aa r X i v : . [ qu a n t - ph ] J u l On photonic controlled phase gates
K. Kieling , J. O’Brien , and J. Eisert , Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany Centre for Quantum Photonics, H. H. Wills Physics Laboratory and Department ofElectrical and Electronic Engineering, University of Bristol, Bristol BS8 1UB, UK and Institute for Advanced Study Berlin, 14193 Berlin, Germany (Dated: November 3, 2018)As primitives for entanglement generation, controlled phase gates take a central role in quantum computing.Especially in ideas realizing instances of quantum computation in linear optical gate arrays a closer look can berewarding. In such architectures, all effective non-linearities are induced by measurements: Hence the probabil-ity of success is a crucial parameter of such quantum gates. In this note, we discuss this question for controlledphase gates that implement an arbitrary phase with one and two control qubits. Within the class of post-selectedgates in dual-rail encoding with vacuum ancillas we identify the optimal success probabilities. We constructnetworks that allow for an implementation by means of todays experimental capabilities in detail. The methodsemployed here appear specifically useful with the advent of integrated linear optical circuits, providing stableinterferometers on monolithic structures.
PACS numbers: 03.67.Lx,42.50.Ex,42.50.Dv
I. INTRODUCTION
Linear optical architectures offer the potential for reliablerealizations of small-scale quantum computing [1] In the re-cent past, numerous proof-of-principle demonstrations haverelied on the precise state manipulation that is available us-ing linear optical elements. The development of better andbrighter sources with good mode quality as well as newtypes of detectors have opened up new perspectives [2] instate preparation and manipulation, for six or more photons.Specifically, integrated optical circuits allow for state manip-ulation with little mode matching problems in interferome-ters [3].Naturally, significant efforts has been devoted towards re-alizing instances of quantum gates. As primitives for suchsmall-scale computing, two-partite quantum gates deliveringa controlled phase-shift of ϕ have already been experimen-tally demonstrated (see Refs. [4–6] for ϕ = π and Ref. [7] forgeneral phases). In this note, we focus on linear optical imple-mentations of phase gates with arbitrary phases. In particular,we will ask when arbitrary phases can be realised in the firstplace, and—one of the main figures of merit in linear opticalapplications—what the optimum probabilities of success are,as any non-linear map is necessarily probabilistic. A realisa-tion of such gates seems interesting from the perspective of(i) gaining an understanding of the probabilistic characterof quantum gates as well as(ii) serving as a proof of principle realisation of a kind ofquantum gate that has several applications in linear op-tical quantum information processing.As far as the first aspect is concerned, one may well ex-pect that there is a trade-off between the notorious problemof having a small probability of success and the phase thatis being realised in the gate. In fact, the study in Ref. [8]suggests exactly such a behavior: the presented upper boundsto the probability of success increase from the minimum at p s ( ϕ = π ) = 1 / to p s (0) = 1 . To investigate such a trade-off is interesting in its own right and helps in building intuitionconcerning the probabilistic behavior of linear optics. Onemay well develop the intuition that “large phases are costly”as far as the probability of success and hence the overhead orrepetition are concerned.Further, concerning the second aspect, there are several ap-plications for which such a trade-off is relevant. In linear op-tical architectures, it may be a good idea to have a smallerphase, if one only has higher success probabilities. The newmeasurement-based quantum computational models [9] forexample offer this perspective: One does not have to have con-trolled π phase gates to prepare cluster states, but one wouldin principle also get away with smaller phases. This may well(but does not have to be) a significant advantage when prepar-ing resources for measurement-based quantum computing dif-ferent from cluster states [9, 10].Of course, in standard gate-based quantum computing, onewill typically encounter all kinds of controlled phase gates.For example, in the quantum Fourier transform [11], one hasto implement several controlled phase gates. They can againbe decomposed into other sets of universal gates (like CNOTor CZ and local unitaries). But, in terms of resource require-ments, it is obviously an advantage to directly implement therelevant quantum gates with phases in the range < ϕ < π .There are also interesting trade-offs between resource require-ments and success probabilities in a number of related con-texts, like non-local gates in distributed quantum computa-tion [12, 13]. Refs. [13], for example, study distributed con-trolled phase-gates which would need less entanglement andsucceed with a higher probability. In the field of linear op-tics gates, numerical results on direct implementation of arbi-trary two-qubit gates are known (see Ref. [14] and referencestherein).Instead of resorting to decompositions in the circuit model,one could gain from implementing unitaries in a fashion “nat-ural” to the respective architecture at hand. In the case oflinear optics it means to leave the computational sub-spacegiven by the encoding of the qubits for the sake of taking a“shortcut” through higher dimensions [7]. Given that the fun-damental information carriers are implemented using bosonicmodes, this will occur when mixing those modes in beamsplitter networks and is an inherent feature of genuine lin-ear optics implementations, in contrast to decompositions intostandard gate sets.Here we will study post-selected gates, so not genuine“event-ready” quantum gates, but—as is common in linear op-tical architectures at least to date—those where one measuresthe output modes and whether the gate actually succeededis determined only a posteriori by accepting only those out-comes which lie in the computational dual-rail subspace ofthe Hilbert space of n photons on n modes. Incorporatingless constraints, these gates concern only a smaller number ofmodes and are still within reach of current experiments. Inprinciple non-demolition measurements of the output wouldbe required for an event-ready gate. II. CONTROLLED PHASE GATESA. Single beam splitter
In a post-selected phase gate on four modes in the standarddual-rail encoding, two of the modes are merely involved as“by-standers”, in that their amplitude is compensated in ex-actly the same fashion as in Refs. [4–6]. In this section, wewill hence concentrate on two modes forming the “core” ofthe scheme, giving rise to a two-qubit dual-rail phase gate onfour physical modes. The core itself may be regarded as asingle-rail phase gate in its own right. Later we will see thatnot breaking the network into a core and by-stander modeswill not give any advantage.Similar to Ref. [4] we will briefly investigate the conse-quences of simply having a single beam splitter forming thecore of the quantum gate. The action on the photonic creationoperators of the two involved modes it is mixing is describedby the matrix U = diag(e ıφ , e − ıφ ) · B · diag(e ıφ , e − ıφ ) (1)with B = (cid:20) sin( ϑ ) cos( ϑ ) − cos( ϑ ) sin( ϑ ) (cid:21) (2)and appropriate phases φ , φ ∈ [0 , π ) and mixing angle ϑ ∈ [0 , π ) . The phases can also be realised deterministi-cally by local operations on the dual-rail qubits, which leavesthe relevant part of the gate U ′ = B . The matrix elementsof the unitary U ′ belonging to U ′ for vacuum, single photonoperation, and the two-photon component read h , |U ′ | , i = 1 , (3) h , |U ′ | , i = A , = sin( ϑ ) , (4) h , |U ′ | , i = A , = sin( ϑ ) , (5) h , |U ′ | , i = per( A ) = cos(2 ϑ ) = 1 − ϑ, (6) respectively. Since we are restricted to n ≤ modes, thesefour quantities determine the action of the core completely.With the constraint − ( ϑ ) = sin ( ϑ ) (7)that ensures equal single- and two-photon amplitudes (equalprobabilities for all dual-rail states), only the two solutions ϑ = ± arcsin(3 − / ) are possible, giving rise to ϕ = 0 , π ,respectively.Hence, one finds that in this way, one can implement quan-tum phase gates, but only two different ones: One is not doinganything, and the other ones effect is a controlled phase of π .This is exactly the gate of Refs. [4, 5]. In other words, with-out invoking at least a single additional mode, one can not gobeyond the known π -phase in this fashion. B. Arbitrary phases
However, we can extend this scheme: The restriction tounitary two-mode beam splitters can be relaxed. Instead ofstarting with U ∈ SU (2) , we use an arbitrary matrix A ∈ C × . Then we will embed the two-mode matrix into a higherdimensional unitary, such that an appropriately rescaled A forms a principle submatrix of a larger unitary matrix A ′ . Theoptimal rescaling is simply dictated by the largest singularvalue of A . We will see that in the two-mode case only asingle additional mode is already the most general extension,so the full set-up would consist of a transformation on threemodes involving at most three beam splitters. In this classof gates, for each ϕ , the one with the optimal probability ofsuccess p s ( ϕ ) can be identified. Observation 1 (Optimal post-selected dual-rail controlledphase gate) . Consider linear optics, an arbitrary number ofauxiliary vacuum modes and photon number resolving detec-tors. When post-selecting the state of the signal modes ontothe computational sub-space and the auxiliary modes onto thevacuum, the optimal network on four modes implementing thegate represented in the computational basis of two dual-railqubits by U = diag(1 , , , e ıϕ ) , ϕ ∈ [0 , π ] , has a successprobability (shown in Fig. 1) of p s ( ϕ ) = (cid:18) (cid:12)(cid:12)(cid:12) sin ϕ (cid:12)(cid:12)(cid:12) + 2 / sin π − ϕ r(cid:12)(cid:12)(cid:12) sin ϕ (cid:12)(cid:12)(cid:12)(cid:19) − . (8) Proof:
In order to find p s —the same for all possible in-put states—we will first construct the linear transformation ofthe relevant creation operators and identify the optimal unitaryextension afterwards. • Two-mode transformation:
The two-mode transforma-tion resulting from solving the equations (3)–(6) im-posed by the gate we want to build is A = p / (cid:20) x (e ıϕ − x/yy/x /x (cid:21) . (9) p s ( ϕ ) ϕ FIG. 1. Optimal success probability p s ( ϕ ) of phase gates with vac-uum ancillas (one vacuum ancilla is already optimal) vs. the phase ϕ (solid line). At ϕ = π the result of Refs. [4, 5], p s ( π ) = 1 / , isreproduced. The intuitive assumption of a monotonous p s ( ϕ ) is notfulfilled: indeed, the success probability is worse than / in the in-terval π/ < ϕ < π . Due to phases ϕ < π not being implementablewith a single beam splitter, the additional unitary extension requiresfurther measurements and therefore decreases the probability of suc-cess near ϕ = π . x and y are free non-zero complex parameters. By writ-ing A = p / diag( a, a − ) · (cid:20) ıϕ −
11 1 (cid:21) · diag( b, b − ) , (10)with a = xy − / and b = y / we see that the singularvalues of A only depend on | a | and | b | , so not on thephases of a , b , and x and y .The general solution to the dual-rail problem is actu-ally composed of the transformation A together withappropriate damping of the by-standers: the probabil-ity of success can not be enhanced by considering a fulltransformation on all four modes. This can be seen bywriting the polynomial system given by the dual-railproblem similar to (3)–(6), consisting of quadraticequations in the matrix elements of B ∈ C × . It turnsout that by permuting modes and appropriate variablesubstitutions all solutions can be brought into the form B ∝ ⊗ A , consisting of the two-mode core given inEqn. (9) and two by-passed modes. • Optimal extension:
Given the × matrix A that re-alises the transformation we are looking for, the optimalunitary extensions can be identified. Let us extend thefirst and second row vectors (denoted by A and A ) todimension by appending A , and A , , respectively,in such a way as to allow for unitarity of the extendedmatrix, A ′ ∈ SU (3) .To see why a dimension of three is already sufficientconsider a linear transformation of the creation opera-tors of n modes, described by its a (not necessarily uni- tary) matrix A . By using the singular value decomposi-tion (SVD) it can be decomposed as A = V · D · W − where V and W are unitary (and therefore have imme-diate interpretations as physical beam splitter matricesthemselves), and D = { d , . . . , d n } is a diagonal ma-trix with real non-negative entries d ≥ d ≥ . . . ≥ d n ,the singular values. In terms of linear optics D can beinterpreted as mixing each mode k = 1 , . . . , n with anadditional mode n + k in the vacuum state which will bepost-selected in the vacuum afterwards [15, 16]. Then, d k describes the transmittivity of the beam splitter usedto couple modes k and n k . Without loss of general-ity one can assume d = 1 , vacuum mixing for thefirst mode. This can be achieved by rescaling with theinverse of the largest singular value, so A d − A ,which implies d k d − d k . Note that such a “global”rescaling of A does not change the post-selected actionon the computational sub-space but only the successprobability of it according to p s d − n p s . Therefore,in general, there are n − additional vacuum modes re-quired to extend an n -mode linear transformation to aunitary, and thus physical, network. Also please note,that the constraint d = 1 has to be taken into accountfor any optimisation of success probabilities of A . Herewe will not explicitly use this decomposition further,but the constraint will be implemented implicitly by re-quiring the n − -mode extension to be unitary.In our specific -mode extension we choose A , and A , such that the new row vectors are orthogonal. Bymultiplying them by the root of the inverse of their re-spective norms, | A ′ | and | A ′ | , they will be normalised.Finding a third orthogonal vector to fill the unitarymatrix can be done with the complex cross product ( A ′ × A ′ ) ∗ , or in general by choosing a vector at ran-dom and orthogonalising it with respect to the givenones.The dependence of the success probability on the exten-sion is p s = ( | A ′ || A ′ | ) − . (11)Therefore, the objective is tominimise f = | A ′ | | A ′ | (12)subject to A ′ ( A ′ ) † = A A † + A , A ∗ , = 0 . (13)The first observation is that the row-scaling by x is al-ready included in the norm of the row vectors, leav-ing us with one parameter less. By using the phaseof y , we can assure that A A † is real and positiveand also arg( A , ) − arg( A , ) = ± π . This con-strained minimisation problem in A , and A , canindeed be solved (by using Lagrange’s multiplier ruleand showing constraint qualification) and we find | y | =(2(1 − cos ϕ )) / . Then an optimal solution (phaseschosen conveniently) is A , = A ∗ , = e ıπ/ (cid:18)r (cid:12)(cid:12)(cid:12) sin ϕ (cid:12)(cid:12)(cid:12) sin π − ϕ (cid:19) / (14) | i| i| i h |h |h || i | i | i | i γγλ λ λ V FIG. 2. Basic spatial modes based set-up obtained from translatingan arbitrary × core into the language of linear optics. The coreextension is provided by mixing with a vacuum mode on the cen-tral beam splitter. This mode, in turn, has to be post-selected in thevacuum state afterwards. The upper and lower beam splitters imple-ment the appropriate compensation by damping the by-passed modes(which is the same for both modes for the optimal solution which weconsider).The labels at the beam splitters will be used to identify them with therespective optical elements in Figs. 3–5. In general, the parameters(i.e., reflectivity and phases) of these elements depend on the gate’sphase ϕ . Further, the notation | i i and | i i is used for the logical / -modes of the i -th qubit to avoid confusion with Fock states. with the probability of success given by p s ( ϕ ) = (cid:16) | y | + | y | p − | y | (cid:17) − (15) = (cid:18) (cid:12)(cid:12)(cid:12) sin ϕ (cid:12)(cid:12)(cid:12) + 2 / sin π − ϕ r(cid:12)(cid:12)(cid:12) sin ϕ (cid:12)(cid:12)(cid:12)(cid:19) − . The reflectivities of the compensating beam splitters inthe by-passed modes have to be chosen such that thesuccess probability is constant for all dual-rail states,i.e., r = p / . (16)The success probability p s ( ϕ ) of this gate is shown inFig. 1. Interestingly—and quite surprisingly—the worst suc-cess probability is not achieved for the sign-flip ( ϕ = π ),but for ϕ ≈ . . This means, gates delivering a phaseshift slightly smaller than π and thereby generating less en-tanglement will not give rise to a larger, but to a smaller suc-cess probability. As expected, the success probability for verysmall phases increases and reaches unity for ϕ = 0 : one canalways do nothing at all with unit success probability. C. Integrated quantum photonics realizations
Sophisticated circuits such as the one shown in Fig. 5 can bebuilt from bulk optical elements (mirrors, beamsplitters, etc.)Such circuits often require implementation of Sagnac inter-ferometers (e.g., Ref. [18]), partially polarizing beam splitters(PPBSs) [6], or beam displacers [19, 30] to achieve interfer-ometric stability. For the most complicated circuits a combi-nation of these elements is required. Indeed a (non-optimal) | i h | | ih | i n o u t in out λ λ PBS PBS γ PPBSFIG. 3. Setup for a controlled phase gate on two polarisation encodeddual-rail qubits. The logical modes of the two qubits are separatedand re-united by means of polarising beam splitters (PBS). Replac-ing the left and right beam splitters in Fig. 2 by wave plates λ and λ , they become easier to tune to different ϕ , and provide better sta-bility. The lower beam splitter, γ , implements compensation of both, | i and | i , modes. λ V is taken care of by the partially polarisingbeam splitter (PPBS, allowing for different reflectivities for the twopolarisations, further explained in Fig. 4) in the centre. Additionally,one of the qubits has to be flipped prior to and after the circuit, whichhere is done by acting with a wave plate on the second qubit. implementation of a two-qubit controlled unitary gate useda combination of beam displacers and PPBSs [7]. However,such circuits are extremely challenging to align, are limitedin performance by the quality of that alignment, and are ulti-mately not scalable.An alternative approach based on lithographically fabri-cated integrated waveguides on a chip has recently been devel-oped [3]. This approach has demonstrated better performance,in terms of alignment and stability, as well as miniaturizationand scalability. The monolithic nature of these devices en-ables interferometers to be fabricated with precise phase andstability, making the Mach-Zehnder interferometer shown inFig. 2 directly implementable without the need to stabilizethe optical phase (either actively, or using the Sagnac-type ar-chitecture of Fig. 5), greatly simplifying the task of makingcomplicated circuits: Essentially the circuit one draws on theblackboard can be directly ‘written’ into the circuit. Indeed,integrated photonics circuits have been used to implement acircuit of several logic gates on four photons, to implementa compiled version of Shor’s quantum factoring algorithm inthis way [20]. In fact laser direct write techniques have beenused to ‘write’ circuits in an even more direct fashion than thelithographic approach.Another key advantage of integrated quantum photonics forthe circuits described here is that on-chip phase control canbe directly integrated with the circuit [21], which could allowmeasurment of the success probability curve (Fig. 1 or Fig. 6for example) to be directly mapped by sweeping the appliedvoltage. D. Experimental issues in free space
In order to render the proposed gates experimentally morefeasible in free space, some simplifications have to be done λ H λ V λ H λ V o u t i n i n o u t in out in out == out in i n o u t FIG. 4. From left to right: (i) A PPBS implementing a beam splitterwith polarisation dependent reflectivity. (ii) It is equivalent to aninterferometer between two PBS where the PPBS’ reflectivities areincorporated by means of wave plates, λ and λ . (iii) By identifyingthe two PBS, the interferometer collapses into a closed loop (whichis more compact and more robust in experimental implementations),leaving only one PBS. In the second and third circuit, a polarisationflip of the second qubit before and after the circuit is added by usinga wave plate. – tailored to the specific physical implementation at hand.Waveguide based setups would not need further simplifica-tions because stablity of the interferometers would be ensuredby the rigid substrate. Implementations more suitable forbeam displacer based setups are known [7]. Influenced by thegate model, those gates include a controlled π -phase gate atthe core. Operating at lower success probabilities, especiallyat phases ϕ approaching , the probability of success does notconverge to .In the following we will discuss simplifications which shallallow for easier free-space implementation of the networks in-troduced above while still preserving optimality with respectto p s . A straightforward set-up on dual-rail encoding that re-alises a three-mode unitary and compensates the amplitudesin the remaining modes is shown in Fig. 2. It includes oneinterferometer, but the whole gate would sit inside a doubleinterferometer, because local unitaries on the input and outputqubits would require classical interference. Thus, the com-plexity of this gate is best described as a nested three-fold in-terferometer. In this first stage, the parameters (reflectivitiesand phases) of all five beam splitters depend on ϕ .To get rid of some of the interferometers, polarisation en-coding is convenient. Two modes can be united in one spatialmode, resulting in inherent stability (neglecting birefringenceof the optical medium) of some interferometers. Rotations onthese modes can be carried out easily using wave plates. Be-cause the core acts on modes coming from different dual-railqubits, they have to be combined into a single spatial modebefore. This is achieved with a PBS, thus permuting H and V modes. Damping of both by-passed modes can be done si-multaneously by a single polarisation insensitive beam split-ter coupled to the vacuum. A straightforward translation ofFig. 2 into polarisation encoding, thereby collapsing the net- work into a single interferometer, is shown in Fig. 3.Due to the asymmetry of the core, still one PPBS is used,the reflectivity of one polarisation component of which actu-ally depends on ϕ , the other one being . Fig. 4 shows howa tunable PPBS can be constructed, introducing another inter-ferometer.Iterating the ideas that led to the compact PPBS implemen-tation once more yields a collapsed form of the phase gatebased on only two PBS and a couple of wave plates. Due tothe paths of light being very similar, this set-up should alsobe more robust. This is illustrated by Fig. 5 since it has anoverall Mach-Zehnder interferometer structure to it, by feed-ing one displaced sagnac loop into another. E. Simple decomposition
Here we want to obtain a simple decomposition and explicit ϕ -dependence of the involved elements of the full × uni-tary transformation U ∈ SU (3) . To do so, we interpret theeffective core, acting on the | i and | i modes in Fig. 2,in a different way: in between U λ (the unitary beam splittermatrix of λ ) and U λ , there acts a diagonal mode transforma-tion such that the first mode is unaffected and the amplitudeof the second one is damped (due to U λ V coupling it with re-flectivity r V to a vacuum mode which will be projected ontothe vacuum).Now considering the singular value decomposition (SVD)of the matrix representing the core transformation, A = V · Σ · W † we can identify V = U λ , W † = U λ and Σ =diag { , r V } . As we have seen earlier, optimal extensions of × cores only require global rescaling, which commuteswith the unitaries involved. Therefore we can use the muchsimpler original form of A in Eqn. (9), and we choose x = 1 and y = −√ e ıϕ − .We find the singular values of Aσ ± = r ϕ ± − ϕ ) / cos ϕ + π . (17)Global rescaling amounts to fixing the largest singular valueto unity, so the new singular values are and r V = σ − /σ + .Due to det U λ = det U λ = 1 we need to attach a phase toone singular value as well in order to apply the identificationof the matrices introduced above. Then the SVD of A yields U λ = 1 √ (cid:20) − − − (cid:21) (18)and U λ = U − λ · e ıφ + σ z with φ ± = arccot " cot ϕ + π ± (cid:18) (2 − ϕ ) / sin ϕ + π (cid:19) − (19)where the order of rows and columns in the matrices is as inFig. 2 from top to bottom.The “complex singular values” are (the first mode isnot affected) and σ − /σ + exp ıφ + + φ − . The latter can beachieved by using the aforementioned coupling to the vac-uum with a reflectivity of r V and a phase upon reflection of φ + + φ − . The further ingredients are the beam splitters re-quired for the “damping” of the by-standers as discussed ear-lier. By confirming /σ = p s in the range ≤ ϕ ≤ π , theoptimality of this construction is assured. III. EVENT-READY GATES
Coming from post-selected gates, the next step towardsscalable quantum computation would be to build gates not re-quiring measurements on the output modes. Intuitively it isclear that the construction of a controlled phase gate in thisclass will be more demanding with respect to the resources(such as the number of auxiliary modes and photons, size andcomplexity of the network) involved.Especially the number of additional photons will changedrastically: having had none in the post-selected case of acontrolled π -phase gate, two are required in the class of event-ready gates. We will use this example as a motivation for adetour to discussing a number of different methods that couldbe useful for handling linear optics state preparation.To do so, we notice that a controlled π -phase gate is moreconstrained than a device that creates EPR pairs from singlephotons. This is meant in the sense that it not only amounts toa state transformation from two single photons to an EPR pair,but a full unitary transformation on the entire computationalstate space in dual-rail encoding (and creating an EPR pairwhen applied to a certain product input corresponding to theproduct state of two photons). In the following two sectionswe will be concerned with different methods to describe linearoptics state preparation and will apply them to the specificexample at hand (i.e., heralded dual-rail EPR pair generationsfrom single photons).It will turn out, that the construction of an EPR pair out ofsingle photons by means of linear optics, vacuum modes, oneadditional photon, and detectors is not possible. Of course,directly solving the polynomial equations in the matrix ele-ments of A (generalisation of Eqns. (3) to (6)) will yield thesame result – no solutions unless two additional photons areinvolved. Having excluded the cases of zero and one auxiliaryphotons, a set-up with two of them is possible, proven by theexistence of such a scheme (EPR construction [22] as well ascontrolled- Z gate [1]). A. State transformations
An obvious way of looking at states of exactly photons in m bosonic field modes is the following. Such a state vectorcan be written as | ψ M i = P ( a † ) | vac i = m X i,j =1 M i,j a † i a † j | vac i = ( a † ) T M a † | vac i , (20) where M is a symmetric m × m matrix [ ? ]. The application ofa unitary mode-transformation U —representing a linear opti-cal network—is reflected by | ψ M i 7→ | ψ M ′ i = ( U a † ) T M ( U a † ) | vac i (21) = ( a † ) T M ′ a † | vac i (22)with M ′ = U T M U clearly again being symmetric. As aspecial case of the singular value decomposition [23], a diag-onal M ′ can be achieved, given an arbitrary input state vector | ψ M i .Now let us choose U such that M ′ is diagonal. Then, la-belled by ν ′ = rank( M ′ ) , (23)there are m different classes of states [16, 17], in each ofwhich is the states are composed by superpositions of pho-tons in either of ν ′ modes. These classes are separated by lin-ear optical mode transformations requiring additional modes.Decreasing the rank is possible by allowing for auxiliary vac-uum modes. However, to increase the rank by one additionalphoton is required.Further, the state matrix M of two single photons on fourmodes has rank ν = 2 while an EPR pair corresponds to amatrix with rank ν = 4 . Therefore, the desired state transfor-mation requires at least two additional single photons. B. Polynomial factorisation
An alternative approach is the following [16]: The polyno-mial describing the objective state vector | ψ i = P ( a † ) | vac i with P ( a † ) = 2 − / (cid:16) a † a † + a † a † (cid:17) (24)does not factorise over C . Using Lemmata 1 and 2 fromRef. [24], the property of factorisation of a bivariate polyno-mial p ( x, y ) = m X i,j =0 p i,j x i y j (25)over C can be tested by assessing the rank of a complex m (2 m − × ( m + 1)(2 m − matrix. Further, apply-ing Lemma 7 of Ref. [25], this technique can be extended tomulti-variate polynomials. Now, a state can be constructedfrom a product state using linear optical gate arrays iff the cor-responding polynomial is factorisable. In the case mentionedbefore (dual-rail EPR pair, so four variables), one can use theresulting × matrix to confirm in the language of polyno-mials of creation operators that additional resources are in factrequired. IV. TOFFOLI GATES
In the same way as above, we can consider a generalisedToffoli gate, the effect of which on the computational basis out in i n o u t λ λ λ V γγ FIG. 5. Compact implementation of a controlled phase gate by usinga single loop to implement the central PPBS and the compensationbeam splitters simultaneously. Additionally, the two PBS are iden-tified, resulting in a second loop. All omitted modes are initialisedin the vacuum and post-selected in the vacuum state (which will beachieved in practice by counting the photons in the other output). p s ( ϕ ) ϕ FIG. 6. Optimal success probabilities of generalised Toffoli gates.The features exhibited by p s ( ϕ ) are similar to the ones observed atthe controlled phase gate (Fig. 1): There is a shallow dip between ϕ = π/ and ϕ = π below p s ( π ) = p s ( π/ and a steep in-cline (more pronounced than for the controlled phase gate) for smallphases towards p s (0) = 1 . realized as dual-rail encodings can be described by the unitary U = diag(1 , , , , , , , e ıϕ ) . (26)The solutions to the polynomial equations describing the ac-tion on the three-mode core—up to mode permutations—canbe parametrised by x, y ∈ R and are given by the matrix A = p / e ıϕ − xy
00 1 yx . (27)This has to be understood similar to the -mode core used bythe controlled phase gate. However, here we do not solve thefull optimisation problem, but only consider global rescaling.For a unitary extension all singular values of A have to beat most . In order to avoid to formulate cubic singular values explicitly, we use the following constraints. Let p AA † ( λ ) =det( AA † − λ ) be the characteristic polynomial of AA † , theroots λ , , of which are the squared singular values of A . Byrequiring p AA † (1) = 0 , one of the singular values has to be . For the roots of p AA † are real-valued and non-negative, thecondition that all other singular values are not larger than isequivalent with the condition that all derivatives of p AA † havethe same sign at λ = 1 . More formally, this results in furtherconstraints of the form ( − α p ( k ) AA † (1) = ( − α d p AA † ( λ )d λ (cid:12)(cid:12)(cid:12)(cid:12) λ =1 ≥ (28)for ≤ k < n where α = 0 , is fixed by the conditionof k = n . For ϕ = π the optimal p s compliant with theseconditions is p s ( π ) = 1 + 3 (cid:16) / − / (cid:17) ≈ / . (29)See Fig. 6 for the maximum success probability in the range ≤ ϕ ≤ π . The corresponding networks could be con-structed in the same way as above. However, they wouldconsist of -mode cores (an interferometer composed of threepartially polarising beam splitters) inside separate interferom-eters for each of the qubits. For free space experiments moreappealing approaches for the specific choice of ϕ = π are pre-sented in Refs. [26, 27], but leading only to success probabil-ities of at most / . V. REMARKS ON PROCESS TOMOGRAPHY
Process tomography amounts to characterizing (or some-times certifying) an unknown physical process. In practice,the task is to identify that completely positive map that isclosest to the date with respect to some meaningful figure ofmerit. To accomplish this task, one has to consider a tomo-graphically complete set of inputs and look at Hilbert-Schmidtscalar products of the output with observables, to faithfully re-construct the matrix form of the channel [30, 31]. Practically,the closest physical process can then be found by solving aconvex optimization problem. Formally equivalently, and ininstances in an experimentally simpler fashion, one can, in-stead of sending in a full set of input states, submit half ofa single fixed maximally entangled state, and hence recon-struct the channel from the Choi matrix, then referrred to asentanglement-assisted process tomography.The latter technique can clearly also be applied in case of apostselected quantum gate like a phase gate. Yet, even withoutentangled inputs and including the actual measurement, onecan reconstruct the resulting POVM elements, to which es-sentially postselected gates amount to when one faithfully in-cludes also the actual measurement in the black-box descrip-tion of the process. If one has a well-characterized source athand, then the statistics of p j,k = tr [ ρ j A k ] , (30)uniquely characterize the process, where { ρ j } form a to-mographically complete well-characterized input set, and A , . . . , A K ≥ constitute a POVM, i.e., K X k =1 A k = (31)(for an experimental realization of such an approach, see Ref.[32]). In practice, one uses methods of convex optimization toidentify the closest physical process to the given data.For the purposes of the present work, one of the outcomes k = 1 , that is, a specific pattern A of detection, then givesrise to the actual postselected linear optical quantum gate.In this way, one can reconstruct postselected quantum gates,without using an ancilla-based approach, even faithfully in-cluding the final measurement as part of the process. VI. CONCLUSION
We have shown how to obtain the maximum probabilityof success of controlled phase- and Toffoli-like gates in the class of post-selected linear optics dual-rail gates without ad-ditional photons. Further, constructions of networks for thesmaller gates suitable for experimental implementation havebeen given, and techniques elaborated upon that allow for theassessment of the possibility of certain linear optical schemes.For further progress concerning the eventual full optimizationof linear optical processes it would be interesting to investi-gate (i) optimal constructions with respect to a class of gatesinherently incorporating such experimental constraints or (ii)identify further decomposition techniques from a given linearoptics mode transformation into suitable physical networks,respecting these constraints.
ACKNOWLEDGEMENTS
For insightful discussions concerning current experimentalabilities we would like to thank M. Duˇsek, W. Wieczorek,C. Schmid, and J. Matthews. KK was supported by MicrosoftResearch through the European PhD Programme and the EU(MINOS), JE by the EU (QAP, COMPAS, MINOS) and theEURYI award. [1] E. Knill, R. Laflamme, and G. J. Milburn, Nature , 46(2001).[2] W. J. Munro, K. Nemoto, T. P. Spiller, S. D. Barrett, P. Kok,and R. G. Beausoleil, J. Opt. B , S135 (2005); J. L. O’Brien,Science , 1567 (2007); I. A. Walmsley, Science , 1211(2008); M. Aspelmeyer and J. Eisert, Nature , 180 (2008).[3] A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, J. L. O’Brien, Science , 646 (2008); B. J. Smith, D. Kundys, N. Thomas-Peter,P. G. R. Smith, and I. A. Walmsley, arXiv:0905.2933.[4] H. F. Hofmann and S. Takeuchi, Phys. Rev. A , 024308(2002).[5] T. C. Ralph, N. K. Langford, T. B. Bell, and A. G. White, Phys.Rev. A , 062324 (2002).[6] N. K. Langford, T. J. Weinhold, R. Prevedel, K. J. Resch,A. Gilchrist, J. L. O’Brien, G. J. Pryde, and A. G. White,Phys. Rev. Lett. , 210504 (2005); N. Kiesel, C. Schmid,U. Weber, R. Ursin, and H. Weinfurter, Phys. Rev. Lett. ,210505 (2005); R. Okamoto, H. F. Hofmann, S. Takeuchi, andK. Sasaki, Phys. Rev. Lett. , 210505 (2005);[7] B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein,T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’Brien, A. Gilchrist,and A. G. White, Nat. Phys. , 134 (2009).[8] J. Eisert, Phys. Rev. Lett. , 040502 (2005); S. Scheel and N.L¨utkenhaus, New J. Phys. , 51 (2004).[9] D. Gross and J. Eisert, Phys. Rev. Lett. , 220503 (2007).[10] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. , 5188(2001).[11] M. A. Nielsen and I. L. Chuang, Quantum computation andquantum information (Cambridge University Press, Cambridge,2000).[12] J. Eisert, K. Jacobs, P. Papadopoulos, and M. B. Plenio, Phys.Rev. A , 052317 (2000).[13] J. I. Cirac, W. Duer, B. Kraus, and M. Lewenstein, Phys. Rev.Lett. , 544 (2001); D. W. Berry, Phys. Rev. A , 032349(2007). [14] D. B. Uskov, A. M. Smith, and L. Kaplan, arXiv:0908.2482.[15] N. M. VanMeter, P. Lougovski, D. B. Uskov, K. Kieling, J. Eis-ert, and J. P. Dowling, Phys. Rev. A , 063808 (2007).[16] K. Kieling, PhD thesis (Imperial College London, 2008).[17] K. Kieling, Linear optics state transformations, in preparation.[18] T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, andS. Takeuchi, Science , 726 (2007).[19] J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, andD. Branning, Nature , 264 (2003).[20] A. Politi, J. C. F. Matthews, and J. L. O’Brien, Science ,1221 (2009).[21] B .J. Smith, D .Kundys, N .Thomas-Peter, P. G. R. Smith, andI. A. Walmsley, Opt. Exp. , 13516 (2009); J. C. F. Matthews,A. Politi, A. Stefanov, and J. L. O’Brien, Nat. Phot. , 346(2009).[22] Q. Zhang, X.-H. Bao, C.-Y. Lu, X.-Q. Zhou, T. Yang, T.Rudolph, and J.-W. Pan, Phys. Rev. A , 062316 (2008).[23] R. A. Horn and C. R. Johnson, Matrix analysis (CambridgeUniversity Press, Cambridge, 1985).[24] W. Ruppert, Journal of Number Theory , 62 (1999).[25] E. Kaltofen, Journal of Computer and System Sciences , 274(1995).[26] T. C. Ralph, K. J. Resch, and A. Gilchrist, Phys. Rev. A ,022313 (2007).[27] J. Fiur´aˇsek, Phys. Rev. A , 062313 (2006).[28] N. L¨utkenhaus, J. Calsamiglia, and K.-A. Suominen, Phys. Rev.A , 3295 (1999).[29] J. Calsamiglia, Phys. Rev. A , 030301 (2002).[30] J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K.Langford, T. C. Ralph, and A. G. White, Phys. Rev. Lett. ,080502 (2004).[31] A. ˇCernoch, J. Soubusta, L. Bartuˇskov´a, M. Duˇsek, and J.Fiur´aˇsek, Phys. Rev. Lett. , 180501 (2008); W. Wieczorek,C. Schmid, N. Kiesel, R. Pohlner, O. Guehne, and H. Wein-furter, Phys. Rev. Lett. , 010503 (2008). [32] J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnell,Ch. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A.Walmsley, Nat. Phys.5