Deep Thermalization and Measurements of Quantum Resources

Abstract

Quantum resource theories (QRTs) provide a unified framework for characterizing useful quantum phenomena subject to physical constraints, but are notoriously hard to assess in experimental systems. In this letter, we introduce a unified protocol for quantifying the resource-generating power (RGP) of arbitrary quantum evolutions applicable to multiple QRTs. It is based on deep thermalization (DT), which has recently gained attention for its role in the emergence of quantum state designs from partial projective measurements. Central to our approach is the use of projected ensembles, recently employed to probe DT, together with new twirling identities that allow us to directly infer the RGP of the underlying dynamics. These identities further reveal how resources build up and thermalize in generic quantum circuits. Finally, we show that quantum resources themselves undergo deep thermalization at the subsystem level, offering a complementary and another experimentally accessible route to infer the RGP. Our work connects deep thermalization to resource quantification, offering a new perspective on the essential role of various resources in generating state designs.

Figures (5)

• Figure 1: (a) Abstract representation of free and non-free elements: $\mathcal{F}$ denotes the set of free states/operators; $U_1$, $U_2$, and $U_3$ are non-free operations mapping $\mathcal{F}$ outside itself, while the free operation $F$ preserves $\mathcal{F}$. (b) Protocol for estimating the resource content of $U$: prepare a random free state $\ket{\psi}$ (e.g., using a free unitary $F_1$ on a fiducial state), apply $U$ followed by a random free operation $F_2$, then perform measurements on a subsystem $B$ and post-process the outcomes. (c) Illustration of the use of the protocol to estimate the Z2-asymmetry generating power $\mathcal{A}_{p}(U)/\overline{\mathcal{A}}$ for $U=u^{\otimes N}$ with $u=\exp{-i\alpha \sigma{x}}$, with $N=8$ and subsystem measurements on $N_B=4$ and $6$ qubits. (d) Estimation of non-stabilizing power $m_p(U)/\overline{m}$ for a two-qubit unitary $U=\exp\{-ic_x \sigma_x\otimes\sigma_x/2\}$ with projective measurements on $N_B=1$ qubit.
• Figure 2: (a) Schematic illustrating repeated free and non-free operations on a fixed free state, followed by projective measurements on a subset of qubits. (b) Deep thermalization of the $Z_2$-asymmetry for the above circuit with $N = 12$ and subsystem sizes $N_A = 2$ to $6$. The close agreement between the curves shows that subsystem asymmetry provides a reliable probe of the AGP. The inset shows relaxation dynamics for system sizes $N = 8, 10,$ and $12$ (light to dark). For every $N$, the relaxation rates corresponding to different $N_A$ nearly coincide. The non-free operation is $U = u^{\otimes N}$ with $u = e^{-i\pi \sigma_x/24}$. (c) Deep thermalization of non-stabilizerness under repeated interspersions of Clifford and non-Clifford operations for the unitary $U = \mathbb{I} \otimes \exp{-i\pi\sigma_x \otimes \sigma_x/8} \otimes \mathbb{I}$, for a fixed $N=12$ and different $N_A$. The figure indicates that $\langle m(t)_{\mathrm{PE}}\rangle/\overline{m_A}$ approach $\langle m_p(U^{(t)})\rangle_{\tilde{C}}$ as $N_A\rightarrow N$. The inset shows data for system sizes $N=8$, $10$, and $12$, with darker colors indicating larger systems. In both (b) and (c), the results are averaged over $\sim 10^3$ random instances of the circuit.
• Figure 3: Deep thermalization, quantified by the trace distance between the moments of the projected ensemble and that of the Haar ensemble, in the QRT of (a) $Z_2$-asymmetry and (b) non-stabilizerness. In both cases, the initial free state evolves under repeated interspersings of random free and non-free operations as depicted in Fig. \ref{['fig:sch-dt2']}(a), followed by measurements in the computational basis. The data are averaged over $\sim 10^2$ realizations of the circuit and the initial free state. The subsystem size is kept fixed at $N_A = 2$. Insets: (a1,b1) The trace distance exhibits exponential relaxation toward its long-time average across all system sizes, consistent with global thermalization. We quantify this relaxation using $1 - \overline{\Delta}^{(k)} / \overline{\Delta^{(k)}(\infty)}$, where $\overline{\Delta^{(k)}(\infty)}$ denotes the long-time average and $k = 2,4$ correspond to the $\mathbb{Z}_2$-AGP and non-stabilizerness, respectively. (b1,b2) The long-time average decays exponentially with increasing system size.
• Figure 4: Thermalization of $Z_2$-asymmetry as characterized by the exponential relaxation of $1-\dfrac{\mathcal{A}_{p}(U^{(t)})}{\overline{\mathcal{A}}}$ for a fixed $U = u^{\otimes N}$ with $u = \exp\{-i\alpha\sigma_x\}$ and $\alpha = \pi/24$. The results are shown for three different system sizes, namely, $N=8$, $10$ and $12$. The numerical results (dots) coincide with the analytical expression in Eq. (\ref{['expsymmain']}) of the main text and Eq. (\ref{['expasym_supp']}) in this supplemental material (dashed lines).
• Figure 5: Schematic illustration for quantifying the entangling power of a bipartite unitary $U_{AB}$ using the protocol outlined in the main text. The measurements at the end of the protocol are performed on the subsystems $A_2\in A$ and $B_2\in B$.

Theorems & Definitions (12)

• Theorem 1
• Theorem 2
• Lemma A1
• proof
• Lemma A2
• proof
• Theorem A3
• proof
• Lemma A4
• proof