Exploring Noisy Quantum Thermodynamical Processes via the Depolarizing-Channel Approximation

TL;DR

The study develops the Global Depolarizing Approximation (GDA) to analytically capture the impact of gate-dependent noise in deep quantum circuits by replacing complex noise with a global depolarizing channel, valid when the circuit depth scales polynomially with system size and the gate set forms an approximate unitary 2-design. Applying the GDA to algorithmic cooling (TSAC) yields a noisy steady state with two dynamic modes, introducing an optimal qubit number and revealing a fundamental limit on cooling efficiency under realistic noise; a corollary for timekeeping noise provides practical predictions for depolarizing strength $\eta$ and optimal performance. The framework is extended to dynamic cooling (DC), where numerical validation shows the GDA predicts final target temperatures within about 1% of gate-level simulations, demonstrating broad applicability to noisy input–output thermodynamic protocols. Overall, GDA offers a scalable, analytically tractable method to bound and optimize finite-resource quantum thermodynamic tasks in the presence of realistic noise, enabling principled resource-error-performance trade-offs for quantum heat engines and related applications.

Abstract

Noise and errors are unavoidable in any realistic quantum process, including processes designed to reduce noise and errors in the first place. In particular, quantum thermodynamical protocols for cooling can be significantly affected, potentially altering both their performance and efficiency. Analytically characterizing the impact of such errors becomes increasingly challenging as the system size grows, particularly in deep quantum circuits where noise can accumulate in complex ways. To address this, we introduce a general framework for approximating the cumulative effect of gate-dependent noise using a global depolarizing channel. We specify the regime in which this approximation provides a reliable description of the noisy dynamics. Applying our framework to the thermodynamical two-sort algorithmic cooling (TSAC) protocol, we analytically derive its asymptotic cooling limit in the presence of noise. Using the cooling limit, the optimal cooling performance is achieved by a finite number of qubits--distinguished from the conventional noiseless TSAC protocol by an infinite number of qubits--and fundamental bounds on the achievable ground-state population are derived. This approach opens new avenues for exploring noisy quantum thermodynamical processes.

Figures (8)

• Figure 1: Global Depolarizing Approximation. Local noises of two-qubit CNOT gates are twirled by the quantum circuit with fixed gates (left panel), making them effectively act as a global depolarizing channel $\hat{\mathcal{U}}'$ on the observable (right panel).
• Figure 2: Summary of optimal performance versus CNOT error probability $p$ (logarithmic scale). (a) Optimal number of total qubits ($n_\text{opt}$) required to maximize the final ground-state population. (b) Corresponding maximally achievable final ground-state population ($P_\text{max}$). All panels compare results from physical noise simulations (circles, dashed lines) and the theoretical GDA model (squares, solid lines). The initial ground-state population is $P_{\text{initial}}=0.85$.
• Figure 3: Verification of the GDA model for the DC mirror protocol across a range of qubit numbers $n$ and CNOT error probabilities $p$. The panels show: (a) the final effective temperature $\mathcal{T}_{\text{sim}}$ obtained from a physical noise simulation; (b) The relative error, $|\mathcal{T}_{\text{sim}}-\mathcal{T}_{\text{model}}|/\mathcal{T}_{\text{model}}$, where $\mathcal{T}_{\text{model}}$ is the result from GDA. Temperatures in (a) are shown in mK; (b) is shown as a percentage. The plots demonstrate agreement of the model with the simulation.
• Figure A1: Numerical validation of the 2-design approximation, Eq. \ref{['eq:2-design']}, for the $n=3$ TSAC circuit. The plot shows the average fidelity $\langle F_k \rangle_k$ between the true averaged noise channel (LHS) and the GDA model (RHS) as a function of circuit depth, simulated by repetition rounds $R$. The initial state is a thermal state with $P=0.8$. The $R=0$ point represents the baseline fidelity without the averaging effect. The rapid convergence to high fidelity for $R \ge 1$ demonstrates the effectiveness of the 2-design assumption.
• Figure A2: Ground state population vs. cooling rounds. Simulations compare the ideal (noiseless) case, the gate-dependent timekeeping (physical) noise simulation, and the global depolarizing model (theoretical). Results are shown for different numbers of qubits ($n$) and error probabilities ($p$): (a) $n=6, p=1 \times 10^{-5}$. (b) $n=4, p=1.5 \times 10^{-4}$.

Theorems & Definitions (11)

• Definition 1: Circuit with incoherent noise
• Theorem 1: Global depolarizing approximation
• Proposition 1: Cooling limit of noisy TSAC
• Corollary 1: GDA parameter for timekeeping noise
• proof
• Lemma 1: Noisy Transition Matrix for TSAC
• proof
• Lemma 2: GDA Parameter for Two-Qubit Bit-Flip Noise
• proof
• Lemma 3: GDA Parameter for Timekeeping Noise