Permanents, Bosons and Linear Optics 2

Home , Computing the permanent

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1 , 2 ,..., d | i | i | i d d 2 ψ = ck k , ck =1. (1) | i | i | | k k X=1 X=1 Group U(d) of unitary matrices describes an evolution of the state. In such description a global phase does not matter [2] and the group SU(d) of unitary matrices with unit determinant may be used instead of U(d) d ′ ′ ψ =u ˆ ψ , c = ujkck, uˆ SU(d). (2) | i | i j ∈ k X=1 Let us denote Sn( ) linear space of symmetric n-tensors, dim Sn( ) = Cd+n−1. H H n Symmetric product of tensors also can be used with similar purpose. The method is related to linear space of polynomials of degree n with d variables [3]. For n = 1 such a space is simply . The Sn( ) can be considered as a state of system with n H H indistinguishable bosons [3]. An element of Sn( ) represented as symmetric product H of n vectors from is called decomposable. It may be treated as a system with n H noninteracting bosons. Two methods of indexing are used for basis of Sn( ) [4]. The ﬁrst one is sequence H j with n indexes

j =(j ,...,jn), 1 j j jn d. (3) 1 ≤ 1 ≤ 2 ≤···≤ ≤ Such notation is obtained from initial basis of tensor product after symmetrization of indexes. It may be rewritten due to polynomial representation [4] j j r1 rd j j j xj1 xj2 ...xjn = x ...x , r + r + + r = n, (4) 1 d 1 2 ··· d where rj 0 is number of indexes k in the sequence j. The numbers rj produce an k ≥ k alternative representation as a vector rj with d elements. The notation is convenient for expression of standard normalization for basic vectors in space of symmetric tensors or polynomials j j r1 rd S x1 ...xd j j xj = , Γj = r1 ! ...rd! (5) Γj and the element of Sn( ) may be considered as a complex vector with Cd+n−1 p H n components in a basis xS . Action of unitary group on each for such normalization | j i H corresponds to an unitary representation on the Sn( ) with respect to usual Hermitian H scalar product and the unitarity [4] is natural due to d n rk rk n! xk x¯k S S xkx¯k = = n! xj x¯j . (6) r ! ...rd! k 1 j∈S X=1 PXrk=n Q X For already mentioned earlier decomposable elements of Sn( ) relations between H the scalar products on Sn( ) and may be expressed using permanents [5]. Let us H H recall, that the permanent of n n matrix A is n ×

per(A)= Aj,σ(j), (7) σ j=1 X Y Permanents, Bosons and Linear Optics 3 where σ denotes all possible permutations of indexes. The only difference with determinant is lack of minus signs for some terms. However, unlike the determinant the permanent is an example of computationally hard problem [6]. ′ d Transformation of product (4) for change of variables xj = k=1 Mjkxk can be expressed with some polynomials (M) of degree n with elements of the matrix M P P ′ ′ ′ k1,...,kn xj1 xj2 ...xjn = (M)j1,...,jn xk1 xk2 ...xkn . (8) k1,...,knP X A simple case suitable for further examples is n = d with consequent indexes k = j = (1, 2,...,n), when the polynomial simply equal to permanent 1,...,n (M) ,...,n = per(M). (9) P 1 More general case should be discussed elsewhere and analogues of (9) also could be expressed using permanents of matrices composed from rows and columns of M. An application of such an abstract model for discussion about permanent and determinant complexity may be found in [7] together with suggestion about ineffectiveness of finding permanents using quantum processes with bosons because of big variance of measurement outcomes.

3. Linear optics

The abstract bosons model described above says a little about physical realization. Alternative way of producing expressions related with permanent may be obtained using quantum model of linear optical networks [8]. It may have relevance with theory of quantum computing [9] and computational complexity of linear optics was discussed further coining the term boson sampling [1] stimulating the series of experiments [10, 11, 12, 13]. An evident distinction in formulation of such a model in comparison with the abstract bosons model from section 2 is definition of some basic transformations using † creation and annihilation operators aˆj,a ˆj, j =1,...,d [8, 9]. The approach is quite natural, because with such a notation linear optical network with conserved total photon number corresponds to transformation [9] d ′ aˆ = Ujkaˆk, U U(d), j,k =1, . . . , d. (10) j ∈ k X=1 Due to some analogy between (10) and (2) they might cause similarity in formal calculations, but meanings of the expressions are quite different. The (10) describes transformation of operators in Heisenberg representation, but (2) is applied to states. Transformation of some operator Aˆ due to (2) in the abstract bosons model from section 2 in Heisenberg representation [14] would be Aˆ′ =u Âûˆ†. (11) Hereu ˆ produces the same result as eiφuˆ reaffirming sufficiency of usingu ˆ SU(d). In ∈ (10) such compensation of phases is dispensable and so the whole unitary group U(d) Permanents, Bosons and Linear Optics 4 may be implemented, but it is rather a hint on more essential difference between the models discussed further. In fact, a model with symmetrization of states very similar with abstract bosons model from section 2 with infinite-dimensional space sometimes used in quantum H optics as well [15]. It can be asked in turn, how to rewrite (10) in a way similar with (11). For such a purpose in next section transformation with conservation of total photon number is considered as particular case of most general linear Bogoliubov transformations [9, 14] without such requirement.

4. Oscillator model

4.1. Schrödinger description The model of quantum harmonic oscillator is recollected below with a brief excursus into theory of symplectic and metaplectic groups necessary for applications to linear optics [16, 17, 18]. Let us consider infinite-dimensional Hilbert space and operators H qˆ,p ˆ of coordinate and momentum. In Schrödinger description the is associated with H space of wave functions ψ(q) L2(R) (square integrable) and the operatorsq ˆ,p ˆ are ∈ defined as qˆ: ψ(q) q ψ(q), pˆ: ψ(q) i∂ψ(q)/∂q (12) → → − with canonical commutation relation (CCR) [ˆp, qˆ]=ˆpqˆ qˆpˆ = i, (13) − − where system of units with ~ = 1 is used for simplicity. The generalization on set of operatorsq ˆk,p ˆk, k =1,...,d with CCR

[ˆpj, qˆk]= iδjk, j,k =1,...,d (14) − is straightforward using space L2(Rd) of wave functions with d variables.

4.2. Symplectic group A real 2d 2d matrix A preserving bilinear form × d

(x, y) = (xkyd k xd kyk) (15) Λ + − + k X=1 is called symplectic. It also has property [19] AT JA = J, (16) where AT is transposed matrix and J is 2d 2d matrix × 0 1 J = d d (17) 1d 0d − ! Permanents, Bosons and Linear Optics 5

with 0d and 1d are zero and unit d d matrices. A composition of such matrices also × satisfies (16) and, so, the symplectic group Sp(2d, R) [16, 17] is defined in such a way. ‡ Due to an analogy with orthogonal matrices R SO(d) satisfying RT R = 1 and ∈ preserving Euclidean norm (x, y)E = xkyk a matrix with property (16) and arbitrary J sometimes is called J-orthogonal. Such a definition includes both orthogonal (for unit P J) and symplectic matrices [19]. Matrices preserving both symplectic (15) and Euclidean forms belong to orthogonal symplectic group defined as intersection Sp(2d, R) SO(2d). The group is isomorphic ∩ with unitary group U(d) [19]. Indeed, if to consider complex variables

zk = xk + ixk d, z¯k = xk ixk d, (18) + − + the real and imaginary part of Hermitian complex scalar product correspond to ′ ′ ′ Euclidean and symplectic forms, respectively z z =(x, x )E + i(x, x ) . h | i Λ To show relation of symplectic group to CCR let us writeq ˆk,p ˆk as formal vector of operators with 2d elements [16]

wˆ =(ˆwk)=(ˆq1,..., qˆd, pˆ1,..., pˆd). (19) Equation (14) can be rewritten in such a case as

[ˆwj, wˆk]= iJjk, j,k =1,..., 2d, (20) − ′ where Jjk are elements of matrix J (17). Due to such property 2d operatorsw ˆj 2d ′ wˆj = Sjkwˆk, j,k =1,..., 2d, (21) k X=1 also satisfy (20) if matrix S Sp(2d, R). ∈ 4.3. Metaplectic group

′ Both the sets of operatorsw ˆj andw ˆj related by (21) satisfy some form of CCR (20) and in agreement with general results about uniqueness of CCR they should be unitary

equivalent, i.e., for any matrix S in (21) some unitary operator ˆS should provide U transformation [16, 17, 18] ′ −1 wˆ = ˆSwˆj ˆ , j =1,..., 2d. (22) j U US iφ Due to (22) ˆS and e ˆS correspond to the same matrix S, but such a phase freedom U U may be withdrawn and the only inevitable ambiguity is a sign ˆS. The group producing ±U such a 2–1 homomorphism on Sp(2d, R) is known as metaplectic, Mp(2d, R) [16, 17, 18]. The unitary representation of Mp(2d, R) used in (22) is not a ﬁnite-dimensional matrix

group, but can be expressed by exponents with appropriate linear combinations ofw ˆjwˆk. The symplectic group is relevant also to classical optics, but metaplectic group is essential in quantum case [16, 17, 18]. The group Mp(2d, R) describes most general linear optical networks with d modes and a subgroup of transformations with conservation of total photon number is discussed below.

Slightly diﬀerent parametrization Sp(d, R) is used in [18, 19]. ‡ Permanents, Bosons and Linear Optics 6

4.4. Annihilation and creation operators The transformations respecting also Euclidean norm, i.e., sum of Hamiltonians of harmonic oscillators d d 1 2 1 Hˆ = wˆ2 = Hˆ , Hˆ = (ˆp2 +ˆq2) (23) 2 k k k 2 k k k k X=1 X=1 correspond to already mentioned orthogonal symplectic group isomorphic with unitary group U(d). Complex coordinates (18) now associated with annihilation and creation (“ladder”) operators

1 † 1 aˆk = (ˆqk +iˆpk), aˆ = (ˆqk iˆpk). (24) √2 k √2 − Equations (23, 24) are widely accepted [16, 17, 18], consistent with rather standard map to U(d) [19] and used further in this work for simplicity instead of more general versions also relevant to quantum optics [14, 20] 1 Hˆ = (ˆp2 + ω2qˆ2), 2 1 1 (25) aˆ = (ωqˆ+iˆp), aˆ† = (ωqˆ iˆp). √2ω √2ω − The oscillator model relates group U(d) treated as a subgroup of Sp(2d, R) with conservation of “total photon number” defined by operator d ˆ ˆ ˆ † N = Nj, Nj =âjaˆj. (26) j=1 X † Instead of (21) already mentioned earlier should be used (10) and expression fora ˆj is obtained using Hermitian conjugation d d ′ ′† ∗ † aˆ = Ujkaˆk, aˆ = U aˆ , U U(d). (27) j j jk k ∈ k k X=1 X=1 Equation (21) rewritten in such a way without requirement about photon number conservation would include termsa ˆ†,a ˆ in both sums (27) reproducing most general Bogoliubov transformations. In such an approach subgroup of Mp(2d, R) corresponding to U(d) sometimes is denoted as MU(d) [21] and can be expressed by exponents with linear combinations of † aˆjaˆk. An analogue of (22) is ′ −1 aˆ = Û aˆj ˆ , j =1, . . . , d, ˆ MU(d). (28) j U UU U ∈ † with conjugated expression fora ˆj. The MU(d) is double cover of U(d) with sign ambiguity inherited from relation between Mp(2d, R) and Sp(2d, R) [18, 21]. The additional subtle problems may appear,

because using some formal manipulations with ˆU an one-to-one map with U(d) could U be obtained, but it cannot be extended to the whole Sp(2d, R) [18]. Permanents, Bosons and Linear Optics 7

The difficulties with unitarity after application of discussed trick to get rid of double cover would produce rather unrealistic model of photons with prohibition to use squeezing transformations. Even if requirement of particle number conservation is justified for model with a massive bosons the approach discussed in this section has other important differences from the abstract bosons model introduced in section 2. A noticeable formal distinction is an additional phase parameter, because action of

U(d) is not reduced to SU(d). The nontrivial structure of ˆU is manifested here, because iφ U (28) is not sensitive to formal phase multiplier e ˆU . So action of phase multiplier U on matrix U in (27) is implemented in alternative way and “embedded” directly into structure of ˆ via additional term proportional to total photon number operator Nˆ (26) U in the exponent with quadratic expressions for elements of group mentioned earlier. Yet another important property of used model with double cover and sign ambiguity due to (22), (28) is similarity with relation between orthogonal and Spin groups [18, 21].

5. Comparison with formal fermionic model

The theory of Spin groups is relevant with question about difference in complexity between bosons and fermionic model often associated with matchgate circuits effectively simulated on classical computer. The matchgate model was introduced in [22] with reformulation using formal fermionic model in [23, 24]. Similar fermionic model and equivalent approach with Spin group and Clifford algebras was also applied earlier to quantum computing problems in [25, 26, 27, 28]. The theme was further developed in works devoted to effective classical simulation of such a class of quantum circuits and they are often described using fermionic † annihilation and creation operators [29, 30, 31]. The notationa ˆj,a ˆj is used here for such operators for distinction from bosonic case. The analogues ofw ˆj (19) here are 2d generators of Clifford algebra † † cˆ j =â +â , cˆ j = i(â aˆ ). (29) 2 −1 j j 2 − j − j Transformation properties operators (29) are similar with (21), (22) and may be written as 2d −1 ˆRcˆj ˆ = Rjkcˆk, (30) S SR k X=1 but Rjk now are elements of orthogonal matrix and ˆR Spin(2d) is corresponding to S ∈ matchgates (or “fermionic”) quantum circuit. The ˆR has matrix representation, but S number of components is exponentially bigger than in matrix R, because it corresponds to quantum network with d qubits. A restricted case of transformation conserving number of fermions discussed in [23] provides even closer analogy with boson case. If the evolution is expressed as † exponent of Hamiltonian with linear combination ofa ˆjaˆk then the fermionic operators Permanents, Bosons and Linear Optics 8

are transformed [23] by direct analogue of (27) d † −1 † Û aˆ ˆ = Ujkaˆ , (31) S jSU k k X=1 with Û is from subgroup of Spin(2d) corresponding to restriction of SO(2d) on S “realification” of SU(d), i.e., orthogonal symplectic group mentioned earlier. Two cases should be taken into account for comparison of complexity fermionic and bosonic models. The single-mode measurement in terminology of [29] is the first one and mainly used in [30, 31, 32]. In the concise form the setup [31] for such experiment is any computational basis state x ...xd as the input and a final measurement of an | 1 i arbitrary qubit in the computational basis as the output. For arbitrary state ψ such a (k) 1 | i measurement may be described by probability p1 to obtain as a result of measurement for a qubit with index k p(k) = ψ aˆ† aˆ ψ (32) 1 h | k k| i and an analogue of such expression in bosonic case † Ψ aˆ aˆk Ψ = Ψ Nˆk Ψ = Nk (33) h | k | i h | | i h i is corresponding to expectation value for number of particles in a mode k. After application of linear optical network new expectation values N ′ are h ki ′ ˆ ′ ˆ† † ˆ Ψ = Ψ , Nk = Ψ aˆ aˆk Ψ | i U| i h i h |U k U| i (34) −1 † −1 = Ψ ˆ aˆ ˆ ˆ aˆk ˆ Ψ . h |U kU U U| i It can be rewritten using (27) and (28) with matrix U corresponding to operator ˆ U d d ′ † ∗ N = Ψ Ukjaˆ U aˆj Ψ h ki j kj j=1 j=1 D X X E (35) d ∗ † = Uk,jU ′ Ψ aˆ aˆj′ Ψ . k,j h | j | i j,j′ X=1 † If an effective way to calculate Ψ aˆ aˆj′ Ψ exists for given input state Ψ then the h | j | i | i N ′ also can be calculated in poly time using (35) and the methods are similar with h ki matchgate model [30, 31]. The Fock states [15] can be considered as an alternative for computational basis for input state for a case with n bosons and d modes † † † aˆj1 aˆj2 aˆjn Ψj = ··· , 1 j1 jn d (36) | i Γj |∅i ≤ ···≤ ≤

with is vacuum state.p The coefficients Γj can be derived from expressions for |∅i ladder operators [15] and coincide with normalization (5) for vector rj = (rj,...,rj) p 1 d defined by (4) for given sequence j =(j1, j2,...,jn) introduced in (3). So, finally j j j † r1 † r2 † rd (â1) (â2) (âd) Ψj = ··· |∅i. (37) | i j j j r1 !r2! ...rd! q Permanents, Bosons and Linear Optics 9

† For such states Ψj aˆ aˆk′ Ψj may be effectively calculated. h | k | i The other case is multi-mode measurement [23, 29]. The analogues of such model † in bosonic case would use instead of (33) polynomials witha ˆk,a ˆk of higher degree. So, different computational complexity of determinant and permanent may be really essential in some examples. ˆ ˆ It may be written = θUˆ with some phase multiplier θUˆ = 1, because U|∅i |∅i † | | U is expressed as an exponent with linear combinations ofa ˆjaˆk and for vacuum state aˆj = 0. Therefore, |∅i ˆ 1 ˆ † † † ˆ−1 ˆ Ψj = aˆj1 aˆj2 aˆjn U| i √Γj U ··· U U|∅i (38) θUˆ ˆ † † † ˆ−1 = aˆj1 aˆj2 aˆjn . √Γj U ··· U |∅i After further application to each multiplier âˆ† ˆ−1 and expansion as sums using (27) U jk U expressions with permanents may be obtained. Anyway, (33) and analogues with small number of ladder operators may hide complexity arising from application of linear optical network to Fock states. An example is (34) or (35) that may be used for effective classical computation of single-mode measurement outcomes (expected average number of photons N ′ for each mode k) h ki after application of linear optical networks to Fock states. The permanent complexity is more essential in expressions for “transition” amplitudes (and probabilities) between two Fock states 2 αkj = Ψk ˆ Ψj , pkj = αkj . (39) h |U| i | | Basic example with consequent indexes in k and j corresponds to (9). For

calculation of probabilities pkj a phase θ ˆ = 1 from (38) should be omitted. An | U | alternative consideration with commutative polynomials resembling an abstract boson model discussed earlier in section 2 may be found in [1, 33]. Analogue models with preserving number of fermions was studied in [23]. The same d-dimensional unitary group (31) again can handle the evolution, but determinants are used instead of permanents and expressions for fermionic amplitudes may be eﬃciently evaluated [23]. The analogue of (39) in fermionic case directly coincides with determinant. So, for fermions multi-mode measurement amplitudes also can be eﬃciently evaluated due to absence of problems with permanent calculation discussed earlier for bosons.

6. Conclusion

Some topics relevant to consideration of complexity of simulation of quantum processes with boson were discussed in this paper. A distinction between an abstract bosons model and more elaborated approach with quantum harmonic oscillator was emphasized. The treatment of real photonic system may require consideration of even subtler problems that should be discussed elsewhere. Permanents, Bosons and Linear Optics 10

Let us recollect similar equations for transformations of some sets of states ψk and | i operatorsω ˆj relevant to many examples discussed above using notation ′ ′ −1 ψ = ˆ ψk , ωˆ = ˆωˆj ˆ = Sjkωˆk. (40) | ki S| i j S S k X For oscillator model of general linear quantum optical network operatorsω ˆj =w ˆj are deﬁned in (19) and the Sjk is symplectic matrix (21) with ˆ is unitary operator ˆS (22) S U from metaplectic group. For an important case with conserving number of bosonsω ˆj

correspond to annihilation and creation operators, ˆ was denoted as ˆU in (28), and S U Sjk is unitary matrix Ujk in (27).

In section 5 about fermionic modelω ˆj =c ˆj (29), ˆ = ˆR Spin(2d) and Sjk S S ∈ corresponds to orthogonal matrix R (30) and conserving number of fermions is taken into account in (31) with unitary matrix. Therefore, for conserving number of boson and fermions evolution can be described by the same unitary group with “reduced” dimension d instead of 2d for most general quantum circuit and significant difference in complexity of amplitudes evaluation for multi-mode measurements looks especially challenging. The fermionic model may be effectively simulated by classical computer producing some controversy with bosonic case. On the other hand average photon numbers in output of each mode can be simply calculated likewise with fermionic case. So, classical computer could effectively simulate output for each mode, but without proper quantum multi-photon correlations. Recent achievements in experiments with photons [34, 35] permit to check such subtleties. For abstract bosons model introduced in section 2 unitary operator ˆ acting on S states could be compared withu ˆ SU(d) in (2), but without analogues of operators ∈ such asω ˆj in (40). A Fock state (36) may be considered as an analogue of (5) in abstract bosons model, but confusing them may produce certain problems. Let us consider simplest case with action of Û on a basis of U H † † † −1 k =â , Û k = Û aˆ = Û aˆ ˆ Û | i k|∅i U | i U k|∅i U kUU U |∅i ∗ † ∗ ∗ = U aˆ θ ˆ = θ ˆ U k = θ Û k , (41) jk j U |∅i U jk| i U | i j j X X where θ ˆ = 1 was already introduced earlier, denotes vacuum state, and k , | U | |∅i | i k =1,...,d correspond to basis of . In such a case in abstract bosons model difference ∗ H between Û and U lacks of proper treatment and simply missed sometimes. On the U over hand (27) and (28) d ′ −1 aˆ = Ujkaˆk = Û aˆj ˆ . (42) j U UU k X=1 ∗ would not make a sense, if Û and U are not distinguished. Indeed, both signs ˆ U ± U lead to the same matrix U in (42) and define of Û as an operator from double cover of U unitary group. Permanents, Bosons and Linear Optics 11

Together with the necessity in consistent mathematical expressions there are also physical reasons to make distinction between two models discussed above. The oscillator model based on quantization of classical linear optics and the abstract bosons model is close related with quantum approach to discrete models such as group of permutations [4]. In such a way the subtle relations between two models may be compared with a wave-particle duality.

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