Quantum error correction is a local subroutine used to protect quantum states against noise, decoherence, and imperfect operations. Unlike classical error correction, it cannot work by directly copying or measuring the protected information. Instead, the quantum state is encoded into a larger Hilbert space so that errors can be detected and corrected without revealing the logical state.

In quantum-network protocols, quantum error correction may be used inside a node to protect stored quantum states, prepare encoded states, process noisy quantum information, or support fault-tolerant operations. It may also appear as part of larger protocols for reliable quantum communication, quantum repeaters, distributed quantum computation, or fault-tolerant quantum computing.

You can find a lot of information about quantum error-correcting codes in The Error Correction Zoo [1].

The subroutine can be described abstractly as follows.

1. Input. A node holds a logical quantum state $\\\rho$ that should be protected.
2. Encoding. The logical state is encoded into a larger physical system using a quantum error-correcting code.
3. Noise. The encoded state may undergo physical errors, such as bit-flip errors, phase-flip errors, erasures, or more general noise.
4. Syndrome extraction. The node measures error information, called the syndrome, without measuring the logical quantum information itself.
5. Recovery. A correction operation is applied depending on the measured syndrome.
6. Output. The output is a recovered logical state, ideally close to the original input state.

• The subroutine assumes that the node can implement the chosen encoding and recovery operations, or at least approximate them with sufficiently low noise.
• The relevant error model should be known or bounded. Different codes are suited to different types of errors, such as Pauli errors, erasures, amplitude damping, or local stochastic noise.
• The syndrome measurement must not reveal the encoded logical state. It should reveal only information about the error.
• The physical error rate must be within the correction capability of the chosen code. If too many errors occur, decoding may fail.
• In fault-tolerant settings, the syndrome extraction and correction operations themselves may also be noisy. In that case, the subroutine must be implemented fault-tolerantly.

• A quantum error-correcting code encoding $k$ logical qubits into $n$ physical qubits is often denoted by $$ [[n,k,d]], $$
  where $d$ is the code distance.
• The code space is a subspace
  $$ \\\mathcal{C} \\\subseteq (\\\mathbb{C}^2)^{\\\otimes n}. $$
• The encoding map is an isometry
  $$ V : (\\\mathbb{C}^2)^{\\\otimes k} \\\rightarrow (\\\mathbb{C}^2)^{\\\otimes n}. $$
• The projector onto the code space is denoted by $P_{\\\mathcal{C}}$.
• If the code has distance $d$, then it can correct up to
  $$ t = \\\left\\\lfloor \\\frac{d-1}{2} \\\right\\\rfloor $$
  arbitrary single-qubit errors under the standard distance-based error model.

• No direct copying. The protected state cannot be copied or measured directly. Error correction must avoid learning the logical state.
• Overhead. Quantum error correction requires additional physical qubits and additional operations.
• Fault tolerance. For long computations or large networks, quantum error correction can be combined with fault-tolerant procedures so that the correction process does not introduce uncontrolled errors.

Let $\\\rho$ be a logical input state on $k$ qubits. An encoding isometry maps it to an encoded state

$$
\\\rho_{\\\mathrm{enc}} = V \\\rho V^{\\\dagger}.
$$

The encoded state is affected by a noise channel $\\\mathcal{N}$, producing

$$
\\\rho_{\\\mathrm{noisy}} = \\\mathcal{N}(\\\rho_{\\\mathrm{enc}}).
$$

A recovery operation $\\\mathcal{R}$ is then applied. The code corrects the noise if

$$
\\\mathcal{R}(\\\mathcal{N}(V\\\rho V^{\\\dagger})) = V\\\rho V^{\\\dagger}
$$

for all logical states $\\\rho$, or approximately corrects the noise if the two states are close.

For an error set ${E_a}$, the Knill-Laflamme condition states that the code corrects these errors if

$$
P_{\\\mathcal{C}} E_a^{\\\dagger} E_b P_{\\\mathcal{C}} = \\\alpha_{ab} P_{\\\mathcal{C}}
$$

for all errors $E_a,E_b$ in the error set, where $\\\alpha_{ab}$ are complex coefficients.

For stabilizer codes, the code space is defined as the simultaneous $+1$ eigenspace of a commuting set of stabilizer generators

$$
S = \\\langle g_1,\\\ldots,g_r\\\rangle .
$$

Syndrome extraction measures the eigenvalues of these generators. If an error has occurred, the syndrome is a string

$$
s = (s_1,\\\ldots,s_r),
$$

where each $s_i$ records the measurement outcome associated with $g_i$.

The decoder uses the syndrome $s$ to choose a correction operation $C_s$. The corrected state is then

$$
\\\rho_{\\\mathrm{corr}} = C_s \\\rho_{\\\mathrm{noisy}} C_s^{\\\dagger}.
$$

If the decoder succeeds, the corrected state is returned to the code space and preserves the original logical information.

Common examples of quantum error-correcting codes include the three-qubit bit-flip code, the three-qubit phase-flip code, Shor’s nine-qubit code, Steane’s seven-qubit code, CSS codes, surface codes, color codes, bosonic codes, and quantum LDPC codes.

Quantum error correction should be distinguished from classical error correction. Classical error correction protects classical strings and can use copying and direct comparison. Quantum error correction protects unknown quantum states and must respect the no-cloning principle and the disturbance caused by measurement.

1. The Error Correction Zoo, “Quantum error-correcting code (QECC).”
2. P. W. Shor, “Scheme for reducing decoherence in quantum computer memory,” Physical Review A, vol. 52, no. 4, 1995.
3. A. M. Steane, “Error correcting codes in quantum theory,” Physical Review Letters, vol. 77, no. 5, 1996.
4. E. Knill and R. Laflamme, “Theory of quantum error-correcting codes,” Physical Review A, vol. 55, no. 2, 1997.
5. D. Gottesman, “Stabilizer Codes and Quantum Error Correction,” PhD thesis, Caltech, 1997.
6. J. Roffe, “Quantum Error Correction: An Introductory Guide,” Contemporary Physics, vol. 60, no. 3, 2019.