Abstract
Coherent errors, which arise from collective couplings, are a dominant form of noise in many realistic quantum systems, and are more damaging than oft considered stochastic errors. Here, we propose integrating stabilizer codes with constantexcitation codes by code concatenation. Namely, by concatenating an [[n, k, d]] stabilizer outer code with dualrail inner codes, we obtain a [[2n, k, d]] constantexcitation code immune from coherent phase errors and also equivalent to a Paulirotated stabilizer code. When the stabilizer outer code is faulttolerant, the constantexcitation code has a positive faulttolerant threshold against stochastic errors. Setting the outer code as a fourqubit amplitude damping code yields an eightqubit constantexcitation code that corrects a single amplitude damping error, and we analyze this code’s potential as a quantum memory.
Introduction
Quantum error correction (QEC) promises to unlock the full potential of quantum technologies by combating the detrimental effects of noise in quantum systems. The ultimate goal in QEC is to protect quantum information under realistic noise models. However, QEC is most often studied by abstracting away the underlying physics of actual quantum systems, and assumes a simple stochastic Pauli noise model, as opposed to coherent errors which are much more realistic.
Coherent errors are unitary operations that damage qubits collectively, and are ubiquitous in many quantum systems. Especially pertinent are coherent phase errors that occur on any quantum system that comprises noninteracting qubits with identical energy levels. In such systems, coherent phase errors can result from unwanted collective interactions with stray fields^{1}, collective drift in the qubits’ energy levels, and fundamental limitations on the precision in estimating the magnitude of the qubits’ energy levels. To address coherent errors, prior work either (1) analyzes how existing QEC codes perform under coherent errors without any mitigation of the coherent errors, (2) uses active quantum control which incurs additional resource overheads to mitigate coherent errors offers partial immunity against coherent errors^{2} or (3) completely avoids coherent errors using appropriate decoherencefree subspaces (DFS)^{3,4,5,6,7,8,9,10,11}. In this paper, we focus on a family of QEC codes that are compatible with approach (3), and discuss performing QEC protocols with respect to this family of QEC codes.
To completely avoid coherent phase errors, quantum information can be encoded into a constantexcitation (CE) subspace^{4,7,11}, which is a DFS of any Hamiltonian that describes an ensemble of identical noninteracting qubits. Given the promise of CE QEC codes to completely avoid coherent phase errors, these codes have been studied within both qubit^{3,4,5,6,7,9,10} and bosonic^{11,12,13,14} settings. Such codes either additionally avoid other types of coherent errors^{4,5}, or can combat against other forms of errors^{3,6,7,9,10,11,12,13,14}. However, qubit CE QEC codes lack a fullfledged QEC analysis, where explicit encoding, decoding circuits, and QEC circuits remain to be constructed. This impedes the adoption of CE codes in a faulttolerant QEC setting.
In this paper, we give an accessible procedure to construct QEC codes that not only completely avoid coherent phase errors, but also support faulttolerant quantum computation. Namely, we concatenate stabilizer codes \({{\mathcal{C}}}_{{\mathtt{Stab}}}\) with a length two repetition code \({{\mathcal{C}}}_{{\mathtt{REP}}2}\), and apply a bitflip on half of the qubits. We can also naturally interpret these codes within the codeword stabilized (CWS) framework^{15,16}, thereby extending the utility of CWS codes beyond a purely theoretical setting.
Amplitude damping (AD) errors model energy relaxation, and accurately describe errors in many physical systems. By concatenating the fourqubit AD code^{17} with the dualrail code^{18}, we construct an eightqubit CE code that corrects a single AD error. We provide this code’s QEC circuits (Figs. 3 and 4), and analyze its potential as a quantum memory under the AD noise model (Fig. 5).
Our work paves the way towards integrating CE codes with mainstream QEC codes. By doubling the number of qubits required, we make any quantum code immune against coherent phase errors. When coherent phase errors are a dominant source of errors, we expect CE codes to significantly reduce faulttolerant overheads.
Results
Hybridizing stabilizer and CE codes
Coherent phase errors can arise from the collective interaction of identical qubits with a classical field. Since the collective Hamiltonian of noninteracting identical qubits is proportional to S^{z} = Z_{1} + ⋯ + Z_{N} where Z_{j} flips the jth qubit’s phase, we model coherent phase errors with unitaries of the form \({U}_{\theta }=\exp (i\theta {S}^{z})\). Here, θ depends on both the interacting field’s magnitude and the qubits’ energy levels.
Using any CE code, we can completely avoid coherent phase errors. This is because such codes must lie within an eigenspace of S^{z}, which is spanned by the computational basis states \(\left{\bf{x}}\right\rangle =\left{x}_{1}\right\rangle \otimes \cdots \otimes \left{x}_{N}\right\rangle\) for which the excitation number, given by the Hamming weight wt(x) = x_{1} + ⋯ + x_{N} of x, is constant. The simplest CE code is the dualrail code^{18}, \({{\mathcal{C}}}_{{\mathtt{KLM}}}\), with logical codewords \(\left{0}_{{\mathtt{KLM}}}\right\rangle =\left01\right\rangle\) and \(\left{1}_{{\mathtt{KLM}}}\right\rangle =\left10\right\rangle\).
However, \({{\mathcal{C}}}_{{\mathtt{KLM}}}\) cannot correct any errors. Therefore, we concatenate it with an [[n, k, d]] stabilizer code \({{\mathcal{C}}}_{{\mathtt{Stab}}}\) to obtain a code \({\mathcal{C}}\) with encoding circuit given in Fig. 1. Then \({\mathcal{C}}\) is an [[2n, k, d]] QEC code that is also impervious to coherent phase errors. Now, concatenating any state \({\sum }_{{\bf{x}}\in {\{0,1\}}^{n}}{a}_{{\bf{x}}}\left{\bf{x}}\right\rangle \in {{\mathcal{C}}}_{{\mathtt{Stab}}}\) with \({{\mathcal{C}}}_{{\mathtt{KLM}}}\) yields \({\sum }_{{\bf{x}}\in {\{0,1\}}^{n}}{a}_{{\bf{x}}}\left\varphi ({\bf{x}})\right\rangle\), where φ((x_{1}, x_{2}, …, x_{n−1}, x_{n})) = (x_{1}, 1 − x_{1}, x_{2}, 1 − x_{2}, …, x_{n−1}, 1 − x_{n−1}, x_{n}, 1 − x_{n}). Since wt(φ(x)) = n for every x ∈ {0, 1}^{n}, it follows that the concatenated state must be an eigenstate of S^{z} with the same eigenvalue. Hence, \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) is a CE code, and therefore avoids coherent phase errors.
The code \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) is very similar to \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\), which is \({{\mathcal{C}}}_{{\mathtt{Stab}}}\) concatenated with a length two repetition code \({{\mathcal{C}}}_{{\mathtt{REP}}2}\) that maps \(\left0\right\rangle\) to \(\left00\right\rangle\) and \(\left1\right\rangle\) to \(\left11\right\rangle\). Since \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}=R{{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\) where R = (I⊗X)^{⊗n}, and I and X denote the identity and bitflip operations on a qubit respectively, \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) is equivalent to \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\) up to the Pauli rotation R and we call \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) a rotatedstabilizer code.
We can also cast \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) within the CWS framework by deriving its word stabilizer and word operators. Since \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) and \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\) are equivalent up to R, it suffices to derive \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\)’s word stabilizer and word operators. Namely, \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) and \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\) have identical word stabilizers generated by the stabilizer and logical Z operators of \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\). Moreover, the word operators \({w}_{1},\ldots ,{w}_{{2}^{k}}\) of \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\) are its logical X operators and the word operators \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) are \(R{w}_{1},\ldots ,R{w}_{{2}^{k}}\). We supply explicit constructs of the word stabilizer and word operators of \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) in “Methods” section.
The code \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) inherits its logical operators from the logical operators of \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\). Given any singlequbit logical operator U on \({{\mathcal{C}}}_{{\mathtt{Stab}}}\), the corresponding unitary L_{REP2}(U) on \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\) is given in Fig. 2a. Then the corresponding logical operator on \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) is \(\tilde{U}=R{{\rm{L}}}_{{\mathtt{REP}}2}(U)R\). Similarly, given an mqubit logical operator U_{m} on \({{\mathcal{C}}}_{{\mathtt{Stab}}}\), the corresponding logical operator on \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\) is L_{REP2}(U_{m}) (Fig. 2b), and the corresponding logical operator on \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) is R^{⊗m}L_{REP2}(U_{m})R^{⊗m}. If U is a tensor product of singlequbit Pauli gates, then \(\tilde{U}\) is also a tensor product of singlequbit Pauli gates. Hence, if \({{\mathcal{C}}}_{{\mathtt{Stab}}}\) has transversal gates comprising of singlequbit Paulis, then \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) also has corresponding transversal gates of the same form. If U_{m} is a diagonal unitary in the computational basis, then \({\tilde{U}}_{m}={\pi }_{m}^{\dagger }({U}_{m}\otimes {I}^{\otimes nm}){\pi }_{m}\) is also the logical operator on \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\).
To design errorcorrection procedures for \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\), we leverage on the errorcorrection procedures of \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\) and the interpretation that \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) is \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\) with an effective R error. We can extract the syndrome of a Pauli error E acting on \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) by measuring eigenvalues of Pauli observables. These Pauli observables can be generators associated with \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\)’s stabilizer, and these generators are derived easily from the generators of \({{\mathcal{C}}}_{{\mathtt{Stab}}}\); if G_{1}, …, G_{n−k} are \({{\mathcal{C}}}_{{\mathtt{Stab}}}\)’s stabilizer’s generators, then \({\bar{G}}_{1},\ldots ,{\bar{G}}_{2nk}\) generate \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\)’s stabilizer, where \({\bar{G}}_{i}={{\rm{L}}}_{{\mathtt{REP}}2}({G}_{i})\) for i = 1, …, n − k and \({\bar{G}}_{nk+j}={Z}_{2j1}{Z}_{2j}\) for j = 1, …, n. We complete the QEC procedure by using measured eigenvalues of \({\bar{G}}_{1},\ldots ,{\bar{G}}_{2nk}\) to estimate the Pauli error \(E^{\prime}\) that could have occurred, and reverse its effect.
The generator \({\bar{G}}_{j}\)’s eigenvalue on \(E\left\psi \right\rangle\) for \(\left\psi \right\rangle \in {{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) when measured is \({\theta }_{j}={(1)}^{{s}_{j}}\) for some s_{j} = 0, 1. Here, s_{j} = 0 when \({\bar{G}}_{j}\) and ER commute and s_{j} = 1 otherwise. Now, denote the eigenvalue of \({\bar{G}}_{j}\) on \(R\left{\psi }_{{\mathtt{KLM}}}\right\rangle \in {{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\) as \({(1)}^{{r}_{j}}\) for some r_{j} = 0, 1. Whenever E = I^{⊗2n}, we have r ⊕ s = 0 where r = (r_{1}, …, r_{2n−k}) and s = (s_{1}, …, s_{2n−k}). Using r ⊕ s, we estimate the error \(E^{\prime}\) that could have occurred on \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\). For this, we use any decoder Dec_{Stab,REP2} that maps a syndrome vector obtained from a corrupted state of \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\) to an estimated Pauli error. Such a decoder Dec_{Stab,REP2} can be a maximum likelihood decoder^{19,20} or a belief propagation decoder^{21,22,23}. Explicitly, our code \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\)’s decoder has the form
and thereby inherits its performance from the decoder Dec_{Stab,REP}_{2} on the stabilizer code \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\).
Now let us introduce some terminology related to the decoding of stabilizer codes. Denoting the singlequbit Pauli operators as I, X, the phaseflip operator Z, and Y = iXZ, the set of nqubit Pauli operators is {I, X, Y, Z}^{⊗n}. Define bin(P) = (a∣b) as a 2nbit binary vector where a = (a_{1}, …, a_{n}) and b = (b_{1}, …, b_{n}) are nbit binary vectors such that \(P=w{X}^{{a}_{1}}{Z}^{{b}_{1}}\otimes \cdots \otimes {X}^{{a}_{n}}{Z}^{{b}_{n}}\) for some w = ±1, ± i. Given any two Pauli matrices P and \(P^{\prime}\) with binary representations bin(P) = (a, b) and \({\rm{bin}}(P^{\prime} )=({\bf{a}}^{\prime} ,{\bf{b}}^{\prime} )\), their symplectic inner product^{24} over \({{\mathbb{F}}}_{2}\) i s defined to be \({\langle {\rm{bin}}(P),{\rm{bin}}(P^{\prime} )\rangle }_{{\rm{sy}}}={\bf{a}}\cdot {\bf{b}}^{\prime} +{\bf{a}}^{\prime} \cdot {\bf{b}}\).
To see how to decode our concatenated code, note that
By linearity of the inner product, it follows that \({r}_{j}\oplus {s}_{j}={\langle {\rm{bin}}({\bar{G}}_{j}),{\rm{bin}}(E)\rangle }_{{\rm{sy}}}\). This shows that \({(1)}^{{r}_{j}\oplus {s}_{j}}\) is equal to the eigenvalue of G_{j} when measured on \(R\left\psi \right\rangle\), the latter of which is a state in \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\), from which we can deduce (1).
When stochastic errors evolve under the influence of \({U}_{\theta }=\exp (i\theta {S}^{z})\), their weight is preserved. First, note that
Then, for any Nqubit Pauli matrix P = P_{1} ⊗ ⋯ ⊗ P_{N}, we have that
When P_{j} = I or Z, we clearly have \(\exp (i\theta Z){P}_{j}\exp (i\theta Z)={P}_{j}\). When P_{j} = X or Y, we have \(\exp (i\theta Z){P}_{j}\exp (i\theta Z)=\exp (2i\theta Z){P}_{j}\). For any value of θ, \(\exp (2i\theta Z)X\) and \(\exp (2i\theta Z)Y\) are never the identity operator. Hence we can see that the weight of \(\tilde{P}\) is identical to the weight of P. By performing stabilizer measurements, the error \(\tilde{P}\) gets projected randomly onto some Pauli of weight equal to the weight of P, if this weight is no greater than half of the code’s distance, it can be corrected according to the earlierdescribed decoding procedure.
We now show that \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) has a positive faulttolerant threshold when \({{\mathcal{C}}}_{{\mathtt{Stab}}}\) is a Calderbank–Shor–Steane (CSS) code^{25,26} that encodes a single logical qubit and has transversal logical Pauli I, X, Y, and Z gates given by \(\bar{I}={I}^{\otimes n}\), \(\bar{X}={X}^{\otimes n}\), \(\bar{Y}={Y}^{\otimes n}\) and \(\bar{Z}={Z}^{\otimes n}\), respectively. (also with transversal Hadamard.) First, \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) has transversal logical Pauli and controllednot (CNOT) gates. Then \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\) has transversal logical X and Z gates given by \({\bar{X}}_{{\mathtt{REP}}2}={\bar{X}}^{\otimes 2}={X}^{\otimes 2n}\) and \({\bar{Z}}_{{\mathtt{REP}}2}=\pi (\bar{Z}\otimes \bar{I}){\pi }^{\dagger }\), respectively, and logical CNOT gate \({\overline{{\rm{CNOT}}}}_{{\mathtt{REP}}2}\) given by 2n transversal CNOT gates. Thus, \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) has its logical X and Z operators given by \({\bar{X}}_{{\mathtt{KLM}}}=R{\bar{X}}_{{\mathtt{REP}}2}R={X}^{\otimes 2n}\) and \({\bar{Z}}_{{\mathtt{KLM}}}=R{\bar{Z}}_{{\mathtt{REP}}2}R={(1)}^{n}{Z}_{{\mathtt{REP}}2}\), respectively. Furthermore, the logical CNOT gate of \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) has the form \({\overline{{\rm{CNOT}}}}_{{\mathtt{KLM}}}=(R\otimes R){\overline{{\rm{CNOT}}}}_{{\mathtt{REP}}2}(R\otimes R)={\overline{{\rm{CNOT}}}}_{{\mathtt{REP}}2}.\) Second, since we can perform these transversal CNOTs and have stabilizers that correspond to a CSS code, we can measure syndromes and logical Paulis faulttolerantly using Steane’s method for CSS codes^{27}. Relying on gateteleportation techniques^{28}, we can implement all Clifford and nonClifford gates faulttolerantly. Since the faulttolerant logical operations will have a finite number of circuit components, using the method of counting malignant combinations in extended rectangles^{29} yields a positive faulttolerant threshold for stochastic noise.
An amplitude damping CE code
The simplest CE code that detects AD errors is the fourqubit \({{\mathcal{C}}}_{{\mathtt{ABC}}+}\) code^{6}. AD errors are introduced by an AD channel \({{\mathcal{A}}}_{\gamma }\), which has Kraus operators \({A}_{0}=\left0\right\rangle \langle 0 +\sqrt{1\gamma } 1\rangle \left\langle 1\right\) and \({A}_{1}=\sqrt{\gamma }\left0\right\rangle \left\langle 1\right\). These Kraus operators model the damping an excited state’s amplitude and the relaxation of an excited state to the ground state with probability γ. While \({{\mathcal{C}}}_{{\mathtt{ABC}}+}\) detects a single AD error, it cannot correct any AD errors. Other CE codes that can correct some AD errors have been designed, but either have overly complicated encoding and QEC circuits^{3}, or lack explicit QEC circuits^{5,7,8,9,10}.
Here, we present a CE code that is the concatenation of the fourqubit AD code \({{\mathcal{C}}}_{{\mathtt{LNCY}}}\)^{17} with \({{\mathcal{C}}}_{{\mathtt{KLM}}}\), and permute the qubits to get \({{\mathcal{C}}}_{{\mathtt{8qubit}}}\) with logical codewords
We elucidate the connection between \({{\mathcal{C}}}_{{\mathtt{LNCY}}}\), \({{\mathcal{C}}}_{{\mathtt{ABC+}}}\), \({{\mathcal{C}}}_{{\mathtt{8qubit}}}\), \({{\mathcal{C}}}_{{\mathtt{KLM}}}\), and \({{\mathcal{C}}}_{{\mathtt{REP2}}}\) in Fig. 3b. We prove that \({{\mathcal{C}}}_{{\mathtt{8qubit}}}\) corrects a single AD error by verifying that the Knill–Laflamme QEC criterion^{30} holds with respect to the Kraus operators K_{1}, …, K_{8} and \({A}_{0}^{\otimes 8}\) where K_{a} denotes an nqubit operator that applies A_{1} on the ath qubit and A_{0} on each of the remaining qubits. The simplicity of \({{\mathcal{C}}}_{{\mathtt{8qubit}}}\) allows for the direct construction of a simple errorcorrection strategy for AD errors, without referring to the properties of \({{\mathcal{C}}}_{{\mathtt{LNCY}}}\), \({{\mathcal{C}}}_{{\mathtt{ABC+}}}\), and \({{\mathcal{C}}}_{{\mathtt{KLM}}}\).
In Fig. 3, we illustrate accessible constructs for \({{\mathcal{C}}}_{{\mathtt{8qubit}}}\)’s encoding circuits and logical computations. In Fig. 4, we give decoding procedures when an AD error is detected. We measure the eigenvalues m_{1}, m_{2}, m_{3}, and m_{4} of the respective operators Z_{1}Z_{2}, Z_{3}Z_{4}, Z_{5}Z_{6}, and Z_{7}Z_{8} to determine if any AD error has occurred. Denoting b_{a} = (1 − m_{a})/2 for a = 1, …, 4, we have five correctible outcomes with respect to the syndrome vector b = (b_{1}, b_{2}, b_{3}, b_{4}). When b = 0, the codespace is damped uniformly and no AD error has occurred. When b has a Hamming weight equal to one, each logical codeword is mapped to a unique product state, and we can ascertain that exactly one AD error must have occurred. When b_{a} = 1 and the other syndrome bits are zero, an AD error must have occurred on either the (2a − 1)th or the (2a)th qubit. Since the effect of an AD error on the (2a − 1)th and (2a)th qubit is identical, this makes \({{\mathcal{C}}}_{{\mathtt{8qubit}}}\) a degenerate quantum code with respect to AD errors, and explains why \({{\mathcal{C}}}_{{\mathtt{8qubit}}}\) has five correctible outcomes as opposed to nine if it were nondegenerate. The elegant structure of the four corrupted codespaces with a single AD error aids our construction of decoding circuits for \({{\mathcal{C}}}_{{\mathtt{8qubit}}}\) (see details in the “Methods” section).
We illustrate \({{\mathcal{C}}}_{{\mathtt{8qubit}}}\)’s performance as a quantum memory assuming perfect encoding and decoding and that AD errors only occur during the memory storage. We calculate probabilities ϵ and ϵ_{base} of having uncorrectable AD errors occurring on \({{\mathcal{C}}}_{{\mathtt{8qubit}}}\) and an unprotected qubit after T applications of \({{\mathcal{A}}}_{\delta }^{\otimes 8}\) and \({{\mathcal{A}}}_{\delta }\) respectively. Since the transmissivity (1 − δ) of an AD channel \({{\mathcal{A}}}_{\delta }\) is multiplicative under composition, (1 − ϵ_{base}) = (1−δ)^{T} and
Whenever \(28{\epsilon }_{{\rm{base}}}^{2}\le {\epsilon }_{{\rm{base}}}\), it is advantageous to use \({{\mathcal{C}}}_{{\mathtt{8qubit}}}\). Hence, whenever T ≤ T*, where
using \({{\mathcal{C}}}_{{\mathtt{8qubit}}}\) is advantageous as compared to leaving a qubit unprotected (Fig. 5).
Discussion
When coherent phase errors occur more frequently than stochastic errors, we expect CE codes to outperform generic QEC codes. For future work, the numerical faulttolerant thresholds of our codes can be calculated when the noise model is a convex combination of stochastic errors and coherent phase errors. In particular, the outer codes could be chosen to be surface codes^{31,32,33}, quantum LDPC codes^{34,35} and Aliferis–Preskill concatenated codes for biased noise^{36}. One can also study other choices for the inner codes in our construction to obtain concatenated codes with different structures and residing in different types of decoherencefree subspaces. For instance, we can consider other CE codes^{37}, quantum codes that avoid exchange errors^{38,39,40,41,42}, and quantum codes that avoid other different errors^{4,5,8,18,43}.
Methods
Our CE code as a CWS code
Here, we derive the word stabilizer and word operators of our CE code \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\). Now denote S_{Stab} as the stabilizer of \({{\mathcal{C}}}_{{\mathtt{Stab}}}\) and G_{1}, …, G_{n−k} as its generators. Then the operators L_{REP2}(G_{i}), Z_{2j−1}Z_{2j} generate \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\)’s stabilizer where i = 1, …, n − k and j = 1, …, n. Denoting the logical X and Z operators of \({{\mathcal{C}}}_{{\mathtt{Stab}}}\) as \({\bar{X}}_{1},\ldots ,{\bar{X}}_{k}\) and \({\bar{Z}}_{1},\ldots ,{\bar{Z}}_{k}\) respectively, the logical X and Z operators of \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\) are given by \({{\rm{L}}}_{{\mathtt{REP}}2}({\bar{X}}_{1}),\ldots ,{{\rm{L}}}_{{\mathtt{REP}}2}({\bar{X}}_{k})\) and \({{\rm{L}}}_{{\mathtt{REP}}2}({\bar{Z}}_{1}),\ldots ,{{\rm{L}}}_{{\mathtt{REP}}2}({\bar{Z}}_{k})\), respectively. Since \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\) is a stabilizer code, its word stabilizer of \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\) is
Since the word stabilizer of \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) is identical to the word stabilizer of \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\), the word stabilizer of \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) is then given by W.
Clearly, the word operators of \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\) are generated by \({{\rm{L}}}_{{\mathtt{REP}}2}({\bar{X}}_{1}),\ldots ,{{\rm{L}}}_{{\mathtt{REP}}2}({\bar{X}}_{k}).\) Hence, the word operators of \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) are
where x_{1}, …, x_{k} = 0, 1.
An amplitude damping CE code: additional details
We now explain the connection between the codes \({{\mathcal{C}}}_{{\mathtt{LNCY}}}\), \({{\mathcal{C}}}_{{\mathtt{ABC+}}}\), \({{\mathcal{C}}}_{{\mathtt{8qubit}}}\), \({{\mathcal{C}}}_{{\mathtt{KLM}}}\), and \({{\mathcal{C}}}_{{\mathtt{REP2}}}\) as illustrated in Fig. 4b. Now recall that the fourqubit amplitude damping code^{17} has logical codewords
Concatenating this with the dualrail code \({{\mathcal{C}}}_{{\mathtt{KLM}}}\) gives the code
It is visually easier to work with a code if we collect the odd and even qubits in separate blocks of four qubits. We can achieve this by applying the permutation π^{†}, which maps qubits 1, 3, 5, 7 to qubits 1, 2, 3, 4 and qubits 2, 4, 6, 8 to qubits 5, 6, 7, 8, to get our code with logical codewords
Note that the above code can be obtained from the fourqubit code \({{\mathcal{C}}}_{{\mathtt{ABC}}+}\) with logical codewords
after concatenation with \({{\mathcal{C}}}_{{\mathtt{REP2}}}\). Note that by concatenating \({{\mathcal{C}}}_{{\mathtt{LNCY}}}\) with \({{\mathcal{C}}}_{{\mathtt{REP2}}}\), we get a concatenated code \({{\mathcal{C}}}_{2{\mathtt{LNCY}}}={{\mathcal{C}}}_{{\mathtt{LNCY}}}\circ {{\mathcal{C}}}_{{\mathtt{REP2}}}\) with logical codewords
Since the stabilizer code \({{\mathcal{C}}}_{2{\mathtt{LNCY}}}\) is equivalent to \({{\mathcal{C}}}_{{\mathtt{8qubit}}}\) up to a Pauli rotation given by X^{⊗4} ⊗ I^{⊗4}, we can interpret \({{\mathcal{C}}}_{{\mathtt{8qubit}}}\) as a rotated concatenated stabilizer code.
To encode an arbitrary singlequbit logical state into \({{\mathcal{C}}}_{{\mathtt{8qubit}}}\), we concatenate the encoding circuits of \({{\mathcal{C}}}_{{\mathtt{LNCY}}}\) and \({{\mathcal{C}}}_{{\mathtt{REP2}}}\), and apply a Pauli rotation. Quantum circuits can be further simplified when encode the logical stabilizer states \(\left{0}_{L}\right\rangle\) and \(\left{+}_{L}\right\rangle =(\left{0}_{L}\right\rangle +\left{1}_{L}\right\rangle )/\sqrt{2}\).
To show that our QEC code spanned by \(\left{0}_{L}\right\rangle\) and \(\left{1}_{L}\right\rangle\), corrects single AD errors, it suffices to verify the Knill–Laflamme QEC conditions. In particular, we show that for i, j = 0, 1 and a, b = 1, …, 8, we have 〈i_{L}∣K_{a}K_{b}∣j_{L}〉 = δ_{i,j}δ_{a,b}g_{a} for some real number g_{a}. Now let us explain the effects of correctible AD errors on \({{\mathcal{C}}}_{{\mathtt{8qubit}}}\). Recall that the correctible AD errors are given by \({K}_{0}={A}_{0}^{\otimes 8}\), \({K}_{1}={A}_{1}\otimes {A}_{0}^{\otimes 7}\), \({K}_{2}={A}_{0}\otimes {A}_{1}\otimes {A}_{0}^{\otimes 6}\), …, \({K}_{7}={A}_{0}^{\otimes 6}\otimes {A}_{1}\otimes {A}_{0}\), and \({K}_{8}={A}_{0}^{\otimes 7}\otimes {A}_{1}\). Then we can see the following.

1.
\({K}_{0}\left{0}_{L}\right\rangle ={(1\gamma )}^{2}\left{0}_{L}\right\rangle\)
\({K}_{0}\left{1}_{L}\right\rangle ={(1\gamma )}^{2}\left{1}_{L}\right\rangle\).

2.
\({K}_{1}\left{0}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1\gamma )}^{3}}\left01110000\right\rangle\)
\({K}_{1}\left{1}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1\gamma )}^{3}}\left01000011\right\rangle\).

3.
\({K}_{2}\left{0}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1\gamma )}^{3}}\left10110000\right\rangle\)
\({K}_{2}\left{1}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1\gamma )}^{3}}\left10000011\right\rangle\).

4.
\({K}_{3}\left{0}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1\gamma )}^{3}}\left11010000\right\rangle\)
\({K}_{3}\left{1}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1\gamma )}^{3}}\left00011100\right\rangle\).

5.
\({K}_{4}\left{0}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1\gamma )}^{3}}\left11100000\right\rangle\)
\({K}_{4}\left{1}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1\gamma )}^{3}}\left00101100\right\rangle\).

6.
\({K}_{5}\left{0}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1\gamma )}^{3}}\left00000111\right\rangle\)
\({K}_{5}\left{1}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1\gamma )}^{3}}\left00110100\right\rangle\).

7.
\({K}_{6}\left{0}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1\gamma )}^{3}}\left00001011\right\rangle\)
\({K}_{6}\left{1}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1\gamma )}^{3}}\left00111000\right\rangle\).

8.
\({K}_{7}\left{0}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1\gamma )}^{3}}\left00001101\right\rangle\)
\({K}_{7}\left{1}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1\gamma )}^{3}}\left11000001\right\rangle\).

9.
\({K}_{8}\left{0}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1\gamma )}^{3}}\left00001110\right\rangle\)
\({K}_{8}\left{1}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1\gamma )}^{3}}\left11000010\right\rangle\).
In the above, we can see that the effect of K_{2j−1} is identical to K_{2j} for j = 1, …, 4. Hence there are only five unique correctible outcomes that correspond to the correctible errors K_{0}, K_{1}, K_{3}, K_{5} and K_{7}. Each of these correctible outcomes are clearly orthogonal. Hence to perform quantum error correction, it suffices to rotate the orthogonal corrupted codespaces back to the original codespace.
Now, to extract the error syndrome, it suffices to measure the stabilizers Z_{2j−1}, Z_{2j} for j = 1, 2, 3, 4. These stabilizer measurements leave the codespace afflicted with correctible AD errors unchanged, and measure the parity of the (2j − 1)th and (2j)th qubits. We can then make the following decisions.

1.
If the parity of the all blocks is even, then we can ascertain that no AD error has occured, which corresponds to the effect of the Kraus operator K_{0}.

2.
If the parity of the first and second qubit is odd, while the parity of the remaining blocks is even, then we can ascertain that either K_{1} or K_{2} has occured.

3.
If the parity of the third and fourth qubit is odd, while the parity of the remaining blocks is even, then we can ascertain that either K_{3} or K_{4} has occured.

4.
If the parity of the fifth and sixth qubit is odd, while the parity of the remaining blocks is even, then we can ascertain that either K_{5} or K_{6} has occured.

5.
If the parity of the seventh and eight qubit is odd, while the parity of the remaining blocks is even, then we can ascertain that either K_{7} or K_{8} has occured.
The structure of the corrupted codespaces allows us to decode them into a physical qubit by first discarding four qubits, and subsequently employing the same decoding circuit up to a permutation.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The code used in this work is available from the corresponding author upon reasonable request.
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Acknowledgements
I acknowledge support from the EPSRC (Grant No. EP/M024261/1) and the QCDA project (Grant No. EP/R043825/1), which has received funding from the QuantERA ERANET Cofund in Quantum Technologies implemented within the European Union’s Horizon 2020 Programme.
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Ouyang, Y. Avoiding coherent errors with rotated concatenated stabilizer codes. npj Quantum Inf 7, 87 (2021). https://doi.org/10.1038/s41534021004298
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