One-way quantum computation with two-photon multiqubit cluster states
We describe in detail the application of four qubit cluster states, built on the simultaneous entanglement of two photons in the degrees of freedom of polarization and linear momentum, for the realization of a complete set of basic one-way quantum computation operations. These consist of arbitrary single qubit rotations, either probabilistic or deterministic, and simple two qubit gates, such as a c-not gate for equatorial qubits and a universal c-phase (CZ) gate acting on arbitrary target qubits. Other basic computation operations, such as the Grover’s search and the Deutsch’s algorithms, have been realized by using these states. In all the cases we obtained a high value of the operation fidelities. These results demonstrate that cluster states of two photons entangled in many degrees of freedom are good candidates for the realization of more complex quantum computation operations based on a larger number of qubits.
Present address: ]Univ. de Genève GAP-Optique, Rue de l’École-de-Mèdecine 20, CH-1211 Genève 4, Suisse
The relevance of cluster states in quantum information and quantum computation (QC) has been emphasized in several papers in recent years Raussendorf and Briegel (2001); Nielsen (2004); Browne and Rudolph (2005); Kiesel et al. (2005); Walther et al. (2005); Bodiya and Duan (2006); Danos and Kashefi (2006); Prevedel et al. (2007); Popescu (2007); Varnava et al. (2008); Lu et al. (2007). By these states novel significant tests of quantum nonlocality, which are more resistant to noise and show significantly larger deviations from classical bounds can be realized Cabello (2005); Scarani et al. (2005); Vallone et al. (2007); Kiesel et al. (2005).
Besides that, cluster states represent today the basic resource for the realization of a quantum computer operating in the one-way model Raussendorf and Briegel (2001). In the standard QC approach any quantum algorithm can be realized by a sequence of single qubit rotations and two qubit gates, such as c-not and c-phase on the physical qubits Nielsen and Chuang (2000); Knill et al. (2001); Kok et al. (2007), while deterministic one-way QC is based on the initial preparation of the physical qubits in a cluster state, followed by a temporally ordered pattern of single qubit measurements and feed-forward (FF) operations Raussendorf and Briegel (2001). By exploiting the correlations existing between the physical qubits, the unitary gates on the so called “encoded” (or logical) qubit Walther et al. (2005) are realized. In this way, non-unitary measurements on the physical qubits correspond to unitary gates on the logical qubits. It is precisely this non-unitarity of the physical process that causes the irreversibility nature (i.e. its “one-way” feature) of the model. By this model the difficulties of standard QC, related to the implementation of two qubit gates, are transferred to the preparation of the state.
The FF operations, that depend on the outcomes of the already measured qubits and are necessary for a deterministic computation, can be classified in two classes:
the intermediate feed-forward measurements, i.e. the choice of the measurement basis;
the Pauli matrix feed-forward corrections on the final output state.
The first experimental results of one-way QC, either probabilistic or deterministic, were demonstrated by using 4-photon cluster states generated via post-selection by spontaneous parametric down conversion (SPDC). Walther et al. (2005); Prevedel et al. (2007). The detection rate in such experiments, approximately 1 Hz, was limited by the fact that four photon events in a standard SPDC process are rare. Moreover, four-photon cluster states are characterized by limited values of fidelity, while efficient computation requires the preparation of highly faithful states.
More recently, by entangling two photons in more degrees of freedom, we created four-qubit cluster states at a much higher level of brightness and fidelity Vallone et al. (2007). Precisely, this was demonstrated by entangling the polarization () and linear momentum (k) degrees of freedom of one of the two photons belonging to a hyperentangled state Barbieri et al. (2005, 2006). Moreover, working with only two photons allows to reduce the problems related to the limited quantum efficiency of detectors. Because of these characteristics, two-photon four-qubit cluster states are suitable for the realization of high speed one-way QC Vallone et al. (2008a, b); Chen et al. (2007).
In this paper we give a detailed description of the basic QC operations performed by using two-photon four-qubit cluster state, such as arbitrary single qubit rotations, the c-not gate for equatorial qubits and a c-phase gate. We verified also the equivalence existing between the two degrees of freedom for qubit rotations, by using either or as output qubit, demonstrating that all four qubits can be adopted for computational applications. Moreover, we also show the realization of two important basic algorithms by our setup, namely the Grover’s search algorithm and the Deutsch’s algorithm. The former identifies the item tagged by a “Black Box”, while the latter allows to distinguish in a single run if a function is constant or balanced.
The paper is organized as follows. In Sec. II we review the one-way model of QC realized through single qubit measurements on a cluster state. We also describe the basic building blocks that can be used in order to implement any general algorithm. In Sec. III we give a description of the source used to generate the two-photon four-qubit cluster state by manipulating a polarization-momentum hyperentangled state. We describe in Sec. IV three basic operations realized by our setup: a generic single qubit rotation, a c-not gate for equatorial target qubit and a c-phase gate for fixed control and arbitrary target qubit. In Sec. V two explicit examples of quantum computation are given by the realization of the Grover’s search algorithm and the the Deutsch’s algorithm. Finally, the conclusions are given in Sec. VI.
Ii One-way computation
Cluster states are quantum states associated to dimensional lattices. It has been shown that two-dimensional cluster states are a universal resource for QC Nielsen (2005). The explicit expression of a cluster state is found by associating to each dot of the lattice (see fig. 1) a qubit in the state and to each link between two adjacent qubits and , a c-phase gate :
Considering a lattice with sites, the corresponding cluster state is then given by the expression:
where . Some explicit examples of cluster states are the 3-qubit linear cluster,
the 4-qubit linear (or horseshoe) cluster
corresponding to four qubits linked in a row (see Fig. 4I)) and the 4-qubit Box cluster
corresponding to four qubits linked in a square (see Fig. 10(left)).
For a given cluster state, the measurement of a generic qubit performed in the computational basis (Fig. 1a)) simply corresponds to remove it and its relative links from the cluster (Fig. 1b)). In this way we obtain, up to possible corrections, a cluster state with qubits:
where if the measurement output is , while if the measurement output is . In the previous equation stands for the set of all sites linked with the qubit . Then, by starting from a large enough square lattice, it is possible to create any cluster state associated to smaller lattices. In the following figures we will indicate with a red cross the measurement of a physical qubit performed in the computational basis, as shown in Fig. 1.
Let’s now explain how the computation proceeds. Each algorithm consists of a measurement pattern on a specific cluster state. This pattern has a precise temporal ordering. It is well known that one-way computation doesn’t operate directly on the physical qubits of the cluster state on which measurements are performed. The actual computation takes place on the so-called encoded qubits, written nonlocally in the cluster through the correlation between physical qubits. We will use for the physical qubits and for the encoded qubits (). Some physical qubits (precisely ) represent the input qubits of the computation (all prepared in the state ) and the corresponding dots can be arranged at the left of the graph. We then measure qubits, leaving physical qubits unmeasured, hence the output of computation will correspond (up to Pauli errors) to the unmeasured qubits. It’s possible to arrange the position of the dots in such a way that the time ordering of the measurement pattern goes from left to right.
The computation proceeds by the measurement performed in the basis
where . Here or if the measurement outcome of qubit is or respectively. The specific choice of for every physical qubit is determined by the measurement pattern. Note that the choice of the measurement basis for a specific qubit can also depend on the outcome of the already measured qubits: these are what we call feed-forward measurements (type i)). In general, active modulators (as for example Pockels cells in case of polarization qubit) are required to perform the FF measurements. In some case, however, when more than one qubit is encoded in the same particle through different degrees of freedom (DOF’s), it is possible to perform FF measurement without the use of active modulators. This will be discussed in Sec. IV, when the measurement basis of the generic qubit , encoded in one particle, depends only on the outputs of some other qubits encoded in the same particle.
One-way computation can be understood in terms of some basic operations, the so-called cluster building blocks (CBB) (see Fig. 2). By combining different CBB’s it becomes possible to perform computation of arbitrary complexity Tame et al. (2005). We introduce here a convenient notation: by writing explicitely a state close to a dot (see Fig. 2), we indicate that the total state could be equivalently obtained by preparing the qubit in the state before applying the necessary gates.
CBB: Qubit rotation. Consider two qubits linked together like CBB shown in Fig. 2. Here the first qubit is initially prepared in the state and the second qubit is arbitrary linked with other dots. By measuring qubit 1 in the basis we remove it from the cluster but we transfer the information into qubit leaving its links unaltered. This corresponds to the following operation on the encoded qubit
where is the Hadamard gate and is a rotation around the axis in the Bloch sphere. The operations depends on the measurement output () of the first qubit. This operation can be understood by noting that by measuring the first qubit of the state in the basis we will obtain .
This simple algorithm can be repeated by using two CBB in a row. By measuring qubit 1 in the bases the encoded qubit then is transformed into and the encoded qubit moves from left to right into the cluster. By measuring qubit 2 in the basis the encoded qubit is now written in qubit 3 as :
In this case the computation can be understood by observing that with the measurement of qubits and of the state in the basis and , we obtain .
CBB: c-phase gate. Consider two qubits linked in a column. This block simply performs a c-phase gate () between the two qubits.
CBB: c-phase gate+rotation. If we have 3 qubits in a column and we measure the second qubit in the basis we remove it from the cluster but the information is transferred to the qubit 1 and 3. Precisely, on the logical qubit, this measurement realizes a c-phase gate followed by two single qubit rotations (see Fig. 2).
By combining these CBB’s we can obtain any desired quantum algorithm, written in general as
The numbers depend on the outcomes of all the single (physical) qubit measurements and determine the FF corrections (type ii)) that must be realized, at the end of the measurement pattern, in order to achieve deterministic computation. We indicate by the symbol that the Pauli matrix acts on the logical qubit . Note that if the output of the algorithm is one among the states of the computational basis, i.e. (), only the ’s of the unitary act non-trivially by flipping some qubits. The Pauli errors are then reduced to
In this way the “errors” can be simply corrected by relabeling the output, and there is no need of active feed-forward corrections on the quantum state. If, by measuring the output qubits, we get the result ( or ) we must interpret it as with the Pauli errors given by the equation (12). This relabeling operation can be performed for example by an external classical computer.
Iii Experimental setup
In our experiment we generated two-photon four-qubit cluster states by starting from polarization()-momentum() hyperentangled photon pairs obtained by SPDC (see Fig. 3). The hyperentangled states were generated by a -barium-borate (BBO) type I crystal pumped in both sides by a cw Argon laser beam () (cfr. 3). The detailed explanation of hyperentangled state preparation was given in previous papers Barbieri et al. (2005); Cinelli et al. (2005), to which we refer for details. In the above expression of we used the Bell states
where correspond to the horizontal () and vertical () polarizations and refer to the left () or right () paths of the photon (Alice) or (Bob) (see Fig. 3). In , the first signs refer to the polarization state and the second ones to the linear momentum state .
Starting from the state and applying a c-phase (CZ) gate between the control () and the target () qubits of the photon A, we generated the cluster state
In the experiment, the CZ gate is realized by a zero-order half wave (HW) plate inserted on the mode, and corresponds to introduce a phase shift for the vertical polarization of the output mode. It is worth noting that, at variance with the four-photon cluster state generation, the state (14) is created without any kind of post-selection111Note that the usual post-selection needed to select out the vacuum state is in any case necessary. This is unavoidable since SPDC is a not deterministic process.. By using the correspondence , , , , the generated state is equivalent to , , or up to single qubit unitaries:
With and we will refer to the cluster state expressed in the “cluster” and “laboratory” basis respectively. The explicit expression of the unitaries depends on the specific ordering of the physical qubits (, , and ) and will be indicated in each case in the following. The basis changing will be necessary in order to know which are the correct measurements we need to perform in the actual experiment. In general if on qubit the chosen algorithm requires a measurement in the basis , the actual measurement basis in the laboratory is given by .
where the are the so called stabilizer operators , (see table 1). The fidelity of the experimental cluster can be measured by
i.e. by measuring the expectation values of the stabilizer operators. In table 1 we report the stabilizer operators for and the corresponding experimental expectation values. The obtained fidelity is , demonstrating the high purity level of the generated state. Cluster states were observed at 1 kHz detection rate. The two photons are detected by using interference filter with bandwidth . In table 1 we use the following notation for polarization
and linear momentum operators
for either Alice (A) or Bob (B) photons.
Iv Basic operations with 2-photon cluster state
In this section we describe the implementation of simple operations performed with the generated four-qubit two-photon cluster state.
iv.1 Single qubit rotations
In the one-way model a three-qubit linear cluster state (simply obtained by the four-qubit cluster by measuring the first qubit) is sufficient to realize an arbitrary single qubit transformation222This is a generic rotation iff the input state is not or . In our case the algorithm is implemented with . In fact three sequential rotations are in general necessary to implement a generic matrix but only two, namely , are sufficient to rotate the input state into a generic state , where and .
The algorithm consists of two CBB’s on a row. Precisely, by using the four-qubit cluster expressed in the cluster basis the following measurement pattern must be followed (see Fig. 4):
A three-qubit linear cluster is generated by measuring the first qubit in the computational basis . As said, in Section II this operation removes the first qubit from the cluster and generates . The input logical qubit is then encoded in qubit 2. If the outcome of the first measurement is then , otherwise .
Measuring qubit 2 in the basis , the logical qubit (now encoded in qubit 3) is transformed into , with .
Measurement of qubit 3 is performed in the basis:
This represents a FF measurement (type i)) since the choice of measurement basis depends on the previous outputs. This operation leaves the last qubit in the state with .
Then, by using some simple Pauli matrix algebra, the output state (encoded in qubit ) can be written as
In this way, by suitable choosing and , we can perform any arbitrary single qubit rotation up to Pauli errors (), that should be corrected by proper feed-forward operations (type ii)) to achieve a deterministic computationPrevedel et al. (2007).
In our case we applied this measurement pattern by considering different ordering of the physical qubits. In this way we encoded the output qubit either in the polarization or in the linear momentum of photon , demonstrating the QC equivalence of the two degrees of freedom. The measurement apparatus is sketched in Fig. 5. The modes corresponding to photons A or B, are respectively matched on the up and down side of a common symmetric beam splitter (BS) (see inset), which can be also finely adjusted in the vertical direction such that one or both photons don’t pass through it. Polarization analysis is performed by a standard tomography apparatus (, and polarizing beam splitter PBS). Depending of the specific measurement the HWs oriented at 22.5 are inserted to perform the Hadamard operation H in the apparatus . They are used together with the in order to transform the states into linearly polarized states. Two thin glass plates before the BS allow to set the basis of the momentum measurement for each photon.
Let’s consider the following ordering of the physical qubits (see Eq. (15)):
The output state, encoded in the polarization of photon B, can be written in the laboratory basis as
where the gate derives from the change between the cluster and laboratory basis. This also implies that the actual measurement bases are for the momentum of photon B (qubit 1) and (i.e. ) for the momentum of photon A (qubit 2). According to the one-way model, the measurement basis on the third qubit () depends on the results of the measurement on the second qubit (). These are precisely what we call FF measurements (type i)). In our scheme this simply corresponds to measure in the bases or , depending on the BS output mode (i.e. or ). These deterministic FF measurements are a direct consequence of the possibility to encode two qubits ( and ) in the same photon. As a consequence, at variance with the case of four-photon cluster states, in this case active feed-forward measurements (that can be realized by adopting Pockels cells) are not required, while Pauli errors corrections are in any case necessary for deterministic QC.
We first realized the experiment without FF corrections (in this case we didn’t use the retardation fiber and the Pockels cells in the setup). The results obtained for (i.e. when the computation proceeds without errors) with are shown in fig. 6(a). We show on the Bloch sphere the experimental output qubits and their projections on the theoretical state for the whole set of and . The corresponding fidelities are given in table 2. We also performed the tomographic analysis (shown in fig 6b),c),d)) on the output qubit for all the possible combinations of and and for the input qubit . The high fidelities obtained in these measurements indicate that deterministic QC can be efficiently implemented in this configuration by Pauli errir active FF corrections.
They were realized by using the entire measurement apparatus of fig. 5. Here two fast driven transverse Pockels cells ( and ) with risetime and are activated by the output signals of detectors () corresponding to the different values of and . They perform the operation on photon B, coming from the output of and transmitted through a single mode optical fiber. Note that no correction is needed when photon A is detected on the output (). Temporal synchronization between the activation of the high voltage signal and the transmission of photon B through the Pockels cells is guaranteed by suitable choice of the delays . We used only one output of photon , namely , in order to perform the algorithm with initial state . The other output corresponds to the algorithm starting with the initial state . By referring to Fig. 5, each detector corresponds to a different value of and . Precisely, corresponds to and and activates the Pockels cell (see eq. 22). Detector corresponds to , i.e. the computation without errors and thus no Pockels cell is activated. Detector corresponds to , and activates , while corresponds to and both and are activated.
In fig. 8 the output state fidelities obtained with/without active FF corrections (i.e. turning on/off the Pockels cells) are compared for different values of and . The expected theoretical fidelities in the no-FF case are also shown. In all the cases the computational errors are corrected by the FF action, with average measured fidelity . The overall repetition rate is about , which is more than 2 orders of magnitude larger than one-way single qubit-rotation realized with 4-photon cluster states.
We also demonstrated the computational equivalence of the two DOF’s of photon by performing the same algorithm with the following qubit ordering:
In this case we used the momentum of photon () as output state. The explicit expression of the output state in the laboratory basis is now . By using only detectors , , , in fig. 5 we measured for different values of (which correspond in the laboratory to the polarization measurement bases ) and (which correspond in the laboratory to the momentum bases ). The first qubit () was always measured in the basis . The tomographic analysis for all the possible values of and are shown in Fig. 7. i.e. for different values of ). We obtained an average value of fidelity (see table 3). In this case the realization of deterministic QC by FF corrections could be realized by the adoption of active phase modulators. The ()-() computational equivalence and the use of active feed-forward show that the multidegree of freedom approach is feasible for deterministic one-way QC.
iv.2 c-not gate
The four qubit cluster allows the implementation of nontrivial two-qubit operations, such as the c-not gate. Precisely, a c-not gate acting on a generic target qubit belonging to the equatorial plane of the Bloch sphere (i.e. a generic state of the form ) can be realized by the four-qubit horseshoe (180 rotated) cluster state (see eq. (4)) Let’s consider Fig. 9(top). The measurement of qubits 1 and 4 realizes a c-not gate (the logical circuit shown in figure) between the control and target qubit. By measuring the qubit 1 in the basis or we realized on the control input qubit the gate or respectively. The measurement of qubit 4 in the basis realizes the gate on the target input qubit . The algorithm is concluded by the vertical link that perform a operation. The input state is transformed, in case of no “errors” (i.e. ), into . This circuit realizes the c-not gate (up to the Hadamard ) for arbitrary equatorial target qubit (since ) and control qubit or depending on the measurement basis of qubit 1.
The experimental realization of this gate is performed by adopting the following ordering between the physical qubits:
In this case the control output qubit is encoded in the momentum , while the target output is encoded in the polarization . In order to compensate the gate arising from the cluster algorithm we inserted two Hadamard in the polarization analysis of the detectors and (see fig. 5). The output state in the laboratory basis is then written as
where all the possible measurement outcomes of qubits 1 and 4 are considered. The Pauli errors are , while the matrix is due to the changing between cluster and laboratory basis.
By measuring in the basis we perform the operation on the control qubit. By looking at eq. (25), this means that if () the control qubit is (), while the target qubit is (). In this case the gate acts on a control qubit or , without any superposition of these two states. We first verified that the gate acts correctly in this situation. In Table 4(top) we report the experimental fidelities () of the output target qubit for two different values of and for the two possible values of . The high values of show that the gate works correctly when the control qubit is or .
The second step was to verify that the gate works correctly with the control qubit in a superposition of and . This was realized by measuring the qubit in the basis and performing the operation on the control qubit. The output state is written (without errors) as . In Table 4(bottom) we show the values of the experimental fidelities of the target qubit , corresponding to the measurement of the output control qubit in the basis when . The results demonstrate the high quality of the operation also in this case.
iv.3 c-phase gate
The four qubit cluster allows also the realization of a c-phase gate for arbitrary target qubit and fixed control . The measurement pattern needed for this gate is shown in Fig. 9(bottom) and consists of the measurements of qubits 1 and 2 in the bases and . These two measurements realize a generic rotation on the input target qubit , as explained in subsection IV.1. The link existing between qubit and in the cluster performs the subsequent c-phase gate between the control qubit and a generic target qubit ,
The experimental realization is done by considering the following ordering between the physical qubits:
We realized a c-phase gate for arbitrary target qubit and fixed control (see Fig. 9(c)) by measuring qubits 1 and 2 of in the bases and respectively. By considering ordering d) we encoded the output state in the physical qubits and . For , by using the appropriate base changing, the output state is written as
Here and the matrix is due to the basis changing. The measured fidelity of the target corresponding to a control () for different values of and are shown in table 5. We obtaining an average value ().
v.1 Grover algorithm
Let’s describe the algorithm in general. Suppose to have elements (encoded in qubits) and a black box (or oracle) that tags one of them. The tagging, denoted as , is realized by changing the sign of the desired element. The goal is to identify the tagged item by repeated query to the black box; the Grover’s algorithm requires operations, while the best classical algorithm takes on average calculations.
The general algorithm starts with the input state prepared as and consists of repeated applications of the Grover operator , given by the oracle tagging followed by the so called inversion about average operation . We can thus write . In general after iterations of the tagged item is obtained at the output of the circuit with high probability.
In the case of qubits the quantum algorithm (shown in Fig. 10(right)) requires just one operation. The four elements are , , and . They are prepared in a complete superposition, i.e. in the state