Constructing 2D and 3D cluster states with photonic modules
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International Journal of Quantum Informationc (cid:13)
World Scientific Publishing Company
CONSTRUCTING 2D and 3D CLUSTER STATES WITHPHOTONIC MODULES
RADU IONICIOIU
Hewlett-Packard Laboratories, Long Down AvenueStoke Gifford, Bristol BS34 8QZ, [email protected]
WILLIAM J. MUNRO
Hewlett-Packard Laboratories, Long Down AvenueStoke Gifford, Bristol BS34 8QZ, [email protected] Institute of Informatics, 2-1-2 HitotsubashiChiyoda-ku, Tokyo 101-8430, Japan
Received Day Month YearRevised Day Month YearLarge scale quantum information processing (QIP) and distributed quantum computa-tion require the ability to perform entangling operations on a large number of qubits.We describe a new photonic module which prepares, deterministically, photonic clusterstates using an atom in a cavity as an ancilla. Based on this module we design a networkfor constructing 2D cluster states and then we extend the architecture to 3D topologicalcluster states. Advantages of our design include a passive switching mechanism and thepossibility of using global control pulses for the atoms in the cavity. The architecturedescribed here is well suited for integrated photonic circuits on a chip and could be usedas a basis of a future quantum optical processor or in a quantum repeater node.
Keywords : Quantum computation; cluster states; photonic modules.
1. Introduction
Cluster state quantum computation1 , ,
4, especially in the context of opti-cal quantum computing.5 , , , , , , , ,
13 There are several advantages of usingphotonic qubits, including low decoherence, free-space propagation, availability ofefficient single qubit gates and the prospect of miniaturization using optical siliconcircuits.14 Cluster states with four15 and six photons16 have been experimentallyprepared and characterized. Recently an 8-qubit photonic cluster state has beendemonstrated in the context of topological quantum error correction.17In order to be useful in quantum algorithms, we need to scale up these promis-ing results to clusters containing tens to hundreds of encoded qubits. One of the ovember 5, 2018 16:26 WSPC/INSTRUCTION FILE cluster R. Ionicioiu and W.J. Munro present roadblocks towards this goal is the probabilistic nature (which implies post-selection) of all the above schemes. Although linear optics schemes (like the KLMmodel18 ,
12) are in theory scalable and universal, they require single photon sources,photon number discriminating detectors, post-selection, fast feed-forward and quan-tum memories – all these put severe experimental constraints. A possible solutionto this problem is to have a deterministic architecture which avoids most of theaforementioned issues and can be easily scaled up.The photonic module concept20 , , ,
23 has been successful in showing howlarge cluster states can be prepared deterministically using a standard buildingblock – an atom in a cavity – and classical switching. The atom in the cavityplays the role of an ancilla and provides the strong interaction required to couplethe photons (the computational qubits). At this stage it is important to exploreseveral designs in order to quantify resource requirements. Indeed, each particulararchitecture will involve complex trade-offs between design simplicity, total numberof elementary operations and their accuracy plus other technological constraints(fabrication methods, operating environment etc).Motivated by these considerations, in this article we explore an alternative archi-tecture for constructing photonic cluster states with photonic modules. The originalphotonic module functions as a parity gate – given n photons as input, it performsa nondestructive parity measurement on the arbitrary photonic state.20 ,
21 Thisoperation determines the blueprint of the optical circuit in terms of the numberof layers and connectivity of basic building blocks, switching sequence, reroutingetc. In this article we examine an alternative photonic module build around thecontrolled- Z gate C ( Z ) instead of the parity gate and see how the design changeswith this choice.The structure of the article is as follows. In Section 2 we begin by discussing thetwo main approaches for building cluster states, using either stabilizer/parity mea-surements or controlled- Z operations. These two paths lead to different photonicmodules which we will call, respectively, the parity module and the CZ module. InSection 3 we explore a new network design of a photonic circuit build around the CZ module and we show how changing the fundamental entangling gate leads to asimplified circuit design for preparing a 2D cluster state. In Section 4 we introducea passive switching mechanism and the corresponding network design. In Section5 we describe how our scheme can be generalized to construct a 3D topologicalcluster state.
2. Cluster states and photonic modules: two approaches
At the core of the photonic module is the interaction between a photon and anatom in a cavity. In the model we are considering here the photons play the role ofthe computational qubits and the atom in the cavity serves as an ancilla mediatingthe coupling between the photons. As in the original photonic module concept, weassume the photon-atom interaction to perform a C ( Z ) gate between the photonicovember 5, 2018 16:26 WSPC/INSTRUCTION FILE cluster Constructing 2D and 3D cluster states with photonic modules = = =
00 0 swap swap a x xx xy n n y n y n y (a) (b) (c) (d) HUUU U Z a Fig. 1. (a)-(d): equivalent quantum networks for a controlled gate C ( U ) acting on n qubits y n .Since the ancilla starts in the | i state, the SWAP gates in (b) are reduced to a pair of CNOTgates as in (c). In (d), the second CNOT in (c) (disentangling the ancilla) can be replaced by ameasurement of the ancilla in the Fourier basis followed by a postprocessing gate Z a on the firstqubit. (computational) and atomic (ancillary) degrees of freedom.19 , ,
21 This gate isthen sufficient to entangle the photonic qubits, as we will discuss in the following.There are two ways of describing a cluster state and each description provides adifferent way of preparing the state in the lab. First, we can view the cluster stateas a stabilizer state, hence we can prepare it by measuring n stabilizer operators,one for each qubit/vertex. The stabilizer operator of vertex i is X i Q j ∈ neigh ( i ) Z j ,where X i , Z j are the Pauli operators of a vertex and the product is over all nearestneighbours; thus for a 2D cluster state each stabilizer involves at most five photons.This is the approach taken in Refs. 21, 22 where a cluster state (two- or three-dimensional) is prepared by sending n unentangled photons through an array ofparity modules (or P -modules). A P -module consists of a cavity with an atom inthe center and performs a nondestructive parity measurement on the photons, i.e.,it projects the initial photonic state onto even (odd) parity states. To prepare a2D cluster state each photon has to pass through five cavities21 (for a 3D clusterstate this number is four22). The architecture of the full circuit is rather complex,consisting of several layers of photonic modules and routing switches directing thephotons in and out of the cavities.In the second description the cluster state is prepared in two steps1: (a) allqubits are initialized in the state | + i ⊗ n , with | + i = ( | i + | i ) / √
2; (b) a controlled- Z operation C ( Z ) = diag(1 , , , −
1) is applied to each pair of qubits sharing a linkin the underlying graph G : Q ( i,j ) ∈ edges ( G ) C ( Z ) ij | + i ⊗ n .This puts into perspective the difference between the two approaches – in thefirst one the central resource is the parity gate, whereas in the second the C ( Z )gate. For photons measuring parity is in general easier than performing a C ( Z )gate. As photons do not interact directly, the usual way to perform a deterministic controlled- U gate C ( U ) between the two photons is to use an ancilla (e.g., an atomin a cavity) coupled to both, as in Fig. 1. The well-known solution is to first swapthe first qubit and the ancilla, perform the C ( U ) gate between the ancilla and thesecond qubit, and then swap back the ancilla and the first qubit; if the ancilla isprepared in the | i state, this sequence requires only two CNOT gates and one C ( U )gate, as in Fig. 1 (a)-(c). This procedure has been used to entangle two photons (theovember 5, 2018 16:26 WSPC/INSTRUCTION FILE cluster R. Ionicioiu and W.J. Munro |+> p=x+y mod 2|+> a xy (ii) Parity module xy (i) CZ module H H Z a H X p Fig. 2. Two types of photonic modules implementing: (i) a C ( Z ) gate, (ii) a parity gate. Thequbits x, y are photons, each interacting with an atom in a cavity ancilla (middle) initialized inthe | + i state. A postprocessing gate is applied to the first qubit depending on the result of themeasurement. In the case of the CZ module the corrective Z a gate can be applied at the end ofthe cluster state preparation since Z commutes with subsequent C ( Z ) gates. qubits) using an atom in a cavity (the ancilla).25 The problem with this schemeis that the first photon has to interact twice with the cavity, first to entangle andsubsequently to disentangle it from the ancilla, Fig. 1(c). This requires a photonicbuffer to store the first photon until the appropriate time and then redirect it tothe cavity, increasing the complexity. For this the reason the parity module wasprefered as the central building block in previous schemes for constructing 2D21and 3D photonic cluster states.22In this article we focus on the second approach of preparing a cluster stateand use the C ( Z ) gate as the main resource – we call this the CZ module. Thefirst step is to notice that the second CNOT gate in Fig. 1(c) is not necessary,and that we can disentangle the first photon and the ancilla by measuring theancilla in the {| + i , |−i} basis, Fig. 1(d). Let’s see how the quantum network inFig. 1(d) works. After the first two gates, CNOT and C ( U ), the initial state istransformed to | x y n i → | xxy n i → | xx i U x | y n i . In order to disentangle the ancillafrom the control qubit, we apply a Hadamard H and then measure the ancilla; theprevious state is first transformed to | x i ( | i + ( − x | i ) U x | y n i (after H ) and thento ( − ax | x i| a i U x | y n i (after measurement, assuming the result is a ). The extraphase is then removed by applying to the first qubit a feed-forward Z a , such thatthe network in Fig. 1(d) performs the following transformation | x y n i → | x i| a i U x | y n i (1)thus proving the circuit to be equivalent to a C ( U ) between x and y n . Note that y n is an arbitrary state of n qubits/qudits. Since Z and C ( Z ) gates commute, wecan apply the corrective Z a action at the very end of the cluster state preparation,further simplifying the network.Fig. 2 shows the difference between the CZ module and the parity module, asdiscussed above. The difference between the two is minimal – only a Hadamard gate H on the ancilla after the first qubit interaction. However, this minimal modificationleads to a simplified circuit implementing a cluster state, as we will describe next.ovember 5, 2018 16:26 WSPC/INSTRUCTION FILE cluster Constructing 2D and 3D cluster states with photonic modules M Q Q M SS QQ M M M M M M M M M M M M M M M M Fig. 3. Left: Building a 2D photonic cluster state. Photons (red dots) enter from the left, preparedin the | + i state. Each photons passes through two M and two M modules. The M ( M ) appliesa C ( Z ) gate between a photon and its left/right (top/bottom) neighbours; these are indicatedby black lines in the final cluster state. Right: The modules can be implemented as Q -switchedcavities. The M module has also two active switches S which redirect the photons to the centralcavity area and then back to their rails after interaction. The two switches S are synchronous andcan be controlled by a single flip-flop circuit.
3. Building a 2D cluster: circuit design
In this section we show how to use the CZ module described above to build a2D square lattice cluster state. As this state is a universal resource for quantumcomputation one can used it to perform an arbitrary quantum algorithm.Each node (qubit) in a square lattice has four neighbours so we need to applyfour C ( Z ) gates to each photon. The circuit architecture is shown in Fig. 3. Photonsare prepared in the | + i state and pass through two M and two M modules.The temporal delay between photons on the same line is T ; photons on adjacentlines of the cluster are delayed by half period T / M interaction region. We assume that T is large enoughsuch that the full cycle of operations of the cavity atom, between initialization andmeasurement, is contained within T . Each module contains an atom in a cavity. The M M initialize in |+>apply Hidleapply H & measure cavity state:
122 1 0 M M
12 1 03 M M
13 2 1 04 M M
13 1 024 M M old link cluster state: new link Fig. 4. Time sequence of the action performed by two M modules. The first (second) moduleapplies a C ( Z ) gate between photon pairs (2 k − , k ) and (2 k, k + 1), respectively. The result isa linear cluster state which is then passed to M modules to complete the 2D structure by addingthe vertical links between qubits. H , are Hadamard gates, see Fig.2(i). ovember 5, 2018 16:26 WSPC/INSTRUCTION FILE cluster R. Ionicioiu and W.J. Munro M modules act on the same line and apply a C ( Z ) gate between a given photonand its left and right neighbours. In Fig. 4 we show a time sequence of this action.The M modules perform the same function between photons on different lines,hence they contain two switches S to direct the photons from, and respectivelyback to, their rails before and after interacting with the cavity. The two switches S are synchronous (there are both up or down at the same time) and can be controlledby a single flip-flop circuit.Let’s see now what are the resource requirements to prepare a m × n cluster state,with n the horizontal dimension of the cluster, equal to the number of time steps. Foreach horizontal line we need two M and one M modules, hence the total numberof M and M modules is, respectively, 2 m and m − m ( n −
1) + n ( m −
1) = 2 mn − m − n .
4. Passive switching
As we discussed before, the network in Fig. 3 uses active switching in the M modules. Although this can be implemented by a classical flip-flop (since bothswitches are synchronous), it requires external pulses and additional wiring, addingan extra layer of complexity. In this section we show how this classical routing canbe eliminated completely by using passive switching .Suppose the photons have an additional, non-computational, degree of freedom.We call it a ’tag’ and for our purpose it is sufficient that it takes only two values.In order to have a passive switch, we need two conditions: (i) the atom-photoninteraction is independent of and does not change this degree of freedom (i.e.,it preserves the value of the tag) and (ii) there is a simple passive device whichroutes the photons according to their tag: say 0 on upper path and 1 on the lowerpath; equivalently, it reflects photons with tag 0 and transmits photons with tag 1.Conceptually, the network in Fig. 3 can be described in the same framework: thephoton time bin is the tag (remember photons on neighbouring rails are temporallyoffset) and the switch is the flip-flop, directing each photon according to their timestamp (but in this case the flip-flop does not qualify as ’passive’).The typical example we will use in the following is the polarization degree offreedom and a polarizing beam splitter (PBS): the PBS transmits H - and reflects V -polarized photons. Another example is a dichroic mirror which reflects, e.g., greenphotons and transmits red ones (however in this case it is difficult to engineer theatom-photon interaction such that the cavity is insensitive to photons’ colour). Athird example is orbital angular momentum and holographic plates which trans-mit/reflect photons according to their orbital angular momentum.24 Of course, onecan imagine various implementations of this scheme using other degrees of freedom.In the following we use the polarization as a tag and the PBS as a passive switch.This requires two things. First, we need to use a mode (i.e., path encoded) qubit asthe computational one. Second, the atom-photon interaction should be polarizationovember 5, 2018 16:26 WSPC/INSTRUCTION FILE cluster Constructing 2D and 3D cluster states with photonic modules M PBSPBS
V VHH M CHHVHV M M M M M M M M M M M M M M M M M Fig. 5. Left: Passive switching architecture for a 2D photonic cluster state. We show only the 1-rails of a dual rail encoding. Photons have an extra degree of freedom (tag) which is orthogonal fornearest-neighbours. If this tag is the polarization, photons are either H - or V - polarized (red andgreen, respectively). Assuming the atom-photon interaction is polarization preserving, switching in M modules can be done passively with polarization beam splitters (PBS). Right: An alternativeway of constructing the M modules by reflecting a photon from a cavity using a circulator C ; thisconstruction, due to Duan and Kimble, eliminates Q -switching of the cavities. preserving; an example of such interaction is e iθ ( n v + n h ) σ z , with n v + n h = n thetotal number of photons.Using path encoding qubits implies the photonic chip has to be placed betweentwo beam-splitters in the 1-arm of a Mach-Zehnder interferometer. In Fig. 5 we showthe new architecture of the chip – note that only the 1-rails in the dual rail encodingare shown; the 0-rails are situated in a parallel plane of the chip. The operations areidentical to the ones described before. The only difference is switching in the M modules which is done passively by the PBSs. Since in this architecture all classicalrouting is done passively (i.e., without external control), the only control signalsare build in M modules, as M modules are nothing but M plus two switches, seeFig. 5.It is worth mentioning another feature of this design. The modules in the two(vertical) layers containing the M modules are synchronized so the control signalsfor the cavities (initialization, Hadamard gates) can be done by a global controlpulse applied to all M modules. The only step in which individual control is stillneeded is the final readout of the cavities at the end of the C ( Z ) gate (see Fig. 2(i)).One can still have a global readout pulse, provided each module has a local 1-bitmemory which stores the measurement result of each cavity. With an appropriatedesign (we need to take into account the offset between green and red photons)one can envisage global control of all modules in the optical chip. In this case wecan eliminate the individual control lines for each module; this becomes especiallyimportant in a 3D layout (see next section), when addressing a particular moduleburied inside the chip is difficult.One can think of two different designs for implementing the C ( Z ) gate betweenthe atom and the photon. The first uses a Q -switched cavity20 ,
21, as in Fig. 3:ovember 5, 2018 16:26 WSPC/INSTRUCTION FILE cluster R. Ionicioiu and W.J. Munro z xy
Fig. 6. Starting with a regular cubic lattice (left), we construct the 3D topological code (right)by removing the blue photons together with all their links. photons enter the cavity through the left, interact with the atom and then are Q -switched out to the right. The second employs the scheme of Duan and Kimble25– photons are reflected from the cavity (the lower mirror is partially reflective)and exit through the same port (Fig. 5, right). In this case an optical circulatorredirects the photons to the exit rail. As a consequence, in this variant of the designwe eliminate both the classical routing and the Q -switching of the cavities.
5. Preparing a 3D topological cluster state
Topological cluster state computation pioneered by Raussendorf, Harrington andGoyal26 ,
27 has attracted recently considerable interest as a fault-tolerant archi-tecture for constructing photonic cluster states.22 Here we briefly discuss how toadapt the previous 2D network design to a 3D setup.The 3D topological cluster state26 ,
27 can be constructed from a regular cubiclattice (Fig. 6, left) by removing the blue qubits together with all their links, as inFig. 6, right. One way of doing this is to construct first a 3D cubic cluster state andsubsequently measure the blue qubits in the S z basis. The measurement eliminatesthe blue photons and their links from the lattice, but is not efficient since it involvesfeed-forward (we have to take into account the result of the measurement).A better way of obtaining the same result is the following. Suppose we havephotonic module which prepares a regular cubic cluster state. We can extend the2D module described in the last section in a straightforward way to a 3D geometryby adding two extra layers of M modules in order to couple each photon to itsnearest neighbours along the third spatial direction, orthogonal to the xz -plane, asin Fig. 7. However, instead of eliminating the blue photons by measurement afterthe photonic modules, we remove the blue photons at the injection stage, i.e., we runthe photonic module with some of the photons missing. Thus the photon sourcesinjecting the green photons run at half the frequency of the sources injecting the redphotons, as in Fig. 7. Moreover, since the green photons don’t have links with othergreen photons, we can eliminate completely the M modules on the green lines.ovember 5, 2018 16:26 WSPC/INSTRUCTION FILE cluster Constructing 2D and 3D cluster states with photonic modules z x xy−plane { HVVHH M M M M M M M M M M M M M M M M M M M M M M Fig. 7. Left: network for a 3D topological cluster state, viewed in the xz -plane; photons flow along x axis. The yellow M modules are oriented in the xy plane and couple two adjacent photons inthe y direction. The green photons are doubly spaced compared to the red ones; note there are no M modules on the green lines. Right: schematics of a 3D topological cluster state; each photonhas only 4 links. Each green photon will pass through four M modules, two for the links in the xz -plan and the other two for the links in the xy -plan. The red photons pass thoroughsix modules, two for each spatial direction, in a straightforward generalization ofthe 2D case. However, since now half of the green photons are missing, each redphoton will have only four links: two links in the x direction (always) and anothertwo with the green photons, either along y or z axis. The resulting state (Fig. 6,right) is exactly the 3D topological cluster state from Ref. 22. Once prepared, the3D topological cluster state can be used as a universal, fault-tolerant resource forQIP.
6. Conclusions
In this article we described a scheme for preparing large scale photonic clusterstates with photonic modules. In our model we implement directly a C ( Z ) gatebetween two photons using as an ancilla an atom in a cavity. Compared to theoriginal photonic module design which uses a parity gate20, this choice of entan-gling gate leads to a simplified architecture with fewer modules (3 m − m for a 2D cluster, with m the width of the cluster) and classical switching.Moreover, if the atom-photon interaction is polarization preserving there is no needfor active switching at all. In this case one can have only passive switching, e.g.,using polarising beam splitters and photons in neighbouring rails having orthogo-nal (H/V) polarization. This passive switching completely eliminates the need ofan active switching mechanism synchronized with the photons, thus reducing thecomplexity and the associated decoherence. Another feature of the present designis the possibility of using global control of all modules in the optical chip. Thisbecomes especially important in a 3D layout, when addressing a particular moduleovember 5, 2018 16:26 WSPC/INSTRUCTION FILE cluster R. Ionicioiu and W.J. Munro buried inside the chip is difficult.The model discussed here paves the way towards integrated photonic circuits14on a chip as a basis for future quantum optical processors. Even with a small tomedium number of photonic qubits available, such a chip will be useful as a quantumrepeater28 , ,
30 or as an element in a future quantum internet31 architecture.
Acknowledgments
We acknowledge financial support from EU projects QAP and HIP and JapanMEXT.
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