Qubus Computation. Gifford, Bristol BS34 8QZ, United Kingdom. Chiyoda-ku, Tokyo , Japan. Queensland, Australia ABSTRACT

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1 Qubus Computation W.J. Munro a,b, Kae Nemoto b,t.p. Spiller a, P. van Loock b, Samuel L. Braunstein c and G. J. Milburn d a Quantum Information Processing Group, Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS34 8QZ, United Kingdom b Quantum Information Science Group, National Institute of Informatics, -1- Hitotsubashi, Chiyoda-ku, Tokyo , Japan c Computer Science, University of York, York YO1 5DD, United Kingdom d Centre for Quantum Computer Technology,Department of Physics, University of Queensland, Australia ABSTRACT Processing information quantum mechanically is known to enable new communication and computational scenarios that cannot be accessed with conventional information technology (IT). We present here a new approach to scalable quantum computing a qubus computer which realizes qubit measurement and quantum gates through interacting qubits with a quantum communication bus mode. The qubits could be static matter qubits or flying optical qubits, but the scheme we focus on here is particularly suited to matter qubits. Universal two-qubit quantum gates may be effected by schemes which involve measurement of the bus mode, or by schemes where the bus disentangles automatically and no measurement is needed. This approach enables a parity gate between qubits, mediated by a bus, enabling near-deterministic Bell state measurement and entangling gates. Our approach is therefore the basis for very efficient, scalable QIP, and provides a natural method for distributing such processing, combining it with quantum communication. Keywords: Universal quantum computation, distributed information processing, geometric two qubit gate, coherent light channel 1. INTRODUCTION Quantum computing has reached a very interesting stage in its development. Over the last decade there have been numerous proposals for qubit realizations. 1, Some of the more mature proposals, such as trapped ions, 3 nuclear spins (in molecules in liquid state) 4 and photonic qubits 5, 6 have now been demonstrated to work in the laboratory at the few-qubit level. Now in terms of their long term prospects for scalability, there is deemed to be considerable promise in solid state qubits, based (directly or indirectly) on fabrication and technologies developed for conventional IT. However, at present such approaches lag behind the more mature ones they are either still on the drawing board, or at the one- or two-qubit demonstration level. The promise of scalability has yet to achieved for these approaches, and over the next few years it will be interesting to see which systems can meet this challenge and which founder. Clearly decoherence and measurement are both important and challenging issues for solid state qubits. With the current emergence of demonstration qubit experiments, there is optimism about these problems being solved to a level that would permit useful small-scale quantum processing. However, even if these problems can be solved, there is still a need for two-qubit quantum gates to be implemented in a manner that enables the addition of more qubits to a system, so there is scalability. This is the main issue that we address in this paper. If these gates are implemented through a direct qubit-qubit interaction (i.e. a direct qubit-qubit coupling term in the basic system Hamiltonian), potential problems with two-qubit gates are: (i) the addition of an extra qubit to a system may disrupt the settings and calibrations that have been put in place for quantum computing with the original system, and the qubits may have to be so close together that individual addressing (both For further information please contact: W.J.M.: bill.munro@hp.com 1

2 for single-qubit gates and measurement) cannot be achieved. Direct qubit interactions may be just fine for demonstrating entanglement between two solid state qubits, but they may not be as good when it comes to building a universal and scalable quantum processor. For example, with just nearest neighbor interactions there is a large SWAP operation overhead to interact chosen qubits, that could be removed through use of a bus to mediate interactions between non-nearest neighbors. One well known technique is to use single photons to mediate this interaction. There have been a number of very elegant proposals focusing on this, but they place highly stringent requirements, for example, on the generation of the single photons or their detection. 7 1 The approach that we present here contains no direct qubit-qubit interactions and does not require the use of single photons. Such interactions are achieved indirectly through the interaction of qubits with a common quantum field mode a continuous quantum variable (CV) 13, 14 which can be thought of as a communication bus 15. Our qubus computer approach 16 brings together the best of both worlds. Static solid state qubits are used where they work best, for processing. Continuous variables are used where they work best, for communication and mediating interactions; they also have the potential to enable interfacing with existing, conventional information technology. We will assume that individual qubits can be prepared, subjected to single-qubit operations and measured. However, as we discuss in order to introduce our approach, interaction of a qubit with a CV bus mode, followed by measurement of the bus mode, can also be used in order to effect quantum non-demolition (QND) measurement of the qubit. 17 This could be the preferred measurement scheme, unless something even better is achievable by other means. Our approach is based on qubits interacting with the bus mode through distinct dipole couplings, such as the electric dipole of a charge qubit coupling to the electric field quadrature, or the dipole of a spin or magnetic moment coupling to the magnetic field quadrature. The approach should be widely applicable in the solid state qubit context and so we present it in a generic fashion without being tied to any specific implementations. As will be seen, our whole approach is based on the idea of a sequence of interactions, or gates, between qubits 16, 1 7 and the bus mode, followed by measurement of the bus in some scenarios, and not in others. The concept is therefore that qubits can be brought into interaction with the bus mode to effect the desired gate sequence, or that (certainly in the scenarios which involve bus measurement) a bus mode pulse can be employed to interact with successive qubits to effect the gate sequence. Our approach is thus to be contrasted with an always on interaction between qubits and a bus mode. In the latter case, such an interaction can in effect mimic a direct qubit-qubit coupling. For example, two qubits simultaneously coupled to a bus through the Jaynes-Cummings 8, 9 interaction behave as if they have a direct exchange interaction in the dispersive limit. Instead, in our approach the qubit-bus interactions are sequential.. THE INTERACTION The physical system we are going to consider is an atom interacting with a light field. Such an interaction can be described by the Hamiltonian H = H + H I (1) where H is the interaction free Hamiltonian given by H = 1 hωa a+ 1 hω σ z and H I the interaction Hamiltonian given by H I = hg ( a σ + aσ + ) () In these equations a, a are the usual creation and destruction operators of the optical field. We describe the matter qubits using the conventional Pauli operators, with the computational basis being given by the eigenstates of σ z 1 1, with z and 1 z. We can now transform to the so called interaction picture using the operator U = exp ( ih t/ h). In this case our Hamiltonian is transformed to H(t) = hg ( e i t a σ + e i t ) aσ + (3) where = ω ω. { ( Now if we consider the time order operator T exp ī t h dt H(t ))} we can make a perturbation expansion leading to { ( T exp ī t )} dt H(t ) = 1 ī h h t dt H(t ) 1 { t h T dt t } dt H(t )H(t ) +... (4)

3 Now if the detuning is large the first term i h t dt H(t ) = ig while the second order terms are of the form { 1 t t } h T dt dt H(t )H(t ) [ (e i t 1)a σ (e i t 1)aσ + ] (5) i g t [ σz a a + σ + σ ] (6) This gives us an effective Hamiltonian of the form 9 H eff = hχ [ σ z a a + σ + σ ] where χ = g, and so when we apply H eff for a time t we obtain the unitary operator 9 3 U eff = exp [ iχt ( σ z a a + σ + σ )] (7) (8) We notice in U eff that there is a σ + σ contribution even when the cavity mode is empty. However as σ + σ commutes with σ z we can remove it by a simple single qubit rotation. This leave us with our desired unitary U eff = exp [ iθσ z a a ] (9) where θ = χt. This is our controlled rotation operation where the probe beam is rotated ±θ depending on the state of the qubit. This is exactly what we need to perform quantum logic. 3. THE PARITY GATE The controlled rotation described above is all that we need to effect universal two-qubit gates. The first one we are going to consider is a parity gate. 1 The circuit diagram for such a gate is depicted in Fig. (1 a) and works as follows: Consider that we have our two atomic qubits initially prepared in the state Ψ = 1 ( ) with the probe beam as the coherent state α. Following the interactions depicted in Fig. (1a) our initial two-qubit-bus state is transformed to Ψ f = 1 ( αe iθ + ( ) α + 11 αe iθ ), (1) where we have assumed both qubits couple to the probe mode with the same strength θ. In Fig. (1 a) we depict the phase space evolution of the bus amplitudes. At this stage we can choose from different types of measurements on the probe beam. The first option we have is to implement a homodyne measurement of some field quadrature X(φ) = (a e iφ + ae iφ ). The easiest measurement to perform would be that of the momentum (P = X(π/)) quadrature. In that case the measurement probability distribution has three peaks with the overlap error between them given by P err = 1 erfc(α sin θ/ ). As long as αθ π this overlap error is small (< 1 3 ) and the peaks are well separated. Now a measurement of the central peak will project the two matter qubits into the entangled state ( )/. We call this our odd-parity state and it occurs with a probability of 1/. 33 Detecting either of the other two side peaks will project the qubits to the known product states or 11. It is of course possible to exceed this 1/ by changing the nature of our measurement. In principle we could achieve a near deterministic gate if we measured the the position (X = X()) instead of the momentum (P ) quadrature. 1 3 For the position quadrature there are two peaks in the measurement result probability distribution, corresponding to the even + 11 and the odd entangled states of the qubits. The overlap between the peaks 1 in this case is given by P err = 1 erfc(α(1 cos θ)/ ) which is an issue because in order to separate the peaks well enough we would require αθ π. This is much more difficult to achieve than αθ π required for the momentum quadrature measurement. 3

4 a) b) (i) Qubit 1 Qubit θσ z1 θσ z Measurement θ (i), α Time Figure 1. a) Circuit diagram for a two-qubit parity gate based on controlled rotations between the qubits and the probe mode. The gate is completed by an appropriate measurement (homodyne for instance). b) Schematic diagram showing the phase space evolution of the bus amplitudes corresponding to the various qubit computational basis states. An alternative strategy for the probe measurement would be to apply an unconditional displacement D( α) on the probe beam followed by a photon number measurement. 4 After the displacement the combined state of the matter qubits and probe beam is Ψ f = 1 ( α(e iθ 1) + 11 α(e iθ 1) + ( ) ) (11) Now a photon number measurement of the bus mode will then either pick out the vacuum state, or project onto two amplitudes α(e ±iθ 1) without distinguishing between them. For an ideal projection onto the number basis n, the state of the two qubits becomes: Ψ f = ( )/ for n = (1) Ψ f = 1 ( + ( 1) n 11 ) for n > (13) with an equal probability of 1/ as long as the coherent amplitudes α(e ±iθ 1) do not contribute significantly to the vacuum. The overlap of these coherent states α(e ±iθ 1) with the vacuum leads to an error probability of P err = e 4 αθ which can be made quite small with a suitable choice of α and θ. 4 For example with θ small we can choose αθ = which leads to an error probability as low as P err Consequently we can obtain a near-deterministic gate if we can implement a photon number measurement. Once we have the parity gate detailed above how shown how we can implement a two qubit operation. This is technically all that is required with single qubit operations to perform universal quantum computation. The parity gates are however not the typical two qubits gates that one generally considers in the standard quantum computational models. The typical two qubit gate generally considered is the CNOT gate. This 1, 34 however can be simply constructed from two parity gates with the additional of one ancilla qubit. This CNOT gate operates as follows: consider our control and target qubits are in the general two qubit state c ct + c 1 1 ct + c 1 ct + c 3 11 ct where the ct labels the control and target modes. We prepare the ancilla qubit in the known state + 1 a. The action of the first parity gate in the {,1} basis between the control and ancilla mode (c and a) conditions the system to c cat + c 1 1 cat + c 11 cat + c cat (14) We then apply a parity gate in the {+, } basis on the ancilla and target modes. This results in the state [(c + c 1 ) c + (c + c 3 ) 1 c ] + + at + [(c c 1 ) c (c c 3 ) 1 c ] at (15) Next performing a measurement of the ancilla mode in the {,1} basis, our control and target qubits are transformed to c ct + c 1 1 ct + c 11 ct + c 3 1 ct for an result, and c 1 ct + c 1 ct + c 1 ct + c 3 11 ct for a 4

5 1 result. The second case can be transformed to the first by a bit flip the target qubit. After all such operations our initial control and target qubits have been transformed to c ct + c 1 1 ct + c 1 ct + c 3 11 ct c ct + c 1 1 ct + c 11 ct + c 3 1 ct which is clearly the result one would expect if a CNOT operation had been performed on the control and target qubits. 4. THE MEASUREMENT FREE GATE In the previous section we showed how the controlled rotation interaction could be used to implement a highly efficient and useful parity gate. The parity gate was then used to implement a CNOT. A critical component of this gate was the measurement of the probe/bus mode. A natural question that arises is whether the measurement is actually needed to implement this gates? In fact this question is easy to answer by considering the following pulse sequence 16 depicted in Fig. () D(i α)e iθˆnσz D( α)e iθˆnσz1 D( i α)e iθˆnσz D( α)e iθˆnσz1 (16) acting on a probe beam initially prepared in the coherent state αe i π 4 with α real. Here the displacement (iv) (i) (iv) (i) (iv) (i) (iv) (i) (v) (vi) (viii) (vii) (v) (vi) (viii) (vii) (vi) (v) (vii) (viii) (vi) (v) (viii) (vii) Figure. Schematic diagram showing the phase space evolution of the bus amplitudes corresponding to the various qubit computational basis states. The small areas shown in grey are traversed in a clockwise sense, giving a negative contribution to the phase acquired. operator is given by D(β) = exp(βa β a). It is tedious but straightforward to show that the four qubit basis states, 1, 1, 11 (including the probe bus) evolve as where the phase shift φ, φ 1, φ 1, φ 11 are given by αe i π 4 e iφ αe i π 4 (17) 1 αe i π 4 e iφ 1 1 αe i π 4 (18) 1 αe i π 4 e iφ 1 1 αe i π 4 (19) 11 αe i π 4 e iφ αe i π 4 () φ = α [7 cos θ cos θ 3 cos 3θ + cos 4θ] + α [sin θ + 5 sin θ sin 3θ + sin 4θ] 4α sin θ (1 + cos θ + sin θ) (cos θ + sin θ) (1) φ 1 = φ 1 = 4α cos θ φ 11 = α [7 cos θ cos θ 3 cos 3θ + cos 4θ] α [sin θ + 5 sin θ sin 3θ + sin 4θ] () 4α sin θ (1 + cos θ sin θ) (cos θ sin θ) (3) Here the requires the additional assumption αθ 1. The transformations represented in () are of the form of a controlled-phase gate. If we remove a global phase factor and perform two single qubits rotations one can 5

6 show that only the 11 obtains a nett phase shift given by φ d = 8α sin θ ( cos θ cos θ). If we work in the regime θ 1 then φ d 8α θ and so we can get our sign shift when αθ = π 8. It is important to mention that there is an intrinsic error in this gate. The bus mode doesn t completely disentangle from the qubits exactly, because the (small) rotations employed are arcs of circles, rather than straight lines. The error is of order αθ, which can clearly be made small (of order 1/α) even for a maximally entangling universal gate by working in the small θ large α limit. 5. A MORE EFFICIENT MEASUREMENT FREE GATE The gate described in the previous section has an intrinsic error in it which limits it use to small θ. This effect can be eliminated by a slightly more complicated circuit. We will show how such a circuit can be constructed, however it is convenient to do this in pieces. Consider the following operation, 35 V D(α cos θ)e iθσza a D( α)e iθσza a D(α cos θ), (4) which we depict in Figure (3) consisting of unconditional displacements and conditional rotations. With this a) P b) D 1 θ σ z D θ σz D 1 Time Qubit X Figure 3. a) Circuit diagram for an effective controlled displacement constructed from uncontrolled displacements and controlled rotations. b) Schematic phase space evolution of a coherent qubus amplitude during the controlled displacement. sequence, a conditional displacement can be realized exactly such that V = exp[iα sin θ σ z (a + a)] D(iα sin θ σ z ). (5) This pulse sequence is exactly what we need to implement a CPhase without constraints on the size of θ. Our two-qubit CPhase gate relies upon the basic principle that a CV mode acquires a phase shift whenever it goes along a closed loop in phase space. This phase shift only depends on the area of the loop and not on its form 36 and it originates from the fact that for any sequence of two displacements, the total displacement operator contains an extra phase factor, D(β 1 )D(β ) = exp [i Im (β 1 β )] D(β 1 + β ). (6) In this sense such a two-qubit gate can be regarded as a geometric phase gate. 36 With controlled displacements available it is straightforward to implement a conditional phase gate, as we now describe. Let us assume that an arbitrary two-qubit state enters the gate such that the total initial state (of the two-qubit-qubus system) may be written as (c 1 + c 1 + c c 4 11 ) qubus, (7) with a qubus-probe mode initially in an arbitrary state qubus. The two-qubit gate follows from four conditional displacements. The sequence of operations is shown in Fig. 4a. This defines the total unitary operator U tot D(iβ σ z )D(β 1 σ z1 )D( iβ σ z )D( β 1 σ z1 ) = exp [i Re(β 1β ) σ z1 σ z ] (8) 6

7 Qubit 1 β 1 σ z1 iβ σ z β 1 σ z1 iβ σ z a) β 1 b) P 1 11 iβ Time Qubit 1 X Figure 4. a) Circuit diagram of a universal two-qubit gate based on controlled displacements between the qubits and the probe bus. b) Schematic phase space evolution of a coherent qubus amplitude, depending on the four basis states of the two qubits. Clearly when this operator acts on the two-qubit-qubus system, the only effect is the generation of phase factors conditional on the two-qubit state. Although it is entangled with the qubits during the gate, the qubus mode finishes in its initial state, disentangled from the qubits. The evolution does not depend on this qubus state. For the case of real β 1 and β, the effect of the total operation on a bus coherent state, conditional on the state of the qubits, is illustrated in Fig. 4b. Now our constructed controlled displacements operations described above have an amplitude d = α sin θ and so if we choose β 1 β = d π/8 a total initial state as in Eq. (7) gives a final pure two-qubit state of e i π 4 U U (c1 + c 1 + c 3 1 c 4 11 ), (9) where U e i π 4 σz. Thus, up to a global phase and local unitaries, a controlled-phase gate results. Such a gate is a universal two-qubit gate. The resultant conditional displacement in Eq. (5) is along the p-axis (in the positive or negative direction, dependent upon the qubit state) by an amount of d = α sin θ. After adding extra unconditional rotations of the probe beam, e.g. a Fourier transform in order to switch between p and x, the entire sequence of Eq. (8) can be achieved solely through uncontrolled qubus operations (displacements and rotations) and controlled rotations of the probe. This provides an exact mechanism to create our controlled phase gate. Assuming β 1 = β = π/8, the strength of the conditional rotations for simulating the conditional displacements are determined by the parameter d = α sin θ = π/8.6. For example, with θ 1, unconditional displacements of about α 1 4 photons are needed. However, we may also satisfy d.6 by using strong non-linearities, θ π/ with weak qubus displacements of the order α 1. We shall now compare this controlled phase gate to the one in the previous section. Two crucial differences exist, both of which highlight the advantages of the new gate. First and foremost the gate in the previous section is only approximate. It has an intrinsic error since the qubus probe does not completely disentangle from the qubits, causing a dephasing effect on the qubits. To keep this error small requires αθ 1, so the gate only works when θ 1. The gate presented here does not have this limitation. In this sense, our scheme here is universal and can be applied to various physical systems, operating in any coupling regime. It works for all θ regardless of whether it is small or large. The second difference is important from a practical point of view and relates to the local single qubit rotations needed to realize the gate in the form of (9). The gate in the previous requires single qubit 7

8 rotations of the form e iα θσ z. This places considerable sensitivity on α and θ, requiring them to be known accurately enough to perform single qubit operations that scale as α θ. In the gate presented here we only require a unitary of the form e i π 4 σz, which is independent of both α and θ and thus much less demanding. 6. DISCUSSION AND CONCLUSION We have presented a new approach to quantum computing a qubus computer which brings together discrete qubits with quantum continuous variables in a single scheme. Through interaction with a common bus mode, it is possible to realize a universal two-qubit gate. We considered three different schemes including: Measurement-based probabilistic but heralded parity gates, Measurement-based near deterministic parity gates and Measurement-free deterministic CPhase gates, with two different interactions (the controlled-displacement and the controlled-rotation) between the discrete qubits and the bus mode. For the latter scheme, no post-interaction measurement is required on the bus mode it effectively plays the role of a catalyst in enabling the gate. All of these approaches are particularly well suited for solid state qubits, which generally have a natural dipole coupling to a common electromagnetic field mode, such as superconducting qubits coupled to a microwave field or an NV diamond center coupled to an optical cavity mode. However the results are also directly applicable to all optical gates. Lastly, our approach does not generally 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. 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 quantum computation. Acknowledgments: We thank R. Van Meter, R. G. Beausoleil, T. Ladd and P. L. Knight for valuable discussions. This work was supported in part by the Japanese JSPS, and MIC research grants, the UK research council EPSRC, the Australian Research Council Centre of Excellence in Quantum Computer Technology and the European Project QAP. REFERENCES 1. Fortschr. Phys. 48, Number 9-11, Special Focus Issue: Experimental Proposals for Quantum Computers, eds. S. Braunstein and H.-K. Lo ().. T. P. Spiller, W. J. Munro, S. D. Barrett, and P. Kok, Contemp. Phys. 46, J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 491 (1995). 4. N. A. Gershenfeld and I. L. Chuang, Science 75, 35 (1997); D. Cory, A. Fahmy, and T. Havel, Proc. Nat. Acad. Sci. 94, 1634 (1997). 5. E. Knill, R. Laflamme and G. J. Milburn, Nature 49, 46 (1). 6. Pieter Kok, W.J. Munro, Kae Nemoto, T.C. Ralph, Jonathan P. Dowling and G.J. Milburn, quantph/ J.I. Cirac, P. Zoller, H.J. Kimble, and H. Mabuchi, Phys.Rev. Lett. 78, 31 (1997). 8. J.I. Cirac, A. Ekert, S.F. Huelga, and C. Macchiavello, Phys. Rev. A 59, 449 (1999). 9. S. Mancini and S. Bose, Phys. Rev. A 7, 37 (4). 1. L.-M. Duan, B.B. Blinov, D.L. Moehring, and C. Monroe, Quant. Inf. Comput. 4, 165 (4). 11. Y. L. Lim, A. Beige and L. C, Kwek, Phys. Rev. Lett. 95, 355 (5). 1. L.-M. Duan, B. Wang and H. J. Kimble, quant-ph/5554; 13. S. L. Braunstein and P. van Loock, Reviews of Modern Physics 77, (5). 14. S. L. Braunstein, Phys. Rev. A 45, 683 (199). 15. S. Lloyd, quant-ph/857. 8

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