Optimal design of error-tolerant reprogrammable multiport interferometers
aa r X i v : . [ phy s i c s . op ti c s ] M a y Optimal design of error-tolerant reprogrammable multiport interferometers
S.A. Fldzhyan, M.Yu. Saygin, ∗ and S.P. Kulik Quantum Technology Centre, Faculty of Physics,Lomonosov Moscow State University, Moscow, Russian Federation
Photonic information processing demands programmable multiport interferometers capable ofimplementing arbitrary transfer matrices, for which planar meshes of error-sensitive Mach-Zehnderinterferometers are usually exploited. We propose an alternative design that uses a single staticbeam-splitter and a variable phase shift as the building block. The design possesses superior re-silience to manufacturing errors and losses without extra elements added into the scheme. Namely,the power transmissivities of the static BSs can take arbitrary values in the range from ≈ / ≈ /
5. We show that the fraction of transfer matrices non-implementable by the interferometers ofour design diminishes rapidly with their size.
I. INTRODUCTION
Photonics is progressively playing a more importantrole in fundamental science and applied areas, motivatedby novel developed approaches to information process-ing which are well-matched with the qualities of optics.Linear transformations between multiple optical channelsare often required by these approaches, thus, making theutilization of multiport interferometer devices a necessity.For example, multiport interferometers are exploited asmode unscramblers [1] and parts of photonic neural net-works [2, 3].In the recent years, optical interferometers have at-tracted appreciable interest by the quantum informationcommunity, because of the promising quantum comput-ing platforms that leverage linear-optics and unique prop-erties of photon discrete variables [4, 5] and field contin-uous variables states [6, 7]. Recent works have demon-strated the versatility of linear-optical quantum systemsand their ability to perform quantum computing tasks,ranging from the algorithms suggested at the dawn ofquantum information theory [8, 9] to more specific onesthat disrupt the landscape nowadays, such as the bo-son sampling algorithm [10–12] and quantum deep neuralnetworks [13, 14].Universal interferometers can be reprogrammed to im-plement an arbitrary linear transformation defined by aspecific transfer matrix. To construct these interferome-ters, decomposition methods are used that represent uni-tary matrices as products of simpler building blocks [15–17]. Among these methods, the most practical are planardecompositions, since they well suit fabrication by themature techniques of integrated photonics enabling mas-sive production of sophisticated optical circuits [2, 5, 18].Today, the most often used methods are those proposedby Reck et al in [15] and Clements et al in [16], which de-compose unitary matrices into planar meshes of variablebeam-splitters (BSs), having triangular and rectangularforms, respectively. In these schemes, each variable BS is ∗ Electronic address: [email protected] conveniently realized by a standard element of the Mach-Zehnder interferometer (MZI), made up of two static bal-anced BSs with variability provided by two phase shifts.Thus, the overall scheme is reprogrammed by setting thephase shifts [18, 19].For these schemes to be universal, it is crucial that thestatic BSs should be balanced. However, this conditioncan not be fully satisfied because of the errors that occurat realization, limiting the scheme’s universality [20, 21].The negative effect of the errors progresses as the inter-ferometer size scales up, effectively imposing stringentrequirements on the fabrication tolerances and makingchallenging the realization of large interferometers.Methods exist that can restore the universality of theMZI-based schemes at the cost of adding extra elementsinto their optical schemes [21, 22]. However, the com-mon drawback of these methods is the burden of auxil-iary control needed to manipulate the additional MZIsand the increased real estate occupied by the scheme onthe chip. Therefore, developing more efficient designs oferror-tolerant inteferometers is highly demanded nowa-days. Here, we propose a new design of planar inter-ferometers, which is error-tolerant to manufacturing er-rors and universal except for a fraction of matrices whichrapidly diminishes with their size.
II. THE MZI-BASED AND BS-BASEDINTERFEROMETERS An N -port interferometer may be described by an N × N transfer matrix acting on vectors of field ampli-tudes according to the relation: a ( out ) = U a ( in ) , where a ( in ) and a ( out ) are the input and output vectors, re-spectively. Provided interferometers are lossless, theirtransfer matrices U are unitary.We first describe interferometers constructed with thecanonical MZI-based design [16] depicted in Fig. 1a. Itis formed by N layers consisting of MZIs, each acting lo-cally only on two neighboring channels. In this scheme,the overall number of MZIs is equal to N ( N − /
2. Ac-counting for the N − N − N × N matrix. a) ... ... ... b) ... ... ... ......1 2 2N3 beam-splitterphase-shifter FIG. 1: Two designs of multiport interferometers: a) the con-ventional universal MZI-based design proposed in [16]. Theelements colored in red are utilized in the rerouting operationfrom the first into the last port. b) the alternative BS-baseddesign proposed in this work.
Therefore, the transfer matrix of the MZI-based inter-ferometer can be written as U MZI = V ( N )MZI · . . . · V (2)MZI V (1)MZI Φ (1)where V ( m )MZI is the transfer matrix of m -th layer, Φ =diag (cid:16) e iϕ ( out )1 , . . . , e iϕ ( out ) N − , (cid:17) is the diagonal matrix with ϕ ( out ) j being the phase-shifts introduced in the end. In(1), layer transfer matrices V ( m )MZI are of the form: V ( m )MZI = Q j ∈ Ω MZI m T ( m )MZI ,j ( ϕ ( m )2 j − , ϕ ( m )2 j ), where T ( m )MZI ,j ( ϕ , ϕ ) = · · · · · · · · · · · · B ( m )1 , B ( m )1 , ...... B ( m )2 , B ( m )2 , ...... . . . ...0 · · · · · · (2)is the block matrix of single MZI placed in the m -th layerbetween channels j and j + 1, Ω MZI m denotes the orderedsequence of MZIs in the layer with index m . Block ma-trix (2) has all diagonal elements 1 except those labeled B ( m )1 , = e i ( ϕ ( m )1 + ϕ ( m )2 ) a ( m ) j and B ( m )2 , = − e iϕ ( m )1 a ( m ) ∗ j ,and all off-diagonal elements equal to 0 except those la-beled B ( m )1 , = e iϕ ( m )1 b ( m ) j and B ( m )2 , = e i ( ϕ ( m )1 + ϕ ( m )2 ) b ( m ) ∗ j ,where we introduced the shorthand notations: a =sin ϕ ( m )1 cos( α ( m )1 − α ( m )2 )+ i cos ϕ ( m )1 sin( α ( m )1 + α ( m )2 ) and b = cos ϕ ( m )1 cos( α ( m )1 + α ( m )2 )+ i sin ϕ ( m )1 sin( α ( m )1 − α ( m )2 ). The variable phase shifts ϕ j are used to reconfigure theinterferometer and have required ranges from 0 to 2 π .Parameters α ( m ) l describe errors caused by the imbal-ances of the static BSs due to non-ideal realization.When α ( m ) l = 0 the MZI-based interferometer is capa-ble of implementing an arbitrary unitary transfer matrix,however, imbalances α ’s undermine its universality. Atrivial example of rerouting from port 1 into port N ofan N -port interferometer is shown in Fig. 1a. Obviously,to attain this transformation all diagonal MZIs should bein the cross state. However, the unit transmissivity of anMZI: τ = | b | = cos ϕ cos ( α + α ) + sin ϕ sin ( α − α ), can be obtained only when α = α = 0.The schematic representation of our design is depictedin Fig. 1b. Our design has rectangular placement ofbuilding blocks, each of which is a single static BS andsingle tunable phase shifter. Hence, the name BS-basedfor our design. The N -port inteferometer of the BS-baseddesign has 2 N layers so that both the scheme depth, asquantified by the maximum static BSs crossed by the sig-nals, and the number of phase shifts are equal to those ofthe MZI-based design. The interferometer transfer ma-trix takes the form: U BS = V (2 N )BS · . . . · V (2)BS V (1)BS Φ , (3)in which layer transfer matrices V ( m )BS = Q j ∈ Ω BS m T ( m )BS ,j ( ϕ ( m ) j ) has the block T ( m )BS ( ϕ ( m ) ) ofform (2), but with B ( m )1 , = e iϕ ( m ) cos( θ + α ( m ) ), B ( m )2 , = cos( θ + α ( m ) ), B ( m )1 , = sin( θ + α ( m ) ) and B ( m )2 , = − e iϕ ( m ) sin( θ + α ( m ) ). Here, angle θ quantifiesthe transmission of the static BSs, which specific valuewill be given below. -15 -10 -5 0 5 10 15 20 25 3010 -15 -10 -5 MZI-based BS-based α (degree) a) - F MZI-based BS-based -15 -10 -5 0 5 10 15 20 25 30 α (degree) -10 -5 b) - F FIG. 2: Infidelity 1 − F as a function of error parameter α for the MZI-based and BS-based interferometers at θ = π/ N = 5 and b) N = 10. For each value of α the infidelity distribution were obtained numerically using aset of 300 unitary matrices drawn randomly from uniformdistribution. The solid curves correspond to the average overall samples size; the lower and upper boundaries of the shadedregions are averages for 10 infidelities with the lowest andhighest values, respectively. a) -9 ( ) p r obab ili t y BS-based b) p r obab ili t y MZI-based
FIG. 3: Normalized histogram of infidelity 1 − F for the BS-based (a) and MZI-based (b) 10-port interferometers at ran-dom errors. The error angles α j were drawn from the distri-bution p ( α ) = exp (cid:0) − α / (cid:1) / √ π ∆ with ∆ = 10 degrees.For the BS-based interferometer the parameter θ = 55 de-grees, roughly corresponding to the center of the high-fidelityplateau, depicted in Fig. 2b. The histograms is the result ofthe optimization of 300 randomly sampled unitary matrices. III. ERROR TOLERANCE
We consider errors as α j = 0 that tune the splittingratios of the static BSs off required values. Firstly, westudy the effect of coherent errors at which α j = α . Forthe schemes manufactured by planar lithography tech-niques this type of errors is linked to the variations ofwaveguide’s material and geometry, which is dominatedas their spatial scale is usually large compared with thearea occupied by the scheme [23]. These arguments canalso be applied to interferometers manufactured by othermethods, for example, femtosecond direct laser writing[19], as well as alternative implementation approachesexploiting repetitively few optical elements to obtain thedesired transformation between multiple modes [24].We evaluate the performance of multiport interferom-eters by calculating the fidelity, defined as: F = (cid:12)(cid:12) Tr( U † U ) (cid:12)(cid:12) N Tr( U † U ) , (4)which compares the target unitary matrix U and theactual transfer matrix U realized by the interferometer,where N is the size of the matrices. Provided that thematrices U and U are equal up to a complex multiplier,the fidelity (4) gets its maximum value of F = 1.Generally, no analytical solution is known to derivephase shifts that maximize the fidelity (4), except for thecase of error-free MZI-based interferometers, for which ananalytical procedure is provided in [16]. Unfortunately,we could not found analogous procedure for the error-freeBS-based interferometers.We used a numerical optimization algorithm based onthe basinhopping algorithm. Given a unitary matrix U ,the algorithm was searching for a global minimum of in-fidelity 1 − F over the space of phase-shifts. To decreasethe chance of falling into local minima, we used multipleruns of the optimization with random initial values of thephases. Each numerical experiment involved optimiza-tion over a series of target matrices U , drawn from the Haar random distribution using the method based on theQR-decomposition of random matrices [25]. With this al-gorithm it took several hours to find optimal phase shiftsfor a single 10 ×
10 transfer matrix, so that a multi-corecomputer has been utilized to derive required dependen-cies. We understand that more efficient numerical algo-rithms can be developed for this specific task [26].The obtained infidelity 1 − F as a function of the errorparameter α is plotted in Fig. 2. The finite accuracy ofthe numerical algorithm sets the minimal infidelity valueof ∼ − − − , which could not be overcome neitherfor the MZI-based interferometer with α = 0 where ex-act zero was expected. The MZI-based interferometersare equally sensitive to both positive and negative valuesof α with the acceptable range of errors is of the orderof several degrees. For the BS-based interferometers, in-fidelity behaves radically different: while at α < α > α aslarge as ∼
20 degrees — several times larger than for theMZI-based interferometers. Fig. 2 suggests that whenthe positive and negative values of α are equiprobable,the optimal choice of the static BSs defined at the designstage is such that θ ≈
55 degree, corresponding to thecenter of the high-fidelity plateau.The superior perfomance of the BS-based interferome-ters at positive α ’s can be attributed to the interplay oftwo competing properties. On the one hand, more trans-missive BSs with α > α = π/ α ’s are distributed at random across the scheme. Theobtained infidelities are shown in Fig. 3. Clearly, the BS-based interferometer design is ultimately tolerant to theincoherent errors.The performed analysis cannot be complete in provingstrict universality of the interferometers, since randomgeneration of target matrices can overlook small sub-sets that are non-implementable error-tolerantly or non-implementable at all. We now show that such matricesdo exist for the BS-based design. For this, we generaterandom matrices of size N = 3 and calculate infidelities.Fig. 4a demonstrates that in this low-dimensional casethe relative volume occupied by the non-implementablematrices is quite large to be caught by random sampling: ≈ − F > − at θ = π/ α = 0. However, taking the results of Fig. 2into account, where not a single non-implementable ma-trix was present, we conclude that the relative volume ofnon-implementable matrices rapidly diminishes with N .Also, we consider some concrete examples which fallout of the picture obtained with random sampling. Oneexample is of the discrete Fourier transform (DFT) -40 -20 0 20 4010 -15 -10 -5 -10 -5 -15 -10 -5 -15 -10 -5 MZI-basedBS-based α (degree) MZI-basedBS-based MZI-basedBS-based α (degree) α (degree) α (degree) - F a) b)c) d) - F - F - F BS-based MZI-based
FIG. 4: Infidelity 1 − F as a function of error parameter α for the BS- and MZI-based interferometers implementing a)300 randomly generated unitary U at N = 3; the lower andupper boundaries of the shaded regions are averages for 10infidelities with the lowest and highest values, respectively.b) DFT matrix at N = 3, c) the swap between port 1 and 2at N = 3, d) Hadamard transformation H of ports 1 and 2at N = 3 (dashed curves) and Hadamard transformation H at N = 4 (solid curves). matrix with elements U ( DF T ) mn = exp( i π ( m − n − /N ) / √ N . Fig. 4b illustrates the resutls for N = 3,showing that both designs are equally tolerant, which isactually true for larger N .We next consider a block-diagonal matrix of size N = 3that comprises the swap block of ports 1 and 2 (Fig. 4c).In contrast to previous results, the MZI-based designworks much better than the BS-based design. Moreover,we have found that the block-diagonal matrices of ar-bitrary sizes are not reproduced error-tolerantly by theBS-based schemes, thus, it represents a class of transfor-mations where the MZI-based design is advantageous.Finally, we consider the Hadamard transformation.Namely, matrix H = √ (cid:18) − (cid:19) as a constituentupper-left block of a 3-by-3 matrix and matrix H = H ⊗ H as a whole 4-by-4 matrix. As can be seenfrom Fig. 4d, both designs are good at implementingthe transformation when it is a part of a larger matrix,while the BS-based design is much better in case of awhole martix. Once again, this is the evidence that theblock-diagonal matrices are better realized by the MZI-based than BS-based interferometers. In addition, wehave found the permutation matrices are also better im-plemented by the MZI-based interferometers rather thanthe BS-based ones. In particular, rerouting operation de-picted in Fig. 1a cannot be implemented perfectly by theBS-based interferometer. IV. LOSS TOLERANCE
In addition, we study the effect of unbalanced lossesthat occur because different paths through the interfer- ometer experience different losses, which can result inpoor fidelity. Typically, static BSs introduce additionallosses due to waveguide bending and scattering, there-fore, in both designs, unbalanced losses occur when thesignals have to pass through side paths, as they containless BSs than the inner paths. To model lossy BSs, each B ( m ) i,j in the MZI-based and BS-based transfer matriceswas multiplied by t and t , respectively, where 0 ≤ t ≤ a) BS-basedMZI-based F N b) F loss per BS (dB) BS-basedMZI-based
FIG. 5: a) average fidelity at a loss of 0 . N , b) average fidelity of a 10-port interferometer as a function of loss introduced by eachstatic BS. In the BS-based design θ = 55 degrees, whileBSs in the MZI-based design are balanced. Each average wascalculated over 50 randomly sampled matrices. V. CONCLUSION
In summary, we proposed a novel design of pro-grammable multiport interferometers, which exhibit su-perior tolerance to errors and losses than the previouslyknown counterparts, while not requiring redundant ele-ments. It is noteworthy that the proposed design is lesscomplex than the counterparts with possible implementa-tion by a variety of experimental platforms. For example,the static interferometers that have been used in experi-ments on boson sampling [12, 27] have suitable placementof the passive BS elements, yet lacking programmability.
Funding.
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