Constant-excitation (CE) code

Constant-excitation (CE) code [1–3]

Description

Code whose codewords lie in an eigenspace of fixed total energy or fixed total excitation number for the underlying quantum system. For qubit codes, such a Hamiltonian is often the total spin Hamiltonian, $H=\sum_i Z_i$. For spin-$S$ codes, this generalizes to $H=\sum_i J_z^{(i)}$, where $J_z$ is the spin-$S$ $Z$-operator. For bosonic (and, similarly, for fermion) codes, such as Fock-state codes, codewords are often in an eigenspace with eigenvalue $N>0$ of the total excitation or energy Hamiltonian, $H=\sum_i \hat{n}_i$.

Protection

CE codewords have to lie in the same excitation subspace in order to protect against changes in the total excitation number.

Fock-state CE codes lie in the constant-excitation Fock space and are in one-to-one correspondence with points on the discrete simplex. They are protected from identical AD acting on all modes because the damping acts on all codewords identically [4,5]. The all-zero AD Kraus operator acts identically on every state and so can be exactly correctable in the case of Fock-state CE codes. For example, this operator’s acting on a Fock state $|\boldsymbol{m}\rangle$ depends only on the total occupation number $|\boldsymbol{m}|=\sum_j m_j$ and not on the individual occupation numbers $m_j$, \begin{align} E_{0}^{\otimes n}|\boldsymbol{m}\rangle=\left(1-\gamma\right)^{|\boldsymbol{m}|/2}|\boldsymbol{m}\rangle~.\tag*{$(1)$} \end{align}This effect extends to the damping portion, $\left(1-\gamma\right)^{\hat{n}/2}$, of any $\ell\neq 0$ AD Kraus operators. Under AD, a Fock-state CE codeword with total excitation $N$ loses at most $t$ particles with probability $p_{N,t}=\mathrm{Pr}[\mathrm{Bin}(N,\gamma)\leq t]$, independent of the code [6; Prop. 3]. An $\epsilon$-approximate Fock-state CE code for the AD channel truncated to at most $t$ losses is a $\sqrt{1-(1-\epsilon^2)p_{N,t}}$-approximate code for the full channel [6; Thm. 5]. Conversely, an $\epsilon$-approximate code for the full channel is an $\epsilon/\sqrt{p_{N,t}}$-approximate code for the truncated one [6; Thm. 5]. A positive-rate family of Fock-state CE codes corrects AD at constant loss rate $\gamma$ with vanishing recovery error if and only if it approximately corrects $\lfloor\delta N\rfloor$ losses for some $\delta>\gamma$ [6; Cor. 6].

In similar fashion, qubit CE codes are protected from coherent noise in the form of transversal $Z$-rotations because such rotations act identically on all codewords [7,8]. In the case of CSS codes, all codes oblivious to such rotations are CE codes [7,8]. Stabilizer codes can be extended to codes that are protected against such coherent noise via an enlargement procedure [8].

Rate

Fock-state CE codes can be used in a protocol that achieves the two-way quantum capacity of the AD Gaussian channel [9]. For every $K,t \geq 2$, there are explicitly constructible $K$-dimensional Fock-state CE codes with $q=N=(K-1)t(t+1)$ modes, total excitation $N$, and distance $t+1$; there also exist families with logical dimension $K = o(2^N)$ and distance of order $o(N/\log N)$ [10].

Decoding

Modified Shor syndrome extraction measures a weight-$2w$ or $(2w-1)$ stabilizer of a CE CSS code with the $w$-CE cat state $\frac{1}{\sqrt{2}}(|01\rangle^{\otimes w}+|10\rangle^{\otimes w})$ [11; Thm. 9]. The $j$th ancilla qubit controls a Pauli on the $j$th data qubit of the support, with zero-controlled gates at even $j$ [11; Alg. 1]. Modified Steane syndrome extraction for dual-rail-concatenated CE CSS codes, in which each transversal CNOT gate is followed by the Pauli correction $\prod_j X_{2j}$ [11; Thm. 12].

Fault Tolerance

Weakly fault-tolerant error correction (in the sense of Ref. [12]) for distance-three CE CSS codes under circuit-level noise interleaved with layers of collective coherent $Z$ rotations [11; Thm. 11]. It combines modified Shor extraction with a conditional control flow that uses at most two syndrome-measurement rounds per stabilizer type [11].

Cousins

Amplitude-damping (AD) code— Fock-state and qubit CE codes exactly protect against the AD Kraus operator $E_{0}^{\otimes n}$ because it acts identically on all Fock (and qubit) states with the same excitation number [4,5].
Qubit CSS code— Qubit CE codes are protected from coherent noise in the form of transversal $Z$-rotations because such rotations act identically on all codewords [7,8]. In the case of qubit CSS codes, all codes oblivious to such rotations are CE codes [7,8]. Any $[[n,k,d]]$ CSS code can be made into an $[[mn,k,\geq d]]$ CE code [7]. Concatenating the dual-rail code with an inner $[[n,k,d]]$ qubit stabilizer code yields a degenerate $[[2n,k,d]]$ constant-excitation stabilizer code that avoids coherent phase errors and is equivalent to a Pauli-rotated repetition-concatenated stabilizer code [13]. CSS structure is preserved when the original code is CSS [11].
$[[5,1,3]]$ Five-qubit perfect code— The five-qubit code can be concatenated with a particular decoherence-free subspace (DFS) [14–17] to yield a 20-qubit CE code [3,18]. Dual-rail concatenation of the five-qubit code yields a $[[10,1,3]]$ CE stabilizer code [11].
Qubit stabilizer code— Concatenating the dual-rail code with an inner $[[n,k,d]]$ qubit stabilizer code yields a degenerate $[[2n,k,d]]$ constant-excitation stabilizer code that avoids coherent phase errors and is equivalent to a Pauli-rotated repetition-concatenated stabilizer code [13]. CSS structure is preserved when the original code is CSS [11].
Constant-weight block code— Constant-weight codes are classical analogues of qubit constant-excitation codes.
Permutation-invariant (PI) code— Modular-qudit PI codes can be converted to constant-excitation Fock-state codes via the simplex mapping [10; Prop. V.2]. Any transversal gates are mapped to Gaussian gates on the Fock-state codes [10].
Fermion code— Fermion codewords lying in a fixed fermion-number subspace have to lie in the same subspace in order to protect against changes in fermion number [19].
Quantum parity code (QPC)— QPCs for even $m_1$ can be made into CE codes by a Pauli transformation (e.g., $XIXI\cdots XI$) applied to each block of $m_1$ qubits.

Member of code lists

Primary Hierarchy

Quantum Domain

Quantum code

Operator-algebra QECC (OAQECC)

Quantum error-correcting code (QECC)

Hamiltonian-based code

Parents

Hamiltonian-based code

Constant-excitation codes are associated with a Hamiltonian governing the total excitations of the system.

Constant-excitation (CE) code

Children

Chuang-Leung-Yamamoto (CLY) code

Chuang-Leung-Yamamoto codewords are constructed out of Fock states with the same total excitation number.

Ouyang-Chao constant-excitation PI code

Icosahedral Fock-state code

$[[12,1,3]]$ CE CSS code

$[[14,3,3]]$ CE phantom code

This code is single-qubit Clifford equivalent to the $[[14,3,3]]$ CE CSS code obtained by dual-rail concatenation of the $[[7,3,2]]$ punctured hypercube code [11].

$((8,2,3))$ Plenio-Vedral-Knight CE code

References

[1]
M. B. Plenio, V. Vedral, and P. L. Knight, “Quantum error correction in the presence of spontaneous emission”, Physical Review A 55, 67 (1997) arXiv:quant-ph/9603022 DOI
[2]
P. Zanardi and M. Rasetti, “Noiseless Quantum Codes”, Physical Review Letters 79, 3306 (1997) arXiv:quant-ph/9705044 DOI
[3]
D. A. Lidar, D. Bacon, and K. B. Whaley, “Concatenating Decoherence-Free Subspaces with Quantum Error Correcting Codes”, Physical Review Letters 82, 4556 (1999) arXiv:quant-ph/9809081 DOI
[4]
D. W. Leung, M. A. Nielsen, I. L. Chuang, and Y. Yamamoto, “Approximate quantum error correction can lead to better codes”, Physical Review A 56, 2567 (1997) arXiv:quant-ph/9704002 DOI
[5]
I. L. Chuang, D. W. Leung, and Y. Yamamoto, “Bosonic quantum codes for amplitude damping”, Physical Review A 56, 1114 (1997) DOI
[6]
D. Elimelech, A. Aydin, and A. Barg, “Asymptotically good bosonic Fock state codes”, Quantum 10, 2240 (2026) arXiv:2603.15190 DOI
[7]
J. Hu, Q. Liang, N. Rengaswamy, and R. Calderbank, “CSS Codes that are Oblivious to Coherent Noise”, 2021 IEEE International Symposium on Information Theory (ISIT) 1481 (2021) DOI
[8]
J. Hu, Q. Liang, N. Rengaswamy, and R. Calderbank, “Mitigating Coherent Noise by Balancing Weight-2 Z-Stabilizers”, IEEE Transactions on Information Theory 68, 1795 (2022) arXiv:2011.00197 DOI
[9]
M. S. Winnel, J. J. Guanzon, N. Hosseinidehaj, and T. C. Ralph, “Achieving the ultimate end-to-end rates of lossy quantum communication networks”, npj Quantum Information 8, (2022) arXiv:2203.13924 DOI
[10]
A. Aydin, V. V. Albert, and A. Barg, “Quantum Error Correction beyond <mml:math xmlns:mml=”http://www.w3.org/1998/Math/MathML” display=”inline”> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:mo stretchy=”false”>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy=”false”>)</mml:mo> </mml:math>  : Spin, Bosonic, and Permutation-Invariant Codes from Convex Geometry”, PRX Quantum 7, (2026) arXiv:2509.20545 DOI
[11]
C.-Y. Lai, P.-H. Liou, and Y. Ouyang, “Fault-Tolerant Quantum Error Correction for Constant-Excitation Stabilizer Codes under Coherent Noise”, (2025) arXiv:2507.10395
[12]
T. Tansuwannont, B. Pato, and K. R. Brown, “Adaptive syndrome measurements for Shor-style error correction”, Quantum 7, 1075 (2023) arXiv:2208.05601 DOI
[13]
Y. Ouyang, “Avoiding coherent errors with rotated concatenated stabilizer codes”, npj Quantum Information 7, (2021) arXiv:2010.00538 DOI
[14]
D. A. Lidar, I. L. Chuang, and K. B. Whaley, “Decoherence-Free Subspaces for Quantum Computation”, Physical Review Letters 81, 2594 (1998) arXiv:quant-ph/9807004 DOI
[15]
D. Bacon, D. A. Lidar, and K. B. Whaley, “Robustness of decoherence-free subspaces for quantum computation”, Physical Review A 60, 1944 (1999) arXiv:quant-ph/9902041 DOI
[16]
D. A. Lidar, D. Bacon, J. Kempe, and K. B. Whaley, “Decoherence-free subspaces for multiple-qubit errors. I. Characterization”, Physical Review A 63, (2001) arXiv:quant-ph/9908064 DOI
[17]
D. A. Lidar, D. Bacon, J. Kempe, and K. B. Whaley, “Decoherence-free subspaces for multiple-qubit errors. II. Universal, fault-tolerant quantum computation”, Physical Review A 63, (2001) arXiv:quant-ph/0007013 DOI
[18]
D. A. Lidar, D. Bacon, J. Kempe, and K. Birgitta Whaley, “Protecting quantum information encoded in decoherence-free states against exchange errors”, Physical Review A 61, (2000) arXiv:quant-ph/9907096 DOI
[19]
A. Schuckert, E. Crane, A. V. Gorshkov, M. Hafezi, and M. J. Gullans, “Fault-tolerant fermionic quantum computing”, (2025) arXiv:2411.08955

Page edit log

Victor V. Albert (2026-10-26) — most recent
Victor V. Albert (2026-10-25)
Victor V. Albert (2026-06-08)
Yinchen Liu (2024-03-15)
Victor V. Albert (2022-03-01)

Cite as:

“Constant-excitation (CE) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/constant_excitation, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/hamiltonian/constant_excitation.yml.