Temperature Scaling in Quantum Memories

• Temperature scaling is a set of techniques for rescaling and analyzing temperature-dependent phenomena in quantum systems, impacting error rates and memory lifetimes.
• The analysis employs scaling laws such as Γ ∼ (ξT e^(–Δ/T)L²/ln L) and crossover temperature T* ≈ Δ/(2ln L) to precisely capture thermal dynamics in finite systems.
• These insights guide fault-tolerant quantum design by balancing system size against thermal fragility, thereby optimizing error mitigation in topological architectures.

Temperature scaling is a collective term denoting a class of theoretical and practical techniques in physics, chemistry, and machine learning for rescaling, analyzing, or modulating temperature-dependent phenomena—either to understand finite-temperature effects, calibrate uncertainty, or achieve system-level control. Across fields, it manifests as explicit scaling laws, dimensionless analysis, or algorithmic transformations, with roles ranging from the regulation of quantum memory lifetimes to the calibration of neural network confidence. The following sections synthesize core principles, methodologies, and consequences of temperature scaling as articulated in the context of topological quantum memories (Freeman et al., 2014).

1. Finite-Temperature Scaling in Many-Body Quantum Systems

Temperature scaling in quantum information systems characterizes how thermal fluctuations drive error processes and limit quantum memory lifetime. In the toric code—a canonical 2D topological quantum memory—thermal excitations (anyon pairs) are generated with Boltzmann-suppressed probability $\exp(-\Delta/T)$, where $\Delta$ is the energy gap. The scaling of the relaxation (decoherence) rate with system size $L$ and temperature $T$ is not trivial in finite systems:

$$\Gamma_{\text{TC}}(\beta, L) \sim \xi\, T\, e^{-\Delta/T} L^2 \, P^{\Omega}_{2D}(L)$$

where $P^{\Omega}_{2D}(L)$ is the probability of a topologically nontrivial random walk on the torus (see Sec. 2), and $\xi$ quantifies system–bath coupling. The crucial scaling variable here is $T$ itself, appearing as both an explicit prefactor and—through the Boltzmann factor—as an exponential suppressor of error rates.

2. Polylogarithmic Finite-Size and Temperature Scaling

In finite lattices, the probability $P^{\Omega}_{2D}(L)$ that a thermally generated anyon pair carries out a nontrivial random walk (leading to a logical error) decreases only polylogarithmically with the linear system size:

$P^{\Omega}_{2D}(L) \propto \frac{1}{\ln L}$

This scaling emerges from the two-dimensional return probability for random walks, which for large step number $n$ behaves as $p^{(p)}(2n) \sim 1/(2n [\ln(2n)]^2)$. By integrating from a cutoff $n \sim L^2$ (the minimum steps required for an odd winding), one obtains the $1/\ln L$ result. This slow decay means that even in large but finite systems, nontrivial errors induced by thermal fluctuations occur at a rate that does not vanish rapidly with $L$.

3. Dynamical Crossover and the Crossover Temperature $T^*$

A key concept is the finite-size “crossover temperature” $T^*$. This is defined as the temperature where the expected number of thermally excited pairs is order unity:

$$N e^{-\Delta/T^*} \sim 1 \implies T^* \sim \frac{\Delta}{\ln N} \sim \frac{\Delta}{2\ln L}$$

When $T < T^*$, the system resides in a regime dominated by single-pair processes and finite-size scaling dominates; above $T^*$, the system is crowded with excitations and scaling crosses over to a different regime with local string-like error processes. Thus, $T^*$ acts as a finite-size boundary separating these dynamical mechanisms. The dependence of $T^*$ on $\ln L$ reveals that maintaining robust quantum memory with increasing $L$ requires exponentially lower temperatures.

4. Lifetime of Topological Memories: Competition Between Scaling Regimes

The inverse of the relaxation rate, $\tau \sim 1/\Gamma_{\text{TC}}$, gives the quantum memory lifetime:

$\tau \sim \frac{\ln L}{\xi T e^{-\Delta/T} L^2}$

This expression reveals a critical competition:

• Thermal fragility: As $L$ increases, the number of creation sites ($L^2$) grows rapidly, and the lifetime decreases unless $T$ is reduced. The $1/\ln L$ factor only partially mitigates this scaling.
• Robustness to unitary perturbations: At $T \to 0$, the lifetime is exponentially protected, favoring large $L$; however, at finite $T$, increasing $L$ can worsen thermal error rates.

Hence, for quantum memory architectures, there is a nontrivial optimization between enlarging $L$ (which suppresses coherent/unitary errors) and controlling $T$ (needed to suppress thermal relaxation). The optimal region is typically for $T < T^*$, in which the scaling penalty is softened, but exponentially low temperatures may be required for large $L$.

5. Physical Interpretation and Broader Consequences

The scaling forms described above delineate two operational regimes:

• Low-temperature (single-pair) regime: Relevant when $T \ll T^*$. The memory’s relaxation time is controlled by rare, but system-size-enhanced, nontrivial quasiparticle walks.
• High-temperature (multi-pair) regime: When $T \gg T^*$, memory fails mainly due to a proliferation of pairs and their random stringlike motion, with relaxation rates growing (at best) linearly with $L$.

The concept generalizes to other topologically protected systems and underlines the necessity for operational temperatures to scale inversely (at least logarithmically) with system size for practical quantum memory deployment. This has direct implications for the engineering of self-correcting qubits and for the thermodynamic design of topologically ordered media.

6. Explicit Formulas and Scaling Laws

Key scaling relations governing the system are summarized in the table below.

Quantity	Scaling Law	Significance
Relaxation rate $\Gamma_{\text{TC}}$	$$\displaystyle \sim \frac{\xi T e^{-\Delta/T} L^2}{\ln L}$$	Governs decoherence; depends nontrivially on $T$ and $L$
Crossover temperature $T^*$	$$\displaystyle \sim \frac{\Delta}{\ln N} \sim \frac{\Delta}{2\ln L}$$	Sets boundary between low- and high-$T$ dynamical regimes
Memory lifetime $\tau$	$$\displaystyle \sim \frac{\ln L}{\xi T e^{-\Delta/T} L^2}$$	Upper bound on robust memory duration at given $T, L$

These relations underline the exponential sensitivity of memory stability to temperature and the requirement for coordinated system design controlling both the energy gap $\Delta$ and operating $T$.

7. Implications for Fault-Tolerant Quantum Architectures

The results from temperature scaling in the toric code model (Freeman et al., 2014) generalize to a broad class of finite-size, gapped, topologically ordered systems coupled to thermal baths. Hardware implementations aiming for fault-tolerance must:

• Ensure operating temperatures are kept below the many-body $T^*$.
• Consider the polylogarithmic suppression of topologically nontrivial error rates in finite geometry.
• Balance increases in $L$ against the corresponding worsening of thermal relaxation (as $L^2/\ln L$).

Consequently, temperature scaling analyses provide concrete guidance for predicting, benchmarking, and ultimately optimizing the longevity of quantum information storage in realistic, nonzero-temperature environments.