Optimal approximation to unitary quantum operators with linear optics

Quantum Information Processing (2021) 20:314 https://doi.org/10.1007/s11128-021-03254-2 Optimal approximation to unitary quantum operators with linear optics Juan Carlos Garcia-Escartin1·Vicent Gimeno2· Julio José Moyano-Fernández3 Received: 6 January 2021 / Accepted: 10 September 2021 / Published online: 21 September 2021 © The Author(s) 2021 Abstract Linear optical systems acting on photon number states produce many interesting evolutions, but cannot give all the allowed quantum operations on the input state. Using Toponogov’s theorem from differential geometry, we propose an iterative method that, for any arbitrary quantum operator Uacting on nphotons in mmodes, returns an operator  Uwhich can be implemented with linear optics. The approximation method is locally optimal and converges. The resulting operator  Ucan be translated into an experimental optical setup using previous results. Keywords Optical implementations of quantum evolutions ·Quantum Optics · Toponogov’s Theorem 1 Introduction Linear optical devices under quantum light show a rich behaviour and have different applications in experiments on the foundations of quantum optics and quantum infor- BJuan Carlos Garcia-Escartin [email protected]a.es Vicent Gimeno [email protected] Julio José Moyano-Fernández [email protected] 1Departamento de Teoría de la Señal y Comunicaciones e Ingeniería Telemática. ETSI de Telecomunicación, Universidad de Valladolid Campus Miguel Delibes, Paseo Belén 15. 47011, Valladolid, Spain 2Universitat Jaume I, Campus de Riu Sec, Departament de Matemàtiques & Institut Universitari de Matemàtiques i Aplicacions de Castelló–IMAC, 12071 Castellón de la Plana, Spain 3Universitat Jaume I, Campus de Riu Sec, Departamento de Matemáticas & Institut Universitari de Matemàtiques i Aplicacions de Castelló–IMAC, 12071 Castellón de la Plana, Spain 123 314 Page 2 of 18 J. C. Garcia-Escartin et al. mation [1–3]. While they can be built with relatively simple optical elements like beam splitters and phase shifters [4–8], their behaviour for photon number states cannot be accurately reproduced by any classical system. One clear example is the boson sampling problem, for which quantum systems can give efficient solutions which cannot be produced by any classical method [9]. Passive lossless linear optical systems where the number of photons is preserved are particularly attractive: they are simple, well understood, and they can be translated to standard experimental setups. There are many results on the synthesis of linear systems from their classical description [4,7,8] and that analyze the evolution of multiple photons in those devices [10–18]. However, there are relatively few methods for the design of tailored quantum evolutions for multiple photons. We have previously presented an inverse method which can tell if any desired quantum evolution on nphotons can be achieved with a linear optical system or not, giving the corresponding system when it is possible [19]. In this paper, we complete this design method with a procedure that gives the best possible approximation to any quantum unitary that cannot be achieved using only photon preserving linear optical systems. The approximation starts from an initial guess and returns a matrix in its neighbourhood which minimizes the matrix distance to the desired unitary but can be achieved using linear optics alone. The result is based on Toponogov’s comparison theorem [20] from differential geometry. Section 2introduces the mathematical description of linear optical systems and previously known results we will use. Section 3describes the structure of the image algebra and its complement and states two theorems that will become useful later. Section 4gives the basic concepts from differential geometry used in the proof and the notation for the rest of the paper. Section 5defines the bi-invariant Riemannian metric in which the results are given. Section 6shows how to apply Toponogov’s theorem to reduce the problem of approximating a unitary to finding a geodesic in the correct manifold. Section 7discusses some tricks related to the generation of random unitaries which are needed to explore the image group and to be able to compute a valid matrix logarithm for any desired U. Section 8describes the iterative method that produces the desired approximation. Section 9gives an application example. Finally, Sect. 10 gives a general overview of the method and comments on some practical problems and possible improvements for the approximation algorithm. 2 Mathematical description of linear optics interferometers All through this paper we restrict to lossless linear optical systems where the total number of photons is preserved. Classically, the evolution of the electrical field in m orthogonal modes going through such a linear optical system is perfectly described by a unitary m×mmatrix, S, called the scattering matrix of the system [21]. Linear optical systems including loss and amplification can be described using quasiunitary matrices [22]. The evolution of nphotons distributed through these mpossible modes is given by an M×Munitary evolution matrix U acting on the M=m+n−1 nstates of the resulting Hilbert space. 123 Optimal approximation to unitary quantum... Page 3 of 18 314 The photonic homomorphism ϕ:U(m)→U(M)gives the evolution matrix U which corresponds to a scattering matrix Sdescribing the photon preserving linear optical system. U=ϕ(S)can be computed from different equivalent methods [10– 12]. Any unitary matrix can be written as an exponential U=eiH for a Hermitian matrix H. In linear optical devices, we will call this matrix the effective Hamiltonian HUof the linear system. Similarly, S=eiH S. The image of the photonic homomorphism, im(ϕ), is a subgroup of U(M)which contains all the quantum evolutions that are allowed for nphotons a linear optical system with mmodes. The image subgroup is a representation of U(m)in U(M)and maps each possible classical scattering matrix Sdescribing a linear system into the quantum evolution U=ϕ(S)it induces for nphotons. The evolution in the corresponding unitary algebras from iH Sto iH Uis given by the differential of ϕ,dϕ:u(m)→u(M), for which there are also explicit expressions [13–18]. From the point of view of system design, a natural question is whether any given quantum evolution U∈U(M)can be realized using only linear optics. From a simple dimensional argument, it is clear that, except when m=1orn=1, there must be some impossible operations [23]. In a previous work, we have given an explicit inverse method to find the Scorresponding to any U∈im(ϕ) which can be implemented using linear optics [19]. Here, we address the problem of approximating U/∈im(ϕ). We give a method to find the linear optics system with an evolution matrix  U∈im(ϕ) which minimizes the distance to Ulocally. 3 The image algebra and its orthogonal complement If we study the induced map dϕ:u(m)→u(M), we can decompose the Lie algebra u(M)orthogonally so that u(M)=im dϕ⊕(im dϕ)⊥,(3.1) where (im dϕ)⊥is the orthogonal complement of im dϕwith respect to the metric u,v=1 2tr(u†v+v†u). (3.2) For this metric, we can prove a couple of useful facts. Theorem 1 For U ∈U(M)such that U /∈im ϕ,letv∈u(M)be the principal logarithm of U. Let v=vT+vN(3.3) be the orthogonal decomposition of v, with a tangent component vT∈im dϕand a normal component vN∈im (dϕ)⊥. Then, 123 314 Page 4 of 18 J. C. Garcia-Escartin et al. 1. Ua=exp(vT)∈im ϕ. 2. U−Ua≤vN. Therefore, for any normalized |ψwith ψ|ψ=1, we have 1≥|Uψ|Uaψ| ≥ 1−vN2 2.(3.4) The proof is given by introducing a bi-invariant metric and reducing the issue to a problem in plane geometry thanks to Toponogov’s comparison theorem [20]. Later, with this theorem, we can give a recursive method to find a locally optimal approximation and show it converges. 4 Prerequisites and Notation For the very basic notions in differential geometry, such as manifold, curve, tangent space, the reader is referred to the books of Do Carmo [24] or Sakai [25]. A Riemannian metric on a differentiable manifold Mis a correspondence which associates to each point pon Man inner product ,pon the tangent space TpM which varies differentiably in the sense that, for any pair of vector fields Xand Ywhich are differentiable in a neighbourhood Vof M, the function X,Yis differentiable on V. The metric with which a Riemannian manifold Mis endowed may come from a distance. Given two points p,q∈M, the distance d(p,q)between them is defined to be the infimum of the lengths of all curves joining pand qwhich are piecewise differentiable. Two fundamental concepts of Riemannian geometry are those of geodesic and curvature. Roughly speaking, a geodesic is a curve minimizing the distance between two nearby points. More precisely, let Ibe a closed interval in R; a parametrized curve γ:I→Mis called a geodesic at t0if the covariant derivative D dt dγ dt vanishes at the point t0(see, i.e. [24], Definition 2.1); if γis a geodesic at tfor all t∈I, then γ is called a geodesic. If [a,b]⊆Iand γ:I→Mis a geodesic, the restriction of γ to [a,b]is called a geodesic segment joining γ(a)to γ(b). By abuse of language it is often referred to the image γ(I)of a geodesic γas a geodesic. Aminimal geodesic between pand qis the shortest one joining pand q. It is easily seen that if there exists a minimal geodesic γjoining pto q, then d(p,q)equals the length (γ ) of γ. This conditional if holds under the hypothesis of completeness: a Riemannian manifold Mis said to be (geodesically) complete if for every p∈Mthe exponential map exppis defined for the whole tangent space TpM, i.e. if any geodesic γ(t)starting from pis defined for all t∈R, and the statement is: Theorem 2 (Hopf-Rinow) Let M be a Riemannian manifold and let p ∈M. Then M is geodesically complete if and only if it is complete as a metric space. Moreover, this implies that for any q ∈M there exists a minimizing geodesic γjoining p to q. An important consequence of Theorem 2is the following, see [25], Corollary 1.4 and Problem 1 of Chapter III: 123 Optimal approximation to unitary quantum... Page 5 of 18 314 Corollary 1 AC∞manifold M is compact if and only if any Riemannian metric on M is complete. On the other hand, the concept of curvature we will refer to is that of sectional curvature. According to Milnor [26] (p. 295), the sectional curvature of the tangential 2- plane spanned by some orthogonal unit vectors uand vcan be described geometrically as the Gaussian curvature, at the point, of the surface swept out by all geodesics having a linear combination of uand vas tangent vector. We are interested in Riemannian manifolds with additional algebraic structure: Lie groups. A Lie group is a group Gwith a differentiable structure such that the mapping G×G→Ggiven by (x,y)→ xy−1,x,y∈G, is differentiable. It follows that translations from the left Lxresp. translations from the right Rxgiven by Lx:G→G,Lx(y)=xy resp. Rx:G→G,Rx(y)=yx are diffeomorphisms. A Riemannian metric on Gis said to be left invariant resp. right invariant if for all p,g∈Gand for all u,v ∈TpGit holds that u,vp=d(Lg)(u), d(Lg)(v)Lg(p)resp. u,vp=d(Rg)(u), d(Rg)(v)Rg(p). A Riemannian metric is called bi-invariant if it is both left and right invariant. Any compact Lie group can be endowed with a bi-invariant metric [24, Exercise 7]. We also consider the Lie algebra Gof G, which consists of the vectors in TeGwith ethe neutral element of Gand with a well-known additional structure provided by a commutator (or Lie bracket) in the usual way. We will focus on the Lie group U(M)as a differentiable manifold, for a positive integer M; its Lie algebra will be denoted by u(M). Identity matrices of any size will be denoted by Id. 5 A bi-invariant Riemannian metric In this section, we endow U(M)with a Riemannian structure which will be useful later on. For u,v ∈u(M), we define an inner product u,v:=1 2tr(u†v+v†u). (5.1) This definition does actually correspond to a positive definite symmetric bilinear form: The bilinearity is an easy exercise, and moreover: 1. Since u,v ∈u(M), then u†=−u,v†=−v, and therefore u,v:=1 2tr(−uv−vu)=−tr(uv). 2. Thesymmetryof(5.1) is clear, since u,v=−tr(uv) =−tr(vu)=v,u. 123 314 Page 6 of 18 J. C. Garcia-Escartin et al. 3. The positive definiteness follows from the fact that u,u=tr(u†u)=||u||2≥0, where ||u|| = i,j|uij|2is the Frobenius norm of u, which is nonnegative and has all the required norm properties [27]. The metric defined above is Riemannian and bi-invariant. Bi-invariant metrics are useful for us because of the following. Theorem 3 (Milnor [26]) Every compact Lie group admits a bi-invariant metric, which has nonnegative sectional curvature. In the case of a bi-invariant metric, the sectional curvature admits an easier formula, see [26, p. 323, Eqn. (7.3)]: κ(u,v)=1 4[u,v],[u,v]. Furthermore, in a Lie group admitting a bi-invariant metric, geodesic curves have an easy description: they coincide with the exponential map. More precisely, for p∈ U(M), a geodesic curve γ:I→U(M)such that γ(0)=pand ˙γ(0)=uis of the form γ(t)=exp (up−1t)·p(5.2) In fact, for p,q∈U(M), there exists a geodesic γjoining pand qwith γ(0)=p such that (γ ([0,t])) =t 0˙γ(t), ˙γ(t)dt =t 0u,udt =t 0udt =ut.(5.3) A geodesic segment γ:[0,1]→U(M)is called minimal if it realizes a distance for any t∈[0,1],i.e., d(γ (0), γ (t)) =(γ ([0,t])). (5.4) By equation (5.2) a geodesic segment γ:[0,1]→U(M)joining γ(0)=pand γ(1)=qcan be always obtained as γ(t)=exp(vt)p,with vsuch that exp(v) =qp−1. The concept of a minimal geodesic is related to the concept of principal logarithm in the following way 123 Optimal approximation to unitary quantum... Page 7 of 18 314 Lemma 1 Let p,q∈U(M),letwbe a principal logarithm of qp−1. Then, the geodesic segment γ:[0,1]→U(M), t→ γ(t)=exp(wt)p is a minimal geodesic segment joining p and q. Recall iK is called a principal logarithm of a unitary matrix M∈U(M)if K†=K,exp(iK)=M,and the eigenvalues of Kare in (−π, π]. There are efficient algorithms that can compute the principal logarithm of a unitary matrix [28]. We choose this definition of principal logarithm over the one for the interval (−π, π) so that there is always a principal matrix logarithm, even for matrices with real negative eigenvalues (-1). Some properties, like infinite differentiability, are lost with this definition, but they are not used in our result. Proof of Lemma 1The length of a geodesic segment γ(t)=exp(vt)pwith exp(v) = qp−1is (by equation (5.3)) (γ ([0,1])) =vp=v. The distance d(p,q)between pand qis the shortest length of the curves joining p and q. In the case of a complete metric this shortest length is attained by a geodesic segment joining pand q. Hence, the geodesic segment γ(t)=exp(vt)pis minimal if and only if v=min{w:exp(w) =qp−1}. But the equation exp(w) =qp−1has the following family of solutions w=U[log(λi)δij]U† with pq−1=UΛU†,Ubeing a unitary matrix, [Λ]ij =(λi)δij being the diagonal matrix of eigenvalues of pq−1, and log(λi)being any logarithm of λi. Observe that w=   M  i=1|log(λi)|2. This expression only depends on the list of eigenvalues of pq−1and their logarithms. Since pq−1is unitary log(λi)=i(ki+2πli), with ki∈(−π,π],and li∈Z. 123 314 Page 8 of 18 J. C. Garcia-Escartin et al. Fig. 1 Geodesic triangle Δ(p1p2p3) Then {w:exp(w) =qp−1}=⎧ ⎨ ⎩   M  i=1|(ki+2πli)|2:li∈Z⎫ ⎬ ⎭, with a minimum when li=0fori=1,...,M, which corresponds to a principal logarithm.  6 An application of Toponogov’s comparison theorem Riemannian manifolds whose curvature is bounded below may be investigated by applying Toponogov’s comparison theorem. We first need to define triangles on the Riemannian manifold. Definition 1 Ageodesic triangle T =Δ(p1p2p3)of a Riemannian manifold Mis a set consisting of three segments of minimal geodesics, which are called the sides of T,say γ1:[0,1]→M,γ 2:[0,1]→M,and γ3:[0,1]→M, such that γi(1)=γi+1(0)for i=1,2, and γ3(1)=γ1(0). The endpoints p1,p2and p3are called the vertices of the triangle. The angle between the tangent vectors to γi−1 and γ−1 i+1at piis called the angle of Tat piand denoted by αi=∠(pi−1pipi+1)or ∠pi. The perimeter is defined as the sum (γ1)+(γ2)+(γ3); if we consider, in addition, that the two sides γ2,γ 3are minimal geodesics, and the side γ1is a geodesic segment, not necessarily minimal, with (γ1)≤(γ2)+(γ3)=d(p1,p3)+d(p1,p2), then the set is said to be a generalized geodesic triangle (Fig. 1). Let us set the part of Toponogov’s comparison theorem we are interested in, cf. [25, Theorem 4.2 in Chapter IV]: Theorem 4 (Toponogov) Let M be a complete Riemannian manifold whose sectional curvatures satisfy κ≥δeverywhere for some constant δ. Denote by M2 δthe 2- dimensional complete simply connected Riemannian manifold of constant curvature 123 Optimal approximation to unitary quantum... Page 9 of 18 314 δ. Consider a generalized geodesic triangle Δ(p1p2p3)such that γ2,γ 3are minimal and (γ1)≤π/√δ. Then the perimeter ≤2π/√δand there exists a geodesic triangle Δ( ˜p1˜p2˜p3)in M2 δwith the same side lengths ( ˜γi)=( ˜γi),fori =1,2,3and satisfying α2≥˜α2and α3≥˜α3. Remark It is to assume that π/√δ=+∞when δ≤0 in the theorem above. Theorem 4allows us to compare triangles of U(M)with triangles in R2by setting δ=0, since in this case M2 0=R2. In our situation it is p1=U,p3=Id and p2=Ua=exp(vT)is our approximation matrix in im(ϕ) (recall that vTis the tangential component of the principal logarithm vof U). Set γ1(t)=Uaexp(−vTt), γ2(t)=exp(vt)and γ3(t)a minimal geodesic segment joining Uwith Ua. Then γ2 and γ3are minimal geodesic segments and γ1is a geodesic segment (not necessarily minimal). Set 1=(γ1([0,1])) =vT,2=(γ2([0,1])) =d(U,Id)and 3=(γ3([0,1])) =d(U,Ua). Observe that 1=vT≤v+d(U,Ua)=2+3. Hence, by Theorem 4, there exists a geodesic triangle in R2with sides 1, 2and 3 of lengths 1, 2resp. 3, such that α2≥˜α2and α3≥˜α3. We want to estimate the distance 3. First of all notice that 3=2−1; hence, the law of cosinus implies that 32=2−12=12+22−212cos ∠(1, 2) and 2 3=2 1+2 2−212cos ˜α3.(6.1) Since cos α3=v,vT vvT=vT v≥0, it follows that ˜α3≤α3≤π 2which implies −cos ˜α3≤−cos α3and, so, 2 3≤2 1+2 2−212cos α3=vT2+v2−2vTvcos α3 =v2+vT2−2v,vT=v−vT2=vN2.(6.2) Figure 2shows a graphical representation of our scenario. Now we want to see the bi-invariant metric in u(M)defined in Sect. 5as a metric in U(M). First observe that the Riemannian manifold (U(M), ,)is a Riemannian subvariety of the manifold Mn(C)of the complex n×n-matrices endowed with the Euclidean inner product. If we write dU(M)for the distance we use on U(M), and dMnfor the one on Mn(C), then 3≥dU(M)(Ua,U)≥dMn(Ua,U). The inequality holds since “distance” between two points is the infimum of the length of any two curves joining the points. Now it is easy to find a minimal geodesic for 123 314 Page 16 of 18 J. C. Garcia-Escartin et al. Fig. 3 Linear optics interferometer approximating the QFT for a system with two inputs and two photons. The system uses two phase shifters (with shifts −π 3and π 3in the upper and lower mode, respectively) and one balanced beam splitter (centre). This configuration gives one possible approximation, U3 a, which has matrix distance 0.85675 to the QFT 9.2 Translation to an optical setup The best approximation to the QFT comes from a scattering matrix S3 awhich corresponds, up to a 5π 12 global phase, to a balanced two input beam splitter with a scattering matrix SBS =1 √21i i1 preceded by a −π 3phase shifter in the first port and followed by a π 3in the second port. The phase shifters have scattering matrices S−π 3=e−iπ 30 01 and Sπ 3=10 0eiπ 3, respectively. The experimental configuration corresponding to this approximation to the QFT is shown in Fig. 3. 10 Summary, recommendations and future improvements We have given an iterative method that finds a linear optical setup that approximates any arbitrary quantum evolution for nphotons in mmodes. The approximation is optimal in the local neighbourhood of the initial guess. Once we have the closest unitary that can be implemented,  U, we can use a previous method [19] to obtain a scattering matrix  Sthat gives the approximated evolution and there are multiple algorithms that give the physical setup corresponding to  Susing only beam splitters and phase shifters [4,7]. This gives a design method which starts from the desired evolution instead of the usual analysis methods which, given a classical description of the linear optical system, find out the evolution it induces on quantum states. The proposed algorithm is based on results from differential geometry, in particular, Toponogov’s theorem. They show the method will converge and find local optima. 123 Optimal approximation to unitary quantum... Page 17 of 18 314 There are a few practical details worth mentioning. First, numerically, we find that if we try the method on an evolution which is possible to obtain from a linear optics system, U∈imϕ, sometimes it will converge to a matrix close the actual solution, but, depending on the local landscape, it might fall into a variety of different matrices all at a similar large distance. In practical applications, the recommendation would be, first, check whether an exact implementation exists (with our previous algorithm [19]) and, if there is none, look for an approximation. There are also some open problems. The presented results are only valid for lossless linear systems. Linear optical systems including losses and squeezing can be described using quasiunitary matrices [13,22], which can also be studied using group theory. However, there appear some technical complications and the theory of these linear system is not as developed as in the lossless case. We leave for future research the adaptation of the presented design methods to these scenarios. Additionally, from the structure of the involved groups and algebras, it is not clear how many local optima exist for any given transformation ϕ. We have proposed a randomized way of exploring the state space of the potential approximation matrices, and, for the limited dimensions that can be numerically explored, it seems to work well, but there is no guarantee the method finds a global minimum. Any further knowledge of the structure of the image group would help in the search for a global minimum or, at least, in finding a probabilistic bound on the optimal approximation. Additionally, a lower bound on U−Ua, as opposed to our upper norm, would help to determine the global optimum. Even in its present form, the proposed algorithm, when combined with previous results, can assist in the design of quantum optical operations and has applications to quantum information and quantum optics experiment design. Acknowledgements The first author has been funded by the Spanish Government and FEDER grant PID2020-119418GB-I00 (MICINN) and Junta de Castilla y León (project VA296P18). The second author has been partially supported by the Research Program of the University Jaume I—Project UJI-B2018- 35, as well as by the Spanish Government and FEDER grants PID2020-115930GA-I00 (MICINN) and MTM2017-84851-C2-2 (MINECO). The third author was partially supported by the Spanish Government, Ministerios de Ciencia e Innovación y de Universidades, grant PGC2018-096446-B-C22, as well as by Universitat Jaume I, grant UJI-B2018-10. Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Loudon, R.: The Quantum Theory of Light, 3rd edn. 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