Cluster state generation and computation
4.1 One way quantum computing with cluster states
certain threshold, the errors in quantum circuits can be caught and corrected using quantum error correction techniques.95 This means that, in principle, arbitrarily long quantum computations can be carried out, although practically speaking the physical re- sources required for useful applications are prohibitively large with current technology.96
Therefore, to enable practical quantum computing, develop- ment of more efficient and scalable ways of producing the re- quired physical resources are required. In particular, the gen- eration of cluster states in simple integrated photonic platforms can facilitate scalable cluster state quantum computing. In this chapter we will explain how cluster states can be generated and encoded in the spatial degree of freedom of the wavefunction in a photonic chip. First we will give a brief introduction to clus- ter state quantum computing in Sec. 4.1. We will then show in Sec. 4.2, how special poling patterns in a nonlinear chip can be designed to generate and switch between a range of cluster states useful for simple quantum algorithms. Finally we will dis- cuss the implementation of quantum algorithms based on cluster states encoded in a waveguide array.
4.1 One way quantum computing with cluster states
Cluster states are highly entangled multi-particle quantum states97 that have drawn significant interest for their potential in quantum information processing.94, 98 These multi-qubit states form a complete basis for one-way quantum computation, where algorithms are carried out by successive measurement of qubits, causing information to flow through the state via entangle- ment.99 Crucially for practical applications, cluster states have been shown to be robust to decoherence and loss of qubits.100 In solid state physics cluster states are naturally produced in spin lattices interacting by an Ising type Hamiltonian,97, 101, 102 but increasingly they are considered useful in quantum photonic systems.
A cluster state is formed by starting with a number of qubits in the |+i = (|0i+|1i)/√2 state, then entangling certain qubits
Qubit a) b) c) d) e) CPHASE gate Measure Measure Feed-forward
Information flow 2-qubit gate
Figure 4.1:a) Diagrammatic representation of the basic elements required for a cluster state, qubits (red circles) joined via CPHASE gates (grey bars). b) A linear four-qubit cluster state, c) a box cluster state and d) a star cluster state. e) Implementation of cluster state algorithm using a sequence of single qubit measurements in bases determined by feed-forward of previous measurement results.
§4.1 One way quantum computing with cluster states 63
with controlled-phase (CPHASE) gates. The CPHASE gate flips the sign of the target qubit in the case where both the target and control qubits are in the state |1i. The action of the gate on a quantum state can be summarized by,
CPHASE(|0it|0ic) = |0it|0ic, (4.1)
CPHASE(|0it|1ic) = |0it|1ic, (4.2)
CPHASE(|1it|0ic) = |1it|0ic, (4.3)
CPHASE(|1it|1ic) = − |1it|1ic. (4.4) The application of the CPHASE gate on two qubits has the effect of entangling those two qubits given they are in certain states, an important example of this being the |+i |+istate. This is rep- resented diagrammatically for a range of different cluster states in Fig. 4.1a)-d), where the balls represent qubits in the |+i state, and the lines joining them represent entangling CPHASE gates that have been applied to create various cluster states.
For quantum computation using cluster states the quantum circuit to be implemented is defined by the structure of entangle- ment links between qubits. The computation proceeds by mea- suring successive qubits in a particular order causing informa- tion to flow through the quantum circuit via the entanglement links. Importantly, for complex circuits, the results pf previous qubit measurements must be fed forward to adjust the choice of measurement basis for subsequent measurements. This feed- forward is the step key that distinguishes cluster state compu- tation from non-universal quantum computation methods such as boson sampling.103 Fig. 4.1b)-e) shows different organizations of the entanglement links for the case of 4-qubit cluster states, which all implement different simple quantum circuits.
Ultimately for non-trivial cluster state quantum computing large complex circuits containing many qubits must be assem- bled, as suggested in Fig. 4.1e). Thus cluster state computing will need reliable multi-photon quantum sources. However, it
is worth noting that in some ways full cluster state computation is wasteful for photonic qubits. This is at least partially because the process of sending information from place to place is easily achieved with photons, with no need to use the ‘quantum-wires’ formed by 1-D cluster states such as Fig. 4.1a). The key bene- fit cluster state brings is in the multi-qubit gates formed when multiple quantum-wires in a cluster state intersect such as in Fig. 4.1c)-e). Thus a practical photonic quantum computer may use some aspects of cluster state computation, but will no doubt combine this with other approaches too.45
4.1.1 Realizations of photonic cluster states
Cluster states based on photonic polarization qubits have been generated in optical systems utilizing nonlinear optics104–109 or periodically driven quantum dots110 to achieve the required en- tanglement between multiple photons. Furthermore basic ele- ments of quantum computation have been demonstrated with these polarization qubit states, including qubit rotation, two- qubit gates and the Grover search algorithm.104 Photonic clus- ter states can also be created using continuous variable quantum entanglement,111–113 where the qubits are encoded in the time de- pendent quadrature of the field. In this chapter we consider the generation of cluster states using a fully spatial encoding of each qubit, which is well suited for on-chip implementation.
Typically photonic cluster states are generated in bulk optical setups by passing a pulsed pump laser twice through a nonlinear crystal, generating a pair of photons by spontaneous parametric down-conversion (SPDC) on each pass giving a four photon po- larization entangled state.104 Alternatively just two photons can be used, since by exploiting hyper-entanglement in spatial and polarization degrees of freedom four quibts can be encoded in the two photons.105, 108 This approach has the advantage of pro- ducing higher photon count rates, while still producing nontriv- ial four qubit cluster states. Ultimately, for quantum computing it is necessary to produce cluster states containing large numbers of qubits. The reason is that cluster state computation relies on successive single photon measurements, so the number of inde-
§4.1 One way quantum computing with cluster states 65
pendent computational steps is limited by the number of qubits in the state. Encoding multiple qubits in a single photon has the potential to significantly increase the size of cluster states that can be realized, and such states can be used for quantum algorithms.105 However, it should be emphasized that encoding multiple qubits into a single photon means that all those qubits must be measured simultaneously in the cluster state algorithm. This is not necessarily possible for every algorithm. Therefore, the choice of which qubits (and how many qubits) to encode into a single photon needs to be carefully considered, depending on which qubits in the specific algorithm can be measured simulta- neously.
So far the realization of cluster state based quantum algo- rithms has been largely restricted to bulk optical setups, but in- evitably cluster state generation will need to be integrated on- chip. This is because the generation of large scale cluster states will require the isolated and stable environment provided by a photonic chip to maintain quantum coherence. Furthermore, in- creasingly complex optical circuitry will be required to produce larger cluster states, which is impractical to build with bulk op- tics.
Cluster states based on hyper-entanglement between polariza- tion and spatial degrees of freedom have been demonstrated on- chip,109 using a linear photonic circuit to shape the two-photon state produced from , but a more natural and convenient realiza- tion would be based on just the spatial degree of freedom, since processing orthogonal polarizations in the same waveguide re- quires highly specialized fabrication platforms. To this end we describe a method for the generation of cluster states within a nonlinear photonic chip with fixed photon polarization, where the state is fully encoded in the spatial properties of the pho- tons. Importantly this method allows switching between differ- ent cluster states all optically, without a need for complex recon- figurable components to be integrated on-chip.
4.2 Generating spatially encoded cluster states in a nonlinear