Computation with Coherent States via Teleportations to and from a Quantum Bus
CComputation with Coherent States via Teleportations to and from a Quantum Bus
Marcus Silva and Casey R. Myers
2, 1 Department of Physics and Astronomy, and Institute for Quantum Computing, University of Waterloo, ON, N2L 3G1, Canada National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan (Dated: November 2, 2018)In this paper we present results illustrating the power and flexibility of one-bit teleportations inquantum bus computation. We first show a scheme to perform a universal set of gates on continuousvariable modes, which we call a quantum bus or qubus , using controlled phase-space rotations,homodyne detection, ancilla qubits and single qubit measurement. The resource usage for thisscheme is lower than any previous scheme to date. We then illustrate how one-bit teleportationsinto a qubus can be used to encode qubit states into a quantum repetition code, which in turn canbe used as an efficient method for producing GHZ states that can be used to create large clusterstates. Each of these schemes can be modified so that teleportation measurements are post-selectedto yield outputs with higher fidelity, without changing the physical parameters of the system.
PACS numbers: 03.67.Lx, 03.67.Mn, 42.65.-k
I. INTRODUCTION
Bennett et al. [1] showed that an unknown quantumstate (a qubit) could be teleported via two classical bitswith the use of a maximally entangled Bell state sharedbetween the sender and receiver. The significance ofteleportation as a tool for quantum information was ex-tended when Gottesman and Chuang [2] showed that uni-tary gates could be performed using modified teleporta-tion protocols, known as gate teleportation , where thetask of applying a certain gate was effectively translatedto the task of preparing a certain state. Since then tele-portation has been an invaluable tool for the quantuminformation community, as gate teleportation was thebasis for showing that linear optics with single photonsand photo-detectors was sufficient for a scalable quantumcomputer [3]. Moreover, Zhou et al. [4] demonstratedthat all previously known fault-tolerant gate construc-tions were equivalent to one-bit teleportations of gates.Recently, the use of one-bit teleportations between aqubit and a continuous variable quantum bus (or qubus )has been shown to be important for fault-tolerance [5].Using one-bit teleportations to transfer between two dif-ferent forms of quantum logic, a fault tolerant methodto measure the syndromes for any stabiliser code withthe qubus architecture was shown, allowing for a linearsaving in resources compared to a general CNOT con-struction. In terms of optics, the two different types ofquantum logic used were polarisation {| (cid:105) = | H (cid:105) , | (cid:105) = | V (cid:105)} and logical states corresponding to rotated coherentstates {| α (cid:105) , | e ± iθ α (cid:105)} , although in general any two-levelsystem (qubit) which can interact with a continuous vari-able mode (qubus) would suffice. The relative ease withwhich single qubit operations can be generally performedprompted the question of whether a universal set of gatescan be constructed with this rotated coherent state logic.In this paper we describe one such construction, whichwe call qubus logic .The fault-tolerant error-correction scheme using aqubus [5] exploits the fact that entanglement is easy to create with coherent cat states of the qubus, suchas | α (cid:105) + | αe iθ (cid:105) , and single qubit operations are easilyperformed on a two-level system. In this paper we de-scribe how these cat sates can be used as a resource toconstruct other large entangled states, such as clusterstates [6, 7, 8], using one-bit teleportations between aqubit and a qubus.Although the average fidelities of qubus logic and clus-ter state preparation are dependent on how strong the in-teraction between the qubit and the qubus can be made,and how large the amplitude α is, these fidelities can beincreased arbitrarily close to 1 through the use of post-selection during the one-bit teleportations, demonstrat-ing the power and flexibility of teleportation in qubuscomputation for state preparation.The paper is organised as follows. First, in Section IIwe revisit one-bit teleportations for the qubus scheme.Next, in Section III we present a technique to performquantum computation using coherent states of the qubusas basis states. To do this we make use of controlled(phase-space) rotations and ancilla qubits. This coherentstate computation scheme is the most efficient to date. InSection IV we show how we can efficiently prepare rep-etition encoded states using one-bit teleportations, andhow such encoders can be used to prepare large clusterstates. II. ONE-BIT TELEPORTATIONS
In the original quantum teleportation protocol an ar-bitrary quantum state can be transferred between twoparties that share a maximally entangled state by usingonly measurements and communication of measurementoutcomes [1]. Modifications of the resource state allowfor the applications of unitaries to an arbitrary state ina similar manner, in what is known as gate teleporta-tion [2]. The main advantage of gate teleportation isthe fact that it allows for the application of the unitaryto be delegated to the state preparation stage. In some a r X i v : . [ qu a n t - ph ] A p r physical realisations of quantum devices, it may only bepossible to prepare these states with some probability ofsuccess. In that case, the successful preparations can stillbe used for scalable quantum computation [2]. Whendealing with noisy quantum devices, it is important toencode the quantum state across multiple subsystems, atthe cost of requiring more complex operations to imple-ment encoded unitaries. In order to avoid the the un-controlled propagation of errors during these operations,one can also employ gate teleportation with the extrastep of verifying the integrity of the resource state beforeuse [2, 3, 4, 9, 10]. In the cases where the teleporta-tion protocol is used only to separate the preparation ofcomplex resource states from the rest of the computa-tion, simpler protocols can be devised. These protocolsare known as one-bit teleportations [4]. Unitaries imple-mented through one-bit gate teleportation can also beused for fault-tolerant quantum computation [4] as wellas measurement based quantum computation [6]. Themain difference between one-bit teleportation and thestandard teleportation protocol is the lack of a maximallyentangled state. Instead, in order to perform a one-bitteleportation it is necessary that the two parties interactdirectly in a specified manner, and that the qubit whichwill receive the teleported state be prepared in a specialstate initially.Some unitary operations on coherent states can be dif-ficult to implement deterministically, while the creationof entangled multimode coherent states is relatively easy.Single qubits, on the other hand, are usually relativelyeasy to manipulate, while interactions between them canbe challenging. For this reason, we consider one-bit tele-portation between states of a qubit and states of a field ina quantum bus, or qubus . The two types of one-bit tele-portations for qubus computation are shown in Fig. (1),based on similar constructions proposed for qubits byZhou et al. [4]. - i √ i + 1 ) FIG. 1:
Approximate one-bit teleportation protocols [4] using con-trolled rotations. Here, the light grey lines correspond to qubits,and the thick red lines correspond to quantum bus modes.
The one-bit teleportation of the qubit state a | (cid:105) + b | (cid:105) into the state of the qubus, in the coherent state basis {| α (cid:105) , | αe iθ (cid:105)} , is depicted in Fig. (1a). The qubit itselfcan be encoded, for example, in the polarisation of aphoton, i.e. | (cid:105) = | H (cid:105) and | (cid:105) = | V (cid:105) . The initial state,before any operation, is (cid:0) a | (cid:105) + b | (cid:105) (cid:1) | α (cid:105) . The controlledphase-space rotation corresponds to the unitary whichapplies a phase shift of θ to the bus if the qubit state is | (cid:105) , and does nothing otherwise . After the controlledrotation by θ the state becomes a | (cid:105)| α (cid:105) + b | (cid:105)| e iθ α (cid:105) . Rep-resenting the qubit state in the Pauli X eigenbasis, this is | + (cid:105) (cid:0) a | α (cid:105) + b | e iθ α (cid:105) (cid:1) / √ |−(cid:105) (cid:0) a | α (cid:105) − b | e iθ α (cid:105) (cid:1) / √
2. Whenwe detect | + (cid:105) we have successfully teleported our qubitinto | α (cid:105) , | e iθ α (cid:105) logic. When we detect |−(cid:105) we have thestate a | α (cid:105) − b | e iθ α (cid:105) . The relative phase discrepancy canbe corrected by the operation ˜ Z , which approximates thePauli Z operation in the {| α (cid:105) , | αe iθ (cid:105)} basis. This correc-tion can be delayed until the state is teleported back toa qubit, where it is more easily implemented.The one-bit teleportation of the state a | α (cid:105) + b | αe iθ (cid:105) of the qubus to the state of the qubit can be per-formed by the circuit depicted in Fig. (1b). That is,we start with the state (cid:0) a | α (cid:105) + b | αe iθ (cid:105) (cid:1) ( | (cid:105) + | (cid:105) ) / √ − θ , the state becomes | α (cid:105) (cid:0) a | (cid:105) + b | (cid:105) (cid:1) / √ (cid:0) b | e iθ α (cid:105)| (cid:105) + a | e − iθ α (cid:105)| (cid:105) (cid:1) / √
2. Pro-jecting the qubus state into the x -quadrature eigenstate | x (cid:105) via homodyne detection, which is the measurementwe depict as (cid:101) Z , we obtain the conditional unnormalisedstate | ψ ( x ) (cid:105)| ψ ( x ) (cid:105) = f ( x, α ) √ a | (cid:105) + b | (cid:105) )+ f ( x, α cos( θ )) √ e iφ ( x ) b | (cid:105) + e − iφ ( x ) a | (cid:105) ) (1)where f ( x, β ) = 1(2 π ) exp (cid:18) − ( x − β ) (cid:19) (2) φ ( x ) = αx sin( θ ) − α sin(2 θ ) , (3)since (cid:104) x | αe ± iθ (cid:105) = e ± iφ ( x ) f ( x, α cos( θ )) and (cid:104) x | α (cid:105) = f ( x, α ) for real α [15, 16].The weights f ( x, α ) and f ( x, α cos( θ )) are Gaussianfunctions with the same variance but different means,given by 2 α and 2 α cos( θ ), respectively. Given x = α (1+cos( θ )), the midpoint between f ( x, α ) and f ( x, α cos( θ )),one can maximise the fidelity of obtaining the desiredstate a | (cid:105) + b | (cid:105) (averaged over all possible values of x )by simply doing nothing when x > x (where f ( x, α ) >f ( x, α cos( θ ))), or applying Z φ ( x ) = exp( − iφ ( x ) Z ), aPauli Z rotation by φ ( x ), followed by a Pauli X , when x ≤ x . For simplicity, the teleportation corrections arenot explicitly depicted in the circuit diagrams. A. Average fidelities
In order to quantify the performance of the protocolsjust described, consider the process fidelity [18, 19, 20]. This can be implemented by an interaction of the Jaynes-Cummings type between the qubit and the qubus, in the dis-persive limit.
The process fidelity between two quantum operations isobtained by computing the fidelity between states iso-morphic to the processes under the Choi-Jamio(cid:32)lkowskiisomorphism. For example, in order to compare a quan-tum process E acting on a D dimensional system to an-other quantum process F acting on the same system, wecompute the fidelity between the states |E(cid:105) = ⊗ E (cid:32) √ d D (cid:88) i =1 | ii (cid:105) (cid:33) (4) |F(cid:105) = ⊗ F (cid:32) √ d D (cid:88) i =1 | ii (cid:105) (cid:33) . (5)In the case of single qubit processes, we just need to con-sider the action of the process on one of the qubits ofthe state √ ( | (cid:105) ± | (cid:105) ). The operational meaning ofthe process fidelity is given by considering the projectionof the first qubit into a particular state a | (cid:105) + b | (cid:105) . Inthis case the second qubit collapses into the state corre-sponding to the output of the process acting on the state a | (cid:105) + b | (cid:105) . Thus a high fidelity between |E(cid:105) and |F(cid:105) implies a high fidelity between the outputs of the E and F .Consider the state produced by the circuit in Fig. (1a) | ψ ± (cid:105) = 1 √ | , α (cid:105) ± | , αe iθ (cid:105) ) , (6)which depends on the qubit measurement outcome. Asthe relative phase is known, and the correction can beperformed after the state is teleported back to a qubit,for each of the outcomes we can compare this state withthe ideal state expected from the definition of the basisstates for the qubus. This results in the process fidelityof 1 for one-bit teleportation into the qubus.For the case where we teleport the state from the qubusback into the qubit, using the circuit in Fig. (1b), weconsider the action of the process on the second mode ofthe state | ψ + (cid:105) from Eq. (6). This is not, strictly speak-ing, the Choi-Jamio(cid:32)lkowski isomorphism, but it gives thesame operational meaning for the process fidelity as aprecursor to the fidelity between the outputs of the dif-ferent processes being compared, as any superpositionof {| α (cid:105) , | αe iθ (cid:105)} can be prepared from | ψ + (cid:105) by projectingthe qubit into some desired state. We expect the outputstate to be √ ( | (cid:105) + | (cid:105) ) from the definition of the ba-sis states, but we instead obtain the unnormalised states | ψ E ( x > x ) (cid:105) = f ( x, α ) √ (cid:18) | (cid:105) + | (cid:105)√ (cid:19) + f ( x, α cos( θ )) √ (cid:18) e − iφ ( x ) | (cid:105) + e iφ ( x ) | (cid:105)√ (cid:19) , (7) | ψ E ( x < x ) (cid:105) = f ( x, α ) √ (cid:18) e − iφ ( x ) | (cid:105) + e iφ ( x ) | (cid:105)√ (cid:19) + f ( x, α cos( θ )) √ (cid:18) | (cid:105) + | (cid:105)√ (cid:19) . (8) The normalised output state, averaged over all x out-comes, is ρ = (cid:90) ∞ x | ψ E ( x > x ) (cid:105)(cid:104) ψ E ( x > x ) | dx + (cid:90) x −∞ | ψ E ( x < x ) (cid:105)(cid:104) ψ E ( x < x ) | dx, (9)so that the average process fidelity for one-bit teleporta-tion into a qubit is F p = 12 + 12 erf (cid:18) x d √ (cid:19) , (10)where x d = 2 α (1 − cos( θ )) ≈ αθ for small θ . Teleporta-tion from the qubus into the qubit is not perfect, even inthe ideal setting we consider, because the states | α (cid:105) and | e iθ α (cid:105) cannot be distinguished perfectly. However, F p canbe made arbitrarily close to one by letting x d → ∞ , or αθ → ∞ if θ (cid:28)
1, as seen in Fig. (2). This correspondsto increasing the distinguishability of the coherent states | α (cid:105) and | e iθ α (cid:105) . x d P r o ce ss fid e li t y FIG. 2:
Fidelity F p of one-bit teleportation from the qubus to aqubit, as a function of x d . B. Post-selected teleportation
In order to improve the average fidelity of the telepor-tations without changing the physical parameters α and θ of the basis states, one can post-select the outcomes ofthe x -quadrature measurements when teleporting statesfrom the qubus mode to a qubit, as these outcomes es-sentially herald the fidelity of the output state with thedesired state. Discarding the states with fidelity belowa certain threshold allows for the average fidelity to beboosted, even in the case where αθ (cid:54)(cid:29)
1, at the cost ofa certain probability of failure. This is particularly use-ful for the preparation of quantum states which are usedas resources for some quantum information processingtasks.Instead of accepting all states corresponding to all x outcomes of the homodyne measurement which imple-ments (cid:101) Z , we only accept states corresponding to out-comes which are far enough away from the midpoint x ,since the state at x has the lowest fidelity with the de-sired state. More explicitly, we only accept states cor-responding to measurement outcomes which are smallerthan x − y or larger than x + y . This post-selectioncan only be performed for one-bit teleportation from thequbus to the qubit, yielding a probability of success givenby Pr( | x − x | > y ) =12 (cid:20) erfc (cid:18) y − x d √ (cid:19) + erfc (cid:18) y + x d √ (cid:19)(cid:21) , (11)and process fidelity conditioned on the successful out-come given by F p,y = erfc (cid:16) y − x d √ (cid:17) erfc (cid:16) y − x d √ (cid:17) + erfc (cid:16) y + x d √ (cid:17) . (12)The effect of discarding some of the states dependingon the measurement outcome for the teleportation inFig. (1b) is depicted in Fig. (3). In particular, we seethat the process fidelity can be made arbitrarily close to1 at the cost of lower probability of success, while α and θ are unchanged, sincelim y →∞ F p,y = 1 . (13)As the probability mass is highly concentrated due tothe Gaussian shape of the wave packets, the probabilityof success drops super-exponentially fast as a function of y . This is because for large z we have [21]2 √ π e − z z + √ z + 2 < erfc( z ) < √ π e − z z + (cid:113) z + π . (14)This fast decay corresponds to the contour lines for de-creasing probability of success getting closer and closerin Fig. (3). Thus, while the fidelity can be increased ar-bitrarily via post-selection (by increasing y ), this leadsto a drop in the probability of obtaining the successfuloutcome for post-selection. Note that, despite this scal-ing, significant gains in fidelity can be obtained by post-selection while maintaining the physical resources such as α and θ fixed, and while maintaining a reasonable prob-ability of success. In particular, if x d = 2 .
5, increasing y from 0 to 1 .
25 takes the fidelity from 0 . .
99 whilethe probability of success only drops from 1 to 0 . x d can bring the fidelity expo-nentially closer to unity, as is evident in Fig. (3). As x d is proportional to the amplitude α of the coherence state,this can be achieved while maintaining θ constant. Since θ is usually the parameter which is hard to increase inan experimental setting, this is highly advantageous. y x d . .
99 0 .
999 0 . . FIG. 3:
Contour lines for post-selected fidelity F p,y of one-bit tele-portation from the qubus to a qubit (blue), and success probabilityfor post-selection (red), as a functions of x d and y . Instead of discarding the outputs with unacceptable fi-delity, one can also use the information that the failure isheralded to recover and continue the computation. In thecase of the one-bit teleportations described here, such anapproach would require active quantum error correctionor quantum erasure codes – the type of codes necessaryfor heralded errors – which have much higher thresholdsthan general quantum error correcting codes [9]. We willnot discuss such a possibility further in this paper, andwill focus instead on post-selection for quantum gate con-struction and state preparation.
III. UNIVERSAL COMPUTATION WITHQUBUS LOGIC
Previous work by Ralph et al. [12, 13] and Gilchrist et al. [14] illustrated the construction of a universalquantum computer using what we call coherent statelogic . In these schemes a universal set of gates is ap-plied to qubit basis states defined as | (cid:105) L = | − α (cid:48) (cid:105) and | (cid:105) L = | α (cid:48) (cid:105) , using partial Bell state measurements andcat states of the form ( | − α (cid:48) (cid:105) + | α (cid:48) (cid:105) ) / √ α (cid:48) ≥ |− α (cid:48) (cid:105) and | α (cid:48) (cid:105) are approximately orthogonal since |(cid:104) α (cid:48) | − α (cid:48) (cid:105)| = e − α (cid:48) ≤ − .Using the one-bit teleportations in Fig. (1) we canalso perform a universal set of gates on a Hilbert spacespanned by the states | (cid:105) L = | α (cid:105) and | (cid:105) L = | e ± iθ α (cid:105) ,which we call qubus logic . As mentioned in the pre-vious section, the two states defined for the logical | (cid:105) L are indistinguishable when we homodyne detectalong the x -quadrature, a fact that will become im-portant later. The overlap between these basis states |(cid:104) α | e ± iθ α (cid:105)| = e − | α | (cos θ − ≈ e −| α | θ (for small θ ) is close to 0 provided αθ (cid:29)
1, so that we mayconsider them orthogonal – e.g. for αθ > .
4, wehave |(cid:104) α | e iθ α (cid:105)| ≤ − . It can be seen that ourbasis states are equivalent to the basis states of co-herent state logic given a displacement and a phaseshifter. That is, if we displace the arbitrary state a | α (cid:105) + b | αe iθ (cid:105) by D ( − α cos ( θ/ e iθ/ ) and apply the phaseshifter e i ( π − θ )ˆ n/ we have a | α sin ( θ/ (cid:105) + be iα sin( θ ) / | − α sin ( θ/ (cid:105) . If we now set α (cid:48) = α sin ( θ/ ≈ αθ/
2, forsmall θ , we see that our arbitrary qubus logical stateis equivalent to an arbitrary coherent state qubit. The e iα sin( θ ) / phase factor can be corrected once we use asingle bit teleportation. If α (cid:48) ≥ αθ ≥
4, which isalready satisfied by the approximate orthogonality con-dition αθ (cid:29)
1. It is important to note that, althoughthe basis states are equivalent, the gate constructions wedescribe for qubus logic are very different than the gateconstructions for coherent state logic.We compare qubus logic and coherent state logic basedon resource usage, i.e. the number of ancilla states andcontrolled rotations necessary to perform each opera-tion. Since the cat state ancillas needed in coherent statelogic, ( | − α (cid:48) (cid:105) + | α (cid:48) (cid:105) ) / √
2, can be made using the cir-cuit in Fig. (1a) with an incident photon in the state( | (cid:105) + | (cid:105) ) / √ α (cid:48) = α (cid:112) (1 − cos( θ )) / ≈ αθ/ , , , −
1) in qubus logic, as this is sufficient foruniversal quantum computation [22].
A. Single Qubit Gates
An arbitrary single qubit unitary gate U can be ap-plied to the state c | α (cid:105) + c | e iθ α (cid:105) by the circuit shownin Fig. (4). We first teleport this state to the qubit us-ing the circuit in Fig. (1b) and then perform the desiredunitary U on the qubit, giving U (cid:0) c | (cid:105) + c | (cid:105) (cid:1) . We canteleport this state back to the qubus mode with Fig. (1a),while the (cid:101) Z correction can be delayed until the next sin-gle qubit gate, where it can be implemented by applyinga Z in addition to the desired unitary. If it happens thatthis single qubit rotation is the last step of an algorithm,we know that this ˜ Z error will not effect the outcomeof a homodyne measurement (which is equivalent to ameasurement in the Pauli Z eigenbasis), so that this cor-rection may be ignored. In total this process requires twocontrolled rotations.Since arbitrary single qubit gates are implemented di-rectly in the two level system, the only degradation inthe performance comes from the teleportation of the statefrom the qubus to the qubit, resulting in the fidelity given - i √ i + 1 ) Uc α + c e iθ α FIG. 4:
A single qubit gate performed on c | α (cid:105) + c | e iθ α (cid:105) . in Eq. (10)In the case that we wish to perform a bit flip on thequbit c | α (cid:105) + c | e iθ α (cid:105) we can simply apply the phaseshifter e − iθ ˆ n to obtain c | e − iθ α (cid:105) + c | α (cid:105) , similarly to thebit flip gate in [13].
1. Post-selected implementation of single qubit gates
The fidelity of single qubit gates in qubus logic can beimproved simply by using post-selected teleportations.For simplicity, if we disregard the second one-bit telepor-tation which transfers the state back to qubus logic, weobtain the probability of success given in Eq. (11) andthe conditional process fidelity given in Eq. (12).
B. Two Qubit Gates
To implement the entangling CSIGN gate we tele-port our qubus logical state onto the polarisation en-tangled state (cid:0) | (cid:105) + | (cid:105) + | (cid:105) − | (cid:105) (cid:1) . The state (cid:0) | (cid:105) + | (cid:105) + | (cid:105) − | (cid:105) (cid:1) = ( ⊗ H )( | (cid:105) + | (cid:105) ) / √ H represents a Hadamard gate, can be producedoffline by any method that generates a maximally en-tangled pair of qubits. As described previously in thecontext of error correction, such a state can be producedwith controlled rotations [5]. If we start with the qubuscoherent state |√ α (cid:105) and an eigenstate of the Pauli X operator ( | (cid:105) + | (cid:105) ) / √ |√ α (cid:105) + |√ e iθ α (cid:105) . Next we put this through a symmet-ric beam splitter to obtain √ (cid:0) | α, α (cid:105) + | e iθ α, e iθ α (cid:105) (cid:1) [14].If we now teleport this state to polarisation logic withFig. (1b) we have, to a good approximation, the Bellstate (cid:0) | (cid:105) + | (cid:105) (cid:1) / √
2, and with a local Hadamard gatewe finally obtain (cid:0) | (cid:105) + | (cid:105) + | (cid:105) − | (cid:105) (cid:1) . To makethis state we have used three controlled rotations and oneancilla photon. Since we are only concerned with prepar-ing a resource state which in principle can be stored, wecan perform post-selection at the teleportations to ensurethe state preparation is of high fidelity, as described inSection II B.After this gate teleportation onto qubits, we teleportback to the qubus modes after a possible X correctionoperation. The overall circuit is shown in Fig. (5). ThisCSIGN gate requires four controlled rotations. As withthe single qubit gates, ˜ Z corrections may be necessaryafter the final teleportations of Fig. (5), but these cor-rections can also be delayed until the next single qubitgate. - c α + c e iθ α - d α + d e iθ α
12 ( 00 + 01 + 10 −
11 )
FIG. 5:
Circuit used to perform a CSIGN between states in qubuslogic.
We can see what affect the condition αθ (cid:54)(cid:29) (cid:0) | , (cid:105)| α, α (cid:105) + | , (cid:105)| α, αe iθ (cid:105) + | , (cid:105)| αe iθ , α (cid:105) + | , (cid:105)| αe iθ , αe iθ (cid:105) (cid:1) . (15)From the basis states we have defined, we expect theoutput | ψ (cid:105) = 12 (cid:0) | , (cid:105)| α, α (cid:105) + | , (cid:105)| α, αe iθ (cid:105) + | , (cid:105)| αe iθ , α (cid:105) − | , (cid:105)| αe iθ , αe iθ (cid:105) (cid:1) . (16)The unnormalised state output from Fig. (5) is | ψ ,o (cid:105) = 14 (cid:110) f ( x, α ) f ( x (cid:48) , α ) [ | (cid:105)| (cid:105) + | (cid:105)| (cid:105) + | (cid:105)| (cid:105) − | (cid:105)| (cid:105) ]+ f ( x, α ) f ( x (cid:48) , α cos( θ )) (cid:104) e − iφ ( x (cid:48) ) ( | (cid:105)| (cid:105) + | (cid:105)| (cid:105) ) + e iφ ( x (cid:48) ) ( | (cid:105)| (cid:105) − | (cid:105)| (cid:105) ) (cid:105) + f ( x, α cos( θ )) f ( x (cid:48) , α ) (cid:104) e − iφ ( x ) ( | (cid:105)| (cid:105) + | (cid:105)| (cid:105) ) + e iφ ( x ) ( | (cid:105)| (cid:105) − | (cid:105)| (cid:105) ) (cid:105) + f ( x, α cos( θ )) f ( x (cid:48) , α cos( θ )) (cid:104) e − i ( φ ( x )+ φ ( x (cid:48) )) | (cid:105)| (cid:105) + e i ( φ ( x (cid:48) ) − φ ( x )) | (cid:105)| (cid:105) + e i ( φ ( x ) − φ ( x (cid:48) )) | (cid:105)| (cid:105) − e i ( φ ( x )+ φ ( x (cid:48) )) | (cid:105)| (cid:105) (cid:105)(cid:111) , (17)where x and x (cid:48) are the outcomes of the (cid:101) Z measurements(top and bottom in Fig. (5), respectively). For simplic-ity, we disregard the final teleportations back to qubusmodes, as we have already discussed how they affect theaverage fidelity of the state in Section II. Since we havetwo homodyne measurements to consider, we need tolook at the four cases: (i) x greater than x and x (cid:48) greaterthan x ; (ii) x greater than x and x (cid:48) less than x ; (iii) x greater than x and x (cid:48) less than x ; (iv) x less than x and x (cid:48) less than x . The necessary corrections for each ofthese cases are (i) ⊗ ⊗ Z φ ( x (cid:48) ) X (iii) Z φ ( x ) X ⊗ Z φ ( x ) X ⊗ Z φ ( x (cid:48) ) X . Integrating over x and x (cid:48) for thesefour different regions, one finds the process fidelity to be F CSIGN = 14 (cid:18) (cid:18) x d √ (cid:19)(cid:19) , (18)which just corresponds to the square of the process fi-delity for a one-bit teleportation into qubits, as the onlysource of failure is the indistinguishability of the basisstates for qubus logic. A plot showing how this fidelityscales as a function of x d is shown in Fig. (6). x d P r o ce ss fid e li t y FIG. 6:
Fidelity F CSIGN of one-bit CSIGN teleportation from thequbus to a qubit, as a function of x d .
1. Post-selected implementation of the entangling gate
We can counteract the reduction in fidelity shown inFig. (6) in a similar way to the single qubit gate case, byonly accepting measurement outcomes less than x − y and greater than x + y . We find the success probabilityand conditional fidelity to be P CSIGN = 14 (cid:18) erfc (cid:18) y − x d √ (cid:19) + erfc (cid:18) y + x d √ (cid:19)(cid:19) (19) F CSIGN ,y = erfc (cid:16) y − x d √ (cid:17) erfc (cid:16) y − x d √ (cid:17) + erfc (cid:16) y + x d √ (cid:17) , (20)respectively. As before, we see that the process fi-delity can be made arbitrarily close to 1 at the cost oflower probability of success. It should also be immedi-ately clear that as y →
0, we have P CSIGN → F CSIGN ,y → F CSIGN .We see the effect of ignoring some of the homodynemeasurements in Fig. (7). Even though performance isdegraded because of the use of two one-bit teleportations,the general scalings of the fidelity and probability of suc-cess with respect to y and x d are similar to the one-bit teleportation. In particular, we see that the fidelitycan be increased significantly by increasing x d (or equiv-alently, α ). y x d . .
99 0 .
999 0 . . FIG. 7:
Contour lines for post-selected fidelity F CSIGN ,y of CSIGNteleportation from the qubus to a qubit (green), and success prob-ability for post-selection (gold), as a functions of x d and y . C. Comparison between Qubus Logic andCoherent State Logic
The total number of controlled rotations necessary toconstruct our universal set of quantum gates on qubuslogic, consisting of an arbitrary single qubit rotation anda CSIGN gate, is nine – the construction of an arbitrarysingle qubit gate required two controlled rotations andthe construction of a CSIGN gate required seven, threefor the entanglement production and four for the gate operation. This is in contrast to the sixteen controlledrotations (where we assume each controlled rotationis equivalent to a cat state ancilla) necessary for auniversal set of gates in coherent state logic [12, 13, 14],where an arbitrary single qubit rotation is constructedvia exp (cid:0) − i ϑ Z (cid:1) exp (cid:0) − i π X (cid:1) exp (cid:0) − i ϕ Z (cid:1) exp (cid:0) i π X (cid:1) ,with each rotation requiring two cat state ancilla, and aCNOT gate requiring eight cat state ancilla.As a further comparison we compare the resource con-sumption of the qubus logic scheme with the recent ex-tension to the coherent state logic scheme by Lund etal. [23] that considers small amplitude coherent states.In this scheme gate construction is via unambiguous gateteleportation, where the failure rate for each teleporta-tion is dependent on the size of the amplitude of thecoherent state logical states. Each gate teleportation re-quires offline probabilistic entanglement generation. Onaverage, an arbitrary rotation about the Z axis wouldrequire three cat state ancilla and both the Hadamardand CSIGN gate would each require 27 cat state ancilla.The scheme proposed here yields significant savingscompared to previous schemes in terms of the numberof controlled rotations necessary to apply a universal setof gates on coherent states. IV. CONSTRUCTION OF CLUSTER STATES
As we have pointed out in the previous section, theGHZ preparation scheme used for fault-tolerant errorcorrection with strong coherent beams [5] can be usedto perform CSIGN gate teleportation. This approachcan be generalised to aid in the construction of clus-ter states [6], as GHZ states are locally equivalent tostar graph states [24, 25]. Once we have GHZ stateswe can either use CNOT gates built with the aid of aqubus [11, 17] to deterministically join them to make alarge cluster state, or use fusion gates [8] to join themprobabilistically.Recent work by Jin et al. [26] showed a scheme to pro-duce arbitrarily large cluster states with a single coher-ent probe beam. In this scheme, N copies of the state( | H (cid:105) + | V (cid:105) ) / √ | H (cid:105) ⊗ N + | V (cid:105) ⊗ N ) / √ N controlled rota-tions and a single homodyne detection . However, the sizeof the controlled rotations necessary scales exponentiallywith the size of the desired GHZ state – the N (cid:48) th con-trolled rotation would need to be 2 N − − θ of order 0 .
1, once N reaches 10 wewould require a controlled rotation on the order of π ,which is unfeasible for most physical implementations. Inthe next section we describe how to prepare GHZ statesthat only require large amplitude coherent states, whileusing the same fixed controlled rotations θ and − θ . A. GHZ State Preparation and RepetitionEncoding
We mentioned a scheme in the previous section to con-struct the Bell state | (cid:105) + | (cid:105) , but this can be gener-alised to prepare GHZ states of any number of subsys-tems. We first start with the state ( | (cid:105) + | (cid:105) ) / √ |√ N α (cid:105) .This will give ( |√ N α (cid:105) + |√ N αe iθ (cid:105) ) / √
2. Sending thisstate through an N port beam splitter with N − | α (cid:105) ⊗ N + | αe iθ (cid:105) ⊗ N ) / √ | (cid:105) ⊗ N + | (cid:105) ⊗ N ) / √
2. The resources thatwe use to make a GHZ state of size N are N +1 controlledrotations, N + 1 single qubit ancillas, a single qubit mea-surement and N homodyne detections.This circuit can also function as an encoder for a quan-tum repetition code, in which case we can allow any inputqubit state a | (cid:105) + b | (cid:105) and obtain an approximation to a | (cid:105) ⊗ N + b | (cid:105) ⊗ N . In order to evaluate the performance ofthis process, we once again calculate the process fidelityby using the input state √ ( | (cid:105) + | (cid:105) ) and acting on thesecond subsystem. Using a generalisation of Eqn. (17)we calculate the effect of αθ (cid:54)(cid:29) N to be F REP = 12 N (cid:18) (cid:18) x d √ (cid:19)(cid:19) N . (21)Again, this corresponds to the process fidelity of a sin-gle one-bit teleportation into a qubit raised to the N thpower. The fidelity of preparing repetition encoded statesdrops exponentially in N . In Fig. (8) we show the fidelityas a function of x d for N = 3 and for N = 9. N=3N=9 x d P r o ce ss fid e li t y FIG. 8:
Process fidelity F REP of repetition encoding as a functionof x d . B. Post-selected Implementation of GHZ StatePreparation and Repetition Encoding
The reduction in fidelity due to αθ (cid:54)(cid:29) P REP = 12 N (cid:18) erfc (cid:18) y − x d √ (cid:19) + erfc (cid:18) y + x d √ (cid:19)(cid:19) N (22) F REP ,y = erfc (cid:16) y − x d √ (cid:17) erfc (cid:16) y − x d √ (cid:17) + erfc (cid:16) y + x d √ (cid:17) N (23)As y → P REP → F REP ,y → F REP .The effect of discarding some of states correspondingto undesired homodyne measurement outcomes can beseen in Figs. (9) and (10). Thus, as discussed in Sec-tion II B, one can prepare a state encoded in the repeti-tion code with an arbitrarily high process fidelity, regard-less of what θ and α are. The expected degradation inperformance due to the additional teleportations is alsoevident in the faster decay of the probability of successwith larger y . y x d . .
99 0 .
999 0 . . FIG. 9:
Contour lines for post-selected process fidelity F REP ,y of3-fold repetition encoding (blue), and success probability for post-selection (red), as a functions of αθ and y . V. DISCUSSION
We have described in detail various uses for one-bitteleportations between a qubit and a qubus. Using theseteleportations, we proposed a scheme for universal quan-tum computation, called qubus logic, which is a signifi-cant improvement over other proposals for quantum com-putation using coherent states. This scheme uses fewerinteractions to perform the gates, and also allows for the y x d . . . . . .
99 0 .
999 0 . . FIG. 10:
Contour lines for post-selected process fidelity F REP ,y of 9-fold repetition encoding (green), and success probability forpost-selection (gold), as a functions of αθ and y . use of post-selection to arbitrarily increase the fidelity ofthe gates given any interaction strength at the cost oflower success probabilities.The one-bit teleportations also allow for the prepara-tion of highly entangled N party states known as GHZstates, which can be used in the preparation of clusterstates. Moreover, the same circuitry can be used to en-code states in the repetition code which is a buildingblock for Shor’s 9 qubit code. In this case, where we areinterested in preparing resource states, the power andflexibility of post-selected teleportations can be fully ex-ploited, as the achievable fidelity of the state preparationis independent of the interaction strength available.The main property of the qubus which is exploited inthe schemes described here is the fact that entanglementcan be easily created in the qubus through the use of abeam splitter. Local operations, on the other hand, areeasier to perform on a qubit. The controlled rotationsallow for information to be transferred from one systemto the other, allowing for the advantages of each physicalsystem to be exploited to maximal advantage.The fidelity suffers as the operations become morecomplex, as can be seen in Figs. (11) and (12). Thisis because multiple uses of the imperfect one-bit tele-portation from qubus to qubit are used. As the processfidelity is less than perfect, error correction would have tobe used for scalable computation. However, as we havediscussed, the fact that the homodyne measurements es-sentially herald the fidelity of the operations, it is possibleto use post-selection in conjunction with error heraldingto optimise the use of physical resources.While the scheme presented has been abstracted fromparticular physical implementations, any physical reali- (a)(b) (c) (d) P r o ce ss fid e li t y x d FIG. 11:
Process fidelity as a function of x d for (a) the qubuslogic single qubit gate ( F p ); (b) the CSIGN teleportation ( F CSIGN );(c) repetition encoding with N = 3 shown in blue ( F REP ); (d)repetition encoding with N = 9 ( F REP ). y x d TELCSIGNREP 3REP 9
FIG. 12:
Contour plot showing the conditional process fidelity(solid curves) as a function of x d and y for F = 0 . N = 3 (blue) and repetition encoding for N = 9(green). The dashed curves are contour curves for the prob-ability of success for post-selection with Pr( | x − x | > y ) = 0 . sations of a qubit and a continuous variable mode wouldsuffice. The only requirements are controlled rotations,along with fast single qubit gates and homodyne detec-tion, which are necessary to enable feed-forward of resultsfor the implementation of the relevant corrections. Acknowledgments
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