CC BY
30
On construction of evolutionary operator for rectangular linear optical multiport
M. M. Lipovich, I. S. Lobanov ITMO University, Kronverkskiy pr. 49, St. Petersburg, 197101, Russia mariel.21@yandex.ru, lobanov.igor@gmail.com
PACS 42.50.-p, 42.50.Ex DOI 10.17586/2220-8054-2015-6-4-537-546
The work of Knill et. al. (2001) established the possibility of nondeterministic realization of certain quantum logic operations using linear optical elements, ancilla photons and postselection techniques. It was also shown that any discrete unitary operator acting on N optical modes can be implemented by a triangular multiport device constructed from a series of beam splitters and phase shifters (see work of Reck, Zeilinger et. al., 1994). Here, we consider the rectangular linear optical multiport that is used for the probabilistic realization of unitary transformations on n qubits. This kind of linear optical scheme is suitable for probabilistic realization of unitary operators using ancilla photons and projective measurements. Qubits are encoded into the bosonic states of optical modes in two possible polarizations, and a number of ancilla photons and photodetectors are used for postselection of the qubits' state, based on the output of the detectors. We derive a procedure of evolutionary operator calculation for schemes of the considered type and present algorithms for their efficient computation on symmetric state space. We also provide complexities for different algorithms for the computation of evolutionary operator and estimate demands of resources in each case. A destructive Toffoli gate, acting on three qubits, using one ancilla photon and a photodetector, is implemented using schemes of the presented type.
Keywords: Quantum computing with linear optics, Projective measurements, Postselection, Photon detectors, Realization of unitary operator, Toffoli gate.
Received: 30 January 2015
Revised: 21 March 2015
1. Introduction
The usage of photons and linear optical elements, such as beam splitters, phase shifters, mirrors, presents the possibility of realizing controllable and scalable quantum computing [1]. Photons demonstrate easily observable, obvious quantum effects and are also able to maintain their coherent states for a long time. Construction of optical devices is comparably easy [1,2]. For instance, these devices do not require low temperatures (except for realization of singlephoton sources). In 2001, E. Knill and R. Lafflame published the work [2] which established the possibility of constructing quantum logic gates based on linear optics. Projective measurements and ancilla photons were used to introduce theinteraction between photons in order to implement non-deterministic realizations of certain quantum operations, like CNOT [3-5].
The experimental realization of unitary operators U(N) transforming N optical modes using beam splitters and phase shifters was proposed in paper [6]. Here we present another type of optical multiport that transforms states of photon qubits and performs quantum teleportation by means of a number of ancilla photons and the detectors provided. The problem is to experimentally realize multiqubit unitary operators using this type of optical schemes. The main result, presented in this paper, is a procedure for calculating the evolutionary operator corresponding to mentioned optical schemes and efficient computational algorithms. This result
introduces a step towards creating a method for the automatic construction of non-deterministic linear optical multiports corresponding to a given quantum circuit.
This paper has five sections. In section 2, we derive the evolutionary operator corresponding to a rectangular optical lattice having two-mode linear optical elements in its nodes. This can be implemented using beam splitters coupled by single-mode optical fibers. For the described network, we learn how to compute the corresponding single-particle evolutionary operator. In sections 3 and 5, we show how to efficiently compute multiparticle evolutionary operator. On the basis of the optical network considered in section 2 and inspired by CNOT gate, presented in [7], we propose a destructive Toffoli gate acting on three qubits (see Sec. 4).
2. Constructing evolutionary operator for rectangle linear optical multiport
Consider the rectangular grid having n rows and m columns (see Fig. 1) in which the nodes are polarizing beam splitters with no phase shift. Here, we compute the single-particle evolutionary operator for this optical grid. Taking into account only the transmitted and reflected modes, an evolutionary operator for this scheme can be implemented by scattering matrix that transforms the amplitudes of the 2(n + m)-mode state. Each beam splitter is associated with a unitary transformation on four optical modes:
( Ha \
va
V v)
where (a, b) and (c, d) denote input and output spatial modes, H and V denote two possible polarizations and parameter 9 describes reflectivity and transmittance of the beam splitter [6,8].
( cos(9) - sin(9)
sin(9) cos(9)
0 0
\ 0 0
00 00
cos(9) - sin(9) sin(9) cos(9)
( h' \
7 c
V
H'd
\ Vc' I
(1)
FIG. 1. Grid network with input ports ci ...cn, ai... am and outputs gi... gn, D ... Dm, for convenience of further calculations some ports are labeled twice
W
aj
\
\
\
\
w.. I.J
\
\
FIG. 2. Subscheme N(1.. .i,j)
First, we consider separately j-th column of length i (see Fig. 2). We denote this subscheme as N(1.. .i, j) and find out how it affects the state of the photon. We imply that initially photon
is in the modes (Haj, Vaj; Hc1,j, Vc1,j,... Hci,j,Vci}j), so the input state for N(1.. .i, j) is the following 2(i + 1)-mode state (order of basis elements is preserved):
Wi...i,j )i
WHa
\Haj) + WVaj \ Va^ + WHbi
°i,j
WHbij-i \HbiJ-1) + ^Vbij-! | -Oij-!
WVb
\Vb
bi,
+... +
\ H\ b
\Hb^ „ ) + Wvb, ,-1 \Vbi which is transformed into an output state of the form:
|W1 ...i,j )out = WHDj \HDj) + WVDj + WHb\ \Hbj + WVbi,j \ H^) + ... +
WHbi,j \ Hbij) + WVbi,j \ Vbij) ,
assuming that finally the photon can be found in the modes (HDj, VDj; Hb1,j, Vb1,j ... Hbi,j, Vbi,j). Input and output states of the particle corresponding to the transformation on 2i modes given by the subscheme N(1.. .i — 1,j) are respectively of the following form:
lWl...i-1,j )in = WHaj \ Haj) + WVaj \ Vaj) + WHb1 — \Hbj-i) + WVbij \Vbj + ... +
^Hbi-ij-i \Hbi-hj-i) + fVbi-ij-i lW1...i-1,j )out = WHwij lHwi,j) + WVwij 1 Vwi,j) + WHbij \Hbj + WVbij \Vbj + ... +
\ Vbi
i,j-i
WHbi-ijj \Hbi-j + WVbi-ijj \ Vbi-ij) .
For each subscheme, we write a transition matrix, that changes the basis of the input states to the basis of the output states and matches the corresponding scattering matrix. Let then be matrix of size 2(i — 1) x 2(i — 1) corresponding to scattering operator of N(1.. .i — 1, j) so |^1...i_1,j) = U(i-1,j) |<W1...i-1,j). If we compare the |w1...i,j)in and |w1...i_,j)in states, then the latter doesn't have a spatial mode bij_1 and has witj instead of Dj. Using this remark and assuming that element in the node (i,j) is associated with scattering matrix T(i,j) (see (1)) with
matrix elements ||t
14
k,l Uk,l=
1 , we get the following system:
( WHwij \ WVwij WHbij WVbij
fHbi-ij
V WVbi-ij J
(
U(i-1,j)
WHa
\
WVa
WHb
i,j-i
WVbi j-i
WHci-ij-i
\ WV,
\
Whd_
WVDj WHbij V WVbijj
T(i,j)
WHwi,j WVwi,j WHbij-i V WVbij-i J
Ci-ij-i
/
^om which the recurrent relation for the scattering matrix U(i,j) of size 2(i + 1) x 2(i + 1), that
transforms | W1...i,j)in into | W1...i,j )out is:
( ¿1,1 ¿1,2
U (i,j)
¿2,1 0
0
¿3,1
¿2,2 0
0
¿3,2
\ ¿4,1 ¿4,2
0 0 1
0
0 0
0 0
0
¿1,3 ¿2,3 0
¿1,4 ¿2,4 0
1 0 0
0 ¿3,3 ¿3,4 0 ¿4,3 ¿4,4
U(i—1,j)
00
0 0
00 010 0 0 1
(2)
where U(1j) = T(1'j). Obviously, each factor of expression (2) can be transformed to blockdiagonal form, where each block is unitary. Then, we conclude that U(i'j) is also unitary, as expected.
To finish the construction of the grid scattering matrix, we consider a rectangular sub-scheme having i rows and j columns with top-left angle matching the one of the large scheme. Let it be denoted as N(1.. .i, 1.. .j) (see Fig. 1). The corresponding input and output states are respectively of the form:
l<Pi...i,i...j )in = ^Haj lHaj) + <fyaj IV a j) + ... + VHai |Hai) + VVai | Vai) + VHci \) + VVci \) +
... + ^Ha |Hci) + ^yCi) | Vci) , \V1...i,1...j)out = VHDj \HDj) + VVDj 1 VDj) + ... + VVDi 1 VD1) + <PHbi,j \Hb1,j) + ^Vbx,j |Vb1,j) + ... + PHbij \Hbi,j) + Wbijj |Vbi,j).
If Y(i,j-1)
is the matrix of the operator corresponding to N(1.. .i, 1.. .j — 1), then using the recurrence for calculating U(i'j), we get the following system:
(
\
VHDj_i WDj-i
VHDi VVDi <£Hbi,j_i fVbij-i
^Hbij-1
\ ^Vbi,j_i J
= y (i,j-1)
VHaj_i ^ ( ^HDj \ VVDj ( V Ha j <PVaj
VVaj_ i
VVai PHbijj VHbij_i
VHci , VVbij = u (i'j) VVbi,j_i
VVci <PHbi,j VHbij_i
PHci \ Wci
\ PVbijj J
\ Wbij-i J
^om which, we eventually derive the following recurrent relation for matrix of operator Y(i'j) of size 2(i + j) x 2(i + j):
/ 1 0
O O
Y(i,j) = 0 1
i, U1i,j) O ufj)
\
1 0 O )
0 1 , (4)
O Y(ij-1) /
where U(i'j) = U(i'j)[1... 2(i+1); 1, 2] and U('j) = U(i'j)[1... 2(i + 1); 3 ... 2(i+1)] (here we denote submatrix of matrix M with rows r1,..., rn and columns c1..., cm by [r1,..., rn; c1..., cm]). Following the same reasoning as for U(i'j), it is also obvious that Y(i'j) is unitary. Hereby, for scattering matrix of the large scheme (see Fig. 1), it is enough to compute Y(m, n).
3. Multiparticle systems
Consider the scheme on Fig. 1 and an arbitrary state of n photons incident to ports c1, ...cn:
|Wo) = £ W(p)\p1i ...Pi> , £ |W(p)|2 = 1,
pen penn
nk = {H,V }xk. (5)
which is a state of n qubits encoded into photon polarization modes H and V. Suppose we define a state of m ancilla photons incident to ports a1,...,am as:
|Wa) = q$ WHj \Haj) + WVj \Vaj) , 1
and input state as |win) = Sym(w0 0 Wa). Then, we can write the output in the following form (see (5)) :
Wout) = £ [Mr )\p 1 ...Pm' ) £ P,P,T )\p1 ...pn' )) + c± №±) ,
T\ T penn, /
Tk = {{DSi}k| Si e 1,... ,k; Si+1 > Si} (6)
where |^±) is a state orthogonal to those having only one photon in each of modes g1,... ,g,, The product of state vectors is defined as their symmetrization:
n •
1 n,\ |a1) ... \°n'+m')
p ... p
P1 pn
£p,P,r = {n(PD ri ,...,PDn:m, ,P1i ,...,pt)}.
Evolutions generated by linear optical elements preserve the photon total number so n' = n + m — m!. Suppose that we detect photons in modes D1,..., Dm. As seen from (6), with the appropriate choice of ancilla photon states and postselection based on polarization and quantity of photons on each detector, we can project the output in modes g1,... ,gn> into the state having pen , |W(p)|2 = 1, thus meaning that we received a state of n' qubits. The scheme is considered non-destructive in case n = n' and destructive if n' < n as we collapsed the state of one or more qubits.
Let scheme with evolutionary operator U transform state of q = n + m input photons. In order to obtain the transformation for bosonic states, we compute the restriction of the multiparticle evolutionary operator U®q on its invariant subspace Spi. Then, the input state space is given by:
Spo = span ({|X 1mi) 0 |X2m2) 0 ... 0 \Xqmq> |Xi e {H, V},
mj e {c1,.. .,cn,a1,.. .,am} , i,j = 1... q}) , (7.1)
where cj are spatial modes corresponding to input ports of the scheme. The restriction operator can be computed as follows, assuming that matrix of U®q is of size 2q x 2q and dim(Spi) = k:
U®q = R\U ®nRi, (7.2)
where Ri is a 2q x k matrix which columns represent basis vectors of Spi. It can also easily be shown that for invariant subspaces, the restriction operator is unitary.
4. Constructing Toffoli gate using linear optics
Inspired by non-destructive CNOT gate presented in work [7] and with use of algorithms given in section 5, it became possible to construct a scheme implementing quantum Toffoli transformation. Although it is known that at least five two-qubit gates are necessary for a nondestructive Toffoli gate [9]. It turns out that for the destructive Toffoli gate on three bosonic qubits only eight polarization modes are needed (see Fig. 3). We use an ancilla photon in equal superposition state and an arbitrary three-qubit state on left input ports of the following scheme:
Fig. 3. Destructive Toffoli gate
№a) = \Ha) + \Va) ,
№0) = «1 \HcHc'Ht) + «2 \HcHc,Vt) + ... + as \VcVc,Vt).
The initial state \^in) = ® is transformed into the output state, which can be
written in the form of equation 6 (see section 3):
\Vw) = 2-2 \Hda) («1 |HdcHdc,Ht) + «2 HHdc, Vdt>) +
1 \Vda) («3 \HdcVdcHdt) + «4 \HdcVdcVdt)) +
2y/2
1 \ Hda) («5 | Hdc, Vdc, Hdt) + «6 | Hdc, Vdc, Vdt)) +
2V2
1 \Hda) («71 Vdc Vdc, Vdt) + «s | Vdc Vdc, Hdt)) + ^ .
2^2 da! \ ^ 7 I r dc v dc, r dt / I ^S | * dcv dcdt // I 2
In the output state, the probability amplitudes of vectors from the following sets are equal to 0 :
S = {|HdcVdc/Hdt) , |HdcVdc/Vdt) , |VdcHdc/Vdt) , |VdcHdc/Hdt)},
S = {\Vdc Hdc Hdt), \Vdc Hdc Vdt), \Vdc Vdc Hdt), \Vdc Vdc Vdt), \Hdc Hdc Hdt), \Hdc Hdc Vdt)},
S3 = {|Vdc/Hdc/Hdt) , |Vdc/Hdc/Vdt) , |Vdc/Vdc/HdJ , |Vdc/ Vdc/Vdt) , |Hdc/Hdc/Hdt) , |Hdc/Hdc/ VdJ}
On construction of evolutionary operator... therefore, when the following events occur:
A1 = {Each spatial mode has only one photon, photodetector registers single H-polarized photon}, P(A1) = 1,
photodetector registers single V-polarized photon},
dc
• A2 = {2 photons in mode
P (A2) = 1,
• A3 = {2 photons in mode dc/, photodetector registers single H-polarized photon},
p (A3) = 8,
we get a quantum Toffoli transformation preserving the state of the controlled qubit, which succeeds with probability P(A1) + P(A2) + P(A3) = 2. The main difficulty concerning implementation of this scheme lies in necessity to count photons using photodetectors, although such devices seem to be available now.
5. Algorithm realization
We present an algorithm for efficient calculation of the evolutionary operator corresponding to a system of p particles and rectangle linear optical multiport considered in section 2 (see Supplementary Materials). This is an optimization of the straightforward algorithm achieved by means of sparse matrices and lazy tensor product calculation. It should be noted that when running procedure 2 (see Supplementary Materials) in parallel, due to the fact that first three nested loops are independent, it becomes possible to obtain a linear performance increase that is proportional to the number of processors in use, whereas number of threads can grow up to Size (Sp)2 (p!).
6. Conclusion
In this paper, a rectangular linear optical grid was considered. Similar to the triangular multiport presented in work [6], in order to implement unitary transformation of N optical modes a number of beam splitters proportional to N2 is required. However, the optical networks considered in the current paper are comparably easy to construct and permit the usage of a sufficient number of ancilla photons and detectors to realize quantum teleportation. The formal procedure of evolutionary operator calculation, corresponding to a given rectangular optical network, introduces a step towards the efficient experimental realization of an arbitrary unitary operator. At present, there are also several other problems of interest: construction of quantum logic gates based on linear optics, the problem of simulation of quantum optical gates with noise and implementation flaws being taken into account. Substantial problems that prevent these schemes from being used in practice are qubit phase drifts and photon loss, which is why effective correcting codes for qubit states are necessary. Unsolved technical problems are the construction of an extremely sensitive and low-latency photodetector and reliable single-photon sources.
7. APPENDIX
Here we present procedures for calculation of evolutionary operator corresponding to system of p particles and rectangle linear optical multiport (see alg. 1, 2, 3).
• @Transition_Matrix_Sparsed (see alg. 1) calculates matrix R and stores it in a sparse matrix.
• @Kron_And_Mult_Sparsed (see alg. 2) calculates U®n (see article, eq. 7.2).
• @Lazy_kron (see alg. 3) is an auxiliary function for @Kron_And_Mult_Sparsed that calculates tensor product matrix elements.
Complexities of simple and optimized procedures for calculation of evolutionary operator are summarized in table 1. In table 2 dimentions of most important subspaces of Sp0 (see article, eq. (7.1)) are given. In table 3 we present mean time and space resource demands for calculation of evolutionary operator on symmetric subspace.
TABLE 1. Complexity of procedures depending on subspace size Size(Sp) and number of particles p
Procedure Time Space
Transition_Matrix_Sparsed O (Size(Sp)(p + 1)!) O(3(2p)p)
Transition_Matrix_Simple O^Size(Sp) (1+ (p +1)!(1-_(2jp)P)+ P!(2P)P)) O(Size(Sp)(2p)p)
Kron_And_Mult_Sparsed O (Size(Sp)2(p!)2p) O (Size(Sp)2)
Kron_And_Mult_Simple o( 1 _ (2P)2p+ OV 1 _ (2p)2 ' O ((2p)2p+
Size(Sp)4(2p)2pj Size(Sp)(2p)p)
TABLE 3. Mean time and space for evolu-TABLE 2. Subspaces tionary operator calculation on symmetric sub-
of Spo. space using an approx. 50 GFLOPS processor
Subspace Size
(Sp) (Size(Sp))
Full (2p)p
Asym (Ï)
Sym (V)
Coinc 2p
Time (s) Space (MB)
p simple sparsed simple sparsed
3 0.46 0.034 0.224 0.014
4 2 ■ 104 25.09 69.16 0.4623
5 1.6 ■ 1010 2.9 ■ 104 38.9 ■ 103 16.43
6 2.09 ■ 1016 4.8 ■ 107 34.15 ■ 106 618.45
Algorithm 1 Transition_Matrix_Sparsed
Require: p > 3, Sp {args: p for number of particles, matrix Sp which rows represent basis vectors of a given subspace. In MATLAB environment function that generates Sp for symmetric subspace may look like a
Sp = combinator(2 * p, p, V, V). }
idx, jdx,vals ^ array((2p)p) cnt ^ 0
for i =1 to len(Sp) do
Perms ^ wniqweperms(Sp(i)) for j = 1 to len(Perms) do pos ^ 1
for k =1 to p do
pos ^ l * (pos — 1) + Perms (j, k) end for
cnt ^ cnt + 1 idx(cnt) ^ i jdx(cnt) ^ pos
vals(cnt) ^ 1/sqrt(len(Perms)) end for end for
T = make_sparse_matrix(idx, jdx, vals)
Algorithm 2 Kron_And_Mult_Sparsed
Require: p > 3, U, Sp {args: p for number of particles, U for single particle evolutionary operator (see article, eq. 4), and matrix Sp for basis vectors of given subspace (see 1)}
T ^ Transiiion_Mairix_Sparsed(Sp,p) m ^ len(Sp) Ur ^ matrix(m, m) for i = 1 to m do tmp_norm_i ^ T(i, 1) for j = 1 to m do tmp_norm_j ^ T(j, 1) for r =1 to len(T(i)) do cr j ^ 0
for r = 1 to len(T(j)) do
up_elem ^ Lazy_kron(T(i,r), T(j,l),p, U)
crj ^ crj + up_elem * tmp_norm
end for
Ur(i, j) = Ur(i, j) + tmp_norm_j * crj
end for end for end for return Ur
ahttp://www.mathworks.com/ matlabcentral/fileexchange/
24325-combinator-combinations-and-permutations
Algorithm 3 Lazy_kron
Require: 1 < i, j < (2p)p,p > 3, U {args: i, j for row and column of element to be calculates, p for tensor product order and U for single-particle evolutionary operator (see article, eq. 4)} res ^ 1 while p > 0 do
ic, jc ^ i % rows(U), j % cols(U)
if ic is 0 then
ic ^ rows(U)
end if
if jc is 0 then
jc ^ cols(U)
end if
res ^ res * U(ic, jc)
i ^ ceil(i/rows(U)) j ^ ceil(j/cols(U)) p ^ p — 1; end while return res
Acknowledgements
This work was partially financially supported by the Government of the Russian Federation (grant 074-U01), by the Ministry of Science and Education of the Russian Federation (GOSZADANIE 2014/190, Project 14.Z50.31.0031), by grant MK-5001.2015.1 of the President of the Russian Federation.
References
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