An efficient, concatenated, bosonic code for additive Gaussian noise
AAn efficient, concatenated, bosonic code for additive Gaussian noise
Kosuke Fukui and Nicolas C. Menicucci Department of Applied Physics, School of Engineering, The University of Tokyo,7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Centre for Quantum Computation & Communication Technology,School of Science, RMIT University, Melbourne, VIC 3000, Australia
Bosonic codes offer noise resilience for quantum information processing. A common type of noise inthis setting is additive Gaussian noise, and a long-standing open problem is to design a concatenatedcode that achieves the hashing bound for this noise channel. Here we achieve this goal using aGottesman-Kitaev-Preskill (GKP) code to detect and discard error-prone qubits, concatenated witha quantum parity code to handle the residual errors. Our method employs a linear-time decoderand has applications in a wide range of quantum computation and communication scenarios.
Introduction. —Quantum-error-correcting codes knownas bosonic codes [1] protect discrete quantum informationencoded in one or more bosonic modes. The infinite-dimensional nature of the bosonic Hilbert space allowsfor more sophisticated encodings than the more com-mon single-photon encoding [2–4] or that of a materialqubit like a transmon [5–7]. The variety of codes al-lows for multiple ways to protect the encoded quantuminformation against decoherence at the physical level,with better than break-even performance demonstratedrecently [8, 9]. Based on the ubiquitous quantum har-monic oscillator, these versatile codes find applicationsin optical, solid-state, and vibrational systems. We callqubits encoded in a bosonic code bosonic qubits .While quantum supremacy has been demonstrated inboth solid-state qubits [10] and optics [11], the ultimategoal of a large-scale, fault-tolerant quantum computerwill require additional innovations, and its ultimate ar-chitecture remains an open question. Such a device’srequirements [12] can be broadly classified into scalabil-ity (many qubits) and fault tolerance (of good quality),and architectures designed to use bosonic qubits as theinformation carriers have recently demonstrated promi-nent advances in both areas.Progress on scalability has been most significant in op-tics through demonstrations of computationally universalcontinuous-variable (CV) cluster states [13, 14] compris-ing ∼ modes [15, 16] and measurement-based imple-mentation of CV quantum gates [17, 18]. When usedto process the bosonic qubits proposed by Gottesman,Kitaev, and Preskill (GKP) [19]—and with high enoughsqueezing—these architectures can be made fault toler-ant [20, 21].The GKP qubit [19] has emerged as a promisingbosonic qubit for fault tolerance due to its excellentperformance against common types of noise when com-pared with other known codes [1]. Experiments involvingtrapped ions [22] and superconducting circuits [23] havedemonstrated a GKP qubit, with the latter boasting asqueezing level close to 10 dB. This level is sufficient forfault tolerance in some proposed architectures [24, 25]and is approaching that required by others [26, 27]. Nu- merous proposals exist to produce these states in optics,as well (see Ref. [28] and references therein). Further-more, the GKP qubit performs well for quantum commu-nication [29–31] thanks to the robustness against photonloss [1]. In fact, recent results show that using GKPqubits may greatly enhance long-distance quantum com-munication [32, 33].A common type of noise in bosonic systems is modelledby the Gaussian quantum channel (GQC) [19, 34], alsoknown as additive Gaussian noise. While the landscapeof Gaussian operations includes other types of chan-nels [35, 36], the GQC is a particularly common one [37],and it is the one that we focus on for this work. The GQCis a simple, canonical type of noise for analyzing bosoniccode performance [19, 34]. Buoyed by the fact that dis-placements form an operator basis, protecting against theGQC allows some level of protection against all types ofbosonic noise [19]. In fact, the GKP encoding is specif-ically designed to protect against the GQC (and thusagainst bosonic noise in general), but despite this, itsperformance “out of the box” as a single-mode code issuboptimal against the GQC [19, 34].A long-standing open problem in CV quantum in-formation is to design a simple and efficient concate-nated code that achieves the hashing bound (discussedbelow) of the GQC [34]. This bound can be achievedby GKP-type codes based on high-dimensional spherepacking [34], but the authors of that work were unsat-isfied with this because such a code is not concatenatedand thus offers no obvious structure that might be ex-ploited [38, 39] to further improve its performance.Analog quantum error correction (QEC) [40] makesstrides toward achieving the goal of Ref. [34] by usingthe real-valued syndrome of a GKP qubit to improve er-ror recovery in a concatenated code. Likelihoods of theerror patterns obtained from the error syndrome are com-pared, and the most likely error pattern is selected. Infact, when used with a suitable qubit code, analog QECcan achieve the hashing bound of the GQC [24, 40]. Thiswould seem to be the end of the story, except for onemajor drawback: The decoder for analog QEC employsa type of belief propagation [41] that becomes unwieldy a r X i v : . [ qu a n t - ph ] F e b as the qubit-level code gets bigger. This is because onemust model the entire probability distribution of the mul-timode code to select the most likely CV error pattern.What we would like instead is a simple CV-level de-coder that generates discrete outcomes that can be feddirectly into a qubit-level code at the next level of con-catenation. This is the key innovation that makes furtherimprovements feasible since more complicated codes oradditional layers of concatenation do not require mod-ifying the CV-level decoding scheme, thus keeping thedecoder simple and efficient.In this work, we achieve this goal. Our innovation usesthe CV-level measurement outcome from GKP error cor-rection to decide whether to keep the qubit or discard itentirely and treat it as a located erasure error. This is asimple, local decoding step and doesn’t require compli-cated modelling of CV-level errors. The quantum paritycode (QPC) [42] is well suited to dealing with the missingqubits [43–45], and we numerically show that concate-nating the GKP code with a QPC achieves the hashingbound of the GQC with a small code and straightforwarddecoding in linear time. GKP qubit. —The GKP code encodes a qubit in anoscillator in a way that protects against errors causedby small displacements in the q (position) and p (mo-mentum) quadratures [19]. (We use conventions a =( q + ip ) / √ , [ q, p ] = i , (cid:126) = 1 , vacuum variance = 1 / .)The ideal code states of the GKP code are Dirac combsin q and in p . Physical states are finitely squeezed ap-proximations to these and are often modelled as a combof Gaussian peaks of width (i.e., standard deviation) σ with separation √ π modulated by a larger Gaussian en-velope of width /σ . Since these approximate states arenot orthogonal, there is a probability of misidentifying | (cid:105) as | (cid:105) (and vice versa) in a measurement of logical- Z ,which is implemented by a q measurement and binning tothe nearest integer multiple of √ π . Similarly, | + (cid:105) and |−(cid:105) may be misidentified when measuring logical- X with a p measurement. A qubit-level measurement error occurswhen the measured outcome is more than √ π/ awayfrom the correct outcome. Gaussian quantum channel. —The GKP code is tai-lored to combat the GQC, which randomly displaces thestate in phase space according to a Gaussian distribu-tion [19, 34]. The GQC is described by the superoperator G ξ acting on density operator ρ as ρ → G ξ ( ρ ) = 1 πξ (cid:90) d α e −| α | /ξ D ( α ) ρD ( α ) † , (1)where D ( α ) = e αa † − α ∗ a is the phase-space displacementoperator. With α = √ ( u + iv ) , the position q and mo-mentum p are displaced independently as q → q + u , p → p + v , where u and v are real Gaussian randomvariables with mean zero and variance ξ . Therefore,the GQC maintains the locations of the Gaussian peaks in the probability for the measurement outcome, but itincreases the variance of each spike by ξ in both quadra-tures. Hashing bound. —The hashing bound, first introducedfor qubit Pauli channels [46–49], may be generalised tothe GQC by maximising the one-shot coherent informa-tion over Gaussian states [34, 50]. The hashing boundis a lower bound on the quantum communication ratethrough a channel having a set level of noise. It is onlyknown to be a tight lower bound for some channels [51],and the GQC is not one of them [36]. Conversely, byfixing the rate, the hashing bound gives a level of noisethat should still allow for quantum communication at thechosen rate using some quantum-error-correcting code.When choosing a rate of zero, the (zero-rate) hashingbound represents a threshold level of noise below whichfinite-rate quantum communication should be possible—that is, it represents a target for a minimum error thresh-old when designing codes for a specific class of channels.When a code displays an error threshold that matchesthe hashing bound, we say the code “achieves the hashingbound” for the channel. In fact, concatenated qubit codesexist that exceed the hashing bound for certain qubitchannels [38, 39]—that is, they have an error thresholdthat is strictly larger than that prescribed by the hash-ing bound—leading to hopes that similar results may betrue for concatenated codes applied to the GQC [34].The (zero-rate) hashing bound for the GQC G ξ , is thestandard deviation [34, 50] ξ HB := 1 √ e ≈ . . (2) Noise model. —GKP error correction, in both its origi-nal [19] (Steane-style [53]) form and in its teleportation-based [54] (Knill-style [55]) form, involve measuring thedeviation of the state’s support in each quadrature ( q, p )away from an integer multiple of √ π . These measure-ment outcomes—each of the form s m = n √ π +∆ m , where n is an integer and | ∆ m | ≤ √ π/ —together form the syn-drome. Normally, each value of ∆ m locally determinesthe displacement to apply in order to correct the error—either snapping back to grid in the original method [19]or applying a logical Pauli in the teleportation-basedmethod [54]. Analog QEC [24, 40] instead feeds all thesereal-valued syndrome data s m directly to a higher-leveldecoder, which makes a global decision. Our proposalkeeps aspects of both approaches. We use ∆ m to locallydecide whether to try to correct the error or to give upand report the qubit as lost to the next-level decoder.We model a damaged GKP codeword as an idealone [19] that has been displaced by a definite (but un-known) amount in each quadrature. This approximatelymodels the errors due to both coherent and incoherentnoise [20, 24, 40, 52] and simplifies the analysis. Dueto the √ π -periodicity of all GKP codewords, displace-ments by u are equivalent to those by u + 2 k √ π for any FIG. 1. Effect of additive Gaussian noise [52] on meassur-ing a GKP qubit. (a) Effect of shift by u (mod 2 √ π ) , dis-tributed according to p ( u ) [Eq. (3)], on an ordinary measure-ment of a GKP qubit [19]. (b) The highly reliable measure-ment (HRM) [24] flags outcomes in the δ -wide “danger zone”(yellow) as unreliable. (c) Postselected error probability of theHRM for several values of δ . (d) Corresponding success prob-ability. Note: ( Squeezing level in dB ) = −
10 log ( σ /σ vac ) ,where the vacuum variance σ vac = . integer k . Thus, given any initial distribution p ( u ) of theunknown displacement u in a single quadrature, its effecton a GKP codeword is captured by folding p ( u ) into the wrapped distribution p ( u ) = (cid:80) k ∈ Z p ( u + 2 k √ π ) , whosedomain is [ −√ π, √ π ) . When p is a zero-mean Gaussianof variance σ , it wraps into p ( u ) = 12 √ π ϑ (cid:18) − u √ π , iσ (cid:19) , (3)where ϑ ( z, τ ) = (cid:80) m ∈ Z exp (cid:2) πi (cid:0) m τ + mz (cid:1)(cid:3) is a Ja-cobi theta function of the third kind. Figure 1(a)shows this distribution and the logical effect of a shiftby u (mod 2 √ π ) on measuring a GKP codeword. Highly-reliable measurement. —Logical errors resultwhen the GKP syndrome value s m , which is wrapped mod √ π , misidentifies u as u ± √ π [19]. The highly-reliable measurement (HRM) [24] buffers against thispossibility by introducing a “danger zone” of out-comes ≤ √ π/ −| ∆ m | < δ for some δ > . Outcomes inthis zone are flagged as unreliable, with δ → recoveringthe usual case [19]. This corresponds to flagging as unre- liable any displacement u (mod 2 √ π ) that falls within δ of a crossover point ±√ π/ , as shown in Fig. 1(b). Whenthe HRM flags a result s m as unreliable, the correspond-ing qubit is discarded and treated as a located erasureerror ( s m → E ), while otherwise the result is kept andbinned as usual ( s m → ± ) [19]. The HRM is thus aternary (3-outcome) decoder for GKP qubits that mapseach raw CV outcome s m from R → {± , E } .Given a definite displacement u ∈ [ −√ π, √ π ) , wedefine probabilities for three cases: the measurementresult is correct, P ( c ) = Pr( | u | < √ π/ − δ ) ; the re-sult is incorrect, P ( i ) = Pr( | u | > √ π/ δ ) ; or theresult is unreliable and the qubit discarded, P ( d ) =Pr( − δ < | u | − √ π/ < δ ) . We further define the suc-cess probability , − P ( d ) , as the probability the qubitwas not discarded and the postselected error probability , P ( i )post = P ( i ) / (1 − P ( d ) ) , as the probability of getting anincorrect outcome within the sample of qubits that arenot discarded. Decreasing the postselected error prob-ability (by increasing δ ) reduces the success probabil-ity [24], as shown in Fig. 1(c,d). Trading errors for loss —The HRM is the key to achiev-ing the hashing bound of the GQC without the compu-tational overhead of analog QEC. Analog QEC requiresmodelling the joint likelihood of real-valued outcomesover multi-mode codewords, while the HRM maps lo-cally detected unreliable results to lost qubits at knownlocations.Loss-tolerant QEC codes were originally proposed toovercome loss of individual photons—the main hurdle inQC based on a photonic qubit. Here, concatenating GKPqubits with one of these codes compensates for the dis-carded (“lost”) qubits due to using the HRM. This trade-off of (unlocated) errors for (located) erasure makes thelogical qubit more robust. In the following, we concate-nate GKP qubits with the QPC proposed by Ralph etal. [42] and implement teleportation-based QEC as pro-posed by Muralidharan et al. [43].The ( n, m ) -QPC [43] is an nm -qubit code built from n blocks of m qubits. Logical basis states are |±(cid:105) L =2 − n/ (cid:0) | (cid:105) ⊗ m ± | (cid:105) ⊗ m (cid:1) ⊗ n . In our code, the physical qubitstates are square-lattice GKP states [19] of a singlebosonic mode—i.e., | (cid:105) = | GKP (cid:105) , and | (cid:105) = | GKP (cid:105) .We implement syndrome measurements by adaptingthe teleportation-based protocol of Ref. [43] to this set-ting, as shown in Fig. 2(a). Using teleportation to do thecorrection guarantees that the output state is already inthe logical subspace, and only logical corrections are re-quired at the end [55]. We use HRMs to implement thebase-level qubit measurements, so we get (after postse-lection) either a qubit value ( ± ) or a located erasure ( E )for every measured qubit. These outcomes are processedinto logical outcomes using the method of Ref. [43], whichis described briefly in Fig. 2(b,c). These logical outcomesdetermine the logical correction to apply at the end. In FIG. 2. Trading errors for loss to improve error correction.(a) Teleportation-based QEC using GKP qubits [19] concate-nated with the quantum parity code (QPC) [42, 43]. Allgates are shown at the logical level. The feedforward op-erations D p and D q implement logical-level X and Z opera-tions via physical-level displacements [19]. The outcomes ofthe logical Bell measurement decide what feedforward is re-quired [43]. (b) Encoded measurement with the ( n, m ) -QPCin the X basis ( n = 5 , m = 4 shown). Performing the HRMin p at the physical level [Fig. 1(b)] gives, for each qubit, ei-ther a binary outcome ( ± ) or a located erasure error ( E ).The latter is indicated in red (with the unreliable binary out-come underneath) and occurs when the CV-level outcome isin the “danger zone” (see Fig. 1) and therefore unreliable. Ahorizontal block is ignored if it has any discarded outcomes.All remaining blocks have their parity (product) taken, afterwhich a majority vote of those parities determines the log-ical outcome—with heralded failure if there is no majority.(c) Encoded measurement with the (5 , -QPC in the Z ba-sis. This is similar to (b), but this time the HRM is done in q ,and majority voting within a block precedes taking the parityof the blocks’ voting outcomes—with heralded failure if anyblock has no majority. See Ref. [43] for more details. Withoutpostselection, both logical outcomes would have been flipped(likely due to uncorrected errors in the discarded values). the example shown, X L = − and Z L = − , so we per-form logical bit- and phase-flip operations on the outputqubit to correct it. At the physical level, these occur viadisplacements by integer multiples of √ π [19]. Numerical simulation. —We evaluate our proposedQEC method using a Monte Carlo simulation of thecircuit in Fig. 2(a). The input state passes through aGQC [Eq. (1)]. Our model simulates code-capacity noise(i.e., assuming no errors aside from the channel noise it-self [56, 57]) in order to evaluate the best possible perfor-mance of our code under the GQC and to compare with
FIG. 3. failure probabilities using the ( n, m ) -QPC, as shownin Fig. 2, for (a) δ = 0 (conventional GKP error correc-tion [19]) and (b) an optimized value of δ = 0 . √ π . The for-mer (a) has a threshold ξ ≈ . (identified by the crossoverpoint), which matches previous work [19, 34]. The latter (b)has a threshold ξ ≈ . , which numerically achieves thehashing bound of the GQC [Eq. (2)]. previous results [19, 24, 34, 40]. We consider two cases:(a) the conventional case, which corresponds to choos-ing δ = 0 , and (b) an optimized choice of δ = 0 . √ π (explained below).A logical- X (- Z ) error occurs when the decoding pro-cedure leads to either (1) the wrong value for X L ( Z L ) or(2) heralded failure of that measurement—see Fig. 2(b,c),caption. While one could treat the heralded failuresin a more sophisticated fashion (concatenating with ahigher-level code, for instance), in our simulations wesimply treat these as logical errors on that outcome.The probability of a logical- X (- Z ) error is denoted E X ( E Z ), and the overall logical error probability p E is − (1 − E X )(1 − E Z ) , which we call the failure probability of the QEC.Figure 3 shows the performance of both methods as afunction of the standard deviation of the GKP qubit forseveral sizes ( n, m ) of the QPC. These sizes are chosento ensure approximately symmetric noise at the output(i.e., E X ∼ E Z ). The conventional method (a), whichuses δ = 0 , gives a threshold of ξ ≈ . , matchingprevious work with concatenated codes and simple de-coding [19, 34]. For our new method (b), we optimizethe value of δ to maximize the threshold value of thestandard deviation, giving δ = 0 . √ π , which gives aloss probability ∼ at threshold. Importantly, thecorresponding threshold numerically achieves the hash-ing bound of the GQC: ξ ≈ . [Cf. Eq. (2)]. Conclusion. —The key insight of this work is thatone does not need to model the full likelihood func-tion [24, 40] to correctly interpret GKP syndrome infor-mation [19] within a concatenated code. Instead, thereal-valued outcomes can be coarse grained to one ofthree qubit-level outcomes through the HRM mapping R → {± , E } , where E represents an untrustworthyvalue. These ternary outcomes suffice to achieve thehashing bound of the GQC by treating E outcomes aserasure errors and concatenating with a qubit-level codedesigned to handle such errors [42–45].The innovation of this work over previous hashing-bound-achieving methods [24, 40] lies in the efficiencyand versatility of the decoder. Respectively, (1) decod-ing happens in linear time since the CV-level decoding isentirely local; and (2) the HRM wraps each GKP qubitin a simple error-detecting code, so concatenating withany qubit-level code designed to handle erasures [58–60]should benefit from this type of outcome mapping. Fur-ther applications and extensions include improved decod-ing in GKP-based architectures (e.g., [26, 27]) and incodes that exploit biased noise (e.g., [57, 61, 62]). Acknowledgments. —We thank Joe Fitzsimons, TimRalph, Ben Baragiola, and Giacomo Pantaleoni for dis-cussions. We acknowledge the organizers of the BBQ2019 workshop, where early results of this work werepresented. This work is supported by the Australian Re-search Council (ARC) Centre of Excellence for QuantumComputation and Communication Technology (ProjectNo. CE170100012). K.F. acknowledges financial supportfrom donations from Nichia Corporation. [1] V. V. Albert, K. Noh, K. Duivenvoorden, D. J. Young,R. Brierley, P. Reinhold, C. 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