All-Gaussian Universality and Fault Tolerance with the Gottesman-Kitaev-Preskill Code

PHYSICAL REVIEW LETTERS 123, 200502 (2019)

Ben Q. Baragiola ,1 Giacomo Pantaleoni,1 Rafael N. Alexander,2 Angela Karanjai,3 and Nicolas C. Menicucci1 1Centre for Quantum Computation and Communication Technology, School of Science, RMIT University, Melbourne, Victoria 3000, Australia 2Center for Quantum Information and Control, Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, USA 3Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia

(Received 4 March 2019; published 13 November 2019)

The Gottesman-Kitaev-Preskill (GKP) encoding of a qubit within an oscillator is particularly appealing for fault-tolerant quantum computing with bosons because Gaussian operations on encoded Pauli eigenstates enable Clifford quantum computing with error correction. We show that applying GKP error correction to Gaussian input states, such as vacuum, produces distillable magic states, achieving universality without additional non-Gaussian elements. Fault tolerance is possible with sufficient squeezing and low enough external noise. Thus, Gaussian operations are sufficient for fault-tolerant, universal quantum computing given a supply of GKP-encoded Pauli eigenstates.

DOI: 10.1103/PhysRevLett.123.200502

Introduction.—The promise of a quantum computer lies non-Gaussian resource (preparation, gate, or measurement) in its ability to dramatically outpace classical computers for for universal QC [16–18]. Further, Gaussian QC alone is certain tasks [1]. Computation using operations restricted to insufficient to correct Gaussian noise [19]. Pauli-eigenstate preparation, Clifford transformations, and Fault tolerance requires discrete quantum information. Pauli measurements—henceforth referred to as Clifford Bosonic quantum error-correcting codes (bosonic codes quantum computing (QC)—cannot outperform classical for short) embed discrete quantum information into CV computation since it is efficiently simulable on a classical systems in a way that maps CV noise into effective computer [2]. Universal quantum computation requires logical noise acting on the encoded qubits [20–23]. Such supplementing Clifford QC by a non-Clifford resource— codes are promising for fault-tolerant computation that is, a preparation, gate, or measurement that is not an [24,25] due to the built-in redundancy afforded by their element of Clifford QC. infinite-dimensional Hilbert space. High-precision con- In the presence of noise, universality is not enough. The trollability of optical-cavity [10,12,13] and vibrational celebrated Threshold Theorem [3] proves that, given low [14] modes further enhances their appeal. With a bosonic enough physical noise, quantum error correction can be code, one may define “logical-Clifford QC,” compri- used to reduce logical noise to arbitrarily low levels—a sing encoded Pauli eigenstates and logical-Clifford property called fault tolerance [4]. Fortunately, Clifford QC operations—allowing error correction at the encoded- provides all the necessary tools for quantum error correc- qubit level. This, too, is efficiently simulable and thus tion. The question then is how to augment Clifford QC such requires additional logical-non-Clifford resources for that the result is both universal and fault tolerant. One way fault-tolerant universality. is to use a non-Pauli eigenstate, referred to as a magic The Gottesman-Kitaev-Preskill (GKP) encoding of a state [5]. qubit into an oscillator [21] is currently experiencing The continuous-variable (CV) analog of Clifford QC is significant theoretical [26–29] and experimental [14,30] Gaussian QC, which includes Gaussian state preparation, interest due to its favorable error-correction properties [31], Gaussian operations (i.e., Hamiltonians quadratic in a;ˆ aˆ †), integration into scalable CV cluster states for measurement- and homodyne detection. CV systems arise naturally based QC [32–34], and all-Gaussian Clifford gates and in many quantum architectures, including optical modes measurements. That is, the GKP encoding is the only [6–9], microwave-cavity modes [10–13], and vibrational known bosonic code for which logical-Clifford QC and modes of trapped ions [14]. Gaussian QC lends itself to error correction require only Gaussian QC along with a optics because the nonlinearities required are limited and supply of logical-Pauli eigenstates, which are non-Gaussian of low order and because homodyne detection is very high [35]. Until now, fault-tolerant universal QC with the GKP efficiency. However, Gaussian QC is efficiently simulable code has required an additional non-Gaussian element— by a classical computer [15] and requires any single cubic-phase gate, cubic-phase state, or logical magic state

[21,36,37]. In this Letter, we show that no such additional We assume that the physical GKP Pauli eigenstates used in non-Gaussian element is required. the following analysis are high quality enough to enable Specifically, we show that high-quality magic states for fault-tolerant GKP Clifford QC. This allows us to approxi- both square- and hexagonal-lattice GKP codes [22] can be mate them as ideal states with noiseless Cliffords for the produced by applying GKP error correction to vacuum or purpose of magic-state preparation [5,39]. We justify this in low-temperature thermal states. The result is that Gaussian the penultimate section. QC and just one type of non-Gaussian resource—a high- Kraus operator for GKP error correction.—In its quality GKP Pauli eigenstate—suffice for both universality original formulation [21], GKP error correction is a and fault tolerance. quantum operation designed to correct an encoded qubit Notation and conventions.—Here we define notation that has acquired some noise (leakage of its state outside and conventions to be used throughout this Letter. We of the logical subspace) by projecting it back into the define position qˆ ≔ 1=p2 aˆ aˆ † and momentum pˆ ≔ GKP logical subspace, possibly at the expense of an ð Þð þ Þ unintended logical operation. Standard implementations −i=p2 aˆ − aˆ † for anyffiffiffi mode aˆ. This means q;ˆ pˆ i, ð Þð Þ 1 ½ ¼ of error correction strive to avoid these unintended logical with a vacuumffiffiffi variance of 2 in each quadrature, and ℏ 1. ˆ ¼ operations (residual errors). In what follows, we apply the The Weyl-Heisenberg displacement operators X s ≔ machinery of GKP error correction to a known Gaussian − ˆ ˆ ˆ ð Þ e isp and Z s ≔ eisq displace a state by s in position and state, which means the outcome-dependent final state is ð Þ þ momentum, respectively. For brevity, we also define a joint known perfectly. ˆ ˆ ˆ T displacement V s ≔ Z sp X sq , where s sq;sp . GKP error correction (EC) [21,40] proceeds in two steps: The functionsðψÞ s ≔ð Þs ψð andÞ ψ˜ s ≔¼ðs ψ denoteÞ First, one quadrature is corrected, then the conjugate ð Þ qh j i ð Þ ph j i position- and momentum-space wave functions for a state quadrature. We define the Kraus operator that corrects just ˆ q ψ , respectively (the tilde indicates momentum space). the q quadrature KEC t via the circuit (read right to left) Anyj i function, including wave functions, can be evaluated ð Þ with respect to position, φ qˆ ≔ dsφ s s s , to pro- ð Þ ð Þj iqqh j duce an operator diagonal in theR position basis—and similarly for momentum. Finally, we define Ш x ≔ Tð Þ Z δ x − nT as a Dirac comb with spacing T. n∈ ð Þ The GKP encoding.—In the original square-lattice GKP ˆ iqˆ⊗qˆ P where the controlled operation is CZ e , and t ∈ R is encoding [21], the wave functions for the logical basis states the measurement outcome. This circuit¼ differs from the 0 ; 1 are Dirac combs in position space with state- fj Li j Lig original [21] in that the correction here is a negative dependent offset: ψ s Ш2 s − jpπ for j ∈ 0; 1 . j;Lð Þ¼ pπð Þ f g displacement by t rather than by t rounded to the nearest Their momentum-space wave functions are also Dirac ffiffi ffiffiffi integer multiple of pπ. The outputs may differ by a logical combs but with no offset, different spacing, and a relative ˆ operation X pπ ,ffiffiffi but this is unimportant because the phase between the spikes: ψ˜ s 1 −1 js=pπШ s . ðÆ Þ j;L p2 pπ input state is known.ffiffiffi ð Þ¼ ð Þ ð Þ ˆ q ˆ Note that the momentum-space spikes for 1L ffiffi alternate Direct evaluation shows K t ψ˜ 0 qˆ X −t .A ffiffi j i ffiffi ECð Þ¼ ;Lð Þ ð Þ sign, and those for 0L are uniform. similar calculation shows that the Kraus operator for j i ˆ ˆ ˆ p ˜ ˆ ˆ GKP logical operators XL and ZL are implemented by correcting the p quadrature is KEC t ψ 0;L p Z −t . displacements Xˆ pπ and Zˆ pπ , respectively, while Applying both corrections (in eitherð Þ¼ order, sinceð Þ ð theyÞ ð Þ ð Þ displacements by integerffiffiffi multiplesffiffiffi of 2pπ in either commute up to a phase) performs full GKP error correction, quadrature leave the GKP logical subspace invariant.ffiffiffi For later use, we define the four GKP-encoded logical Paulis Kˆ t Kˆ p t Kˆ q t Πˆ Vˆ −t ; 3 ECð Þ¼ ECð pÞ ECð qÞ¼ GKP ð Þ ð Þ μ μ T σˆ ≔ σ jL kL ; 1 with measurement outcomes t t ;t . This Kraus L jkj ih j ð Þ ¼ðq pÞ Xjk operator (i) displaces the state by an outcome-dependent amount Vˆ −t , and then (ii) projects it back into the GKP μ σμ where σjk is the jkth element of Pauli matrix (with ð Þ ˆ logical subspace with ΠGKP [41]. σ0 I ˆ μ ˆ ). Note that σL have support only on the GKP logical Applying K t to an input state ρˆ produces the ¼ ˆ ˆ ECð Þ in subspace, while XL and ZL have full support and act both ¯ˆ t ˆ t ˆ ˆ † t unnormalized state ρ KEC ρinKEC , where the within and outside of the GKP subspace. We denote the bar indicates lack ofð Þ¼ normalization.ð Þ Theð Þ joint prob- (rank-2) projector onto the square-lattice GKP logical ability density function (PDF) for the outcomes PDF t subspace [21,38] ð Þ¼ Tr ρ¯ˆ t normalizes the output state: ρˆ t ρ¯ˆ t =PDF t . ½ ð Þ ð Þ¼ ð Þ — ð Þ ˆ 0 Bloch vector for the error-corrected state. Using Π ≔ σˆ ψ˜ 0 qˆ ψ˜ 0 pˆ ψ˜ 0 pˆ ψ˜ 0 qˆ : 2 GKP L ¼ ;Lð Þ ;Lð Þ¼ ;Lð Þ ;Lð Þ ð Þ the logical basis in Eq. (1) we represent the output state

200502-2 PHYSICAL REVIEW LETTERS 123, 200502 (2019)

ρˆ t 1 r t σˆ μ by a four-component Bloch vector Equation (5) can be evaluated analytically when the ð Þ¼2 μ μð Þ L r t withP outcome-dependent coefficients rμ t ≔ input state is Gaussian, ð Þ μ ð Þ Tr ρˆ t σˆ . For the unnormalized state r¯0 t PDF t , ½ ð Þ L ð Þ¼ ð Þ 1 l¯ and for the normalized state r0 t 1. In what follows, we −1 μ ¯ t 1 v Θ v ð Þ¼ rμ G0; 4πΣ − μ μ ; τ ; 6 use the notation r r0; r⃗ , where r⃗ is the ordinary (three- ð Þ¼4π ½ ð Þ ð Þ þ 2 ð Þ component) Bloch¼ð vectorÞ within r. i −1 1 We employ the Wigner functions for the logical basis where τ Σ , v τ l − 1=pπ x0 t , and the ¼ 2 μ ¼ ½2 μ ð Þð þ Þ states [21], shown in Fig. 1(a), to find the Wigner functions Riemann (also known as Siegel) theta functionffiffiffi is defined as for the GKP-encoded Pauli operators and the GKP logical 1 T T H Θ z; τ ≔ m∈Zn exp 2πi 2 m τm m z for τ ∈ n. identity, Eq. (1). Their explicit form is ð Þ H ½ ð þ Þ — The set nPdenotes the Siegel upper half-space i.e., the set of all complex, symmetric, n × n matrices with positive- n l¯ · μ l −1 2 μ definite imaginary part (see Ref. [42], for example). The Wσμ x ð Þ δð Þ x − n pπ ; 4 L 2 2 overall coefficient 1=4π ensures that PDF t is normalized ð Þ¼ 2 þ ð Þ nX∈Z ffiffiffi over a single unit cell. ð Þ GKP magic states from error correction.—GKP error T T T where x q; p , l0 0; 0 , l1 1; 0 , l2 correction of a Gaussian state yields a known, random state T ¼ð Þ T ¼ð¯ Þ ¼ð Þ ¼ 1; 1 , l3 0; 1 , and l is just l with its entries encoded in the GKP logical subspace. Unless that state ð Þ ¼ð Þ μ μ swapped. The Wigner functions are shown in Fig. 1(b). is highly mixed or too close to a logical-Pauli eigenstate, Πˆ ˆ μ Πˆ ˆ μ Since GKPσL GKP σL, we skip the projection using it can be used as a (noisy) magic state along with ˆ ¼ GKP Clifford QC for fault-tolerant universal QC [5]. ΠGKP and directly calculate the unnormalized Bloch-vector components from the overlap of the unnormalized error- Reference [38] suggested coupling a vacuum mode to an corrected state ρ¯ˆ t with the logical Paulis. We find the external qubit to perform GKP error correction and then t 0 overlaps in the Wignerð Þ representation: postselecting an outcome close to ≈ to produce a logical H-type state [5]. In fact, neither postselection nor interaction with a material qubit is required. ¯ t ¯ˆ t ˆ μ ˆ −t ˆ ˆ † −t ˆ μ rμ Tr ρ σL Tr V ρinV σL With access to a supply of 0 states, there is no need for ð Þ¼ ½ ð Þ ¼ ½ ð Þ ð Þ j Li 2 any resources beyond Gaussian QC, since nearly any 2 x x t μ x π d Win Wσ ; 5 t ¼ ZZ ð þ Þ L ð Þ ð Þ outcome from applying GKP error correction to the vacuum state produces a distillable H-type magic state ˆ [5,43], as shown in Fig. 2(a). This is because there are where Win x is the Wigner function of the input state ρin. ð Þ ˆ 12 H-type magic states (all related by Cliffords to H ), Note that r¯0 t Tr ρ¯ t PDF t , which is normalized L and any of them will do the job [5]. The relevantjþ quantityi over a unit cellð Þ¼ of size½ ð2Þp ¼π × 2ðpÞπ (since the full PDF ð Þ ð Þ is the fidelity F to the closest H-type state [43]. Without is periodic). The normalizedffiffiffi Blochffiffiffi four-vector is r t ≔ ð Þ loss of generality, assume this is HL , whose Bloch r¯ t =r¯0 t . ⃗ 1 2 1 0 1jþ i ð GKPÞ ð errorÞ correction of Gaussian states.—In what three-vector is rH =p ; ; . (If not, apply GKP ¼ð Þð Þ ˆ t follows, we apply GKP error correction to a general Cliffords until it is.) Then,ffiffiffi F HL ρ HL 1 1 ⃗ ⃗ t ¼hþ j ð Þjþ i¼ Gaussian state—i.e., an input state whose Wigner function 2 rH · r . States of sufficient fidelity can be twirled ½ þ ð Þ is W x Gx Σ x , where Gx Σ is a normalized Gaussian onto the HL axis, depolarized to make them identical, and inð Þ¼ 0; ð Þ 0; with mean vector x0 and covariance matrix Σ. then distilled [5,44]. Input-state purity is not required either. Applying GKP (a) (b) error correction to a thermal state also produces a distillable mixed state with nonzero probability as long as its mean ≕ occupation number n<¯ 0.366 n¯ thresh;H; see Fig. 2(c).[A 1 thermal state is Gaussian with x0 0 and Σ n¯ I, ¼ ¼ð þ 2Þ which we plug into Eq. (6) to produce this plot.] Most high- purity Gaussian states can be GKP error corrected into a distillable magic state because most states do not preferentially error correct to a Pauli eigenstate. For the vacuum, PDF t is always between 0.066 and 0.094—i.e., ð Þ FIG. 1. Wigner-function representations of the square-lattice all outcomes, and thus a wide variety of states, are roughly GKP (a) Pauli eigenstates and (b) logical-Pauli operators in a equally likely. single unit cell of phase space with dimensions 2pπ × 2pπ . Hexagonal-lattice GKP code.—Our results can be ð Þ ð Þ The states are normalized to one over one unitffiffiffi cell, whichffiffiffi extended to the hexagonal-lattice GKP code [31] by simply determines the coefficients c. modifying the Gaussian state to be error corrected as follows.

200502-3 PHYSICAL REVIEW LETTERS 123, 200502 (2019)

(a) (c)

(b)

(e) (d)

FIG. 2. (a) GKP error correction of the vacuum: outcome-dependent fidelity F with the nearest H-type magic state. The outcomes that do not yield a distillable magic state are marked with a white “x” (these yield GKP Pauli eigenstates). Bloch vectors for the representative outcomes (A–D) are shown on the GKP Bloch sphere in (b). (c) Probability of producing an H-type resource state of at least fidelity F using a thermal state of mean occupation n¯. Resource states with fidelity higher than the distillation threshold, F>0.853 (i.e., outside the stabilizer octahedron, or Paulihedron), can be distilled into higher-quality HL states [43,44]. Distillation is possible ˆ jþ i for n<¯ n¯ thresh;H 0.366. (d) U maps logical square-lattice GKP states to equivalent logical states in the hexagonal-lattice GKP encoding [31]. A¼ vacuum state on the hexagonal lattice (1-σ error ellipse shown in blue) is mapped to a squeezed state on the square ˆ † hex hex lattice under U . (e) Probability of producing a resource state distillable to TL of at least fidelity F by performing GKP error correction on a thermal state of mean occupation n¯. Resource states whose Blochjþ vectorsi lie on or within the Paulihedron (F ≤ 0.789) cannot be distilled, which occurs at n¯ bound;T 0.468. For T-type resource states, a distillation threshold has been proven for F>0.8273 [45], which occurs for n¯ 0.391. ¼ thresh;T ¼

ˆ ˆ square Define U as the Gaussian unitary such that U ψ L similar to what GKP proposed [21] but with photon counting hex j i¼ replaced by heterodyne detection, which is Gaussian. To see ψ L , where the logical state is the same although the encodingj i differs. Let ρˆ be a Gaussian state to be GKP error this, note that a Bell state can be written (ignoring normali- x 0 zation) as η σˆ μ ⊗ σˆ ν ,whereη diag 1; −1; −1; −1 . corrected using the hexagonal lattice, with 0 and μν μν L L ¼ ð Þ covariance Σ. Then, the equivalent state to be GKP¼ error A coherent-stateP measurement on the first mode with outcome ˆ μ ˆ ν ˆ ˆ † ˆ ˆ α produces μν ημνTr α α σ σ on the second mode, corrected on the square lattice is ρin0 U ρinU,whichis ðj ih j LÞ L ¼ −1 −T ˆ t Gaussian with x0 0 and covariance Σ0 S ΣS [46], which agreesP with Eq. (5) using ρin as vacuum and ¼−1 2 −1 ¼ 2 T ¼ where S 2p3 2 .Thismappingisshownfor −p Re α; Im α but with the opposite sign of the resulting ¼ð Þ ð0 p3Þ ⃗ ð Þ ρˆ vac vac ffiffiffiin Fig. 2(d). r. Intuitively,ffiffiffi this is just Knill-type error correction [47], in ffiffi Using¼j thisih mapping,j we can get results for hexagonal- which involves teleporting the state to be corrected through an encoded Bell pair and reinterpreting vacuum teleportation as lattice GKP error correction by reusing Eq. (6) with the heterodyne detection. modified state. Vacuum and thermal states are biased toward Noise and imperfections.—We employ ideal GKP the xz plane of the Bloch sphere in the square-lattice Clifford QC in our analysis because the fidelity requirements encoding but unbiased with respect to all three Pauli axes for fault-tolerant Clifford QC (which we have access to in the hexagonal-lattice encoding. Thus, in Fig. 2(e), we plot by assumption) are orders of magnitude stricter than those the fidelity of hexagonal-lattice GKP error correction of a for magic-state distillation [5], so small additional noise will thermal state with T-type magic states [5] such as Thex , jþ L i not qualitatively change our main result [39]. Using finite- which has Bloch three-vector r⃗ 1=p3 1; 1; 1 . precision GKP states for error correction causes uncertainty T ¼ð Þð Þ Error correction as heterodyne detection.ffiffiffi —We note that in the measurement outcome t [40], which can be modeled an alternate description of what we are proposing is to as additive Gaussian noise on the input state—i.e., by perform heterodyne detection (measurement in the coher- replacing n¯ ↦ n¯ Δn¯, where Δn¯ equals the q or p variance ent-state basis) on half of a GKP-encoded Bell pair. This is of an individual GKPþ spike. Δn¯ is between 0.05 and 0.016

200502-4 PHYSICAL REVIEW LETTERS 123, 200502 (2019)

1 10 −rdB= STOC 97 Proceedings of for 10–15 dB GKP states (Δn¯ 2 10 , where rdB is the [3] D. Aharonov and M. Ben-Or, squeezing measured in dB) ¼[24], and fault tolerance is the Twenty-Ninth Annual ACM Symposium on Theory of unlikely to be possible with lower-quality states than these Computing, El Paso, Texas, USA (ACM, New York, 1997), – [24,48,49]. Since n¯ > 0.36, this additional noise is pp. 176 188. thresh [4] D. Gottesman, An introduction to quantum error correc- qualitatively unimportant to our main result. Furthermore, tion and fault-tolerant quantum computation, arXiv: the resulting magic states will be the same quality as the 0904.2557v1. GKP ancillas [40], which are sufficient for fault tolerance by [5] S. Bravyi and A. Kitaev, Universal quantum computation assumption. Having established our result’s robustness to with ideal Clifford gates and noisy ancillas, Phys. Rev. A 71, imperfections, we leave detailed elaboration to further work. 022316 (2005). Discussion.—We have deployed GKP error correction in [6] M. Chen, N. C. Menicucci, and O. Pfister, Experimental a nonstandard way to extract the magic from easy-to- Realization of Multipartite Entanglement of 60 Modes of a prepare Gaussian states that extend into the “wilderness Quantum Optical Frequency Comb, Phys. Rev. Lett. 112, space” outside a bosonic code’s logical subspace. The 120505 (2014). wilderness space may be rich in other resources, too—e.g., [7] S. Yokoyama, R. Ukai, S. C.Armstrong, C.Sornphiphatphong, providing the means to produce other logical states or T. Kaji, S. Suzuki, J.-i. Yoshikawa, H. Yonezawa, N. C. perform logical operations more easily than would be Menicucci, and A. Furusawa, Ultra-large-scale continuous- possible by restricting to the logical subspace. This feature variable cluster states multiplexed in the time domain, Nat. Photonics 7, 982 (2013). is likely to extend beyond GKP to other bosonic codes such [8] J.-i. Yoshikawa, S. Yokoyama, T. Kaji, C. Sornphiphatphong, as rotation-symmetric codes [23] (including cat [50,51] and Y. Shiozawa, K. Makino, and A. Furusawa, Invited Article: binomial [52] codes), bosonic subsystem codes [53], and Generation of one-million-mode continuous-variable cluster multimode codes [22,31,54]. state by unlimited time-domain multiplexing, APL Photonics Our result is an example of the fact that two efficiently 1, 060801 (2016). simulable subtheories (GKP Clifford QC and Gaussian QC [9] J. Roslund, R. M. De Araujo, S. Jiang, C. Fabre, and N. here) together can contain all the ingredients for univer- Treps, Wavelength-multiplexed quantum networks with sality. For qubits, this is straightforward [55,56]: combine ultrafast frequency combs, Nat. Photonics 8, 109 (2014). Clifford QC from different Pauli frames since stabilizer [10] N. Ofek et al., Extending the lifetime of a quantum bit states of one are magic states for the other. In CV systems, with error correction in superconducting circuits, Nature dual-rail photonic qubits [57] also exhibit this feature: (London) 536, 441 (2016). Clifford QC (requiring several non-Gaussian elements) and [11] S. Rosenblum et al., A CNOT gate between multiphoton Gaussian single-qubit gates together give universality. qubits encoded in two cavities, Nat. Commun. 9, 652 (2018). The GKP encoding stands out among bosonic codes as [12] L. Hu et al., Quantum error correction and universal gate set the only known code for which Clifford QC is implemented operation on a binomial bosonic logical qubit, Nat. Phys. 15, entirely with Gaussian operations given a supply of 503 (2019). encoded Pauli eigenstates. We show that once high-quality [13] Y. Y. Gao, B. J. Lester, K. S. Chou, L. Frunzio, M. H. GKP Clifford QC is achieved—a challenging task already Devoret, L. Jiang, S. M. Girvin, and R. J. Schoelkopf, in progress [14,29,30]—then fault-tolerant universality is Entanglement of bosonic modes through an engineered just a trivial Gaussian state away. This means there is no exchange interaction, Nature (London) 566, 509 (2019). longer any need to pursue creating cubic-phase states for [14] C. Flühmann, T. L. Nguyen, M. Marinelli, V. Negnevitsky, the GKP encoding. Focus on making high-quality GKP K. Mehta, and J. P. Home, Encoding a qubit in a trapped-ion Pauli eigenstates, and the rest is all Gaussian. mechanical oscillator, Nature (London) 566, 513 (2019). [15] S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. We thank Andrew Landahl for discussions. R. N. A. is Nemoto, Efficient Classical Simulation of Continuous supported by National Science Foundation Grant No. PHY- Variable Quantum Information Processes, Phys. Rev. Lett. 1630114. A. K. is supported by the Australian Research 88, 097904 (2002). Council Centre of Excellence for Engineered Quantum [16] S. Lloyd and S. L. Braunstein, Quantum Computation over Systems (Project No. CE170100009). This work is sup- Continuous Variables, Phys. Rev. Lett. 82, 1784 (1999). ported by the Australian Research Council Centre of [17] M. Walschaers, S. Sarkar, V. Parigi, and N. Treps, Tailoring Excellence for Quantum Computation and Communi- Non-Gaussian Continuous-Variable Graph States, Phys. cation Technology (Project No. CE170100012). Rev. Lett. 121, 220501 (2018). [18] Y.-S. Ra, A. Dufour, M. Walschaers, C. Jacquard, T. Michel, C. Fabre, and N. Treps, Non-Gaussian quantum states of a multimode light field, arXiv:1901.10939. [1] M. A. Nielsen and I. L. Chuang, Quantum Computation [19] J. Niset, J. Fiurášek, and N. J. Cerf, No-Go Theorem for and Quantum Information (Cambridge University Press, Gaussian Quantum Error Correction, Phys. Rev. Lett. 102, Cambridge, England, 2000). 120501 (2009). [2] D. Gottesman, The Heisenberg representation of quantum [20] P. T. Cochrane, G. J. Milburn, and W. J. Munro, Macroscop- computers, arXiv:quant-ph/9807006. ically distinct quantum-superposition states as a bosonic

200502-5 PHYSICAL REVIEW LETTERS 123, 200502 (2019)

code for amplitude damping, Phys. Rev. A 59, 2631 [40] S. Glancy and E. Knill, Error analysis for encoding a qubit (1999). in an oscillator, Phys. Rev. A 73, 012325 (2006). [21] D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in [41] A. Ketterer, A. Keller, S. P. Walborn, T. Coudreau, and P. an oscillator, Phys. Rev. A 64, 012310 (2001). Milman, Quantum information processing in phase space: [22] V. V. Albert et al., Performance and structure of single-mode A modular variables approach, Phys. Rev. A 94, 022325 bosonic codes, Phys. Rev. A 97, 032346 (2018). (2016). [23] A. L. Grimsmo, J. Combes, and B. Q. Baragiola, Quantum [42] B. Deconinck, M. Heil, A. Bobenko, M. van Hoeij, and computing with rotation-symmetric bosonic codes, arXiv: M. Schmies, Computing Riemann theta functions, Math. 1901.08071. Comput. 73, 1417 (2004). [24] N. C. Menicucci, Fault-Tolerant Measurement-Based Quan- [43] B. W. Reichardt, Quantum universality from magic states tum Computing with Continuous-Variable Cluster States, distillation applied to CSS codes, Quantum Inf. Process. 4, Phys. Rev. Lett. 112, 120504 (2014). 251 (2005). [25] S. Rosenblum, P. Reinhold, M. Mirrahimi, L. Jiang, L. [44] E. T. Campbell and D. E. Browne, Bound States for Magic Frunzio, and R. J. Schoelkopf, Fault-tolerant detection of a State Distillation in Fault-Tolerant Quantum Computation, quantum error, Science 361, 266 (2018). Phys. Rev. Lett. 104, 030503 (2010). [26] K. R. Motes, B. Q. Baragiola, A. Gilchrist, and N. C. [45] T. Jochym-O’Connor, Y. Yu, B. Helou, and R. Laflamme, Menicucci, Encoding qubits into oscillators with atomic The robustness of magic state distillation against errors in ensembles and squeezed light, Phys. Rev. A 95, 053819 Clifford gates, Quantum Inf. Comput. 13, 361 (2013). (2017). [46] N. C. Menicucci, S. T. Flammia, and P. van Loock, Graphi- [27] K. Fukui, A. Tomita, and A. Okamoto, Analog Quantum cal calculus for Gaussian pure states, Phys. Rev. A 83, Error Correction with Encoding a Qubit into an Oscillator, 042335 (2011). Phys. Rev. Lett. 119, 180507 (2017). [47] E. Knill, Quantum computing with realistically noisy [28] C. Vuillot, H. Asasi, Y. Wang, L. P. Pryadko, and B. M. devices, Nature (London) 434, 39 (2005). Terhal, Quantum error correction with the toric Gottesman- [48] K. Fukui, A. Tomita, A. Okamoto, and K. Fujii, High- Kitaev-Preskill code, Phys. Rev. A 99, 032344 (2019). Threshold Fault-Tolerant Quantum Computation with [29] Y. Shi, C. Chamberland, and A. W. Cross, Fault-tolerant Analog Quantum Error Correction, Phys. Rev. X 8, 021054 preparation of approximate GKP states, New J. Phys. 21, (2018). 093007 (2019). [49] B. W. Walshe, L. J. Mensen, B. Q. Baragiola, and N. C. [30] P. Campagne-Ibarcq et al., A stabilized logical quantum bit Menicucci, Robust fault tolerance for continuous-variable encoded in grid states of a superconducting cavity, arXiv: cluster states with excess antisqueezing, Phys. Rev. A 100, 1907.12487. 010301(R) (2019). [31] K. Noh, V. V. Albert, and L. Jiang, Quantum capacity [50] A. P. Lund, T. C. Ralph, and H. L. Haselgrove, Fault- bounds of gaussian thermal loss channels and achievable Tolerant Linear Optical Quantum Computing with Small- rates with Gottesman-Kitaev-Preskill codes, IEEE Trans. Amplitude Coherent States, Phys. Rev. Lett. 100, 030503 Inform. Theory 65, 2563 (2019). (2008). [32] W. Asavanant et al., Generation of time-domain-multiplexed [51] M. Mirrahimi, Z. Leghtas, V. V Albert, S. Touzard, R. J. two-dimensional cluster state, Science 366, 373 (2019). Schoelkopf, L. Jiang, and M. H. Devoret, Dynamically [33] O. Pfister, Continuous-variable quantum computing in the protected cat-qubits: a new paradigm for universal quantum quantum optical frequency comb, arXiv:1907.09832. computation, New J. Phys. 16, 045014 (2014). [34] S. Takeda and A. Furusawa, Toward large-scale fault- [52] M. H. Michael, M. Silveri, R. T. Brierley, V. V. Albert, J. tolerant universal photonic quantum computing, APL Salmilehto, L. Jiang, and S. M. Girvin, New Class of Photonics 4, 060902 (2019). Quantum Error-Correcting Codes for a Bosonic Mode, [35] Note: These states cannot be prepared using the Gaussian Phys. Rev. X 6, 031006 (2016). measurements that correspond to logical-Pauli measure- [53] G. Pantaleoni, B. Q. Baragiola, and N. C. Menicucci, ments because these measurements are destructive. A. Modular bosonic subsystem codes, arXiv:1907.08210. Karanjai, J. J. Wallman, and S. D. Bartlett, Contextuality [54] V. V. Albert, S. O. Mundhada, A. Grimm, S. Touzard, bounds the efficiency of classical simulation of quantum M. H. Devoret, and L. Jiang, Pair-cat codes: autonomous processes, arXiv:1802.07744. error-correction with low-order nonlinearity, Quantum Sci. [36] K. Marshall, R. Pooser, G. Siopsis, and C. Weedbrook, Technol. 4, 035007 (2019). Repeat-until-success cubic phase gate for universal con- [55] C. D. Hill, A. G. Fowler, D. S. Wang, and L. C. L. tinuous-variable quantum computation, Phys. Rev. A 91, Hollenberg, Fault-tolerant quantum error correction code 032321 (2015). conversion, Quantum Inf. Comput. 13, 439 (2013). [37] F. Arzani, N. Treps, and G. Ferrini, Polynomial approxi- [56] A. Paetznick and B. W. Reichardt, Universal Fault- mation of non-Gaussian unitaries by counting one photon at Tolerant Quantum Computation with Only Transversal a time, Phys. Rev. A 95, 052352 (2017). Gates and Error Correction, Phys. Rev. Lett. 111, 090505 [38] B. M. Terhal and D. Weigand, Encoding a qubit into a cavity (2013). mode in circuit QED using phase estimation, Phys. Rev. A [57] E. Knill, R. Laflamme, and G. J. Milburn, A scheme for 93, 012315 (2016). efficient quantum computation with linear optics, Nature [39] P. Brooks, Quantum error correction with biased noise, (London) 409, 46 (2001). Ph.D. thesis, California Institute of Technology, 2013.

200502-6