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Implementation-independent sufficient condition of the Knill-Laflamme type for the autonomous protection of logical qudits by strong engineered dissipation

Autonomous quantum error correction utilizes the engineered coupling of a quantum system to a dissipative ancilla to protect quantum logical states from decoherence. We show that the Knill-Laflamme condition, stating that the environmental error operators should act trivially on a subspace, which then becomes the code subspace, is sufficient for logical qudits to be protected against Markovian noise. It is proven that the error caused by the total Lindbladian evolution in the code subspace can be suppressed up to very long times in the limit of large engineered dissipation, by explicitly deriving how the error scales with both time and engineered dissipation strength. To demonstrate the potential of our approach for applications, we implement our general theory with binomial codes, a class of bosonic error-correcting codes, and outline how they can be implemented in a fully autonomous manner to protect against photon loss in a microwave cavity.


I. INTRODUCTION
In the majority of practically realized experiments, the unavoidable coupling to an environment leads to non-unitary evolution and introduces noise and dissipation into the system [1]. This is often regarded as detrimental and hindering the control of the experiment. In the last two decades however, the at first sight counterintuitive fact has been established that dissipation can in fact support various quantum information processing tasks, such as quantum computation [2][3][4][5], entanglement generation [6][7][8][9], and the preparation of quantum many-body states [10,11].
Quantum error correction (QEC) is a method of correcting for effects of noise in quantum computers [12] by quantum control [13][14][15][16][17][18][19]. Conventional measurement-based QEC utilizes periodic measurement of error syndromes and an appropriate unitary evolution that eliminates the error. This measurement-based QEC scheme can correct errors iff encoded logical quantum states satisfy the Knill-Laflamme condition [20,21], which predicates that errors can be corrected iff an environment causing the error does not gain any information about the encoded states [22].
In what follows, we identify an implementation independent, sufficient condition for AutoQEC, of the Knill-Laflamme type. In particular, we prove its sufficiency by providing an explicit Markovian engineered dissipation and by deriving a rigorous upper bound of logical error probability in terms of the engineered dissipation strength. In addition, we do not assume any structure of the error-correcting codes, and thus our theory also applies to the codes which are beyond the paradigm of stabilizer-based qubit codes. We illustrate this by proposing a fully autonomous implementation of the recently proposed binomial codes [40], which protects encoded information against photon loss in a microwave cavity.

II. AUTOQEC THEOREM
Consider an open quantum system evolving according to a Markovian master equation: where D(A)ρ := AρA † − 1 2 (A † Aρ + ρA † A). H sys and F sys , and H eng and F eng are Hamiltonian and Lindbladian operators that are intrinsic to the system and imposed by dissipative engineering, respectively; we set = 1. We omit the subscript sys from the intrinsic Lindblad operators in the following. We assume that intrinsic dissipation is uncontrollable, while engineered dissipation, in Markovian form, can be artificially generated and increased at will. In the limit M 1 we consider, the engineered dissipation strength parametrized by M is much larger than its intrinsic counterpart. We discuss the practical feasibility of strong Markovian dissipation for a concrete implementation below.
We seek a condition for the code space C ⊂ H (dim H = n < ∞, with H the total Hilbert space), under which there exist an engineered Hamiltonian H eng and Lindblad jump operators {F eng,i } i=1,··· ,L which autonomously protect the code space against the Markovian error generated by the intrinsic jump operators {F k } k=1,··· ,N . That a code space is protected means that for sufficiently strong engineered dissipation, any logical states in the d-dimensional code subspace C are left invariant until a given time T and up to an arbitrarily small error , which is defined in Eq.
(3) below. We find that the Knill-Laflamme condition [20,21], when applied to the set of intrinsic error operators E = {1, F k } k=1,...,N , is sufficient for the autonomous protection of the qudit code subspace C = span{|W µ } µ=0,··· ,d−1 . The condition states that for all E l , E l ∈ E and l, l ∈ {0, ..., N }, where W µ |W µ = δ µµ , Π C := d−1 µ=0 |W µ W µ | and c k k are constants. Our main technical result, then, is as follows: Theorem (AutoQEC condition). If a code space C satisfies the Knill-Laflamme condition for the error set E = {1, F k } k=1,...,N (see Eq. (2)), there exists a set of engineered dissipative jump operators F eng,i such that, for any initial density matrix ρ C in the code space, where γ := N k=1 F k 2 denotes the measure of intrinsic dissipation strength. Throughout, we use X := sup v =1 Xv as the norm of an operator X, where v are Hilbert space vectors, while superoperators have norm Y := sup X =1 Y(X) .
Summary of the proof. We first note that the intrinsic Hamiltonian H sys can be eliminated by setting H eng = −H sys . Consider a code space C satisfying the AutoQEC condition in Eq. (2). The density matrix generated by this code space is represented by the cross-striped areas in Fig. 1. The intrinsic jump operators F k corrupt the states in C and move them into the corrupted code space S ccs (cf. green dashed line in Fig.  1). We explicitly construct a set of engineered jump operators F eng,i (see Eq.(5)) which pump the corrupted states back into the code space C without loss of coherence (blue double line in Fig. 1). We then show that the d × d blocks of grayshaded and cross-striped areas in Fig. 1 form a noiseless subsystem [21,[41][42][43][44], if the intrinsic dissipation operator is limited to act on the code subspace, cf. Appendix A. Finally, we show that the remaining dissipation increases the error only to O(γT /M ), as indicated on the right-hand side of Eq. (3). For details on the proof, we refer the reader to Appendix B.
We observe that the relation (3) implies protection of a logical qudit, since one can choose M γT / to obtain any desired error less than . Also, we remark that Eq. (2) was previously identified as the necessary and sufficient condition for the elimination of the first order error contribution at short times in [45]. Although related, our approach notably differs from the earlier one in that we construct an explicit recovery of the Lindbladian form to achieve AutoQEC (see Eq. (5) below), and consider the limit of arbitrarily long times relevant for quantum computing applications.
FIG. 1: Schematic of density matrix , Lindblad jump operators, and noiseless subsystem symmetry. The total Hilbert space is divided into d × d arrays of d 2 large blocks. The logical code space C is formed by the cross-striped areas. The arrows represent dissipative jump operators, and the gray-shaded areas represent the space of corrupted states, connected via Lindblad operator f (green dashed, Eq. (A1)) to the code states. Engineered dissipation F eng (blue double line, Eq. (5)) pumps the corrupted states back to the code space, and in consequence each large block supports a steady state U µ ρ 0,st U † ν , and S ccs forms a noiseless subsystem. U µ intertwines between S 0 and S µ , and is defined in Eq. (A4). The remaining intrinsic dissipation f (red-dotted, Eq. (A1), for clarity only one such process is shown) breaks this noiseless subsystem symmetry and induces an undesired error.
We design the engineered dissipation as follows: where L = n−m(d−1)−1 and |Φ p (p = 0, · · · , n−md−1) can be any state in S ccs . We term the first m − 1 jump operators corrective jumps because they pump erroneous states |W µ ; i back into the code state |W µ coherently, similarly to measurement-based QEC. The remaining jumps (i.e., F eng,i with m ≤ i ≤ L) we term preventive jump operators: They force states in H − S ccs to decay into S ccs , preventing further uncorrectable errors. The necessity of corrective jump operators is clear from the perspective of measurement-based QEC. However, it might not be immediately obvious whether the preventive jump operators are essential for a successful AutoQEC. In this regard, we emphasize that the preventive jump operators are indeed crucial to achieve the desired error scaling in Eq. (3) since otherwise any leakage to H − S ccs cannot be recovered (see Fig. 2 (a) and the discussion in subsection IV B for an illustration).
In addition, while it is essential that the corrective jumps are given by the first line of Eq. (5), preventive jump operators can be chosen arbitrarily as long as im(F eng,i ) ⊆ S ccs ∀i ∈ {m, · · · , L} and ker( L i=m F † eng,i F eng,i ) = S ccs hold, where im(A) and ker(A) represent image space and null space (kernel) of A, respectively (cf. subsection IV C).

A. 3-bit repetition code against bit-flip error
In order to show that earlier results can be recovered from our general design, we consider the 3-bit repetition code |W 0 = |000 , |W 1 = |111 in the presence of bit flip error: where X k represents the Pauli operator X (the bit flip by σ x ) acting on the k th qubit (e.g., X 1 = XII, where I is the identity operator). In this case, the AutoQEC error set is given by E = {III, XII, IXI, IIX}, and the 3-bit repetition code satisfies the Knill-Laflamme condition for this error set. Then, following our jump operator design principle, we find |W µ ; i = X i |W µ (i = 1, 2, 3), and thus which are consistent with the earlier designs in [23][24][25]35]. Note that all these jump operators are corrective, and we do not need any preventive jump operators because µ=0,1 spans the entire Hilbert space (i.e., H = S ccs ).

B. Binomial code against photon loss error
In order to exhibit the generality of our approach, we discuss the autonomous protection of the binomial code |W 0 = (|0 + |4 )/ √ 2 , |W 1 = |2 (a recently proposed bosonic code; see [40]) against the error caused by photon loss, i.e., we consider where a is the annihilation operator of the single bosonic mode. In this case, the AutoQEC error set is given by E = {I, a}, and the binomial code satisfies the Knill-Laflamme condition for this error set. Upon a single photon loss, logical states of the binomial code are corrupted as a|W 0 = √ 2 |3 and a|W 1 = √ 2 |1 , and thus we have |W 0 ; 1 = |3 and |W 1 ; 1 = |1 . The four states {|W µ , |W µ ; 1 } µ=0,1 however do not span the entire Hilbert space H 4 = span{|0 , · · · , |4 } due to the residual basis vector |φ 1 = (|0 − |4 )/ √ 2 . Then, we get the following engineered jump operators from Eq. (5): where |Φ can be any state in H 4 orthogonal to |0 − |4 . Note that F eng,1 is a corrective jump operator while F eng,2 is a preventive jump operator. We stress that the binomial code is not described by a set of stabilizers, and the proposed engineered jump operators in Eq. (9)and, in particular, the preventive jump operator F eng,2 , are not derivable from the earlier framework laid down, e.g., in [27,28]. To illustrate that the preventive jump operator indeed plays a crucial role, we plot in Fig. 2 the average fidelity of 6 representative logical states of the autonomously protected binomial code with and without the preventive jump operator. The fidelity is defined as ψ in |e Lt (|ψ in ψ in |)|ψ in , for an input state |ψ in .
As can be seen from Fig. 2 (a), the autonomously protected binomial code without the preventive jump operator F eng,2 (or |Φ = 0) can barely perform comparably to the unprotected physical qubit with the least energy (i.e., |W phys 0 = |0 , |W phys 1 = |1 possessing the longest physical qubit lifetime), even in the M → ∞ limit. On the other hand, if we add a preventive jump operator (Fig. 2 (b); |Φ = (|0 + |4 )/ √ 2 ), the autonomously protected binomial code begins to perform comparably to the physical qubit at M 10 (i.e., at the break-even point; see [18,19]), and allows for logical qubit lifetimes much longer than those of the physical qubit if M 100, as guaranteed by the AutoQEC Theorem in section II. Thus, the preventive jump operator F eng,2 is essential for the superior performance of the binomial code. We also remark that this superior performance (in particular, the scaling (T ) = O(γT /M )) does not depend on the choice of |Φ as long as Φ|Φ = O(1) (cf. Fig. 2 (b,c)).

C. Comparison with measurement-based QEC
The Lindbladian generator D(A) consists of the jump term A • A † and the no-jump evolution term − 1 2 {A † A, •}. We identify that the corrective jump operators suppress the adverse effects of jump-type errors, while the preventive jump operators suppress the effects of the no-jump evolution. Since the jump-type error induces population transfer from a code state |W µ to a corrupted state |W µ ; i , it is essential to have the corrective jump operators in the exact form as given in the first line of Eq. (5). On the other hand, since the no-jump evolution only accumulates an undesirable coherence between a code state |W µ and a residual state |φ p without direct population transfer, any preventive jump operators emptying such coherence is sufficient. Thus, we are presented with a flexibility in choosing |Φ p for preventive jump operators, which may be exploited for practical experimental implementation (see, for example, the discussion below Eq. (15)).
In measurement-based QEC, on the other hand, the undesired coherence caused by no-jump evolution is fixed by a unitary rotation conditioned on not detecting the jump-type errors. The rotation angle should however be fine-tuned to precisely counter the accumulation of the undesired coherence.
To make our discussion more concrete, let us briefly review the measurement-based QEC of the binomial code outlined in [40], and compare it with the AutoQEC scheme presented above. In measurement-based QEC of the binomial code, the error syndrome is extracted by a photon-number parity measurement [47,48]. If the measured parity is odd, we infer that photon loss occurred, and apply the corrective unitary opera- cf. the text below Eq.
(2) in [40]. On the other hand, if the measured parity is even, we infer that no loss error has happened. In this case, in order to counter the undesired accumulation of coherence between |W 0 = (|0 + |4 )/ √ 2 and |φ 1 = (|0 −|4 )/ √ 2 (caused by the no-jump evolution), we need to apply the following corrective unitary rotation where ∆t is the waiting time between the syndrome measurements and U res 2 is an arbitrary unitary operator on the subspace span{|1 , |3 } (see Eqs. (16), (17) in [40]).
The corrective unitary operator U 1 corresponds to the corrective jump operator F eng,1 in Eq.(9), and U 2 corresponds to the preventive jump operator F eng,2 . Notably, the corrective unitary operator U 2 should have the definite form above in the subspace span{|0 , |2 , |4 }. (Especially, the rotation angle γ∆t needs to be fine-tuned.) By contrast, for the preventive jump operator F eng,2 , we can choose any |Φ of order unity norm orthogonal to |φ 1 .

A. Physical Requirements
We now discuss the practical feasibility of the M → ∞ limit. In principle, any desired Markovian engineered dissi-pation can be realized by modulating the Hamiltonian coupling between the system and fast decaying ancillary qubits [3, 49,50]. Consider to this end the master equation with . Then, in the weak coupling and fast decay limit (i.e., λ i κ i ), the entire system approximately evolves as ρ T (t) = ρ(t) ⊗ L i=1 |g i g i |, and the system density matrix ρ(t) obeys the desired Lindbladian master equation (see Proposition 3 in [50]; for the bound of deviation from the approximation, see Proposition 1 therein). The dimensionless engineered dissipation strength M then scales as where γ is the strength of intrinsic dissipation. Thus, the M → ∞ limit can be achieved if , when the ancillary qubits' decay rate is much larger than the intrinsic one of the system. AutoQEC by engineered dissipation was experimentally realized in circuit QED systems [38,39]. In both experiments, quantum information was encoded in the photon mode of a microwave cavity, using the two-component cat states |C ± α ∝ |α ± | − α as the logical qubit basis. Stabilization of the two-component cat code (in a high-Q cavity mode) was achieved by an engineered dissipation D(a 2 −α 2 ), which was implemented by coupling the high-Q cavity mode (decay rate γ) to a fast-decaying low-Q cavity mode (decay rate κ) via interaction H coupl = λ((a 2 − α 2 )b † + h.c.). In the earlier experiment [38], λ 700 kHz, κ 40 MHz, γ 50 kHz, and thus M 1 was achieved, and more recently λ 900 kHz, κ 3 MHz, γ 10 kHz, and thus M 100 was realized [39].
We note that the realized two-component cat code is not robust against photon loss (a dominant error source in a high-Q cavity mode), since a single loss causes an irreversible logical bit-flip: a|C ± α α|C ∓ α . There have been many proposals to leverage the capability of the cat code, either based on variations [32,37] or concatenation [36] of the two-component cat code, such that the logical information is protected against the photon loss errors. Below, we propose an alternative Au-toQEC scheme, tailored to the same physical platform, based on the binomial code discussed above and briefly compare it with the cat code schemes.

B. Implementation of binomial code AutoQEC
The engineered dissipation given in Eq. (9) fully protects the binomial code space against the photon loss error in the M → ∞ limit (see Fig. 2). In principle, such an engineered dissipation can be realized by coupling a high-Q cavity mode to two fast-decaying transmon qubits [51,52] or low-Q cavity modes via the interaction where |g i and |e i are the ground and excited states of the i th qubit (or low-Q cavity mode) for i = 1, 2 (cf. Eqs. (12), (13)), respectively.
Generating higher (than second) order interaction between a cavity mode and a qubit (or another cavity mode) is a challenging task (especially at strong coupling), both in the photon number selective and non-selective cases. We however remark that photon-number non-selective three quanta exchange a 2 b † was realized in two previous experiments [38,39], and there is a concrete theoretical proposal for the photon number nonselective six-quanta exchange a 4 b †2 [53].

VI. CONCLUSION
In summary, we have provided a sufficient condition of the Knill-Laflamme type, under which the code space manifold can be autonomously protected against intrinsic errors in the limit of large engineered dissipation. We have constructed explicit engineered jump operators to achieve AutoQEC, and derived the temporal error bounds up to which the information initially stored in the code space can be preserved. We have briefly compared AutoQEC with measurement-based QEC, and clarified the difference between them as regards the protection against no-jump evolution. Based on our general theory of AutoQEC, we have proposed an autonomous implementation of the binomial code, which may allow for significantly longer logical qubit lifetimes than those of the most stable uncorrected physical qubits. In the intrinsic dissipation operator, we identify two separate parts by using the projection operator Π C : The part f k = F k Π C is the component of F k that is applied on the code subspace. We first show that, if the remaining termf k is absent, all S µ are collecting subspaces: For each µ ∈ {0, · · · , d − 1}, an arbitrary initial state in S µ ⊆ H never leaves S µ during the time evolution. This holds because for each Lindblad operatorF ∈ {f k , F eng,i } := F, the following general conditions hold (see Lemmas 9 and 11 of [54]) where Π Sµ is the projection operator onto S µ and H = 0 in our case since we set H eng = −H sys . It then follows that each S µ supports a unique steady state, ρ µ,st for each µ ∈ {0, · · · , d − 1}.
In addition to these d steady states, there exist d 2 − d stationary phase relations between all S µ and S ν with µ = ν: Note that the following unitary intertwiner It commutes with all F ∈ F, due to µ-independence of W µ ; i|F k |W µ . Then, Proposition 16 of [54] implies that all steady states ρ µ,st are unitarily connected by U µ , i.e., ρ µ,st = U µ ρ 0,st U † µ , and furthermore U µ ρ 0,st U † ν establishes a stationary phase coherence between S µ and S ν for µ = ν -cf. Fig. 1. Therefore, for any d × d qudit density matrix ω, are left invariant under the Lindbladian dynamics generated by the jumps in F. Thus S ccs = ∪ d−1 µ=0 S µ forms a noiseless subsystem [21,[41][42][43][44] if the intrinsic dissipation operator is limited to act on the code subspace. Note that states outside the corrupted code space are omitted in Fig. 1 and Eq. (A5), since they are not occupied in the steady states.
Appendix B: Long-time limit for large engineered dissipation Given the noiseless subsystem symmetry derived in Appendix A, it is natural to divide the Lindbladian L in (1) into three parts as follows In the above, L 0 is the engineered part while L 1 is the intrinsic dissipation that follows the noiseless subsystem symmetry and leaves ω ⊗ ρ st invariant: Finally, L 2 breaks this noiseless subsystem symmetry and brings about evolution for the otherwise steady state, L 2 [ω ⊗ ρ st ] = 0. Following the decomposition in (B1), we define new Lindbladians, with properly chosen scaling factors and derive the main result Eq. (3) by proving an equivalent statement that a time evolution generated by the rescaled Lindbladian L/M leaves states in C approximately invariant in the same sense up to time M T . We use the perturbation theory of Lindbladian superoperators [55], which has found applications so far when the magnitude of a weakly unitary or dissipative operator is the small parameter of the perturbation series [4, 50,56,57]. In our approach, on the other hand, the inverse of the magnitude of strong engineered dissipation, 1/M , is the small parameter. Formally, we establish the following bound of the error (t) in the interval 0 ≤ t ≤ M T , for states projected to the code subspace C: where P C := lim t→∞ e tL0 is the superoperator projection to the code space C and τ = O(M 0 ) is of order the inverse of the smallest real part of nonzero eigenvalue of L e /M , as discussed below. The above bound (B3) leads to our AutoQEC Theorem as represented by Eq. (3). We now proceed to prove the bound in the second line of Eq. (B3). First, we denote the rescaled Lindbladian superoperator as L = L/M and, similarly, L 2 and L e . The error (t) as defined in Eq. (B3) satisfies the following inequalities: (t) = e tL P C − P C ≤ e tPLP P e − P e + (e tL − e tPLP )P e + e tL +1 P C − P e ≤ e tPLP P e − P e +( e tL + e tPLP ) P − P e + e tL +1 P C − P e . (B4) The triangle inequalities A + B ≤ A + B and (e tL − e tPLP )P = 0 were used.
The kernel of L e is d 2 -dimensional because we have a d-dimensional noiseless subsystem. Let P be the direct sum of the projection onto eigenspaces of L, which perturbatively originate from the kernel of the unperturbed superoperator L e . Then, from the perturbation theory of linear operators [55], PLP = P e L 2 P e − P e L 2 SL 2 P e − P e L 2 P e L 2 S − SL 2 P e L 2 P e + O(τ 2 L 2 3 ) (B6) holds. Here, S is the pseudo-inverse of L e , which satisfies SL e = L e S = 1 − P e . Then, τ = S is of the order of the inverse of the smallest nonzero eigenvalue of L e and is of order O(M 0 ). This is proven as follows: P C , P e denote projection superoperators that project the linear operators of the Hilbert space H onto the kernel of the superoperators L 0 , L e , respectively. For example, P C projects operators onto the kernel ker L 0 = {ρ : L 0 [ρ] = 0}, which is indeed the code subspace. Since ∪ d−1 µ=0 S µ forms a noiseless subsystem of L e , Appendix A, ker L e is a projection onto the subspace composed by metastable states, ω ⊗ ρ st . Using the perturbation theory of linear operators viewing L 1 /M as a perturbation to the Lindbladian L e = L 0 + L 1 /M , one finds [55] P e − P C = − 1 M (P C L 1 S 0 − S 0 L 1 P C ) + 1 M 2 O( L 1 2 ), (B7) where S 0 denotes the pseudo-inverse of L 0 , which satisfies S 0 L 0 = L 0 S 0 = 1 − P C . This implies P C − P e = O(1/M ). Now, by the definition of f k : f k Π C = 0, it follows that P C L 2 P C [ρ] = Π C L 2 [Π C ρΠ C ]Π C = 0. This entails P C L 2 P C = 0. Putting the results together using the tri-angle inequality A + B ≤ A + B , we get P e L 2 P e ≤ (P e − P C ) L 2 P e + P C L 2 (P e − P C ) + P C L 2 P C = O(