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Channel-resolved memory time of Haah's cubic code under reversible rate filtering and temporally correlated baths

This study simulates Haah's cubic code under various non-Markovian and filtered bath conditions to demonstrate that suppressing the zero-energy diffusion channel significantly extends memory time as a prefactor gain rather than improving asymptotic scaling, while also revealing the negligible impact of pair-creation channels and the sensitivity of memory performance to energy rescaling and bath correlations.

Original authors: Hikaru Wakaura

Published 2026-10-12

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hikaru Wakaura

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Storing information in the quantum world is a race against time. Unlike a classical computer, which keeps data safe by simply holding a charge or a magnetic orientation, a quantum computer stores information in fragile states that collapse the moment they interact with their surroundings. To keep this information alive, engineers usually rely on active error correction: a constant, high-speed cycle where the system measures its own health, a classical computer calculates what went wrong, and a correction is applied before the damage spreads. This works, but it requires massive amounts of extra hardware and incredibly fast processing, creating a heavy overhead that limits how large these machines can become. Scientists have long dreamed of a "self-correcting" memory, a system that protects itself without needing a constant external decoder, much like how a biological cell repairs its own DNA. The leading candidate for this in three dimensions is a specific arrangement of quantum bits known as Haah's cubic code. While theory suggests this code could hold information for a long time, it was unclear how the environment around it—specifically the heat and noise from the surrounding material—would help or hinder its ability to store data.

A new study by Hikaru Wakaura at QuantScape Inc. takes a deep dive into this question, simulating how this cubic code behaves under different environmental conditions. The researchers treated the code as a system of tiny switches that can flip due to thermal noise, creating defects that threaten the stored information. They asked a simple but profound question: if we could tune the way the environment interacts with these switches, could we make the memory last significantly longer? To test this, they built a detailed simulation of the code and introduced three specific "knobs" to control the environment. First, they adjusted how easily defects could be created or destroyed. Second, they changed how quickly defects could move around without changing the energy of the system. Third, they introduced a time delay, making the environment's noise fluctuate in a correlated way rather than being completely random. They ran these simulations on systems containing up to over 13,000 quantum bits to see how the memory time changed as the system grew larger.

The results revealed a surprising hierarchy of influence among these environmental factors. The most powerful lever turned out to be the rate at which defects could move around without costing any energy. In a standard, flat environment where all rates are equal, the memory time grows as the system gets larger, but only up to a certain point before it starts to decline. However, when the researchers slowed down this zero-energy movement by a factor of one thousand, the memory time exploded, lasting eighty to one hundred twenty times longer than before. Crucially, this improvement acted as a massive multiplier rather than changing the fundamental rule of how the memory scales with size. It also pushed the point at which the memory begins to fail much further out, allowing the system to grow larger before the benefits of size were lost to entropy. In contrast, changing the rate at which new defects were created had almost no effect on the memory time, provided the creation rate was not impossibly low. The environment's ability to create new errors was not the bottleneck; the bottleneck was how easily existing errors could wander and spread.

The study also explored what happens when the environment's noise is not random but has a memory of its own, fluctuating in a predictable pattern over time. When the researchers introduced a correlation time, meaning the noise stayed in a particular state for a while before changing, the memory time improved by a factor of two to eleven, depending on the strength of the fluctuation. This gain was consistent with a statistical average of how often the system faced a dangerous state. Interestingly, the researchers tested a theoretical model where the environment might act like a local filter that changes the energy of the defects themselves. They found that while such a filter could change the speed at which the system relaxes, it did not actually lower the population of defects in the way a true energy shift would. In fact, in some cases, it made the defects effectively cheaper to create, which would be counterproductive. This ruled out the idea that a simple, local interaction with a structured environment could act as a uniform cooling mechanism that boosts the memory's size scaling.

Perhaps the most significant finding was that a uniform increase in the energy cost of defects, which the researchers simulated by artificially raising the energy barrier by twelve percent, did change the scaling law. This adjustment caused the memory time to grow much more steeply as the system size increased, shifting the mathematical exponent that describes this growth. However, the researchers noted that no physical environment they tested in the simulation actually produced this specific, uniform energy shift. The study concludes that while we cannot yet engineer a passive memory that scales perfectly, we now know exactly which environmental properties matter most. Suppressing the movement of errors without energy cost is the most effective strategy, offering gains of up to two orders of magnitude. Temporal correlations in the noise offer a smaller but reliable boost. The path to a truly self-correcting memory does not lie in changing how errors are created, but in carefully engineering the environment to freeze the movement of the errors that do occur, buying time for the system to remain stable. This work provides a concrete benchmark for future engineers, showing that the environment is not just a source of noise to be fought, but a set of variables that can be tuned to extend the life of quantum information.

Technical Summary: Channel-Resolved Memory Time of Haah's Cubic Code

Problem Statement
Self-correcting quantum memories aim to store logical information without real-time decoding by relying on the Hamiltonian to confine thermally created defects. Haah's cubic code is the primary three-dimensional candidate for such a memory. While its behavior under a flat, Markovian (Davies) bath is established—exhibiting a memory time that grows as a power law of the linear size $L$ up to an entropy-determined optimum—the dependence of this memory time on more complex environmental structures remains unmapped. Specifically, it is unknown how the memory time scales when the bath possesses structured spectral weights at different energy channels, temporal correlations, or when the code energies are effectively renormalized. Without separating these effects, it is impossible to determine which environmental properties should be engineered to improve passive memory performance.

Methodology
The author performs kinetic Monte Carlo simulations of Haah's cubic code (Code 1) on an $L \times L \times L$ periodic lattice with two qubits per site ( $N=2L^3$ ). The system is modeled as a classical continuous-time Markov chain on error configurations, driven by a Davies generator with single-qubit $X$ jump operators. The study focuses on the finite-size regime ( $L=5$ to $19$) at inverse temperatures $\beta = 4.5$ and $5.0$.

The methodology introduces three phenomenological environmental controls to decouple specific bath properties:

Reversible Rate Filtering: The transition rates for the three allowed energy changes ( $|\Delta E| \in \{4, 2, 0\}$ ) are multiplied by independent factors $s_4$ (pair creation/annihilation), $s_2$ (barrier crossing), and $s_0$ (zero-energy diffusion). This preserves detailed balance and the Gibbs distribution while allowing the spectral weight at specific energy channels to be tuned.
Temporally Correlated Baths: The zero-energy diffusion rate $s_0$ is modulated by an external Ornstein–Uhlenbeck process with a tunable correlation time $\tau_c$ and variance $\sigma^2$ , introducing non-Markovian temporal correlations without altering the mean rate or equilibrium distribution.
Uniform Energy Rescaling: The code energies are uniformly rescaled to simulate an effective inverse temperature $\beta' = c\beta$ , serving as a benchmark for what a uniform cooling effect would achieve.

Additionally, a one-defect pseudomode model (a qubit coupled to a damped bosonic mode) is solved exactly via the Lindblad equation to analyze the relationship between structured spectral densities, defect populations, and relaxation rates.

Memory time ( $T_{mem}$ ) is defined as the first-failure time of a specific renormalization-group (RG) decoder. A run fails if the decoder cannot clear all defects or returns a logical error. The study reports sample means over 60–100 runs per data point.

Key Results

Channel Sensitivity: The memory time is overwhelmingly controlled by the zero-energy diffusion channel ( $s_0$ ).
- Pair Creation ( $s_4$ ): Over the range $s_4 \in [0.01, 10]$ , the pair-creation channel has no measurable effect on $T_{mem}$ . Only when $s_4$ drops below a crossover threshold ( $s_4^* \approx 3.8 \times 10^{-4}$ at $L=9$ ) does the memory time become limited by the time to the first error, following an additive form.
- Barrier Channel ( $s_2$ ): This channel shows weak sensitivity ( $d \ln T_{mem} / d \ln s_2 \approx -0.29$ ).
- Zero-Energy Channel ( $s_0$ ): This channel dominates with a sensitivity of $\approx -0.60$ . Suppressing diffusion by a factor of $10^3$ ( $s_0 = 0.001$ ) increases the memory time by a factor of 80–120 for $L=7–13$ . Crucially, this gain acts as a large prefactor rather than a steeper size exponent; the size exponent remains consistent with the flat bath within declared tolerances. Furthermore, suppressing diffusion shifts the optimal size $L^*$ outward, allowing the filtered memory to outlive the flat-bath optimum (e.g., at $L=19$ , the filtered bath is 45 times longer than the flat bath, which has already decayed).
Temporal Correlations: Introducing a bath correlation time $\tau_c$ yields a gain in memory time consistent with hazard averages.
- In the annealed regime ( $\tau_{cluster} \lesssim \tau_c \ll T_{mem}$ ), the gain is described by an annealed hazard average, yielding a factor of $\approx 2$ for $\sigma=2$ .
- In the quenched regime ( $\tau_c \gg T_{mem}$ ), the gain increases to a factor of $\approx 7$ (for $\sigma=2$ ), consistent with a quenched hazard average. The lifetime distribution in this regime is compatible with a quenched power-law model.
Energy Rescaling: A uniform 12% rescaling of code energies (equivalent to increasing $\beta$ ) raises the fitted size exponent from 2.7 to 4.3, aligning with the Bravyi–Haah scaling law ( $T_{mem} \propto L^{2.93\beta - 10.52}$ ). However, the author demonstrates via a two-defect pseudomode model that a locally damped structured mode does not realize this uniform rescaling. Instead, sub-gap spectral weight lowers the mean-force energy of defects (making them "cheaper") and induces defect-defect interactions, failing to act as a simple cooling mechanism.

Significance and Claims
The paper establishes the first channel-resolved finite-size benchmark for a passive three-dimensional quantum memory. Its primary contributions are:

Identification of the Control Lever: It identifies the zero-energy spectral weight as the critical environmental parameter for engineering passive memories, showing that suppressing diffusion provides a massive prefactor gain without altering the asymptotic scaling law (within the tested window).
Quantification of Correlation Benefits: It quantifies the benefit of temporal correlations, showing a factor of 2 to 7 improvement depending on the correlation time relative to the memory time, consistent with annealed and quenched hazard averages.
Limitations of Local Models: It clarifies that locally damped structured modes do not provide a uniform energy rescaling (cooling) for the code; they alter populations and induce interactions in ways that do not simply shift the Bravyi–Haah exponent.

The author explicitly states that these results do not establish asymptotic scaling laws (as data is limited to $L < L^*$ for most filters) nor do they demonstrate an advantage over active decoding. The work serves as a guide for bath engineering, indicating that while suppressing zero-frequency noise and introducing temporal correlations are beneficial, achieving a uniform energy rescaling via local structured baths remains an open challenge. The study is limited to the X-sector, specific decoders, and phenomenological rate filters, noting that efficient decoders for fracton codes are largely unknown and that the reported failure times characterize the specific RG decoder used.

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