Experimental One-Way Quantum Computing

Experimental One-Way Quantum Computing

highly-entangled cluster state. From this point, the quantum computation proceeds by a sequence of single-qubit measurements with classical feedforward of their outcomes. Because of the essential role of measurement, a one-way quantum computer is irreversible. In the one-way quantum computer, the order and choices of measurements determine the algorithm computed. We have experimentally realized four-qubit cluster states encoded into the polarization state of four photons. We fully characterize the quantum state by implementing the first experimental four-qubit quantum state tomography. Using this cluster state, we demonstrate the feasibility of one-way quantum computing through a universal set of one- and two-qubit operations. Finally, our implementation of Grover's search algorithm demonstrates that one-way quantum computation is ideally suited for such tasks.

The quantum computer is a powerful application of the laws of quantum physics. Such a device will be far more efficient at factoring or database searches compared to its classical counterparts. Considerable effort has been directed toward understanding the role of measurement and entanglement in quantum computation. A significant step forward in our understanding was the introduction of the "one-way" quantum computer which highlights the importance of both measurement and entanglement in a striking way. In this model, all of the entanglement is provided in advance through a highly-entangled multi-particle cluster state. The quantum computation on the cluster state proceeds via local, single-qubit projective measurements with the outcomes potentially affecting those measurement settings that follow. It is a strength of the cluster state model that the intrinsic randomness of quantum measurement results creates specific types of errors which can be corrected through this classical feedforward. Most importantly, feedforward makes cluster state quantum computation deterministic. In the present, proof-of-principle experiment we perform measurements using fixed single-port polarizers making our computations probabilistic. Different algorithms require only a different pattern of adapted single-qubit operations on a sufficiently large cluster state. Since it is entirely based on single-particle measurements instead of unitary evolution, the computation is inherently not time-reversible - it is one-way. Most importantly, cluster state quantum computation is universal in that any quantum circuit can be implemented on a suitable cluster state. Open theoretical questions remain about the scalability under realistic noise conditions required for fault-tolerant one-way quantum computation. Although a threshold has been proven to exist, it is unknown whether cluster state quantum computation will be more or less sensitive to noise than the standard model.

The one-way quantum computer does not perform quantum logic on the individual qubits of the cluster state. In order to describe the computational circuit, we need to distinguish between the physical qubits, in our case the polarization state of photons, which make up the cluster state and on which actual measurements are carried out, and encoded qubits, on which the computation is actually taking place. Due to the specific entanglement of the cluster state, no individual physical qubit carries any information about an input state. Therefore, each encoded qubit is written on the cluster state non-locally, i.e. the information is carried by the correlations between the physical qubits. As the quantum computation proceeds, the encoded input qubits are processed in the imprinted circuit, whose output is finally transferred onto physical readout qubits. Interestingly, while the entanglement between the physical qubits in general decreases as a result of the measurement sequence, the entanglement between encoded qubits may increase.

A cluster state can be thought of as emerging from an array of equally prepared independent qubits, which then interact via controlled-phase gates with their nearest neighbors. Specifically, a cluster state can be built up as follows: A large number of physical qubits are each prepared in the superposition state

|+>=(|0)+|1))//2, where |0 and |1 are the computational basis of the physical qubits. A controlled-phase operation is then applied between pairs of neighboring, connected qubits and effectively generates entanglement between them. The choice which physical qubit neighbors are entangled by the controlled-phase operations, drawn as connecting "bonds," determines the structure of the cluster state which defines the basic type quantum circuit it can implement. This construction provides an intuitive understanding for the graphical representation of cluster states as connected arrays of physical qubits, in which each line corresponds to a previous nearest-neighbour interaction. We will later demonstrate how the highly entangled cluster states can be generated in a different way, directly from nonlinear optical processes.

Given a cluster state, two basic types of single-particle measurements suffice to operate the one-way quantum computer. Measurements in the computational basis have the effect of disentangling, i.e., removing, the physical qubit j from the cluster leaving a smaller cluster state. Such operations can be used to modify the structure of the cluster and thus the imprinted circuit. The measurements which perform the actual quantum information processing are made in the basis where a is a real number. The choice of measurement basis determines the single-qubit rotation, followed by a Hadamard operation, on encoded qubits in the cluster. Combinations of rotations about the z-axis and Hadamard operations can implement rotations through the matrix identity. Any quantum logic operation can be carried out by the correct choice of on a sufficiently large cluster state. We define the outcome s, of a measurement on the physical qubit j zero if the measurement outcome is and one if the outcome is. In those cases where the zero outcome is found, the computation proceeds as desired.

However, in those cases where the one outcome is found, a well-defined Pauli error is introduced. Feedforward, such that the output controls future measurement, compensates for these known errors.

For the implementations of single- and two-qubit quantum logic, we post-select only those cases where the zero outcome is found and the computation proceeds error free. In the final section, where we report the implementation of Grover's search algorithm, the feedforward determines the final, classical, measurement. There we measured all possible combinations of the measurement results individually and applied the feedforward relation in such a way that the earlier measurements define the physical meaning of the final ones.

Even a small cluster state suffices to demonstrate all the essential features of one-way quantum computing. Each of the three- and four-particle cluster states shown in Figure one can implement the quantum circuit shown to its right that consist of a series of single- and two-qubit quantum gates. The computation proceeds via single-particle measurements carried out from the left side of the cluster to the right side, where the final readout takes place. The important feature of the quantum circuits is that the output of one circuit can be fed into the input of a subsequent one if their cluster states are bonded together by controlled-phase operations. Thus these small circuits, which form a universal set of logic gates, can be used as subunits for a fully-functional quantum computer.

Four. As an example, consider the four-particle box cluster state on a two- to four-dimensional lattice. The encoded input to the two-qubit quantum circuit is the product state |Yin)=|+)|+)|sub E, where the numerical subscript labels the qubit and the subscript E is used to distinguish encoded qubits from physical qubits. The circuit processes this pair of encoded qubits through a sequence beginning with a CPhase gate,

followed by single-qubit rotations R two negative a and R sub negative on encoded qubits one and two, then a Hadamard operation, H, on both qubits, ending with a second CPhase operation. The values for a and B of the rotation gates are set by the choice of measurement bases B(a) and B four(B) on the physical qubits one and four respectively. The output of the quantum computation is transferred onto physical qubits two and three. This kind of two-qubit quantum gate is essential for universal quantum computation since it can generate entanglement between the encoded qubits.

On the other hand, by changing the geometry of the cluster state to the one- dimensional cluster lin four one, it now results in a different quantum circuit corresponding to a set of single-qubit rotations on one encoded qubit. Consecutive measurements B one(a), B two(B), and B three(y) on the physical qubits one, two, and three transform the input state, in our case |Yin=+ one, to |Yout)= H R sub negative Y R sub negative B R sub negative a |Yin) and store the output on qubit four.

In order to demonstrate all the circuits shown in Figure one a through one e, it is sufficient to first produce a linear cluster state of four qubits. The particular circuit implemented is then determined by the order of the measurements performed. Specifically, the one- dimensional linear structure is implemented by sequentially measuring qubits one, two and three, with the final result then being available at qubit four. The two-dimensional horseshoe structures are implemented by measuring either qubits two and three or one and four with the final result then being available at qubits one and four or two and three, respectively. The four qubit box cluster can be obtained from the four-qubit linear cluster by Hadamard rotations and by swapping (i.e., relabeling) the physical qubits two and three. All this will be described in more detail in the next section and in the section "two-qubit gates".

The difficulty of one-way quantum computer lies with the cluster-state preparation. Cluster states naturally arise in spin chains or spin lattices via nearest- neighbor Ising interaction, a well-known interaction model in solid-state physics. Therefore, the first proposals to achieve cluster states were based on analogues in dipole-dipole coupling between atoms in optical lattices. Although photon-photon interactions are negligible, recent proposals have nevertheless shown that optical systems may be well-suited for implementing the cluster state model. These schemes utilize sequences of probabilistic quantum-logic gates based on linear optical elements to construct large photonic cluster states. These optical one-way quantum computation proposals are less demanding on resources than the comparable optical implementation in the standard model.

In the present work, we have employed nonlinear optics to directly produce four- photon cluster states. This method exploits a mode- and polarization-entangled four- photon state produced in pulsed-pump spontaneous parametric down conversion (SPDC). We reconstructed the density matrix of the four-qubit cluster state using quantum state tomography and studied the state's entanglement properties relevant for quantum computation. We then implement all of the quantum circuits shown in Figure one a through one e and demonstrate a two-qubit quantum search algorithm. In doing so, we have demonstrated the first universal set of gates and an important algorithm in a one-way quantum computer.

Creation and Characterization of the Cluster State

Our cluster state is produced experimentally using the mode- and polarization-entangled output of nonlinear spontaneous parametric down-conversion and linear optical elements as described in detail in the Methods section. When four photons are emitted into the output modes of the polarizing beam-splitters one, two, three, and four, they are in the highly-entangled cluster state,

One where | H and | V represent horizontally- and vertically-polarized photon states and the subscript labels the spatial mode. This state, | Phi sub cluster is equivalent to the four-qubit linear cluster, | Phi sub lin four, and the horse-shoe cluster states, | Phi sub subset four and | Phi sub superset four under the local unitary operation H sub one tensor I sub two tensor I sub three tensor H sub four on the physical qubits, where H; I sub i is a Hadamard (Identity) operation on qubit i. The four-qubit linear cluster can easily be reduced to a three-qubit linear cluster by measuring qubit one in the computational basis of the cluster and thus disentangling it from the rest. The state, | Phi sub d l a f e r, can be converted to the box cluster state by the local unitary operation H sub one tensor H sub two tensor H sub three tensor H sub four and a swap (or relabeling) of qubits two and three. Note that the four-qubit cluster state thus realized is also the smallest cluster state that represents a new kind of entanglement, while the two-qubit and three-qubit cluster states are locally-equivalent to a maximally-entangled Bell state and the three-qubit GHZ state, respectively.