

D. Bonneau, E. Engin, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, J. L. O’Brien, and M. G. Thompson, “Quantum interference and manipulation of entanglement in silicon wire waveguide quantum circuits,” New J. Phys. 14, 045003 (2012).
[Crossref]

Ferrari, G.

F. Morichetti, A. Annoni, S. Grillanda, N. Peserico, M. Carminati, P. Ciccarella, G. Ferrari, E. Guglielmi, M. Sorel, and A. Melloni, “4-channel all-optical mimo demultiplexing on a silicon chip,” in Optical Fiber Communication Conference (2016), paper ThE3.7.

Flamini, F.

A. Crespi, R. Osellame, R. Ramponi, M. Bentivegna, F. Flamini, N. Spagnolo, N. Viggianiello, L. Innocenti, P. Mataloni, and F. Sciarrino, “Suppression law of quantum states in a 3D photonic fast Fourier transform chip,” Nat. Commun. 7, 10469 (2016).
[Crossref]

M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1, e1400255 (2015).
[Crossref]

N. Spagnolo, C. Vitelli, M. Bentivegna, D. J. Brod, A. Crespi, F. Flamini, S. Giacomini, G. Milani, R. Ramponi, P. Mataloni, R. Osellame, E. F. Galvão, and F. Sciarrino, “Experimental validation of photonic boson sampling,” Nat. Photonics 8, 615–620 (2014).
[Crossref]

Galvão, E. F.

M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1, e1400255 (2015).
[Crossref]

N. Spagnolo, C. Vitelli, M. Bentivegna, D. J. Brod, A. Crespi, F. Flamini, S. Giacomini, G. Milani, R. Ramponi, P. Mataloni, R. Osellame, E. F. Galvão, and F. Sciarrino, “Experimental validation of photonic boson sampling,” Nat. Photonics 8, 615–620 (2014).
[Crossref]

Garcia-Ripoll, J. J.

B. Peropadre, A. Aspuru-Guzik, and J. J. Garcia-Ripoll, “Spin models and boson sampling,” arXiv preprint arXiv:1509.02703 (2015).

Gasulla, I.

J. Capmany, I. Gasulla, and D. Perez, “Microwave photonics: the programmable processor,” Nat. Photonics 10, 6–8 (2016).
[Crossref]

Gates, J. C.

B. J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X.-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Quantum teleportation on a photonic chip,” Nat. Photonics 8, 770–774 (2014).
[Crossref]

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
[Crossref]

Gaylor, T. K.

T. K. Gaylor, E. I. Verriest, and M. M. Mirsalehi, “Integrated optical givens rotation device,” U.S. patent4950042 A (August21, 1990).

Giacomini, S.

N. Spagnolo, C. Vitelli, M. Bentivegna, D. J. Brod, A. Crespi, F. Flamini, S. Giacomini, G. Milani, R. Ramponi, P. Mataloni, R. Osellame, E. F. Galvão, and F. Sciarrino, “Experimental validation of photonic boson sampling,” Nat. Photonics 8, 615–620 (2014).
[Crossref]

Grillanda, S.

F. Morichetti, A. Annoni, S. Grillanda, N. Peserico, M. Carminati, P. Ciccarella, G. Ferrari, E. Guglielmi, M. Sorel, and A. Melloni, “4-channel all-optical mimo demultiplexing on a silicon chip,” in Optical Fiber Communication Conference (2016), paper ThE3.7.

Guerreschi, G. G.

B. Peropadre, G. G. Guerreschi, J. Huh, and A. Aspuru-Guzik, “Microwave boson sampling,” arXiv preprint arXiv:1510.08064 (2015).

Guglielmi, E.

F. Morichetti, A. Annoni, S. Grillanda, N. Peserico, M. Carminati, P. Ciccarella, G. Ferrari, E. Guglielmi, M. Sorel, and A. Melloni, “4-channel all-optical mimo demultiplexing on a silicon chip,” in Optical Fiber Communication Conference (2016), paper ThE3.7.

Hadfield, R. H.

D. Bonneau, E. Engin, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, J. L. O’Brien, and M. G. Thompson, “Quantum interference and manipulation of entanglement in silicon wire waveguide quantum circuits,” New J. Phys. 14, 045003 (2012).
[Crossref]

Hall, E.

L.-N. Chen, E. Hall, L. Theogarajan, and J. Bowers, “Photonic switching for data center applications,” IEEE Photon. J. 3, 834–844 (2011).
[Crossref]

Harris, N. C.

N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Bosonic transport simulations in a large-scale programmable nanophotonic processor,” arXiv preprint arXiv:1507.03406 (2015).

Harrold, C.

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. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711 (2015).
[Crossref]

Hashimoto, T.

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. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711 (2015).
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Smith, P. G. R.

B. J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X.-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Quantum teleportation on a photonic chip,” Nat. Photonics 8, 770–774 (2014).
[Crossref]

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
[Crossref]

Sorel, M.

F. Morichetti, A. Annoni, S. Grillanda, N. Peserico, M. Carminati, P. Ciccarella, G. Ferrari, E. Guglielmi, M. Sorel, and A. Melloni, “4-channel all-optical mimo demultiplexing on a silicon chip,” in Optical Fiber Communication Conference (2016), paper ThE3.7.

Spagnolo, N.

A. Crespi, R. Osellame, R. Ramponi, M. Bentivegna, F. Flamini, N. Spagnolo, N. Viggianiello, L. Innocenti, P. Mataloni, and F. Sciarrino, “Suppression law of quantum states in a 3D photonic fast Fourier transform chip,” Nat. Commun. 7, 10469 (2016).
[Crossref]

M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1, e1400255 (2015).
[Crossref]

N. Spagnolo, C. Vitelli, M. Bentivegna, D. J. Brod, A. Crespi, F. Flamini, S. Giacomini, G. Milani, R. Ramponi, P. Mataloni, R. Osellame, E. F. Galvão, and F. Sciarrino, “Experimental validation of photonic boson sampling,” Nat. Photonics 8, 615–620 (2014).
[Crossref]

Sparrow, C.

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. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711 (2015).
[Crossref]

Spring, J. B.

B. J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X.-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Quantum teleportation on a photonic chip,” Nat. Photonics 8, 770–774 (2014).
[Crossref]

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
[Crossref]

N. Thomas-Peter, N. K. Langford, A. Datta, L. Zhang, B. J. Smith, J. B. Spring, B. J. Metcalf, H. B. Coldenstrodt-Ronge, M. Hu, J. Nunn, and I. A. Walmsley, “Integrated photonic sensing,” New J. Phys. 13, 055024 (2011).
[Crossref]

Stabile, R.

R. Stabile, A. Albores-Mejia, A. Rohit, and K. A. Williams, “Integrated optical switch matrices for packet data networks,” Microsyst. Nanoeng. 2, 15042 (2016).
[Crossref]

Steinbrecher, G. R.

N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Bosonic transport simulations in a large-scale programmable nanophotonic processor,” arXiv preprint arXiv:1507.03406 (2015).

Suda, S.

Sugaya, T.

Suzuki, K.

Suzuki, N.

D. Bonneau, E. Engin, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, J. L. O’Brien, and M. G. Thompson, “Quantum interference and manipulation of entanglement in silicon wire waveguide quantum circuits,” New J. Phys. 14, 045003 (2012).
[Crossref]

Tanizawa, K.

Tanner, M. G.

D. Bonneau, E. Engin, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, J. L. O’Brien, and M. G. Thompson, “Quantum interference and manipulation of entanglement in silicon wire waveguide quantum circuits,” New J. Phys. 14, 045003 (2012).
[Crossref]

Theogarajan, L.

L.-N. Chen, E. Hall, L. Theogarajan, and J. Bowers, “Photonic switching for data center applications,” IEEE Photon. J. 3, 834–844 (2011).
[Crossref]

Thomas-Peter, N.

B. J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X.-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Quantum teleportation on a photonic chip,” Nat. Photonics 8, 770–774 (2014).
[Crossref]

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
[Crossref]

N. Thomas-Peter, N. K. Langford, A. Datta, L. Zhang, B. J. Smith, J. B. Spring, B. J. Metcalf, H. B. Coldenstrodt-Ronge, M. Hu, J. Nunn, and I. A. Walmsley, “Integrated photonic sensing,” New J. Phys. 13, 055024 (2011).
[Crossref]

N. Thomas-Peter, “Quantum enhanced precision measurement and information processing with integrated photonics,” Ph.D. thesis (Balliol College, University of Oxford, 2012).

Thompson, M. G.

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. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711 (2015).
[Crossref]

D. Bonneau, E. Engin, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, J. L. O’Brien, and M. G. Thompson, “Quantum interference and manipulation of entanglement in silicon wire waveguide quantum circuits,” New J. Phys. 14, 045003 (2012).
[Crossref]

Toyama, M.

Vanacker, L.

A. Ribeiro, A. Ruocco, L. Vanacker, and W. Bogaerts, “Demonstration of a 4 × 4-port universal coupler,” in Group IV Photonics Conference (2016), paper PD5.

Verriest, E. I.

T. K. Gaylor, E. I. Verriest, and M. M. Mirsalehi, “Integrated optical givens rotation device,” U.S. patent4950042 A (August21, 1990).

Viggianiello, N.

A. Crespi, R. Osellame, R. Ramponi, M. Bentivegna, F. Flamini, N. Spagnolo, N. Viggianiello, L. Innocenti, P. Mataloni, and F. Sciarrino, “Suppression law of quantum states in a 3D photonic fast Fourier transform chip,” Nat. Commun. 7, 10469 (2016).
[Crossref]

M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1, e1400255 (2015).
[Crossref]

Vitelli, C.

M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1, e1400255 (2015).
[Crossref]

N. Spagnolo, C. Vitelli, M. Bentivegna, D. J. Brod, A. Crespi, F. Flamini, S. Giacomini, G. Milani, R. Ramponi, P. Mataloni, R. Osellame, E. F. Galvão, and F. Sciarrino, “Experimental validation of photonic boson sampling,” Nat. Photonics 8, 615–620 (2014).
[Crossref]

Walmsley, I. A.

B. J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X.-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Quantum teleportation on a photonic chip,” Nat. Photonics 8, 770–774 (2014).
[Crossref]

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
[Crossref]

N. Thomas-Peter, N. K. Langford, A. Datta, L. Zhang, B. J. Smith, J. B. Spring, B. J. Metcalf, H. B. Coldenstrodt-Ronge, M. Hu, J. Nunn, and I. A. Walmsley, “Integrated photonic sensing,” New J. Phys. 13, 055024 (2011).
[Crossref]

Weinfurter, H.

G. Alber, T. Beth, M. Horodecki, P. Horodecki, R. Horodecki, M. Rötteler, H. Weinfurter, R. Werner, and A. Zeilinger, Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments (Springer, 2003), Vol. 173.

Werner, R.

G. Alber, T. Beth, M. Horodecki, P. Horodecki, R. Horodecki, M. Rötteler, H. Weinfurter, R. Werner, and A. Zeilinger, Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments (Springer, 2003), Vol. 173.

Williams, K. A.

R. Stabile, A. Albores-Mejia, A. Rohit, and K. A. Williams, “Integrated optical switch matrices for packet data networks,” Microsyst. Nanoeng. 2, 15042 (2016).
[Crossref]

Yokoyama, N.

Yoshida, H.

D. Bonneau, E. Engin, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, J. L. O’Brien, and M. G. Thompson, “Quantum interference and manipulation of entanglement in silicon wire waveguide quantum circuits,” New J. Phys. 14, 045003 (2012).
[Crossref]

Zeilinger, A.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref]

G. Alber, T. Beth, M. Horodecki, P. Horodecki, R. Horodecki, M. Rötteler, H. Weinfurter, R. Werner, and A. Zeilinger, Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments (Springer, 2003), Vol. 173.

Zhang, L.

N. Thomas-Peter, N. K. Langford, A. Datta, L. Zhang, B. J. Smith, J. B. Spring, B. J. Metcalf, H. B. Coldenstrodt-Ronge, M. Hu, J. Nunn, and I. A. Walmsley, “Integrated photonic sensing,” New J. Phys. 13, 055024 (2011).
[Crossref]

Zhang, Z.

C. Shen, Z. Zhang, and L.-M. Duan, “Scalable implementation of boson sampling with trapped ions,” Phys. Rev. Lett. 112, 050504 (2014).
[Crossref]

Zwiller, V.

D. Bonneau, E. Engin, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, J. L. O’Brien, and M. G. Thompson, “Quantum interference and manipulation of entanglement in silicon wire waveguide quantum circuits,” New J. Phys. 14, 045003 (2012).
[Crossref]

IEEE Photon. J. (1)

L.-N. Chen, E. Hall, L. Theogarajan, and J. Bowers, “Photonic switching for data center applications,” IEEE Photon. J. 3, 834–844 (2011).
[Crossref]

Microsyst. Nanoeng. (1)

R. Stabile, A. Albores-Mejia, A. Rohit, and K. A. Williams, “Integrated optical switch matrices for packet data networks,” Microsyst. Nanoeng. 2, 15042 (2016).
[Crossref]

Nat. Commun. (3)

A. Crespi, R. Osellame, R. Ramponi, M. Bentivegna, F. Flamini, N. Spagnolo, N. Viggianiello, L. Innocenti, P. Mataloni, and F. Sciarrino, “Suppression law of quantum states in a 3D photonic fast Fourier transform chip,” Nat. Commun. 7, 10469 (2016).
[Crossref]

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
[Crossref]

A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’Brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun. 2, 224 (2011).
[Crossref]

Nat. Photonics (3)

B. J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X.-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Quantum teleportation on a photonic chip,” Nat. Photonics 8, 770–774 (2014).
[Crossref]

N. Spagnolo, C. Vitelli, M. Bentivegna, D. J. Brod, A. Crespi, F. Flamini, S. Giacomini, G. Milani, R. Ramponi, P. Mataloni, R. Osellame, E. F. Galvão, and F. Sciarrino, “Experimental validation of photonic boson sampling,” Nat. Photonics 8, 615–620 (2014).
[Crossref]

J. Capmany, I. Gasulla, and D. Perez, “Microwave photonics: the programmable processor,” Nat. Photonics 10, 6–8 (2016).
[Crossref]

New J. Phys. (2)

D. Bonneau, E. Engin, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, J. L. O’Brien, and M. G. Thompson, “Quantum interference and manipulation of entanglement in silicon wire waveguide quantum circuits,” New J. Phys. 14, 045003 (2012).
[Crossref]

N. Thomas-Peter, N. K. Langford, A. Datta, L. Zhang, B. J. Smith, J. B. Spring, B. J. Metcalf, H. B. Coldenstrodt-Ronge, M. Hu, J. Nunn, and I. A. Walmsley, “Integrated photonic sensing,” New J. Phys. 13, 055024 (2011).
[Crossref]

Opt. Express (2)

Optica (1)

Photon. Res. (1)

Phys. Rev. Lett. (2)

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref]

C. Shen, Z. Zhang, and L.-M. Duan, “Scalable implementation of boson sampling with trapped ions,” Phys. Rev. Lett. 112, 050504 (2014).
[Crossref]

Sci. Adv. (1)

M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1, e1400255 (2015).
[Crossref]

Science (1)

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. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711 (2015).
[Crossref]

Other (11)

N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Bosonic transport simulations in a large-scale programmable nanophotonic processor,” arXiv preprint arXiv:1507.03406 (2015).

T. K. Gaylor, E. I. Verriest, and M. M. Mirsalehi, “Integrated optical givens rotation device,” U.S. patent4950042 A (August21, 1990).

“Birth of the programmable optical chip,” Nat. Photonics10, 1 (2016), Editorial.

F. Morichetti, A. Annoni, S. Grillanda, N. Peserico, M. Carminati, P. Ciccarella, G. Ferrari, E. Guglielmi, M. Sorel, and A. Melloni, “4-channel all-optical mimo demultiplexing on a silicon chip,” in Optical Fiber Communication Conference (2016), paper ThE3.7.

A. Ribeiro, A. Ruocco, L. Vanacker, and W. Bogaerts, “Demonstration of a 4 × 4-port universal coupler,” in Group IV Photonics Conference (2016), paper PD5.

G. Alber, T. Beth, M. Horodecki, P. Horodecki, R. Horodecki, M. Rötteler, H. Weinfurter, R. Werner, and A. Zeilinger, Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments (Springer, 2003), Vol. 173.

B. Peropadre, A. Aspuru-Guzik, and J. J. Garcia-Ripoll, “Spin models and boson sampling,” arXiv preprint arXiv:1509.02703 (2015).

B. Peropadre, G. G. Guerreschi, J. Huh, and A. Aspuru-Guzik, “Microwave boson sampling,” arXiv preprint arXiv:1510.08064 (2015).

N. Thomas-Peter, “Quantum enhanced precision measurement and information processing with integrated photonics,” Ph.D. thesis (Balliol College, University of Oxford, 2012).

These random unitary matrices were generated using the QR decomposition of random matrices, following the code developed by Toby Cubitt, available on www.dr-qubit.org/Matlab_code.html .

http://dx.doi.org/10.5287/bodleian:6gMmJYmab .

Supplementary Material (1)

NameDescription
» Supplement 1: PDF (601 KB)      A. Method for characterizing elements of a realistic circuit. B. General decomposition procedure.

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Figures (3)

Fig. 1.
Fig. 1. Universal N-mode multiport interferometer (shown here for N=9) can be implemented using a mesh of N(N1)/2 beam splitters such as (a) the one proposed by Reck et al. or (b) the one we demonstrate in this paper. As shown in (c), a line corresponds to an optical mode, and crossings between two modes correspond to a variable beam splitter described by a Tm,n(θ,ϕ) matrix, which can be implemented by a Mach–Zehnder interferometer consisting of two 50:50 directional couplers, preceded by a phase shift at one input port. Although the total number of beam splitters in both interferometers is identical, our scheme clearly has a much shorter optical depth and therefore suffers less propagation loss. This reduction in optical depth stems from the fact that each mode crosses its nearest neighbor at the first possible occasion in contrast to the Reck scheme, where the top modes must propagate for some distance before interacting with other modes. Furthermore, the high symmetry inherent to our design improves the loss tolerance of the interferometer, as we show in the main text.
Fig. 2.
Fig. 2. Illustration of the algorithm for programming a universal multiport interferometer for a 5×5 interferometer. The left-hand side presents our decomposition procedure, and the right-hand side shows how our decomposition corresponds to building up the corresponding interferometer. (1) We start with any random unitary matrix U, and a blank interferometer. (2) We first null the bottom-left element of U with a T1,21 matrix, which causes the first two columns of U to mix. This corresponds to adding the top-left beam splitter in the interferometer. (3) We then null the next two elements of U using a T3,4 matrix followed by a T4,5 matrix, which correspond to the two bottom-right beam splitters in the interferometer. T3,4 mixes rows 3 and 4, and T4,5 mixes rows 4 and 5. Since both the (4,1) and (5,1) elements of U had been nulled, they are not affected by T4,5. (4) and (5) At every step in the algorithm, we null a successive diagonal of the updated U matrix by alternating between Tm,n and Tm,n1 matrices, which corresponds to adding diagonal lines of beam splitters to the interferometer. Tm,n (resp. Tm,n1) matrices of a given color cause the rows (resp. columns) m and n, which are shown in the same color, to mix and null the corresponding element of that color in U. It is clear from this process that once a matrix element has been nulled, no subsequent operation can modify it. (6) After step 5, U is a lower triangular matrix, which by virtue of its unitarity must be diagonal. As explained in the main text, we can then write U in the way shown here, which by construction exactly corresponds to the desired interferometer.
Fig. 3.
Fig. 3. Left: average fidelity for an interferometer with a constant loss of 0.2 dB per beam splitter (as in the universal multiport interferometer in [6]) for interferometers built according to the Reck design (black) and our design (blue), for different interferometer sizes. Inset: close-up of the fidelity in our design. Right: fidelity as a function of loss for interferometers implementing 20×20 transformations. We see from our results that our design is much more loss-tolerant than the Reck design and maintains high fidelity with the target unitary even in the case of high loss. This is because the mismatched path lengths in the Reck design causes the loss to severely affect the resulting interference. See [24] for underlying data.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

Tm,n(θ,ϕ)=[10001eiϕcosθsinθeiϕsinθcosθ10001],
U=D((m,n)STm,n),
T4,5T3,4T2,3T1,2T4,5T3,4UT1,21T3,41T2,31T1,21=D,
U=T3,41T4,51T1,21T2,31T3,41T4,51DT1,2T2,3T3,4T1,2.
U=DT3,4T4,5T1,2T2,3T3,4T4,5T1,2T2,3T3,4T1,2,
F(Uexp,U)=|tr(UUexp)Ntr(UexpUexp)|2,

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