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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 constant-excitation codes by code concatenation. Namely, by concatenating an [[_n_, _k_, _d_]] stabilizer outer code with dual-rail inner

codes, we obtain a [[2_n_, _k_, _d_]] constant-excitation code immune from coherent phase errors and also equivalent to a Pauli-rotated stabilizer code. When the stabilizer outer code is

fault-tolerant, the constant-excitation code has a positive fault-tolerant threshold against stochastic errors. Setting the outer code as a four-qubit amplitude damping code yields an

eight-qubit constant-excitation code that corrects a single amplitude damping error, and we analyze this code’s potential as a quantum memory. SIMILAR CONTENT BEING VIEWED BY OTHERS PHASE

TRANSITION IN MAGIC WITH RANDOM QUANTUM CIRCUITS Article 23 September 2024 STABILIZER CODES FOR OPEN QUANTUM SYSTEMS Article Open access 29 June 2023 DECODING GENERAL ERROR CORRECTING CODES

AND THE ROLE OF COMPLEMENTARITY Article Open access 10 January 2025 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

non-interacting qubits with identical energy levels. In such systems, coherent phase errors can result from unwanted collective interactions with stray fields1, 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 errors2 or (3) completely avoids coherent errors using appropriate decoherence-free 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 constant-excitation (CE) subspace4,7,11, which is a DFS of any Hamiltonian that describes an ensemble of identical non-interacting

qubits. Given the promise of CE QEC codes to completely avoid coherent phase errors, these codes have been studied within both qubit3,4,5,6,7,9,10 and bosonic11,12,13,14 settings. Such codes

either additionally avoid other types of coherent errors4,5, or can combat against other forms of errors3,6,7,9,10,11,12,13,14. However, qubit CE QEC codes lack a full-fledged QEC analysis,

where explicit encoding, decoding circuits, and QEC circuits remain to be constructed. This impedes the adoption of CE codes in a fault-tolerant 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 fault-tolerant 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 bit-flip on half of the qubits. We can also naturally interpret

these codes within the codeword stabilized (CWS) framework15,16, 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 four-qubit AD code17 with the dual-rail code18, we construct an eight-qubit 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 fault-tolerant 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 non-interacting identical qubits is proportional to _S__z_ = _Z_1 + ⋯ + _Z__N_ where

_Z__j_ flips the _j_th 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 dual-rail code18, \({{\mathcal{C}}}_{{\mathtt{KLM}}}\), with logical codewords

\(\left|{0}_{{\mathtt{KLM}}}\right\rangle =\left|01\right\rangle\) and \(\left|{1}_{{\mathtt{KLM}}}\right\rangle =\left|10\right\rangle\). 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 [[2_n_, _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 \(\left|0\right\rangle\) to \(\left|00\right\rangle\) and \(\left|1\right\rangle\) to \(\left|11\right\rangle\). 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 bit-flip 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 rotated-stabilizer 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 single-qubit logical operator _U_ on \({{\mathcal{C}}}_{{\mathtt{Stab}}}\), the corresponding

unitary LREP2(_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 _m_-qubit logical operator _U__m_ on

\({{\mathcal{C}}}_{{\mathtt{Stab}}}\), the corresponding logical operator on \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}\) is LREP2(_U__m_) (Fig. 2b), and the corresponding logical

operator on \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) is _R_⊗_m_LREP2(_U__m_)_R_⊗_m_. If _U_ is a tensor product of single-qubit Pauli gates, then \(\tilde{U}\) is also a tensor

product of single-qubit Pauli gates. Hence, if \({{\mathcal{C}}}_{{\mathtt{Stab}}}\) has transversal gates comprising of single-qubit 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

error-correction procedures for \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\), we leverage on the error-correction 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 \({\bar{G}}_{1},\ldots ,{\bar{G}}_{2n-k}\) 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

\({\bar{G}}_{n-k+j}={Z}_{2j-1}{Z}_{2j}\) for _j_ = 1, …, _n_. We complete the QEC procedure by using measured eigenvalues of \({\bar{G}}_{1},\ldots ,{\bar{G}}_{2n-k}\) 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 _E__R_ 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_⊗2_n_, we have R ⊕ S = 0 where R = (_r_1, …, _r_2_n_−_k_) and S = (_s_1, …, _s_2_n_−_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 DecStab,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 DecStab,REP2 can be a maximum likelihood decoder19,20 or a belief

propagation decoder21,22,23. Explicitly, our code \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\)’s decoder has the form

$${{\mathtt{Dec}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}({\bf{s}})={{\mathtt{Dec}}}_{{\mathtt{Stab}},{\mathtt{REP}}2}({\bf{r}}\oplus {\bf{s}}),$$ (1) and thereby inherits its performance from the

decoder DecStab,REP2 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 single-qubit Pauli operators as _I_, _X_, the phase-flip operator _Z_, and _Y_ = _i__X__Z_, the set of _n_-qubit Pauli operators is {_I_, _X_, _Y_, _Z_}⊗_n_. Define bin(_P_) = 

(A∣B) as a 2_n_-bit binary vector where A = (_a_1, …, _a__n_) and B = (_b_1, …, _b__n_) are _n_-bit 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 product24 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 $$\begin{array}{lll}{r}_{j}\,=\,{\langle

{\rm{bin}}({\bar{G}}_{j}),{\rm{bin}}(R)\rangle }_{{\rm{sy}}},\\ {s}_{j}\,=\,{\langle {\rm{bin}}({\bar{G}}_{j}),({\rm{bin}}(E)+{\rm{bin}}(R))\rangle }_{{\rm{sy}}},\end{array}$$ (2) 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 $${U}_{\theta

}=\mathop{\prod }\limits_{j=1}^{N}\exp (-i\theta {Z}_{j})=\exp {(-i\theta Z)}^{\otimes N}.$$ (3) Then, for any _N_-qubit Pauli matrix _P_ = _P_1 ⊗ ⋯ ⊗ _P__N_, we have that

$$\tilde{P}={U}_{\theta }P{U}_{\theta }^{\dagger }=\mathop{\bigotimes }\limits_{j=1}^{N}\exp (-i\theta Z){P}_{j}\exp (i\theta Z).$$ (4) 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 earlier-described decoding procedure. We now show that \({{\mathcal{C}}}_{{\mathtt{Stab}},{\mathtt{KLM}}}\) has a positive fault-tolerant threshold when

\({{\mathcal{C}}}_{{\mathtt{Stab}}}\) is a Calderbank–Shor–Steane (CSS) code25,26 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 controlled-not (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 2_n_ 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 fault-tolerantly using Steane’s method for CSS codes27. Relying on gate-teleportation techniques28, we can implement all Clifford and non-Clifford gates fault-tolerantly. Since the

fault-tolerant logical operations will have a finite number of circuit components, using the method of counting malignant combinations in extended rectangles29 yields a positive

fault-tolerant threshold for stochastic noise. AN AMPLITUDE DAMPING CE CODE The simplest CE code that detects AD errors is the four-qubit \({{\mathcal{C}}}_{{\mathtt{ABC}}+}\) code6. AD

errors are introduced by an AD channel \({{\mathcal{A}}}_{\gamma }\), which has Kraus operators \({A}_{0}=\left|0\right\rangle \langle 0| +\sqrt{1-\gamma }| 1\rangle \left\langle 1\right|\)

and \({A}_{1}=\sqrt{\gamma }\left|0\right\rangle \left\langle 1\right|\). 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 circuits3, or lack explicit QEC circuits5,7,8,9,10. Here, we present a CE code that is the concatenation of the four-qubit

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

$$\begin{array}{l}\left|{0}_{L}\right\rangle =(\left|11110000\right\rangle +\left|00001111\right\rangle )/\sqrt{2}\\ \left|{1}_{L}\right\rangle =(\left|00111100\right\rangle

+\left|11000011\right\rangle )/\sqrt{2}.\end{array}$$ (5) 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 criterion30 holds with respect to the Kraus operators _K_1, …, _K_8 and \({A}_{0}^{\otimes 8}\) where _K__a_ denotes an

_n_-qubit operator that applies _A_1 on the _a_th 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 error-correction 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 (2_a_ − 1)th or the (2_a_)th

qubit. Since the effect of an AD error on the (2_a_ − 1)th and (2_a_)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 non-degenerate. 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 $$\epsilon =1-{(1-{\epsilon }_{{\rm{base}}})}^{8}-8{\epsilon }_{{\rm{base}}}{(1-{\epsilon }_{{\rm{base}}})}^{7}\le 28{\epsilon

}_{{\rm{base}}}^{2}.$$ (6) Whenever \(28{\epsilon }_{{\rm{base}}}^{2}\le {\epsilon }_{{\rm{base}}}\), it is advantageous to use \({{\mathcal{C}}}_{{\mathtt{8qubit}}}\). Hence, whenever _T_ ≤

 _T_*, where $${T}^{\star }=\frac{{\rm{log}}(27/28)}{{\rm{log}}(1-\delta )},$$ (7) 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

fault-tolerant 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 codes31,32,33, quantum LDPC codes34,35 and Aliferis–Preskill concatenated codes for biased noise36. 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 decoherence-free subspaces. For instance, we can consider other CE codes37, quantum

codes that avoid exchange errors38,39,40,41,42, and quantum codes that avoid other different errors4,5,8,18,43. 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 LREP2(_G__i_), _Z_2_j_−1_Z_2_j_ 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

$$W=\left\{{S}_{{\mathtt{Stab}},{\mathtt{REP}}2}^{a}\mathop{\prod }\limits_{j=1}^{k}{{\rm{L}}}_{{\mathtt{REP}}2}{({\bar{Z}}_{j})}^{{z}_{j}}:a,{z}_{1},\ldots ,{z}_{k}=0,1\right\}.$$ (8) 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

$${w}_{({x}_{1},\ldots ,{x}_{k})}=R\mathop{\prod }\limits_{j=1}^{k}{{\rm{L}}}_{{\mathtt{REP}}2}{({\bar{X}}_{j})}^{{x}_{j}}$$ (9) 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 four-qubit amplitude damping code17 has logical codewords

$$\left|{0}_{{\mathtt{LNCY}}}\right\rangle =(\left|0000\right\rangle +\left|1111\right\rangle )/\sqrt{2}$$ (10) $$\left|{1}_{{\mathtt{LNCY}}}\right\rangle =(\left|1100\right\rangle

+\left|0011\right\rangle )/\sqrt{2}.$$ (11) Concatenating this with the dual-rail code \({{\mathcal{C}}}_{{\mathtt{KLM}}}\) gives the code

$$\left|{0}_{{\mathtt{LNCY}},{\mathtt{KLM}}}\right\rangle =(\left|01010101\right\rangle +\left|10101010\right\rangle )/\sqrt{2}$$ (12)

$$\left|{1}_{{\mathtt{LNCY}},{\mathtt{KLM}}}\right\rangle =(\left|10100101\right\rangle +\left|01011010\right\rangle )/\sqrt{2}.$$ (13) 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 $$\left|{0}_{L}\right\rangle =(\left|00001111\right\rangle +\left|11110000\right\rangle )/\sqrt{2}$$ (14)

$$\left|{1}_{L}\right\rangle =(\left|11000011\right\rangle +\left|00111100\right\rangle )/\sqrt{2}.$$ (15) Note that the above code can be obtained from the four-qubit code

\({{\mathcal{C}}}_{{\mathtt{ABC}}+}\) with logical codewords $$\left|{0}_{{\mathtt{ABC}}+}\right\rangle =(\left|0011\right\rangle +\left|1100\right\rangle )/\sqrt{2}$$ (16)

$$\left|{1}_{{\mathtt{ABC}}+}\right\rangle =(\left|1001\right\rangle +\left|0110\right\rangle )/\sqrt{2},$$ (17) 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

$$\begin{array}{lll}\left|{0}_{2{\mathtt{LNCY}}}\right\rangle &=&(\left|00000000\right\rangle +\left|11111111\right\rangle )/\sqrt{2},\\ \left|{1}_{2{\mathtt{LNCY}}}\right\rangle

&=&(\left|00110011\right\rangle +\left|11001100\right\rangle )/\sqrt{2}.\end{array}$$ (18) 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 single-qubit 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. \({K}_{1}\left|{0}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1-\gamma )}^{3}}\left|01110000\right\rangle\) \({K}_{1}\left|{1}_{L}\right\rangle

=\sqrt{\gamma }\sqrt{{(1-\gamma )}^{3}}\left|01000011\right\rangle\). * 3. \({K}_{2}\left|{0}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1-\gamma )}^{3}}\left|10110000\right\rangle\)

\({K}_{2}\left|{1}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1-\gamma )}^{3}}\left|10000011\right\rangle\). * 4. \({K}_{3}\left|{0}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1-\gamma

)}^{3}}\left|11010000\right\rangle\) \({K}_{3}\left|{1}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1-\gamma )}^{3}}\left|00011100\right\rangle\). * 5. \({K}_{4}\left|{0}_{L}\right\rangle

=\sqrt{\gamma }\sqrt{{(1-\gamma )}^{3}}\left|11100000\right\rangle\) \({K}_{4}\left|{1}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1-\gamma )}^{3}}\left|00101100\right\rangle\). * 6.

\({K}_{5}\left|{0}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1-\gamma )}^{3}}\left|00000111\right\rangle\) \({K}_{5}\left|{1}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1-\gamma

)}^{3}}\left|00110100\right\rangle\). * 7. \({K}_{6}\left|{0}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1-\gamma )}^{3}}\left|00001011\right\rangle\) \({K}_{6}\left|{1}_{L}\right\rangle

=\sqrt{\gamma }\sqrt{{(1-\gamma )}^{3}}\left|00111000\right\rangle\). * 8. \({K}_{7}\left|{0}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1-\gamma )}^{3}}\left|00001101\right\rangle\)

\({K}_{7}\left|{1}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1-\gamma )}^{3}}\left|11000001\right\rangle\). * 9. \({K}_{8}\left|{0}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1-\gamma

)}^{3}}\left|00001110\right\rangle\) \({K}_{8}\left|{1}_{L}\right\rangle =\sqrt{\gamma }\sqrt{{(1-\gamma )}^{3}}\left|11000010\right\rangle\). In the above, we can see that the effect of

_K_2_j_−1 is identical to _K_2_j_ 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_2_j_−1, _Z_2_j_ for _j_ = 1, 2, 3, 4. These stabilizer measurements leave the codespace afflicted with correctible AD

errors unchanged, and measure the parity of the (2_j_ − 1)th and (2_j_)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

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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. AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Department of Physics and Astronomy, University of Sheffield, Sheffield, UK Yingkai Ouyang Authors *

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_npj Quantum Inf_ 7, 87 (2021). https://doi.org/10.1038/s41534-021-00429-8 Download citation * Received: 21 October 2020 * Accepted: 19 April 2021 * Published: 02 June 2021 * DOI:

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