Demonstration of a Controlled-Phase Gate for Continuous-Variable One-Way Quantum Computation
Ryuji Ukai, Shota Yokoyama, Jun-ichi Yoshikawa, Peter van Loock, Akira Furusawa
aa r X i v : . [ qu a n t - ph ] J u l Demonstration of a Controlled-Phase Gatefor Continuous-Variable One-Way Quantum Computation
Ryuji Ukai , Shota Yokoyama , Jun-ichi Yoshikawa , Peter van Loock , , and Akira Furusawa Department of Applied Physics, School of Engineering, The University of Tokyo,7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Optical Quantum Information Theory Group, Max Planck Institute for the Science of Light,G¨unther-Scharowsky-Str.1/Bau 26, 91058 Erlangen, Germany Institute of Theoretical Physics I, Universit¨at Erlangen-N¨urnberg, Staudstr.7/B2, 91058 Erlangen, Germany
We experimentally demonstrate a controlled-phase gate for continuous variables in a fullymeasurement-based fashion. In our scheme, the two independent input states of the gate, encodedin two optical modes, are teleported into a four-mode Gaussian cluster state. As a result, one of theentanglement links present in the initial cluster state appears in the two unmeasured output modesas the corresponding entangling gate acting on the input states. The genuine quantum character ofthis gate becomes manifest and is verified through the presence of entanglement at the output for aproduct two-mode coherent input state. By combining our controlled-phase gate with the recentlyreported module for universal single-mode Gaussian operations [R. Ukai et al. , Phys. Rev. Lett. ,240504 (2011)], it is possible to implement universal Gaussian operations on arbitrary multi-modequantum optical states in form of a fully measurement-based one-way quantum computation.
The one-way model of measurement-based quantumcomputation (QC) [1] is a fascinating alternative to thestandard unitary-gate-based circuit model, for discrete-variable systems such as qubits as well as for continuous-variable (CV) encodings on quantized harmonic oscil-lators [2–4]. Such one-way computations are realizedthrough single-qubit or single-mode measurements, andsome outcome-dependent feedforward operations, on apre-prepared multi-party entangled state, the so-called“cluster state”. By choosing an appropriate set of mea-surement bases on a sufficiently large cluster state, anarbitrary unitary operation can be implemented for thecorresponding encoding.Towards CV QC, initially, a CV analogue to qubit clus-ter states was proposed [2]. Subsequently, the notion ofan in-principle universality of CV one-way quantum com-putation was then proven, relying upon the asymptoticassumptions of sufficiently long measurement-based gatesequences [5] and perfectly squeezed optical cluster-stateresources as well as the inclusion of at least one nonlin-ear, non-Gaussian measurement device [3]. Only shortlythereafter, by using squeezed vacuum states and beamsplitters [6], various cluster states were generated in thelab [7, 8]. Among these was the four-mode linear clusterstate which would allow to implement arbitrary single-mode Gaussian operations [9, 10]. However, in order todemonstrate such single-mode gate operations on arbi-trary input states, the input mode has to be attachedto the cluster state. For a single quadratic gate, this canbe accomplished using two squeezed-state ancillae, as de-scribed in Ref. [11]. A much simpler and more generalsolution [10], however, would employ a multi-mode mea-surement such as a Bell measurement, similar to standardCV quantum teleportation [12]. By using a four-modelinear cluster state and a Bell-measurement-based inputcoupling, a typical set of single-mode Gaussian opera- tions such as squeezing and Fourier transformations wasrecently experimentally demonstrated [13].The final missing element towards implementing ar-bitrary multi-mode Gaussian transformations in a one-way fashion [10] is a universal two-mode entangling gate.In fact, universal multi-mode operations (even includingnon-Gaussian ones) are, in an asymptotic sense, realiz-able when universal single-mode gates (including at leastone non-Gaussian gate) are combined with any kind ofquadratic (Gaussian) interaction gate [5]. More specif-ically, an arbitrary multi-mode Gaussian operation canbe exactly recast as a finite decomposition into single-mode Gaussian gates and a quadratic (Gaussian) two-mode gate [5, 10, 14, 15]. The most natural and eas-ily implementable two-mode gate in this setting wouldbe that corresponding to a vertical link between two in-dividual modes of a CV cluster state – the controlled-phase ( C Z ) gate [3]. Very recently, Wang et al. reportedan attempt to experimentally demonstrate this gate [16].However, the cluster state in that experiment was not ofsufficient quality in order to operate the gate as a genuinenonclassical entangling gate; in fact, the two-mode inputquantum state was degraded by such a large excess noisethat no entanglement at all was present at the output.In this paper, we demonstrate a CV cluster-based C Z gate operating in the quantum realm. In order to ver-ify its nonclassicality, we show that the two-mode outputstate is indeed entangled when the input state is a prod-uct of two single-mode coherent states. Furthermore, sev-eral manifestations of the general input-output relationof the gate are realized, using various distinct coherentstates as the input of the gate. The resource state is afour-mode linear cluster state, as illustrated in Fig. 1(a).The two input coherent states are prepared indepen-dently of the cluster state and subsequently coupled to itthrough quantum teleportations. Note that a similar ex- Bell measurementBell measurementcluster stateInput
Bell E O M x E O M E O M x E O M
50 %RLO LO LO50 %R gateverification
OPOOPOOPOOPO Bell measurementBell measurementcluster stateInput
FIG. 1: (color online) (a) An abstract illustration of our ex-periment. (b) Input coupling through quantum teleportationfor larger one-way quantum computations. (c) A schematicof our experimental setup. OPO: optical parametric oscilla-tor, LO: local oscillator for homodyne measurement, EOM:electro-optical modulator, HD: homodyne detection, Bell:Bell measurement. periment in the qubit regime was reported recently [17].Our CV gate can be directly incorporated into a one-way quantum computation of larger scale, see Fig. 1(b)[18]. In particular, as noted above, when combined withuniversal single-mode gates, arbitrary multi-mode oper-ations are in principle realizable; and certain nonlinearmulti-mode Hamiltonians such as a fairly strong two-mode cross-Kerr interaction (a universal entangling gatefor photonic qubits [19] when effectively enhanced) wouldonly require applying tens of quadratic and cubic single-mode gates, in addition to the two-mode C Z gate [20].In the following, we shall use the canonical positionand momentum operators, ˆ x j and ˆ p j , where the subscript j denotes an optical mode and [ˆ x j , ˆ p k ] = iδ jk /
2. TheCV C Z gate corresponds to the unitary operator ˆ C Zjk = e i ˆ x j ˆ x k with the input-output relation, ˆ ξ ′ jk = ( I SS I ) ˆ ξ jk ,where ˆ ξ jk = (ˆ x j , ˆ p j , ˆ x k , ˆ p k ) T , S = ( ), and I is the2 × C Z gate by using a four-modelinear cluster state [ C C C C p Cj − P k ∈ N j ˆ x Ck ( ≡ ˆ δ j ). Here, N j denotes the set of nearest-neighbor modes of mode j , when thestate is represented by a graph, see Figs. 1(a-b). Thefour-mode linear cluster state can be interpreted as twoEinstein-Podolsky-Rosen (EPR) pairs ( C C C C
4) with a C Z interaction between them ( C C α and β are teleported to modes C C C Z interaction between the two EPR pairs is teleported ontothe two input states [18].Let us describe the above procedure for a non-ideal,finitely squeezed cluster state corresponding to non-zerovariances for the operators ˆ δ j . When a four-mode linearcluster state is generated by using four finitely squeezedstates and three beam splitters as in Fig. 1(c), theexcess noises are as follows, ˆ δ = √ e − r ˆ p (0)1 , ˆ δ = − √ e − r ˆ p (0)3 − √ e − r ˆ p (0)4 , ˆ δ = √ e − r ˆ p (0)1 − √ e − r ˆ p (0)2 ,and ˆ δ = −√ e − r ˆ p (0)4 , where e − r ˆ p (0) j is the squeezedquadrature of the j -th squeezed state. Here, we assumeidentical squeezing levels with parameter r for simplicity.Note that the limit r → ∞ corresponds to an ideal clusterstate. Two pairs of modes, ( α, C
1) and ( β, C p α − ˆ x C , ˆ x α − ˆ p C , ˆ p β − ˆ x C , and ˆ x β − ˆ p C aremeasured, giving the measurement results t α , t , t β , and t , respectively. Then the corresponding feedforward op-erations ˆ X C ( t ) ˆ Z C ( t α + t ) ˆ X C ( t ) ˆ Z C ( t β + t ) are per-formed on modes C C
3, where ˆ X j ( s ) = e − is ˆ p j andˆ Z j ( s ) = e is ˆ x j are the position and momentum displace-ment operators. The resulting position and momentumoperators of modes C C µ and ν , can then be written as,ˆ ξ µν = (cid:18) I SS I (cid:19) ˆ ξ αβ + ˆ δ . (1)This completes the C Z operation. Here, ˆ δ = ( − ˆ δ , − ˆ δ +ˆ δ , − ˆ δ , − ˆ δ + ˆ δ ) T represents the excess noise of our C Z gate. In the ideal case with r → ∞ , the noiseterm ˆ δ vanishes and a perfect C Z operation is achieved.As the C Z gate is an entangling gate, the presence ofentanglement at the output for a product input state(in spite of the excess noise ˆ δ ) is crucial to prove thenonclassicality of our gate implementation. A suffi-cient condition for inseparability of a two-mode state is (cid:10) ∆ ( g ˆ p µ − ˆ x ν ) (cid:11) + (cid:10) ∆ ( g ˆ p ν − ˆ x µ ) (cid:11) < g for some g ∈ R [21–23]. We will show that this inequality is satisfiedat the output for a two-mode vacuum input. In ourcase, g = 3 / e − r < /
5, corresponding to approximately − . P o w e r [ d B ] ˆ x µ ˆ p µ ˆ x ν ˆ p ν P o w e r [ d B ] ˆ x µ ˆ p µ ˆ x ν ˆ p ν ˆ x µ ˆ p µ ˆ x ν ˆ p ν ˆ x µ ˆ p µ ˆ x ν ˆ p ν ˆ x µ ˆ p µ ˆ x ν ˆ p ν (a) Input: vac. (b) Input: ˆ x α -coh. (c) Input: ˆ p α -coh. (d) Input: ˆ x β -coh. (e) Input: ˆ p β -coh. FIG. 2: (color) Powers at the outputs. (a) shows variances of the output quadratures for vacuum inputs. The black and redtraces correspond to the shot noise levels (SNLs) and output quadratures, respectively. The green lines show the theoreticalprediction when no resource squeezing is available, while the cyan traces show the theoretical prediction for an ideal C Z gate.(b), (c), (d), and (e) show the powers of the output quadratures where ( h ˆ x α i , h ˆ p α i , h ˆ x β i , h ˆ p β i ) are ( a ,0,0,0), (0, a ,0,0), (0,0, b ,0),and (0,0,0, b ), and where a and b correspond to 21.5 dB and 21.2 dB above the SNL, respectively. The blue lines show thetheoretical prediction based on (a) and different input coherent amplitudes. Vac.: vacuum state. Coh.: coherent state. employ the experimental techniques described in Refs. [8]and [24] for the generation of the cluster state and thefeedforward process, respectively. The resource squeezingis − C Z gate, the variances of ˆ x µ and ˆ x ν remain unchanged and thus are equal to the shotnoise level (SNL), while those of ˆ p µ and ˆ p ν are 3 dB abovethe SNL (two times the SNL) as shown by the cyan lines.When the resource squeezing is finite, the output statesare degraded by excess noise. We show as a reference thetheoretical prediction for a vacuum resource [ r = 0 inEq. (1)] by green lines, where the variances of ˆ x µ and ˆ x ν are 4.8 dB above the SNL (three times the SNL), whilethose of ˆ p µ and ˆ p ν (cid:10) ∆ ˆ x µ (cid:11) , (cid:10) ∆ ˆ p µ (cid:11) , (cid:10) ∆ ˆ x ν (cid:11) , and (cid:10) ∆ ˆ p ν (cid:11) , shown by the red traces, are be-low the green lines due to the finite resource squeezing.These correspond to 2.4 dB, 4.6 dB, 2.2 dB, and 4.6 dBabove the SNL, respectively. These results are consistentwith the resource squeezing level of − x µ , ˆ x ν and 4.7 dB for ˆ p µ , ˆ p ν above the SNL.In order to verify the general input-output relations, weemploy coherent input states [Figs. 2(b-e)]. The powers of the amplitude quadratures are measured in advance,corresponding to 21.5 dB for mode α and 21.2 dB formode β , respectively, compared to the SNL.Fig. 2(b) shows the powers of the output quadraturesas red traces when the input α is in a coherent state witha nonzero amplitude only in the ˆ x α quadrature; the input β is in a vacuum state. We observe an increase in powersof ˆ x µ and ˆ p ν caused by the nonzero coherent amplitude.On the other hand, ˆ p µ and ˆ x ν are not changed comparedto the case of two vacuum inputs. In the same figure, thetheoretical prediction is shown by blue lines. Clearly, theexperimental results are in agreement with the theory.Similarly, Figs. 2(c-e) show the results with a nonzerocoherent amplitude in the ˆ p µ , ˆ x ν , and ˆ p ν quadratures,respectively. We see the expected feature of the C Z gatethat the quadratures in modes α and β are transmit-ted to modes µ and ν with unity gain and ˆ x α and ˆ x β are transferred to ˆ p ν and ˆ p µ . We believe that the smalldiscrepancies between our experimental results and thetheoretical predictions are caused by propagation lossesand imperfect visibilities.For assessing the entanglement at the output, we usetwo input states in the vacuum. The two homodynesignals are added electronically with a ratio of g : 1and 1 : g in power, by which (cid:10) ∆ ( g ˆ p µ − ˆ x ν ) (cid:11) and (cid:10) ∆ ( g ˆ p ν − ˆ x µ ) (cid:11) are measured.Fig. 3(a) shows the theoretical and experimental re-sults for (cid:10) ∆ ( g ˆ p µ − ˆ x ν ) (cid:11) + (cid:10) ∆ ( g ˆ p ν − ˆ x µ ) (cid:11) with severalgains g . The sufficient condition for entanglement is that (cid:10) ∆ ( g ˆ p µ − ˆ x ν ) (cid:11) + (cid:10) ∆ ( g ˆ p ν − ˆ x µ ) (cid:11) is less than g , shownby line (iv), for some g ∈ R . When g = 0 .
63, 0 . .
89, this criterion is satisfied in the experiment.The results without and with resource squeezing roughlycoincide with the theoretical curves (i) and (ii), respec-tively. In particular, Figs. 3(b-c) show the results forthe optimal gain g = 3 /
4. Traces (vi) show the refer-ence for normalization when the two homodyne inputsare vacuum states. These levels correspond to 1 + g P o w e r [ d B ] (b) ∆ ˆ p µ − ˆ x ν (c) ∆ ˆ p ν − ˆ x µ
012 0 0.5 1 P o w e r g (vi)(vii)(viii) (vi)(vii) (viii) (a) gain curve (i) (ii)(iii)(iv)(v) FIG. 3: (color online) Entanglement at the output. (a) showsentanglement verification with several gains g . (i) withoutresource squeezing, (ii) with − (cid:10) ∆ ( g ˆ p µ − ˆ x ν ) (cid:11) and (cid:10) ∆ ( g ˆ p ν − ˆ x µ ) (cid:11) at g = 3 /
4, respectively. 0 dB corresponds to the SNL. (vi)the reference where two homodyne inputs are vacuum states,(vii) the measurement results, and (viii) sufficient conditionfor entanglement. times the SNL. Traces (vii) show the measurement re-sults for (cid:10) ∆ ( g ˆ p µ − ˆ x ν ) (cid:11) and (cid:10) ∆ ( g ˆ p ν − ˆ x µ ) (cid:11) , which are − . ± .
02 dB and − . ± .
02 dB relative to traces(vi), respectively. Note that the error in determining theSNL is included in the above errors. Lines (viii) showthe sufficient condition for entanglement, correspondingto the theoretical prediction with about − . D ∆ ( √ g ˆ p µ − √ g ˆ x ν ) E + D ∆ ( √ g ˆ p ν − √ g ˆ x µ ) E = 0 . ± . < , at g = 3 / . (2)Note that traces (vi), (vii), and line (viii) correspondto curves (v), (ii), and line (iv) at g = 3 /
4, respectively.In conclusion, we have experimentally demonstrateda fully cluster-based C Z gate for continuous variables.In our scheme, the two-mode input state was coupledto a four-mode resource cluster state through quantumteleportations. For a product input state, entanglementat the output was clearly observed, verifying the essential property of the C Z gate. In combination with our recentwork on the experimental demonstration of single-modeGaussian operations, all components for universal multi-mode Gaussian operations are now available in a one-way configuration. The quality of our C Z gate is onlylimited by the squeezing level of the resource state, andthe recently reported, higher levels of squeezing [25, 26]would even allow to realize multi-step multi-mode one-way quantum computations. To achieve full universalitywhen processing arbitrary multi-mode quantum opticalstates, the only missing ingredient is a single-mode non-Gaussian gate.This work was partly supported by SCF, GIA, G-COE, APSA, and FIRST commissioned by the MEXT ofJapan, ASCR-JSPS, and SCOPE program of the MIC ofJapan. R.U. acknowledges support from JSPS. P.v.L. ac-knowledges support from the Emmy Noether programmeof the DFG in Germany. [1] R. Raussendorf and H.J. Briegel, Phys. Rev. Lett. ,5188 (2001).[2] J. Zhang and S.L. Braunstein, Phys. Rev. A , 032318(2006).[3] N.C. Menicucci et al. , Phys. Rev. Lett. , 110501(2006).[4] A. Furusawa and P. van Loock, Quantum Teleportationand Entanglement, Wiley-VCH, Berlin (2011).[5] S. Lloyd and S.L. Braunstein, Phys. Rev. Lett. , 1784(1999).[6] P. van Loock, C. Weedbrook, and M. Gu, Phys. Rev. A , 032321 (2007).[7] X. Su et al. , Phys. Rev. Lett. , 070502 (2007).[8] M. Yukawa et al. , Phys. Rev. A , 012301 (2008).[9] P. van Loock, J. Opt. Soc. Am. B , 340 (2007).[10] R. Ukai et al. , Phys. Rev. A , 032315 (2010).[11] Y. Miwa et al. , Phys. Rev. A , 050303(R) (2009).[12] A. Furusawa et al. , Science , 706 (1998).[13] R. Ukai et al. , Phys. Rev. Lett. , 240504 (2011).[14] S. L. Braunstein, Phys. Rev. A , 055801 (2005).[15] M. Reck et al. , Phys. Rev. Lett. , 58 (1994).[16] Y. Wang et al. , Phys. Rev. A , 022311 (2010).[17] W.B. Gao et al. , PNAS , 20869 (2010).[18] For deriving this fact, we must only consider the com-mutation relation between the C Z interaction and thefeedforward displacements in the teleportations.[19] I. L. Chuang and Y. Yamamoto, Phys. Rev. A , 3489(1995).[20] S. Sefi and P. van Loock, arXiv:1010.0326v2.[21] L. M. Duan et al. , Phys. Rev. Lett. , 2722 (2000).[22] P. van Loock and A. Furusawa, Phys. Rev. A , 052315(2010).[23] J. Yoshikawa, et al. , Phys. Rev. Lett. , 250501 (2008).[24] M. Yukawa, H. Benichi, and A. Furusawa, Phys. Rev. A , 022314 (2008).[25] M. Mehmet et al. , Phys. Rev. A et al. , Optics Express15