Weak nonlinearities: a new route to optical quantum computation

Quantum information processing (QIP) offers the promise of being able to do things that we cannot do with conventional technology. Here we present a new route for distributed optical QIP, based on generalized quantum non-demolition measurements, providing a unified approach for quantum communication and computing. Interactions between photons are generated using weak nonlinearities and intense laser fields—the use of such fields provides for robust distribution of quantum information. Our approach only requires a practical set of resources, and it uses these very efficiently. Thus it promises to be extremely useful for the first quantum technologies, based on scarce resources. Furthermore, in the longer term this approach provides both options and scalability for efficient many-qubit QIP.


Introduction
It is well known that using quantum information can enable certain communication and computation tasks that cannot be performed with conventional IT, or improve some that can [1]. Quantum computers (or smaller processors) for instance can search faster and simulate better than their classical counterparts, and factor large numbers efficiently [2,3]. Quantum cryptography enables secure communication, based on the laws of physics [4]. There is now considerable research activity devoted to finding the best routes to realizing new quantum information technologies. Here we present a new paradigm for all-optical quantum information processing (QIP), bringing together communication and computation in a single approach. This has many appealing features and significant promise as a technology route: • The approach is all-optical, so qubit interconversion is not a prerequisite for combining communication and processing. • Quantum information is distributed robustly, using intense laser pulses. • Our approach only requires a practical set of physical resources. In particular, we do not assume the existence of single photon sources and detectors, but describe how these can be constructed from the underlying resources. • Moving on from quantum cryptography, the next technologies-based on limited resourcesare likely to involve few-qubit processing and some distribution of quantum information. Our approach thus provides for this very efficiently. • Our approach provides adaptable building blocks, so the computational approach is not fixed.
For example, gate-or measurement-or cluster-state-based QIP can be performed, depending upon the scale and application. • A longer term aim is for technologies based on many-qubit and distributed QIP. Our approach gives an efficient and scalable route for this.
Optical QIP is already a very active research area. Knill et al [5] provided an important theoretical breakthrough, showing that in principle universal quantum computation is possible with linear optics, quantum gates effectively being introduced through photon bunching effects 3 Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT and measurement. Furthermore, there have been a number of recent impressive experimental demonstrations of the building blocks for such optical QIP [6]- [8]. However, despite all this, gates in linear optical QIP are intrinsically probabilistic, because they are based on photon bunching and measurement. This means that the scheme is practically rather inefficient (in terms of photon resources) to implement. Even with the application of cluster-state methods, on average over 100 photons are needed for a single two-qubit gate [9,10]. In addition, all these photons have to be identical enough for bunching effects to occur, practically a very onerous requirement. So this road to actual devices and technology looks tough.
An alternative method for realizing a gate between two photons is to get them to interact in a nonlinear medium. An example is a cross-Kerr nonlinearity, with an interaction Hamiltonian H ck =hχn anp (wheren a =â † aâ a andn p =â † pâ p are the photon number operators for the two interacting modes and χ is the interaction strength). In order to produce sufficient interaction for useful quantum gates directly between photons, strong nonlinearities are needed, with θ ≡ χt ∼ π, t being the interaction time. Unfortunately, in practice such nonlinearities are not available. In effect, an approach is needed that can 'amplify' the effect of the rather weaker nonlinearities that are available (with θ 1), to enable QIP [11]- [15]. A key feature of our work is to do just this, by using intense coherent states of light as a 'bus' to mediate interactions between photon qubits. The strength of the coherent states can offset the weakness of the nonlinearities. Two other important advantages arise from this approach. Firstly, photonic qubits never talk to each other directly and so we do not rely on two-photon effects between identical photons. Our photon qubits do not therefore have to be perfectly indistinguishable, as they do in linear optical schemes. Secondly, the coherent states used as bus modes also provide for the natural communication and distribution of quantum information, enabling gates between photonic qubits that never meet, and that are separated by distances up to the scale over which quantum communications work. As some quantum communication tasks are more simply and robustly implemented with coherent states (continuous variables (CV), as opposed to qubits), it is natural to combine both qubit and CV resources in a truly distributed computing scheme.
Our distributed optical QIP scheme requires various fundamental physical resources. Some of these are linear in nature, but for resource efficiency nonlinear elements are also necessaryfor these we will specify their nature and quantify the strength required. In our proposal the fundamental physical resources required are the following: 1. Sources of large coherent states (which can clearly be filtered to make weak ones). 2. Highly efficient homodyne/heterodyne detection. 3. Standard linear optical elements, such as beam splitters, phase shifters and the ability to perform fast classical feed-forward. 4. Weak cross-Kerr nonlinearities.
With a view towards actual technology, this list is deliberately practical, so it only includes coherent states of light-a standard optical resource-of the form |α = e −|α| 2 /2 n α n / √ n!|n and weak nonlinearities (with θ 1). Quantum buffers or storage for the qubits are not absolutely necessary; however, adding them to the list renders actual implementations much simpler and even more efficient.
These core physical resources are all the elements needed to perform any distributed single photon computation and communication task. All the other necessary elements (photon sources, The presence of photons in mode a causes a phase shift on the coherent state |α p directly proportional to n a which can be determined with a quadrature measurement. The single photon source (b) uses the QND detector from (a) with a weak coherent state (mean photon number |α a | 2 ∼ 1) input to the signal mode. The homodyne measurement of the probe then allows the signal mode to be projected into a definite number state. detectors and gates) needed to enable the information processing can be created from these core resources. Very importantly, no further hidden source of nonlinearities are needed in any of the elements, so it is straightforward to cost the actual nonlinear resources needed for any given QIP task.

Single photon sources and detectors
Our scheme embodies the actual quantum information in single photon qubits, encoding into polarization of the photon (|H , |V ), or which of two paths it takes. (These encodings are equivalent, and physical transformation between the two is achieved through a linear polarizing beam splitter.) So single photons need to be created from the core physical resources detailed above. We begin with a brief discussion of how these core resources can be used to create a high efficiency quantum non-demolition single photon detector [11,17,18], which can be used to condition incoming photon states and thus also serve as a single photon source. In the quantum non-demolition detector (depicted schematically in figure 1(a)) there are two optical modes, a signal mode a and a probe mode p. In our scheme the signal mode is some superposition of Fock states and the probe beam is an intense coherent state |α p . The signal and probe modes interact via a weak cross-Kerr nonlinearity that generates a unitary evolution U ck = exp [iθn anp ], where θ is the total strength of the nonlinearity. If the signal mode is initially prepared in the state |ψ = c 0 |0 a + c 1 |1 a + c 2 |2 a , the cross-Kerr interaction causes the combined signal and probe system to evolve to Now a highly efficient homodyne/heterodyne measurement of the probe field will effectively project this onto some state of a chosen field quadrature x(ξ) = a p e iξ + a † p e −iξ , and the signal mode into a corresponding definite number state |n a [11,16]. There is of course an error in the discrimination of the |n a states due to the fact that the measured probe quadrature probability distributions for the different n have overlapping tails-the phase-shifted coherent states of the probe beam are not completely orthogonal to |α p . In terms of the measured quadrature value, the natural discrimination boundary for adjacent n values is the mid-point between the probability peaks. For the case of just two peaks (c 2 = 0 in state (1)), real α and measurement of x(π/2), the discrimination error is the probability in the tails of the distributions on the wrong side of the mid-point, which is 1 2 . With more peaks the total discrimination error does not exceed P err = erfc[|α| sin θ/ √ 2] which can be made small by ensuring that αθ 1. In fact for αθ ∼ π this discrimination error P err ∼ 10 −3 , which is likely to be smaller than other error and noise processes, so this approach enables accurate non-absorbing photon measurement.
Next we focus on the generation of single photons on demand. The QND detector is the required element for such a source. Consider that the signal mode a is now injected with a weak coherent state |α a (α a real). Such states are straightforward to prepare, and are in our list of core resources. The QND detector with an appropriate homodyne/heterodyne measurement of the probe mode will project the signal mode into a photon number state |n a . The value of n is identified by the probe measurement result. So production of a single photon state |1 a is heralded and occurs with probability P = exp [ − α 2 a ]α 2 a which has a maximum value of 1/e ∼ 0.367 when α a ∼ 1. Several such driven sources can therefore give a very high probability for the heralded generation of a single photon. This photon then simply needs routing to where it is required. Clearly, the addition of a controlled quantum buffer or storage device enables a probabilistic heralded single photon source to be turned into an 'on-demand' single photon source. Essentially, a probabilistic source can be used repeatedly until a single photon is generated; it can then be held in a buffer until it is required.

The distributed parity gate
We have described how to produce both single photon sources and means to detect them from our core physical resources. Now we turn our attention to photon qubit interactions. There are obviously a number of techniques or methods we could use to enable interaction between single photon qubits, such as the approach of Knill et al [5]. However, we propose using weak cross-Kerr nonlinearities and strong coherent states, as this makes efficient use of resources from our core list and also enables a naturally distributed quantum gate. Clearly, a single weak nonlinearity (θ 1) is not sufficient to enable a maximally entangling gate (such as CPhase) directly between two single photon qubits-a great many nonlinearities would be needed with this approach. However, in conjunction with an intense probe beam just two weak nonlinearities are sufficient to implement a two-qubit parity gate [13]. Such a gate is illustrated in figure 2. This projects the two-qubit state into a subspace of even or odd parity.
The parity gate works as follows: consider that the two parties Alice and Bob have the shared two polarization qubit state β 0 |HH ab + β 1 |HV ab + β 2 |VH ab + β 3 |VV ab with each qubit at a different spatial location (and potentially stored or buffered). This state may be separable or entangled depending on whether Alice and Bob have interacted previously. The first action of the parity gate is to entangle the probe beam |α p with Alice's polarization qubit. The |H a component of Alice's qubit causes a θ phase on the probe beam while the |V a component leaves the probe beam unchanged. After this interaction, the probe beam is transmitted to Bob, who also uses a weak cross-Kerr nonlinearity θ to interact his |V b component with the probe field. This entangles Bob's qubit to the probe field and Alice's qubit. A linear phase shift of −θ is then applied to the probe beam. The resulting three-party state (Alice, Bob and probe) is We notice immediately that depending upon the type of measurement on the probe beam p, the state of Alice's and Bob's system can be conditioned into a number of distinct pieces. A parity measurement could be effected by distinguishing the probe state |α p from |α e iθ p and |α e −iθ p , but not (even in principle) distinguishing |α e iθ p from |α e −iθ p . There are various measurement strategies for the probe beam that can realize this state conditioning. The simplest is a high efficiency x(0) quadrature homodyne measurement [13,14]. Whilst this is in principle a straightforward measurement to implement, it is by no means optimal. A near-optimal measurement is preferable as this enables the strength of the cross-Kerr nonlinearities used to be near-minimal. To this end, we propose performing a QND photon number measurement on the probe beam. However, during the action of the gate the probe beam has to have a large mean photon number in order to 'amplify'the effect of the weak nonlinearities.

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Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Therefore, before measurement the probe beam must be displaced by an amount D(−α). 4 This results in the three mode state Note that the |HV ab and |VH ab amplitudes have picked up a phase shift due to the displacement. These phase shifts are unwanted but can be simply removed by static phase shifters (no feedforward is required). The overlap between the probe components of the even parity (|0 p ) and odd parity (|α(e ±iθ − 1) p ) amplitudes is very small if αθ ∼ π. In this case the mean photon number of the odd parity components is not large, being approximatelyn p,odd ∼ 10. Hence a measurement of n p with a QND photon number resolving detector cannot distinguish the |α(e ±iθ − 1) p components from each other, but can distinguish these from the |0 p components. This results in Alice's and Bob's combined state being conditioned to where φ(n p ) = n p tan −1 [cot(θ/2)]. For θ 1, φ(n p ) can be simplified to φ(n p ) = −n p π 2 which is in effect a sign flip for n p odd and no change for n p even. This phase shift φ(n p ) can then simply be eliminated via a classical feed-forward operation (a phase shift dependent on the result of the measurement) as the QND measurement gives n p and θ is known. The classical feed-forward operation is needed because a different operation is needed depending on whether n p is even or odd. In many computational circuits this feed-forward operation (determined by the result of the QND measurement) can be delayed and performed at the final measurement stage for the qubits. We also need to point out that the error in discriminating the two components (even and odd parity states) is approximately P err ≈ 10 −4 for αθ ∼ π. This is a near-optimal measurement.

Decoherence
The distributed parity gate approach is clearly a very appealing method for the remote creation of entangled states which, as we shall explain, can be extended to perform distributed quantum computation. However, it is necessary to examine the effects of noise and decoherence on this distributed approach to judge its real practicality. One of the main sources of decoherence in the transmission of the probe field from Alice to Bob is likely to be amplitude damping or photon loss in the channel. For instance, if the channel is a fibre then photons from the probe beam will be absorbed as it is transmitted between the remote locations. There are a number of ways to treat this loss, with the simplest being to model it via a beam splitter of reflectivity η which discards a portion of the probe beam while it is being transmitted between the remote locations. It is assumed that η does not vary with time, and that it can be measured in advance through suitable test experiments, so its value is known.
Consider now the distributed parity gate, but with such loss on the probe beam. Alice and Bob prepare their qubits as | a = c + |H a + c − |V a and The parity gate is performed as before, but now, crucially, with a slightly reduced displacement 4 A displacement operationD p (−α) = exp(α * â p − αâ † p ) can be achieved by inputing the probe beam to a highly reflective beam splitter, with a large (compared to the probe beam) coherent state on the second input.

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Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT of D(−α 1 − η 2 ) applied to the probe beam, due to the fact that some of this beam has been lost during transmission and a phase correction η 2 |α| 2 sin θ applied to one of the two qubits. The beam loss (the discarded output from the model beam splitter) has to be traced over, which leaves Alice and Bob with a mixed state. We focus on an example case, the regime with (1 − η 2 )|α| 2 [1 − cos θ] ∼ 10, which can certainly be attained with physically reasonable parameters. Then the resulting mixed state is where There are now two interesting points; firstly, the effect of loss in the probe beam leads to mixing of the two qubits (rather than the loss of the qubit if it has been transmitted), and secondly, the probe beam even with finite loss can project the qubits into parity states. We obviously want to operate in the regime of small γ as we then effectively have the pure state | + ab , | + ab . However, with moderate loss, our two qubits can be heavily mixed but still retain some degree of entanglement which can be purified/distilled using standard techniques on the photonic qubits (the effect of loss on the probe causes a dephasing error in the photonic qubits). Next, to be able to perform the parity measurement, we require (1 − η 2 )|α| 2 [1 − cos θ] ∼ 10 to have a low failure rate. If this cannot be achieved then the parity measurement gives one of three possible results: even parity, odd parity or indeterminate. In the indeterminate case the gate fails, but the information encoded in the photonic qubits is not necessarily lost and could be recovered. Alternatively the gate could be attempted again with fresh/re-prepared qubits.
There will obviously be other forms of error that one needs to deal with (for instance loss of photons in the qubits), but these can be dealt with using the standard techniques available for linear optical quantum computation.

Techniques of computation
We have indicated how to produce the components needed for optical QIP-sources of single photons, detectors of single photons and two-photon parity measurements-from our list of fundamental resources. We now consider the potential models for distributed optical processing. There are a number of computational models available; here we consider the standard gate-based schemes and computation by measurement. This illustrates the flexibility of our approach, as these two schemes can be realized from the same basic ingredients.

Gate-based computation
The standard gate-based model of universal computation requires sources of single photons, single qubit rotations, an entangling operation and projective measurements on these photons. For polarization or which path encoded qubits, the single qubit operations are essentially trivial and can be implemented using linear elements; beam splitters, phase shifter, etc. Historically, the tricky operation has been the entangling (two-qubit) gate. For distributed processing we propose using coherent state bus modes, which can suffer some loss, and thus avoid sending single photons between remote locations. We propose an efficient distributed gate based on an application of the linear optical CNOT ideas of Franson and co-workers [19]. Our new gate is depicted in figure 3 and comprises several local parity gates plus a shared maximally entangled Bell state. This distributed maximally entangled Bell state can be created by an application of the parity gate to photonic qubits initially prepared as |H + |V and, crucially, just involves the transmission of a coherent state between the two parties. Consider the control and target qubits initially in the state β 0 |HH ct + β 1 |H V ct + β 2 |VH ct + β 3 |V V ct . With a maximally entangled ancilla Bell state |HH ab + |V V ab the action of the control location (left) parity gate and subsequent ancilla qubit measurement projects and conditions the remaining three photons into β 0 |HH ct |H a + β 1 |H V ct |H a + β 2 |V H ct |V a + β 3 |V V ct |V a . Here a bit flip has been applied on the remaining ancilla qubit for an odd parity gate measurement result and a sign flip on the |V a ancilla photon amplitude for aD ancilla qubit measurement result. Application of the target location (right) parity gate, with the standard PBS replaced with 45 PBS, to the target and remaining ancilla qubit followed by a measurement on the ancilla qubit, conditions the control and target qubits to Here an odd parity measurement result conditions a bit flip on the ancilla qubit and a sign flip on the |V amplitude of the control qubit. Similarly, if the QND measurement on the ancilla yields V then a bit flip is performed on the target qubit. The action of all these measurements and corrections implements a CNOT operation on the distributed initial state without transmitting single photons between the remote locations. Using similar ideas it is straightforward to show how three-qubit gates such as the Toffoli and Fredkin gates can be constructed. It is obvious that these can be constructed directly from their logical breakdown into two-qubit gates. However, it is far more efficient to construct them directly from the fundamental resources, using extensions of the primitives employed in the parity and CNOT gates and only transmitting coherent probe (bus) states between separated locations.

Local Clusters
Distributed Parity Gate Figure 4. Schematic diagram of a distributed cluster formed by joining two local cluster states with the parity detector.

Computation by projective measurements
An alternative approach to performing quantum computation is through direct measurements applied to a collection of qubits. One of the most well-known examples is the cluster-state approach [21]. This requires the creation of a specific type of entangled state-a cluster stateof the qubits as an initial resource. The computation is then accomplished by sets of single qubit measurements applied to this entangled state. In linear optics this scheme can be implemented using the Browne and Rudolph fusion techniques [10] and recently demonstrated in [20]. A significant resource saving can be achieved by replacing the beam splitter-based fusion gates with the parity gate. This means that, on average, only one photon is needed to add one photon to the cluster state (rather than 45 Bell pairs in the original Browne and Rudolph scheme). The addition of qubits to the cluster is thus near-deterministic in nature. Also, because of the distributed nature of the parity gate it is possible to envisage situations in which local microclusters are generated and then joined via distributed entangling operations ( figure 4). Clearly, the single qubit measurements, needed to accomplish the computation once the cluster state is constructed, can be made within our proposed framework.
Other measurement-based approaches to quantum computation are also available. Single qubit measurements first need the construction of a cluster state. However, if any nondestructive two-qubit (or more) projective measurements can be implemented these can facilitate computation directly using the techniques of Nielsen [22] and Leung [23]. A specific example is that of near-deterministic Bell state measurements, which do not absorb the photon qubits. Such measurements can be implemented in our framework by two applications of the parity gate, with the second parity gate operating in a 45 • rotated basis compared to the first. The implementation of distributed projective measurements in an entangled basis is another very appealing feature of our weak nonlinearity approach.
Clearly, the use of weak nonlinearities enables several different approaches to perform very efficient distributed optical computation. The choice of approach will depend on many factors, such as the scale and architecture of the processor, how distributed it is, and the actual physical resources available. It is quite possible that a hybrid approach, combining various computational techniques, will be the most effective.

Discussion
We have presented a new paradigm for all-optical quantum processing, based on generalized quantum non-demolition measurements. The starting point is a practical list of resources (coherent states, homodyne measurements, linear optical elements and weak nonlinearities) and we have shown how everything that is needed can be built from these resources. As the approach is all-optical, communication and computation blend together seamlessly with no need for qubit interconversion. Distributed processing is enabled naturally and, as coherent states mediate the quantum information over distance, it is robust to photon loss. Furthermore, the use of QND measurements makes our approach very efficient in terms of its use of the fundamental resources.
The two basic QND building blocks in our approach are single photon QND detectors and distributed two-qubit parity gates. The photon detectors can be made of very high fidelity and can also be used to generate single photon qubit resources. The parity gate is near-optimal in terms of its measurement scheme. Both detector and parity gate require θα ∼ π; this scaling is very important as it enables weak nonlinearities (θ ∼ 10 −5 ) to be used with modest coherent states in the probe or bus modes. The detectors and gates are not deterministic, but near-deterministic. However, the errors arise due to overlapping tails of different probe coherent states and these can be made very small.
Our approach does not force a choice of computation scheme and processor architecture; rather it provides building blocks which can be put together to suit the task at hand. This, along with the very high resource efficiency, makes our approach extremely flexible. In the short-term, it provides a new way forward to distributed few-qubit technologies based on scarce resources and in the longer term it provides a new approach to efficient, scalable optical quantum computing either between distinct nodes or within a node.