A Note on Linear Optics Gates by Post-Selection

Recently it was realized that linear optics and photo-detectors with feedback can be used for theoretically efficient quantum information processing. The first of three steps toward efficient linear optics quantum computation (eLOQC) was to design a simple non-deterministic gate, which upon post-selection based on a measurement result implements a non-linear phase shift on one mode. Here a computational strategy is given for finding non-deterministic gates for bosonic qubits with helper photons. A more efficient conditional sign flip gate is obtained.


II. PRELIMINARIES
The physical system of interest consists of bosonic modes, each of whose state space is spanned by the number states | | |0 , | | |1 , | | |2 , . . .. If more than one mode is used, they are distinguished by labels. For example, | | |k r is the state with k photons in the mode labeled r. The hermitian transpose of this state is denoted by r k| | |. The vacuum state for a set of modes has each mode in the state | | |0 and is denoted by | | |0 .
The anniliation operator for mode r is written as a (r) and the creation operator as c (r) = a (r) † . Recall that c (r) | | |m = √ m + 1| | |m + 1 . Labels are omitted when no ambiguity results. Hamiltonians that are at most quadratic in creation and annihilation operators generate the group of linear optics transformations.
Among these, the ones that preserve the particle number are called passive linear. Every passive linear optics transformation can be achieved by a combination of beam splitters and phase shifters. If U is passive linear, then U c (r) | | |0 = s u sr c (s) | | |0 , where u sr defines a unitary matrixû. Conversely, for every unitary matrixû, there is a corresponding passive linear optics transformation [10]. For the remainder of this note, all linear optics transformations are assumed to be passive.

III. CONDITIONAL PHASE SHIFTS
A conditional phase shift by θ on two modes is the map CS θ : | | |ab → e i(ab)θ | | |ab for 0 ≤ a, b ≤ 1.
These phase shifts can be used to implement conditional phases on two bosonic qubits. A bosonic qubit Q r,s is defined by identifying logical | | |0 with | | |01 rs and logical | | |1 with | | |10 rs . The modes r and s can be two distinct spatial modes or the two polarizations of one spatial mode. To realize the conditional sign flip between Q 1,2 and Q 3,4 , apply CS 180 • to modes 1 and 3. The bosonic qubit controlled-not can then be implemented using conditional sign flips and single qubit rotations, which are realizable with beamsplitters.
In [1], conditional sign flips were implemented indirectly using a non-deterministic realization of the map that succeeds with probability 1/4. This realization requires one helper photon and two ancilla modes.
The goal is to implement CS θ more efficiently directly using two helper photons. One helper photon can be shown to be insufficient by means of the same algebraic method about to be used. Let modes 1 and 2 contain the state to which CS θ is to be applied. The basic scheme is to start with two ancilla modes 3 and with the linear optics transformation, with u sr the entries ofû. The post-selected final state is determined completely by the 4 × 4 upper left submatrix V ofû with entries V rs = u sr for s, r ≤ 4.
It is necessary to consider the effects of the scheme on the initial states | | |00 Since photon number is conserved, we have, without renormalization: To be successful, the amplitudes have to satisfy The amplitudes are polynomials of the coefficients of V . For example α 0000 = v 33 v 44 + v 34 v 43 . More generally, define p s = r v rs c (r) . If the initial state in mode s has d s photons, then the output state is given by P = s p ds s . Thus, P is a polynomial of the c (r) . If β is the coefficient of the monomial t c (t) mt in P , then the output amplitude for having m t photons in mode t is given by t m t !β. This shows that the amplitudes α abcd are polynomials of the v rs .
The first step for constructing CS θ is to solve Eqs. 6-8, which are polynomial identites in the v rs . Before showing how to reduce the difficulty of doing that, let us see how to proceed from there. Since there are 16 free complex variables, the solution will have a number of remaining free variables that must be chosen to optimize the probability of success (|α 0000 | 2 ) and to satisfy one more constraint: The solution is an (explicit) matrix V that needs to be extended to a unitary matrixû. This is possible if and only if the maximum singular value (that is the square root of the maximum eigenvalue of V † V ) is at most one. The extension is not unique. One can set the first four columns ofû to the matrix with orthonormal columns and then complete the last four columns with any orthonormal basis of the orthogonal complement of the space spanned by the columns of X. The maximum singular value constraint is needed to be able to compute the square root in the expression for X. If some of the singular values of V are equal to one, then fewer than four additional columns and rows can be used The singular value constraint cannot be easily achieved using algebraic methods. In principle, one can reparametrize the matrix V to guarantee the constraint, for example by using the polar decomposition and an Euler angle representation of unitary matrices. In the case where CS θ is to be applied to the "left" modes of a pair of bosonic qubits, the singular value constraint can be removed by exploiting a rescaling symmetry. Now there are two additional modes to complete the bosonic qubits. The total number of photons is always four. Let V be a matrix whose coefficients satisfy the identities for the α abcd . Let λ = λ(V ) be the maximum singular value of V and consider the matrix where the first two indeces are associated with the two other ("right") qubit modes. V e has maximum singular value 1 and can be extended to a unitaryû e as before. The claim is that if the resulting optics Briefly, the function to be optimized takes as input the remaining free complex variables (v 14 , v 23 , v 34 , l 1 ), and a non-redundant subset of the scaling variables. To avoid infinities, one can provide the logarithms of the scaling variables as inputs. The scales can be taken to be real since phases have no effect on the probability of success. The function can then be optimized using random starting points. With the optimization procedures provided by Matlab, it was found useful to randomly perturb the point returned and repeat until the solution no longer changes significantly. This procedure routinely finds the same optimum.
For θ = 180 • , it was possible to guess the algebraic numbers that it converged to. Here is a version of the matrix found, which turns out to be unitary: The probability of success is 2/27. The matrix can be systematically decomposed into elementary beam splitter and phase shift operators [10]. An optical network realizing it is shown in Fig. 1. The implementation uses fewer elements (4 beam splitters, 4 modes, 2 photon counters, probability of success 2/27) than the solution in [1] (6 beam splitters, 6 modes, 2 photon counters, 2 photo-detectors, probability of success 1/16). As before, the counters must be able to distinguish between zero, one, or more than one photons. A matrix that can be extended to obtain CS 90 • by post-selection was also obtained: It is an open problem to determine whether the above solutions are indeed optimal as is suggested by the results of the numerical experiments.

IV. BOUNDS ON CONDITIONAL PHASE SHIFTS?
To obtain bounds on the probability of success of a phase-shift gate implemented with helper photons, one can attempt to characterize the states obtained in the output modes after tracing out the helper modes.
There is some choice of the initial state of the modes that the gate is applied to. Assume that this is a state obtained by applying linear optics to prepared single photons. In this case, the final state after a linear optics transformation is given by The goal is to show that after tracing out modes m ′ +1, . . . , m, the state in the remaining modes is a mixture of states of the form In fact, this is the case if the final state before tracing out is also of this form. To be explicit, add to the factors in the expression for | | |ψ f any constant terms α k0 so that .
Iterating this procedure proves the desired result.
Consider the conditional sign-flip gate. With this gate and using a few beam splitters, one can map the state | | |1100 to the state 1 , a well-known entangled photon state. By the above, before post-selection on a measurement of the other modes and with n helper photons, the state can be written as a mixture of products of linear expressions in the creation operators. To obtain a bound on the probability of success, it suffices to obtain a bound for the overlap of (normalized) such states with the Bell state. Because the normalized overlap of (c (1) + c (3) )(c (2) + c (4) ) with the Bell state is 1/ √ 2, the bound on the probability of success thus obtained can be no smaller than 1/2. It is clear that the probability of success cannot be made equal to one: The polynomial xy + uv associated with the creation operators in the Bell state cannot be factored.
A problem suggested by the above is: Problem. What is the maximum probability of success for implementing CS θ using linear optics with at most k independently prepared helper photons and post-selection from photon counters without feedback?