Quantum Coherence and Anomalous Work Extraction in Qubit Gate Dynamics

TL;DR

This paper deploys Kirkwood-Dirac quasiprobabilities to quantify how coherence contributes to work extraction in cyclic quantum evolutions, revealing that KDQ negativity (via Margenau-Hill quasiprobabilities) signals anomalous, coherence-enabled energy exchanges. It develops a gate-decomposition framework that expresses the KDQ of a deep circuit as a weighted sum of its constituent gates plus a correction term that encodes incompatibility with energy projectors; this yields conditions under which complex circuits exhibit nonclassical work statistics even when individual gates do not. The study analyzes single- and two-qubit gates (notably Hadamard, pi/8, and CNOT) and demonstrates through concrete examples that coherence can enable positive work beyond classical limits, with the Hadamard-like time evolution and the minimal HTH circuit serving as key nontrivial illustrations. The results illuminate the thermodynamic relevance of circuit structure in quantum computation and provide a versatile framework for exploring nonclassical work statistics in broader quantum dynamics. Overall, the KDQ framework offers a principled route to assess when quantum coherence in circuits can yield thermodynamic advantages, potentially impacting the design of energy-aware quantum protocols and the study of nonclassical thermodynamics in complex quantum systems.

Abstract

We develop a framework based on the Kirkwood-Dirac quasiprobability distribution to quantify the contribution of coherence to work extraction during generic, cyclic quantum evolutions. In particular, we focus on ``anomalous processes'', counterintuitive scenarios in which, due to the negativity of the quasiprobability distribution, work can be extracted even when individual processes are associated with energy gain. Applying this framework to qubits undergoing sequences of single- and two-qubit gate operations, we identify specific conditions under which such anomalous work exchanges occur. Furthermore, we analyze the quasiprobabilistic structure of deep quantum circuits and establish a compositional relation linking the work statistics of full circuits to those of their constituent gates. Our work highlights the role of coherence in the thermodynamics of quantum computation and provides a foundation for systematically studying potential thermodynamic relevance of specific quantum circuits.

Figures (6)

• Figure 1: Panel (a): sketch of the protocol for a two-qubit system, whose state is transformed by a unitary $\hat{\mathcal{U}}$ decomposed into elementary gates constituting a quantum circuit. We explore the role of the coherences and of the quantum circuit in the extractable work and in its nonclassical statistics. Panel (b): Pictorial representation of population and coherent parts of the KDQs in a single-qubit system. The blue arrows refer to the population part, the red arrows to the coherent part that potentially lead to nonclassicality.
• Figure 2: Real part of $q_{\downarrow\uparrow}$ and work extracted from a pure state $\ket{\psi(p,\varphi)}=\sqrt{p}\ket{\uparrow}+e^{i\varphi}\sqrt{1-p}\ket{\downarrow}$ that undergoes a transformation described by $\hat{\mathcal{H}}_h=h\left(\hat{X}+\hat{Z}\right)$, with $h>0$. Panel (a): the population is fixed to $p=\frac{1}{2}$ and different values of the phase $\varphi$ are considered. Panel (b): the phase is fixed to $\varphi=\pi/2$ and different values of the population $p\in[0,0.5]$ are considered.
• Figure 3: Panel (a): Sketch of the KDQs of a unitary transformation decomposed in multiple gates, with a pictorial representation of the full KDQ $q_{if}^{U_N \ldots U_1}(\hat{\rho})$ and the three possible KDQs associated to the constituent gates: (i) $q_{if}^{U_{j+1}}(\hat{\rho}_j)$ with $\hat{U}_{j+1}$ not in the border of $\mathcal{U}$, (ii) $q_{if}^{U_1}(\hat{\rho})$, (iii) $q_{if}^{U_{N}}(\hat{\rho}_{N-1})$. Panel (b): pictorial representation of the structure of the operator $\hat{\mathcal{M}}_{if}^{U_{j+1}}$ whose derivation is given in Eq. \ref{['eq:M_operator_main']}. The yellow block represents the operator $\hat{U}_N \ldots \hat{U}_1$, the blue block represents the operator $\hat{U}_j \ldots \hat{U}_1$, The red block represents the operator $\hat{U}_{N} \ldots \hat{U}_{j+2}$ and the green block is the constituent gate $\hat{U}_{j+1}$.
• Figure 4: MHQs associated to the $\hat{H}\hat{T}\hat{H}$ circuit that contribute to the work extraction for an input state $\hat{\rho}(\theta,\varphi)=\ket{\psi(\theta,\varphi)}\bra{\psi(\theta,\varphi)}$ with $\ket{\psi(\theta,\varphi)}=\cos(\theta/2)\ket{\downarrow}+e^{i\varphi}\sin(\theta/2)\ket{\uparrow}$. The two panels report the MHQs associated to the two different processes $\ket{\uparrow}\rightarrow\ket{\downarrow}$ (panel (a)) and $\ket{\downarrow}\rightarrow\ket{\uparrow}$ (panel (b)). The red dot indicates the values of $(\theta,\varphi)$ for which an in-depth analysis is performed in the main text.
• Figure 5: Thermodynamic features of the circuit $\hat{\mathcal{V}}=\hat{U}_{\rm CNOT}\hat{H}^{\otimes 2}$ for an input state $\ket{\Psi(\theta,\varphi)}=\ket{\psi(\theta,\varphi)}^{\otimes 2}$, with $\ket{\psi(\theta,\varphi)}=\cos(\theta/2)\ket{\downarrow} + e^{i\varphi}\sin(\theta/2)\ket{\uparrow}$ for different values of $\theta$ and $\varphi$ parameters. Panels (a),(b).(c) and (d) report the norms in Eq. \ref{['eq:norms']} that are defined from the vectors \ref{['eq:r_i<f']} and \ref{['eq:r_i>f']}. Panel (e) reports the total extractable work in the same range of input parameters and panel (f) reports the coherent part of the work. The red line indicates the values of $(\theta,\varphi)$ for which an in-depth analysis is performed in the text.