Six-qubit two-photon hyperentangled cluster states: characterization and application to quantum computation

Six-qubit two-photon hyperentangled cluster states: characterization and application to quantum computation

Giuseppe Vallone http://quantumoptics.phys.uniroma1.it/ Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi, Via Panisperna 89/A, Compendio del Viminale, Roma 00184, Italy Dipartimento di Fisica, Università Sapienza di Roma, Roma 00185, Italy    Gaia Donati http://quantumoptics.phys.uniroma1.it/ Dipartimento di Fisica, Università Sapienza di Roma, Roma 00185, Italy    Raino Ceccarelli http://quantumoptics.phys.uniroma1.it/ Dipartimento di Fisica, Università Sapienza di Roma, Roma 00185, Italy    Paolo Mataloni http://quantumoptics.phys.uniroma1.it/ Dipartimento di Fisica, Università Sapienza di Roma, Roma 00185, Italy Istituto Nazionale di Ottica Applicata (INOA-CNR), L.go E. Fermi 6, 50125 Florence, Italy
July 12, 2019
Abstract

Six-qubit cluster states built on the simultaneous entanglement of two photons in three independent degrees of freedom, i.e. polarization and a double longitudinal momentum, have been recently demonstrated. We present here the peculiar entanglement properties of the linear cluster state related to the three degrees of freedom. This state has been adopted to realize various kinds of Controlled NOT (Cnot) gates, obtaining in all the cases high values of the gate fidelity. Our results demonstrate that a number of qubits 10 in cluster states of two photons entangled in multiple degrees of freedom is achievable. Furthermore, these states represent a promising approach towards scalable quantum computation in a medium term time scale. The future perspectives of a hybrid approach to one-way quantum computing based on multi-degree of freedom and multi-photon cluster states are also discussed in the conclusions of this paper.

I Introduction

Multiqubit graph states Hein et al. (2004) are a basic resource for a number of important quantum information applications. These states have been proposed in particular for advanced tests of quantum nonlocality in which the violation of local realism increases exponentially with the number of qubits Mermin (1990); Gühne et al. (2005); Cabello and Moreno (2007); Cabello et al. (2008), and for the realization of quantum computation algorithms of increasing complexity in the one-way model Raussendorf and Briegel (2001); Briegel and Raussendorf (2001). Other application fields deal with quantum communication Cleve et al. (1999) and quantum error correction Schlingemann and Werner (2001).

In recent years, photon cluster states of four, six and up to ten qubits have been realized by different approaches and used to deeply investigate the peculiar properties of high dimensional entanglement Briegel et al. (2009) and to perform basic quantum computation algorithms Chen et al. (2007); Vallone et al. (2008).

Figure 1: (a) Graph associated to the hyperentangled state . Each set represents a photon and every vertex is associated to a qubit. Qubits and are encoded in the DOF, qubits and in polarization and qubits and in the DOF. See text for further details. (b) Graph associated to the two-photon six-qubit linear cluster state . can be obtained from by application of two CZ operations between qubits belonging to different DOFs.

Two strategies are generally used to create multiqubit cluster states: one consists of increasing the number of entangled photons Zhao et al. (2003); Walther et al. (2005a, b); Kiesel et al. (2005); Prevedel et al. (2007), the second one is based on the encoding of more qubits in different degrees of freedom of the particles Chen et al. (2007); Vallone et al. (2008, 2007); Gao et al. (2008). By the first approach, some examples of four and six photon Zhao et al. (2003); Walther et al. (2005a, b); Kiesel et al. (2005) cluster states have been experimentally demonstrated, up to now, with very low rates. The second approach, which is based on two-photon hyperentanglement, has been used to create two-photon four-qubit cluster states Vallone et al. (2007); Gao et al. (2008); Tokunaga et al. (2008); Mair et al. (2001); Cinelli et al. (2005); Barbieri et al. (2005); Barreiro et al. (2005); Schuck et al. (2006); Park et al. (2007); Lanyon et al. (2009); Vallone et al. (2009). By using hyperentanglement, five photons have been recently entangled in ten qubits encoded in the polarization and longitudinal momentum degrees of freedom (DOFs) Gao et al. (2008).

The advantages of the hyperentangled state approach, as far as generation/detection rate and fidelity of the states are concerned, have been already demonstrated Chen et al. (2007); Vallone et al. (2008). These properties have been very recently confirmed by the realization of the linear 2-photon 6-qubit cluster state starting from the triple entanglement of two photons in three independent DOFs Ceccarelli et al. (2009), namely the polarization and a double longitudinal momentum. The is the only distribution of six qubits between two particles whose perfect correlations have the same nonlocality as those of the six-qubit Greenberger-Horne-Zeilinger state Cabello et al. (2008), but only requires two separated carriers Cabello and Moreno (2007).

In this paper we give a detailed characterization of the state realized by using the triple hyperentanglement of two photons and demonstrate its feasibility for one-way quantum computation by the high fidelity realization of different kinds of Cnot gates.

The paper is organized as follows. In Sec. II we describe the realization of the six-qubit linear cluster state, derived from the application of suitable Cphase gates to a six-qubit hyperentangled state. Sec. III reports on the characterization of the state by a sequence of quantum tomographic reconstructions performed in the three DOFs. Sec. IV describes how the Cnot gate has been efficiently realized with six qubits. Finally, the future perspectives of the realization of multiqubit cluster states built on an increasing number of photon DOFs are discussed in the conclusions of Sec. V.

Ii Generation of the Six-Qubit Cluster State

Figure 2: Setup of the experiment. (a) Source (green box) of the 8-mode hyperentangled state. A detailed description of the source is given in Barbieri et al. (2005); Cinelli et al. (2005); Ceccarelli et al. (2009). (b) Mode labelling: upper (lower) modes correspond to Alice (Bob) photon. For each photon we indicate with () the right (left) modes and with () the internal (external) modes. We also show the two half wave plates () used to transform the hyperentangled state to the cluster state . The on the modes of photon is oriented at while the on the modes of photon is oriented at . (c) Measurement scheme: the momentum measurement setup consists of two chained interferometers, the first () measuring the qubit, while the second ( and ) measuring the qubit. Polarization analysis is performed by standard waveplates and polarizing beam splitters (PBS). Path delays and , are varied to obtain the optimal temporal superposition of the modes respectively in the first and second interferometer.

Cluster states are peculiar entangled states associated to -dimensional lattices where each vertex represents a qubit and connections between vertices correspond to Ising interactions between the two-level quantum systems. Two-dimensional lattices have proved to be a universal resource for Quantum Computation (QC) Raussendorf and Briegel (2001); from here on, we shall then restrict ourselves to the case . The explicit expression of a cluster state is obtained by preparation of each qubit in the state and subsequent application of a Cphase gate, , between two adjacent vertices and . We have

(1)

where is the identity operator. From now we will use the following simplified notation for the Pauli operators: and analogous relations for and .

For a lattice with sites, the corresponding cluster state can then be written as

(2)

where .

In general, the cluster state associated to a specific graph can be equivalently defined as the only state satisfying the eigenvalue equations

(3)

for every lattice vertex , where the operators

(4)

are known as the stabilizer generators for the cluster state. is the set of vertices connected with the vertex .

The linear cluster state is the state associated to the lattice shown in Fig. 1(b). We generated a six-qubit two-photon linear cluster state , equivalent to up to single qubit unitary transformations, starting from the hyperentangled state and exploiting the three degrees of freedom (DOFs) of polarization and two different kinds of longitudinal momentum. To show that the cluster state obtained in the laboratory is equivalent to , we start describing the source of the hyperentangled state , the first step for the generation of the linear cluster .

The two-photon six-qubit source, extensively described elsewhere Barbieri et al. (2005); Cinelli et al. (2005); Ceccarelli et al. (2009), consists of a continuous wave (cw), vertically-polarized Ar laser beam (, ) interacting through spontaneous parametric down-conversion (SPDC) with a Type I, 0.5 thick -Barium-Borate (BBO) crystal. The nonlinear interaction between the laser beam and the BBO crystal produces degenerate photon pairs at wavelength , entangled in polarization and belonging to the surfaces of an emission cones. Referring to Fig. 2(a), the insertion of a holed mask allows us to select four pairs of correlated spatial modes from the conical surface, which is all we need for the creation of the hyperentangled state . The labels used to identify the selected modes require some explanations [cfr. Fig. 2(b)]: the distinction between left and right modes provides us with the first longitudinal momentum DOF (, also known as the linear momentum ), while distinguishing between external and internal modes supplies the second momentum DOF (). Moreover, the conical emission of the BBO crystal can be divided into an “up” circular half and a “down” one with respect to an ideal horizontal line passing through the center of the mask. Every mode belonging to the “up” half shall be associated to carrier photon ; an analogous correspondence is adopted for the “down” half and the second carrier photon . By doing so we have at our disposal two SPDC photons, and , to each of which we associate three different qubits corresponding to the three DOFs (polarization, first, and second momentum) introduced above.

By appropriately setting the phase of each pair of modes, the source generates the hyperentangled state , explicitly written as

(5)

It comes out that the state is given by a tensor product of three maximally entangled state, one for each DOF.

By setting the following correspondences between physical and computational qubits,

(6a)
(6b)
(6c)
(6d)
(6e)
(6f)

we can express the state (5) as

(7)

where is the state associated to the graph shown in Fig. 1(a) and is the Hadamard operator acting on qubit . From the definition of graph states in eq. (2), is obtained from the graph state by the application of the two-qubit gates and .

We build the state by applying the gates and to the hyperentangled state . The gate CX is defined as . We are now in the position to state the relation between the state , and the state :

(8)

The previous relations can be easily demonstrated by using the property . We thus see that the generated cluster state is equivalent to the linear six-qubit two-photon cluster state up to the unitary transformation consisting of single qubit unitaries. In the generated state, qubits and are encoded in the longitudinal momentum DOF, qubits and in the polarization variable and qubits and in the momentum DOF (see Fig. 1). Specifically, the relation given in (8) between and implies that is the only common eigenstate of the generators obtained from by changing , , , and .

Starting from Eq. (8), we can write the following explicit expressions for the generated state by differently factoring the terms referring to the three considered DOFs:

(9a)
(9b)
(9c)

where we omitted the subscripts . The states and are the four polarization Bell states, while the states and are the standard Bell states encoded in the and degrees of freedom, respectively (the “” subscript standing for “cone”).

The realization of the two-qubit gates responsible for the transformation of the hyperentangled state into the cluster state in terms of optical components was made possible by the insertion of two wave-plates after the holed mask; since qubits and belong to photon , the first gate was realized by means of a wave-plate oriented at and intercepting the two internal modes (see Fig. 2(b) and Eq. (6a)). Analogously, the gate was obtained thanks to a second wave-plate oriented at and intercepting the two left modes (see Fig. 2(b) and Eq. (6f)). It actually proved convenient to have two separated wave-plates on the left modes, but this was a choice uniquely related to our specific experimental setup.

Iii Characterization of the Six-Qubit Cluster State

Let us refer to Fig. 2(c). The two chained interferometers, whose core elements are the three symmetric beam splitters , and , allow the simultaneous measurement of the three single-qubit compatible observables associated to both particles and . The modes are made indistinguishable (in space as well as in time) from the ones on , while and modes are matched on or depending on which photon they refer to. By means of a trombone mirror assembly in each of the two interferometers, it is possible to act on the optical path delays, and , and find the optimal temporal superposition conditions for both of the interference phenomena. Let us now refer to the : we set and , for , as its input and output states. The insertion of a thin glass plate intercepting two right modes (one internal and one external) transforms the input states in the following way: and , for external and internal modes. By detecting the photons on the or the output we are projecting the input state respectively into or . An analogous glass plate intercepts the left modes111In this case the projection is performed into the states and .

Two more such phase shifters, intercepting the external A and B modes, are inserted in the second interferometer before and . Four single-photon detectors , , and receive the radiation belonging to the “up” and “down” output modes (see Fig. 2(b)), which we can label as for . In the presence of the glass plates cited above, the following input-output transformations concerning and hold: and . Finally, a polarization analyzer constituted of a wave-plate, a wave-plate and a polarizing beam splitter (PBS) is added in front of each detector. In these conditions, we recorded nearly coincidences per second.

separable basis output DOF output state Fidelity
Table 1: Fidelities of Bell states for each DOF. We selected a separable state for two DOFs (first column) and performed a tomographic reconstruction of the density matrix of the remaining DOF (second column). The expected output states and the relative fidelities are shown in the last two columns.
Figure 3: Tomographic reconstruction of the four polarization states in Eq. (9) (real parts). The imaginary components are negligible. The corresponding theoretical Bell states are: (a) , (b) , c) , d) .
Figure 4: Tomographic reconstruction of the four states encoded in the DOF (real parts). The imaginary components are negligible. The corresponding theoretical Bell states are: 1) and 3) , 2) and 4) .
Figure 5: Tomographic reconstruction of the state encoded in the DOF (real part) corresponding to the spatial mode pair and the polarization state . The imaginary components is negligible. The corresponding theoretical state is .

The characterization of the generated state relies on a tomographic reconstruction technique followed by a “maximum likelihood” method James et al. (2001). Particularly, we aim at recovering Eq. (9), which shows three alternative and perfectly equivalent ways of writing the cluster state . Indeed, Eq. (9) is important to prove since it highlights the inner structure of the generated state. As we see, each of the expressions (9) is obtained by writing the states of four qubits corresponding to two DOFs in a separable basis, and expressing the remaining couple of qubits in the appropriate entangled Bell basis; for example, the first relation shows the four polarization Bell states. Equation (9a) shows that the state is obtained by a coherent superposition between four terms, each of them referring to a specific pair of correlated modes. We first demonstrated that the four polarization states corresponding to the different pairs of modes are given by the Bell states. The coherence between them can be shown by using equations (9b) and (9c). It is easy to show that the first two terms in (9b) arise from the superposition between the first two terms in (9a), and the same applies for the last two terms. By selecting the appropriate separable basis in two DOFs we performed the tomographic reconstructions to recover the Bell states encoded in the remaining degree of freedom. As a consequence, these measurements prove not only the presence of the various terms appearing in Eq. (9), but also implicitly tell us about the coherences between the states involved.

Stabilizer Experimental value
Stabilizer Experimental value
Table 2: Experimental results: measurement of the 64 stabilizers of , i.e., all the products of the generators . Last three columns indicate in which Bell inequality test each experimental value was used.

The reconstruction concerning the polarization variable exactly followed the strategy presented in James et al. (2001), while the complete sets of tomographic analysis states associated to the two longitudinal momentum DOFs were established combining the known complete set of polarization states (as given in James et al. (2001)) with the stated correspondence between physical and computational qubits (see equations (6)).

The experimental density matrix reconstructions are shown in Fig. 3 for the polarization variable, in Fig. 4 for the linear momentum and in Fig. 5 for the DOF. The fidelities associated to the considered tomographic analysis are listed in Table 1. As we see, most of these values exceed and some get above ; the lowest experimental fidelity corresponds to the tomographic reconstruction associated to the DOF.

We attribute this to the difficulty to achieve perfect mode matching in the second interferometer due to mode divergences. Nevertheless, the obtained results represent a first evidence of the correct generation of the cluster state . We also measured the state fidelity to give further informations on the state preparation.

As said, the reported tomographic reconstructions allow us to test the validity of Eq. (9); this approach is naturally connected to the first definition of cluster states recalled in this paper (see Eq. (2)).

We can then refer to Eq. (4) instead, which gives the characterization of cluster states in terms of their stabilizer generators, and adopt a complementary point of view (with respect to the one condensed in Eq. (9)) leading to a more complete characterization of the cluster state . Actually, its stabilizer generators generate the so-called stabilizer group

(10)

where is a subset of . The elements are known as the stabilizing operators of , and satisfy the relation .

It can be shown that

(11)

The fidelity of the experimental cluster state, whose density matrix is , can then be calculated as

(12)

i.e., by measuring the expectation values of the stabilizing operators of the generated cluster state. We obtained . The experimental expectation values are shown in Table 2.

We tested the genuine six-qubit entanglement of the created cluster state by evaluation of an appropriate entanglement witness, defined as Tóth and Gühne (2005)

(13)

There is entanglement whenever

(14)

We found , which being negative by standard deviations proves the existence of a genuine six-qubit entanglement.

The data present in Table 2 were also used for a nonlocality test of quantum mechanics Ceccarelli et al. (2009). Any local theory in which every single-qubit Pauli observable can be interpreted as an element of reality as intended by EPR satisfies the following inequality:

(15)

where is defined as

(16)

We tested the Bell inequality (15) and obtained (see checked rows in the third column of Table 2); this result implies a degree of nonlocality equal to . We also tested the persistency of entanglement of against the loss of two qubits. This property can be investigated by considering two alternative Bell inequalities with respect to (15): in the first one qubits and are ignored, while in the second inequality we trace out qubits and :

(17a)
(17b)

By using the measurements given in Table 2 we found

(18a)
(18b)

showing violations of the Bell inequalities (17a) and (17b). See Ceccarelli et al. (2009) for more details concerning Bell inequalities with the 2-photon 6-qubit cluster state.

Iv Experimental Realization of the Cnot Gate

Figure 6: Graph associated to the six-qubit horseshoe cluster state, equivalent to the generated linear cluster state.

Let us now turn to the one-way model of QC Briegel et al. (2009). Given a cluster state, it can be useful to think of the distinct horizontal qubits as “the original [logical] qubit at different times” Nielsen (2004), with the temporal axis oriented from left to right (a choice made possible by appropriately designing the lattice); single-qubit gates are represented by pairs of horizontally adjacent qubits, while vertical connections play the role of Cphase gates. Each computation process is then obtained as a sequence of single-qubit projective measurements performed on the so-called physical qubits, simultaneously determining the propagation of information through the cluster and the loss of entanglement in the original state Raussendorf and Briegel (2001); Nielsen (2004).

This last feature is responsible for the irreversibility of the process and explains why we speak of one-way computation. The difference existing between physical and encoded qubits deserves a deeper understanding. Physical qubits in the initial cluster state represent an entanglement resource; encoded (or logical) qubits constitute the quantum information being processed Raussendorf et al. (2003).

Pattern Qubit [DOF] Measurement CB Measurement LB
111See Eq. (19). 111See Eq. (19).
111See Eq. (19). 111See Eq. (19).
Table 3: Measurement bases for the different considered patterns. For each pattern we indicate the measured qubit (and the DOF in which the qubit is encoded) and the corresponding measurement in the cluster (CB) and laboratory bases (LB).
Figure 7: Measurement pattern I: we indicate by a cross a measurement in the bases and by and a measurement in the basis and . The output state is encoded in qubits 5 and 2. The circuit associated to the considered measurement is shown. We first indicate the circuit obtained by directly following the one-way rules and then the equivalent circuit composed by single-qubit gates and a two-qubit Cnot gate. The gates indicated by ”Change of basis” are due to the change between the computational and laboratory basis.
Figure 8: Measurement patterns , and and the corresponding circuit representations. Each circuit is composed by single-qubit gates and a two-qubit Cnot gate.

Let be the number of physical qubits and the number of encoded qubits, with . input cluster qubits, all prepared in the state , are usually positioned on the left of the two-dimensional graph. The single-qubit measurements involve qubits. Consequently, the output of the computation can be read on the unmeasured qubits up to local Pauli errors, as will be specified later on in this paper. More precisely, the measurements driving the computation are performed in the following basis:

(19)

with . If we take as signalling the presence of a Pauli error, we usually associate to the measurement outcome (error-free case) and to . The choice of (and the consequent possible errors occurring in the computation) depends on the algorithm to be implemented. Measuring a qubit in the computational basis has a completely different effect on the cluster, in that it removes the measured qubit and leads to the cluster state

(20)

where is the set of vertices connected to site .

The generated six-qubit cluster allows the implementation of non-trivial two-qubit operations such as the Cnot gate. For this purpose, it is convenient to think of a horseshoe ( rotated) six-qubit cluster instead of the one depicted in Fig.1(b); the two are physically equivalent, but the horseshoe one reveals easier to translate into a circuit representation of the Cnot gate. Let us consider Fig. 6. Since we realize our computation within the one-way model, we perform simultaneous single-qubit measurements on qubits and and on qubits and and then read the corresponding output on qubits and , both encoded in the polarization DOF.

We pointed out four possible measurement patterns in order to accomplish different logical operations, depending on the bases chosen for the single-qubit measurements. From now on, when referring to a given measurement basis we will always think of the so-called “laboratory basis” (LB), which differs from the “cluster basis” (CB) because of the presence of the local operations affecting qubits () and () (see Eq. (8)). The four considered measurement patterns, both in the cluster and in the laboratory bases, are listed in Table 3.

For each pattern a corresponding computational circuit can be derived. In Fig. 7 we show the detailed derivation of the corresponding circuit for the first considered pattern: the measurements implement the “Cluster algorithm” (see figure) and the change between the CB and the LB corresponds to the final gates (labeled as “Change of basis” in the figure). The circuit can be equivalently written as shown in the right part: it consists of two single qubit rotations and a Cnot gate. The Pauli errors, as usual, depend on the measurement results of qubits 3, 4, 6 and 1. In Fig. 8 we show the equivalent circuits corresponding to the other three measurement patterns we have considered.

By taking into account their circuit representations shown in Fig. 7 and Fig. 8, we can write the expected output state, encoded in the physical qubits 5 (photon , control) and 2 (photon , target), for each measurement pattern:

(21a)
(21b)
(21c)