Resource reduction for distributed quantum information processing using quantum multiplexed photons
Nicolo Lo Piparo, Michael Hanks, Claude Gravel, Kae Nemoto, WIlliam J. Munro
aa r X i v : . [ qu a n t - ph ] M a y Resource Reduction For Distributed Quantum Information Processing UsingQuantum Multiplexed Photons
Nicolò Lo Piparo, ∗ Michael Hanks, Claude Gravel, Kae Nemoto, and William J. Munro
2, 1 National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan. NTT Basic Research Laboratories & NTT Research Center for Theoretical Quantum Physics,NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa, 243-0198, Japan. (Dated: May 29, 2020)Distributed quantum information processing is based on the transmission of quantum data overlossy channels between quantum processing nodes. These nodes may be separated by a few micronsor on planetary scale distances, but transmission losses due to absorption/scattering in the channelare the major source of error for most distributed quantum information tasks. Of course quantumerror detection (QED) /correction (QEC) techniques can be used to mitigate such effects but errordetection approaches have severe performance limitations due to the signaling constraints betweennodes and so error correction approaches are preferable — assuming one has sufficient high qualitylocal operations. Typically, performance comparisons between loss-mitigating codes assume oneencoded qubit per photon. However single photons can carry more than one qubit of information andso our focus in this work is to explore whether loss-based QEC codes utilizing quantum multiplexedphotons are viable and advantageous, especially as photon loss results in more than one qubit ofinformation being lost. We show that quantum multiplexing enables significant resource reduction:in terms of the number of single photon sources while at the same time maintaining (or evenlowering) the number of two-qubit gates required. Further, our multiplexing approach requires onlyconventional optical gates already necessary for the implementation of these codes.
There are many active approaches being pursued in thedevelopment of quantum technologies, including those as-sociated with imaging and sensing [1–3], communication[4–9] and computation [10–15]. What has become clear isthat many of these will be distributed in nature [5] and,as such, it will be essential to share quantum informationbetween the remote nodes, regardless of whether thosenodes are separated on the atomic or planetary scales[16–18]. This distributed nature means we are going torequire both a quantum interface between matter & pho-tonic qubits and a photonic bus to transfer such informa-tion between nodes [19]. However, real implementationswill suffer from losses, which will dramatically affect theperformance of the quantum protocols in which such de-vices are being used. Mechanisms must be developed tomitigate such detrimental effects.There are quite a number of routes available to offsetloss effects, ranging from the development of lower lossfibers to more efficient quantum information coding. Thelatter route is quite appealing as it can be used with cur-rent technology and is likely to be more compatible withour existing infrastructure. There is a well known set ofloss based quantum detection and error correction codesthat can be used in this situation. In [20] they discuss asimple quantum network scenario in which the quantummultiplexing (QMu) of photonic degrees of freedom al-lows one to design a single-step combined entanglementdistribution and error detection protocol with improvedentanglement generation rates, using fewer physical (pho-tons and quantum memories) and temporal resources. ∗ [email protected] However, their performance is still limited by the proba-bilistic nature of the various quantum operations and theresulting necessary heralding signals.Quantum error correction codes (ECCs) naturallyavoid a heralding bottleneck, with example loss basedcodes including the quantum parity [21], cat [22], bi-nomial [23], Reed-Solomon [24], surface [25] and GKPcodes [26]. They allow us to approach the deterministictransmission of quantum information over a lossy chan-nel, as long as those total losses do not exceed a certainthreshold (50% at most) [27, 28]. Typical encodings useeither the polarization or time bin degrees of freedombut are not particularly resource efficient as they requirea large number of single photons. The creation of reli-able single photon sources has proved challenging sincethe generation probability does not exceed 70% [29–32],whereas, on the contrary, single-qubit gate fidelities canreach 99% [33–35]. Another limiting factor comes fromtwo qubit gates, which require longer times with fideli-ties below 90% [36–38]. However, single photons have thepotential to carry much more information using differentdegrees of freedom (see Supplemental Material for fur-ther detail). Hence the natural question is whether usingmultiple degrees of freedom is advantageous, in terms ofreducing the number of photons while maintaining thesame number of two qubit gates.Here we investigate the potential of quantum multi-plexing to reduce the resources required to implementloss based error correction codes. We take as a centralfigure of merit the required number of single photonsas well as qubits. We analyze two well known ECCs,the redundant quantum parity [21] and quantum Reed-Solomon codes [24], determining the number of photonsand qubits required to reach a threshold success proba-bility with the multiplexing method.Let us begin by exploring the redundant quantumparity code [21] in a photon transmission regime forwhich both the number of qubits (memories within thenode) and the number of photons can be reduced us-ing our quantum multiplexing approach, all while main-taining the near deterministic transmission of informa-tion between the two nodes. In the redundant quantumparity code the information α, β in our encoded state | ψ i ( n, m ) = α | + i ... | + i n + β |−i ... |−i n (each blockterm |±i i = | H i ⊗ m ± | V i ⊗ m containing m photons) issuccessfully transmitted over the channel when at leastone block of m qubits arrives intact (no losses) and eachother block retains at least one photon (see Figure 1ainset). The success probability is given by [39] P S = [1 − (1 − p t ) m ] n − [1 − p mt − (1 − p t ) m ] n , (1)where p t is the single photon transmission probabilitythrough the channel. Our first observation is that thisconcatenated code is not particularly resource efficient asthe number of qubits at the first logical layer, m , growsinversely with the transmission probability p t . Further, n grows inversely with p mt and so ( m, n ) grow exponentiallywith distance between nodes. Our quantum multiplexeris a natural solution [20]: here we encode multiple qubitsonto a single photon, meaning less photons in total needto be transmitted. More specifically, we enact a two qubitgate between the first degree of freedom (polarization)and a second photonic or matter qubit. Then, swappingthe polarization of the initial photon with another de-gree of freedom (time bin in this case), a third systemcan then interact independently with the the polariza-tion of this same photon (further details in SupplementalMaterial). The quantum mutliplexer has many potentialbenefits including deterministic operations between dif-ferent degrees of freedom — especially important whensingle photon sources are probabilistic in nature.Let us explore this in a little more detail. In the inset ofFig. (1a.i) we illustrate a six photon redundant quantumparity code realization without the use of quantum mul-tiplexing in which 3 blocks of 2 photons each are used.After the photons are transmitted over the lossy chan-nel, the code is successful if at least one block containstwo photons and the other two blocks each contain oneor more photons. One can think of substituting those 6photons with three quantum multiplexed photons eachcarrying two qubits of information. In Fig. (1a.ii) theseare represented by the colored lines connected to the dotscontained in the blocks (see Supplemental Material forfurther details). In this case, the ECC code only toleratesthe loss of one photon. Therefore it would seem logicalthat we can reduce the number of photons by using themultiplexing approach, provided that the success prob-ability is above the desired threshold value. This raisesthe question as to what the success probability P QMu S willbe in this quantum multiplexed approach. One can show for n tot transmitted photons that P QMu S = n ∗ X i =0 h(cid:16) n tot i (cid:17) − ( U i + E i ) i p n tot − it (1 − p t ) i , (2)where U i , E i are the number of events in which losing i photons will leave none of the blocks with the initialnumber of qubits or at least one empty block and n ∗ isthe number of lost photons the ECC code can tolerate.We need to determine both U i and E i , which are highlydependent on how the quantum multiplexed photons areconnected to the blocks (see Fig. (1b)). Different config-urations lead to different success probabilities. We canalso release the constraints of all blocks having to havethe same number of qubits (an unbalanced configura-tion), which is typically not utilized in error correctionschemes. This enables us to further reduce the numberof qubits (and photons) even in the non-multiplexed case(see Supplemental Material).In Figure (1a) we plot the overall success prob-ability P S versus p t for two non-multiplexed (equaland unbalanced) configurations alongside one quantum-multiplexed situation with a minimum threshold successprobability requirement of P S = 0 . (typical for manyquantum computation based tasks). It is clear that our 3photon quantum multiplexed case (3 blocks with 2 qubits/ photon) dramatically outperforms the traditional 6photon non-multiplexed case (3 blocks with 2 qubits,photons each). In the region . . p t . . the6 photon case does not reach our threshold target, whilethe 3 photon multiplexed approach does. The 7 photonconfiguration (with the first block containing 3 photonswhile the second and third blocks contain 2 photons each)performs slightly better than the multiplexed case. How-ever both are above the threshold and the multiplexedsituation uses fewer photons, qubits and two qubit gates.The multiplexed approach also halves the number of pho-tons in the region . . p t ≤ . . These are criticalresource savings.It is clear that the lower p t is, the more qubits (twoqubit gates) and photons we will need to reach P S . Itis important, in reducing the total numbers of these re-sources, to also explore unbalanced quantum multiplex-ing configurations. In Figure (2a) we plot the minimalnumber of qubits, N min , and photons, n min , versus p t for resource-optimal configurations with 2 and 4 qubitsper photon. Quantum multiplexed systems utilize fewerphotons, however the number of qubits is either the sameor slightly higher, except in a small region near p t ∼ . (Fig. (1a)). In fact we can almost halve the numberof photons being transmitted over the channel — quitean advantage, especially as single photon sources are cur-rently not as efficient as quantum gates or measurements.The number of qubits can be maintained equal to thenon multiplexing case, while reducing the number of pho-tons, with a mixed strategy, in which each photon cancarry an arbitrary number of qubits (from 1 to 4). Ta-ble I shows the total number of photons and the total S u cc e ss p r obab ili t y p t p t P S Q M u (a) (b) Figure 1. (a) Plot of the overall success probability versus photon transmission probability p t of the redundant quantum paritycode with (blue curve) and without (red and yellow curves) quantum multiplexed photons. The inset of (a) depicts a schematicillustration of a particular instance of the six-qubit quantum redundancy parity code in which each photon carries one qubit(i) and three photons carry two qubits of information each (2q/p) (ii). Similarly in (b), we show the success probability P QMu S versus p t for three different configurations of a quantum multiplexed system. Here 6 photons carry 3 qubits each, distributedover 6 blocks (each block containing 3 qubits). p t N m i n , n m i n p t (a) (b) N min n min Figure 2. Plot of the minimum number of qubits (solid lines) and photons (dotted lines) for the (a) redundant quantum parityand (b) Reed-Solomon codes required to reach a threshold success probability of P S = 0 . versus the photon transmissionprobability p t using a quantum multiplexed encoding of 2 − ∗ (mixed) 1511 (2q/p) 2211, 10, 9, 8 (mixed) 217 (3q/p) 21Table I. Minimum number of photons and qubits requiredto reach our overall information transfer success probabilitythreshold of P S = 0 . with p t = 0 . . Similar results areseen for most values of p t . The star corresponds to the optimal case, in which, by using the mixing strategy, for a given N min we reach the lowest n min for a specific value of p t . number of qubits needed for reaching P S at p t = 0 . using the pure and the mixed strategies. We observethat we can reach the required P S with a lower num- ber of photons (12) given the same number of qubits(15) when we apply the mixed strategy. The numberof two qubit gates required is therefore the same as forthe non-multiplexing case, even for bigger codes. Thisfurther highlights the potential advantages of quantummultiplexing. Can these improvements be generalized toother loss based quantum error correction codes?In the quantum Reed–Solomon [[ d, k − d, d − k + 1]] d code information is encoded in d qudits, with the codefailing on the loss of d − k + 1 out of d qudits. Forcomparative purposes, we will express the degree of mul-tiplexing as q qubits of information per photon. Whenwe encode the qudits in these q degrees of freedom ofquantum multiplexed photons, any qudit of informationdepends upon the successful transmission of ⌈ log ( d ) /q ⌉ photons [24]. The probability of failure is therefore P F ail = d X j = d − k +1 (cid:18) dj (cid:19) (1 − p ⌈ log2( d ) q ⌉ t ) j p ⌈ log2( d ) q ⌉ ( d − j ) t . (3)In this code the block is given by the total number of pho-tons encoding a single qudit, and if a block is incomplete,the qudit is not successfully transmitted. Therefore, theperformance can be improved by maintaining indepen-dence between these blocks, and by reducing the chancesfor loss events within any single block. Adding additionalquantum multiplexing will help so long as it preservesindependence between qudit loss events. For the quan-tum Reed-Solomon code we can also determine the lowestnumber of qubits and photons required to reach P S , asshown in Fig. 2b. Here, the advantage of using quantummultiplexed photons is evident in terms of a reductionof the number of qubits, two qubit gates and photonscompared to the no quantum multiplexing case. In par-ticular, the higher is the quantum multiplexing degreethe less qubits and photons we require. For instance,at p t = 0 . , we have that for q = 4 , N min ≃ and n min ≃ , whereas when no quantum multiplexed pho-tons are in use, we have that both N min and n min are over1000. As p t gets lower, the number of photons and qubitsincreases considerably, hence, we need to use higher de-grees of quantum multiplexing. Furthermore, by com-paring Fig. (2a) with Fig. (2b), we infer that there isalways a specific value of q for which the Reed-Solomoncode requires a lower number of resources compared tothe parity code (for q = 4 , at p t = 0 . , N min ( n min ) is72%( 75%) lower for the Reed-Solomon code than theparity code). For other error correction codes based onthe transmission of qudits we expect the same reductionin the number of qubits, two qubit gates and photonswhen quantum multiplexing is in use.There are other loss codes based on encoding informa-tion in superposition of photon number (bosonic [23] andGKP [26] for instance), in which quantum multiplexingis ineffective. In these cases this would correspond to theassignment of information about multiple excitation tothe various degrees of freedom of a single mode. How-ever, any quantum multiplexed photon mode is equiva-lent, in this case, to a no quantum multiplexed mode.There is always therefore a code using fewer excitationsand a higher number of modes than the original that willperform as well as the quantum multiplexed case.It is essential to compare these quantum multiplexedcodes to the best loss codes currently known - namelythe GKP codes [40, 41]. In particular, in [41] the au-thors show that the hexagonal GKP code is the optimalamong all single-mode bosonic codes against loss errorsexpressed as a Gaussian displacement channel. In Fig.(2a, b) we plot (black curves) the average number ofphotons for the hexagonal GKP code [42]. This suggeststhat there are regions where the GKP code has a bet-ter performance and other regions where this is reversed.The multiplexed codes operate better in the higher lossregimes. Further, a critical consideration has to be thenear deterministic implementation of the code itself. Ourquantum multiplexing approach requires the same basic two qubit/qudit gates needed for quantum logic (and theoriginal codes themselves) with the addition of high effi-ciency optical switches to swap state between the differ-ent degrees of freedom. On the other hand, the initializa-tion of the GKP code is quite demanding to achieve in anear-deterministic way and necessitates a more complexcontinuous variable procedure, though Gaussian opera-tions are sufficient for subsequent qubit control. Generat-ing such codes in a heralded but probabilistic fashion hasbeen achieved but unfortunately increases the resourcesrequired [43, 44] (see Supplemental Material). This in-dicates that additional resources will be required at theend nodes to process the quantum data transmitted overthe communication channel. We note that a proof-of-principle demonstration of deterministic preparation wasperformed in [44], and look forward to the developmentof this promising approach going forward.To summarize, we have shown how quantum multi-plexed loss codes have the potential to significantly de-crease the resources required to transfer quantum infor-mation between two adjacent nodes. This is achievedwhile maintaining or even lowering the required numberof two-qubit gates. Two primary error correction codeswere considered: the redundant quantum parity codeand the quantum Reed-Solomon code. For the former,we found that the total number of single photons thatneed to be transmitted through the channel can be dra-matically reduced (near 50 percent) without significantlyincreasing the number of qubits. Further, we found itadvantageous for individual photons to have different de-grees of quantum multiplexing, as well as for blocks tocontain different numbers of qubits. The quantum Reed-Solomon code significantly outperforms the redundantquantum parity code and, using quantum multiplexedqudits, has the potential to reduce simultaneously thenumber of photons, qubits and gates used. These im-provements should be possible in many (but not all) ofthe other loss based error correction codes when quantummultiplexing is used. Quantum multiplexing has the po-tential to be a new resource saving tool especially fornear term implementations. Our findings can be appliedto any communication system that needs error correctionto improve its efficiency, such as in quantum repeaters,quantum computation and quantum sensing. We thank Koji Azuma for useful discussions. Thisproject was made possible through the support of theMEXT KAKENHI Grant-in-Aid for Scientific Researchon Innovative Areas “Science of Hybrid Quantum Sys-tems” Grant No. 15H05870, the MEXT QuantumLeap Flagship Program (MEXT Q-LEAP) Grant No.jp-mxs0118069605, and a grant from the John Temple-ton Foundation (JTF [1] D. S. Simon, G. Jaeger, and A. V. Sergienko, Int. J.Quantum Inform. , 1430004 (2014).[2] L. A. Lugiato, A. gatti, and E. Brambilla, J. Opt. B ,176 (2002).[3] C. L. Dogen, F. Reinhard, and P. Cappellaro, Rev. Mod.Phys. , 035002 (2017).[4] C. H. Bennett and G. Brassard, Theoretical computerscience , 7 (2014).[5] W. J. Munro, K. Azuma, K. 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