Randomly Monitored Quantum Codes

• Randomly monitored quantum codes are quantum error-correcting protocols that use random measurements to assess the resilience and recoverability of encoded information.
• They employ diverse measurement strategies—including varied bases and strengths—to analyze error thresholds and stability in quantum systems.
• Applications span fault-tolerant computing, holographic coding, and multi-user quantum protocols, leveraging techniques like decoupling and percolation theory.

Randomly monitored quantum codes are quantum error-correcting codes (QECCs) or information-processing protocols in which physical qubits, code stabilizers, or encoded quantum systems are measured at random locations, in random bases, or with random measurement strengths—with the objective of studying the resilience, recoverability, and functionality of the quantum information under such partial monitoring. Contrary to the conventional view that measurements irreversibly collapse quantum information, it has been demonstrated that for a broad class of quantum codes and monitored circuits, logical information can remain robust even when a majority of constituents are measured, and that measurement-induced phenomena can exhibit deep links with quantum entanglement, scrambling, and statistical mechanics (Lee et al., 2024, &&&1&&&, Berthusen et al., 2023, Foletto et al., 2020).

1. Fundamental Definitions and Measurement Models

An [[n, k, d]] stabilizer or CSS code encodes $k$ logical qubits into $n$ physical qubits with minimum distance $d$. The stabilizer group $S \subset \text{Pauli}_n$ has order $|S| = 2^{n-k}$; the logical operator group $L = \mathcal{C}(S) \setminus S$ contains $2^{2k}$ elements. In a prototypical random measurement process, each physical qubit $i$ is selected independently with probability $p = m/n$ to be measured in a random basis $P_i \in \{X_i, Y_i, Z_i\}$ with specified frequencies $(\alpha_X, \alpha_Y, \alpha_Z)$ summing to 1. The post-measurement stabilizer group is updated as $S' = \langle S \cap \mathcal{C}(M),\ M \rangle$, where $M$ is the abelian group generated by the measured Pauli operators $P_j$ (possibly with outcome signs $m_j$), and $\mathcal{C}(M)$ is the centralizer of $M$ in $\text{Pauli}_n$ (Lee et al., 2024).

In random syndrome measurement models relevant for qLDPC and hypergraph-product codes, each check operator is included in the measurement mask $D$ (i.e., unmeasured) independently with probability $p_{\mathrm{mask}} = 1 - p$, and only the partial syndrome is available for decoding (Berthusen et al., 2023).

Sequential quantum random access codes (QRACs) extend this paradigm to weak measurements of a single encoded qubit by multiple parties, tuning the measurement strength $s$ to trade off information gain and disturbance (Foletto et al., 2020).

2. Threshold Conditions and Resilience to Monitoring

For stabilizer codes, the preservation or destruction of logical information under random monitoring is determined by the interplay between the measured group $M$ and the logical group $L$. Logical information is preserved (i.e., one can still recover $k$ logical qubits) if and only if no nontrivial logical operator is entirely contained in the measured group: $M \cap L \subseteq S$. Equivalently, for all nontrivial logicals $\ell \in L \setminus S$, there exists an equivalent representative $\ell' = \ell \cdot s$ ($s \in S$) that commutes with all measured operators: $[\ell', M] = 0$ (Lee et al., 2024). The transition point is quantified by the measurement threshold $p_m^{\mathrm{th}}$:

• $p < p_m^{\mathrm{th}}$: logical information is retained with high probability as $n \to \infty$;
• $p > p_m^{\mathrm{th}}$: logical information is lost.

The threshold satisfies $p_e^{\mathrm{th}} \leq p_m^{\mathrm{th}} \leq 1$, where $p_e^{\mathrm{th}}$ is the standard erasure threshold. Critical behaviors include:

• Cat/repetition codes: any single Z measurement destroys all logical information ($p_m^{\mathrm{th}} = 0$).
• Toric codes: for $X$ or $Z$ measurements, $p_m^{\mathrm{th}} = 1/2$ (bond-percolation threshold); for uniform Pauli measurements, $p_m^{\mathrm{th}} \to 1$.
• Haar-random codes: $p_m^{\mathrm{th}} = 1$; logical information persists until only $k+o(n)$ qubits remain unmeasured.

In hypergraph-product codes, for random masking of syndrome measurements, a threshold $p_0$ exists such that for $p > p_0$, the logical error probability $P_{\mathrm{fail}}$ is exponentially suppressed in code size. Empirically, thresholds persist up to $p_{\mathrm{mask}} \approx 0.4 - 0.5$ for certain schedules and decoders (Berthusen et al., 2023).

3. Analytical Techniques and Proof Strategies

Measurements effect a stabilizer code update through $S' = \langle S \cap \mathcal{C}(M), M \rangle$; the loss of logical information corresponds to the presence of a logical operator within $M$. The cleaning lemma demonstrates that, when $M \cap L = \emptyset$, all logical operators may be relocated to reside exclusively on unmeasured qubits by multiplication with stabilizer elements.

For topological codes (e.g., toric code), percolation-theoretic arguments map $X$-measurements to deleted edges of the dual lattice. As long as the measured edge fraction $p_X < 1/2$, the system remains in the subcritical percolation regime; logical loops continue to exist in the punctured lattice, preserving code integrity.

In Haar-random codes, decoupling techniques based on the 2-norm distance of the post-measurement reference state indicate that as long as $k + \epsilon$ qubits are unmeasured, the remainder remains nearly maximally entangled with the reference and thus recoverable. Similar decoupling appears in monitored Clifford circuits, where the entanglement structure and code distance can be inferred from the structure of spacetime-ordered measurement sequences (Yoshida, 2021).

For randomly masked syndrome measurement in qLDPC codes, the analysis employs the local-stochastic noise framework, cluster expansions, and small-set-flip (SSF) decoding guarantees. The probability of a logical error is bounded by percolation arguments that prevent large masked clusters from bridging logical operators (Berthusen et al., 2023).

4. Examples and Quantitative Results

Selected paradigmatic behaviors are summarized in the following table:

Code Type	$p_m^{\mathrm{th}}$ (Critical Measurement Fraction)	Key Features
Repetition (cat)	0 (any nonzero Z measurement fatal)	Logical destroyed by a single Z measurement
2D toric code	1 ($X$, $Y$, $Z$ uniform < 1; $X$, $Z$ < 1/2)	Threshold set by bond percolation; Y-logical loops rare
Haar-random CSS	1 (survive until $k+o(n)$ qubits remain)	Decoupling ensures robustness to maximal random monitoring
Hypergraph-product	$\approx 0.5$ (fraction of syndrome checks masked)	Exponential suppression of logical error until threshold

For hypergraph-product codes with random mask fraction $p_{\mathrm{mask}}$, numerical results reveal that the logical error rate decays as $e^{-\beta(p)\,d}$, where $d \sim \sqrt{n}$ is the code distance; the breakdown occurs at $p_{\mathrm{mask}}\approx0.5$. A single-shot small-set-flip decoder is essential to maintain this threshold (Berthusen et al., 2023).

Monitored circuits with Clifford dynamics map entanglement properties to dual classical codes and relate quantum code distance to subleading entropy scaling and entanglement recoverability. The code distance in 1D Clifford circuits scales sublinearly as $d_{\mathrm{code}} \sim L^\gamma$ with $\gamma \approx 0.36$–$0.38$ (Yoshida, 2021).

Weak-sequentially monitored QRACs demonstrate that for measurement strengths $s$ in the interval $(1/\sqrt{2}, \sqrt{2\sqrt{2}-2})$, two parties can both achieve success probabilities exceeding the optimal classical bound of $3/4$ when extracting information from the same qubit (Foletto et al., 2020).

5. Applications and Operational Implications

Randomly monitored quantum codes establish that quantum information, when suitably encoded, is extraordinarily robust against measurement-induced decoherence. This insight underpins various proposed and demonstrated protocols:

• Quantum fault-tolerance schemes that interleave random measurements, entangling gates, and error correction, leveraging measurement-induced transitions and robust logical preservation.
• "Decoding by measurement" and measurement-based entanglement distillation or verification schemes, particularly in the context of monitored Clifford circuits where reconstructing entanglement maps to classical codeword recovery (Yoshida, 2021).
• Holographic quantum codes and quantum gravity models where bulk information remains reconstructible unless the measured boundary fraction exceeds unity, highlighting the contrast between measurement and erasure thresholds (Lee et al., 2024).
• Multiuser quantum information tasks in QRACs, including semi-device-independent randomness certification, quantum key distribution, and sequential device self-testing enabled by weak sequential measurements (Foletto et al., 2020).

For hypergraph-product codes and related qLDPC architectures, random partial monitoring of syndrome checks enables practical constant-overhead fault-tolerant protocols even when certain nonlocal measurements are infeasible or must be measured with reduced frequency, as long as the measurement rate remains above the established threshold (Berthusen et al., 2023).

6. Decoupling, Universality, and State-Independence

A notable universal feature is the decoupling of local subsystems from the initial state and remote measurement outcomes below the code-distance scale. In monitored circuits, for subsystems smaller than $d_\mathrm{code}$, entanglement and coherent information become nearly independent of past history—a phenomenon supported by both Clifford and generic scrambling unitary dynamics. The code distance thus establishes a characteristic scale beyond which state dependence and recoverability become nontrivial (Yoshida, 2021).

A plausible implication is that the design of robust quantum codes and protocols for distributed or monitored architectures ought to prioritize code families (such as random CSS, concatenated, and certain topological codes) with maximally large measurement thresholds, as these maintain entanglement and logical integrity despite aggressive or unavoidable monitoring. Conversely, specific code classes (e.g., repetition or Bacon–Shor codes) exhibit pathological fragility; even a single measurement can disrupt logical information (Lee et al., 2024).

7. Outlook and Future Directions

Future developments may focus on:

• Generalizations to higher-dimensional codes, multi-party monitoring, and more elaborate measurement scheduling (iterative or adaptive masks).
• Integration of weak measurement protocols on photonic or solid-state platforms for flexible, distributed, or multiuser quantum communication applications (Foletto et al., 2020).
• Exploration of the percolation and entanglement dynamics in non-Clifford and genuinely interacting monitored circuits, where current decoupling arguments and classical code mappings provide a suggestive but incomplete framework (Yoshida, 2021).
• Theoretical and experimental formulation of robust self-testing and measurement incompatibility certification schemes, particularly in the presence of adversarial monitoring or hardware-induced syndrome masking.

Randomly monitored quantum codes offer a unifying perspective on measurement-induced decoherence, entanglement, fault tolerance, and the operational limits of quantum information retrieval under partial or sequential observation, establishing sharp boundaries between fragility and robustness within the landscape of quantum information science.