Optimal randomized measurements for a family of non-linear quantum properties

TL;DR

An upper bound for ORM's sample complexity is established and its optimality for observables with a large trace-norm is shown, including Pauli and local observables, closing a gap in the literature.

Abstract

Quantum learning encounters fundamental challenges when estimating non-linear properties, owing to the inherent linearity of quantum mechanics. Although recent advances in single-copy randomized measurement protocols have achieved optimal sample complexity for specific tasks like state purity estimation, generalizing these protocols to estimate broader classes of non-linear properties without sacrificing optimality remains an open problem. In this work, we introduce the observable-driven randomized measurement (ORM) protocol enabling the estimation of ${\rm Tr}(Oρ^2)$ for an arbitrary observable $O$ -- an essential quantity in quantum computing and many-body physics. We establish an upper bound for ORM's sample complexity and show its optimality for observables with a large trace-norm, including Pauli and local observables, closing a gap in the literature. For these observables, ORM admits an efficient implementation with Clifford circuits. Numerical experiments validate that ORM requires substantially fewer state samples to achieve the same precision compared to classical shadows. Additionally, we introduce a braiding randomized measurement protocol for multiple low-rank non-linear observables, reducing circuit complexities in practical applications.

Figures (5)

• Figure 1: The framework of observable-driven randomized measurement protocol. To estimate $\Tr(O\rho^2)$ for arbitrary observables $O$, the ORM protocol first decomposes $O$ into a few dichotomic observables. For each dichotomic observable, one rotates the target state into the eigenbasis of the observable, evolves it with a block-diagonal random unitary, measures in the computational basis, and post-processes the measurement results to estimate its non-linear expectation value.
• Figure 2: Protocol for measuring any Pauli observable. Here, $V_C$ is a Clifford unitary that rotates $O$ to the Pauli-$Z$ operator on the first qubit. Depending on the measurement outcome of the first qubit, we then apply $U_i$ to the remaining part of the system, which can be either a global random unitary or a tensor product of single-qubit random unitaries.
• Figure 3: Virtual cooling of 6-qubit thermal states using GORM and LORM protocols. The black dashed curve shows the theoretical value $\Tr[Z_1Z_2\,\rho(2\beta)]$ on cooled states, while the grey dashed curve shows the value $\Tr[Z_1Z_2\,\rho(\beta)]$ on original thermal states. Markers give ORM estimates of $\Tr\left[Z_1Z_2\,\rho(\beta)^{2}\right] / \Tr\left[\rho(\beta)^{2}\right]$ for five values of $\beta$ and are slightly displaced along the horizontal axis to enhance visibility. Each point is the mean of ten independent runs with $N_{U}=10$ random unitaries (drawn from the global Haar measure or the tensor product of single-qubit Haar measure) and $N_{M}=1000$ measurements per unitary, with error bars denoting the standard error. Both protocols accurately reproduce the cooled-state expectation values.
• Figure 4: Number of state copies needed to estimate $\Tr(Z_1Z_2\,\rho^{2})$ with $\text{MSE}\le$ 0.01 averaged over 100 experiments, as a function of system size $L$. Within both the global-unitary and local-unitary settings, ORM requires far fewer copies than classical shadows, offering a significant advantage in sample complexity with increasing system size.
• Figure 5: Decomposition of a dichotomic observable. On the left, an observable $O$ with $d_- < \frac{d}{4}$ is depicted, where the green and blue blocks represent the eigensubspaces corresponding to $\Pi_-$ and the three partitions of $\Pi_+$, denoted by $\Pi_+^{(1)}$, $\Pi_+^{(2)}$, and $\Pi_+^{(3)}$, respectively. The $\pm$ signs within each block denote the corresponding diagonal entries of $\pm1$. On the right, the decomposition $O=\frac{1}{2}(\mathbb{I}+O_1+O_2+O_3)$ is shown: the first matrix represents the identity $\mathbb{I}$, while the remaining matrices correspond to $O_1$, $O_2$, and $O_3$, each satisfying $d_{\pm}^{(i)}\geq \frac{d}{3}$.

Theorems & Definitions (38)

• Lemma 1: Estimating dichotomic observables with block-diagonal unitary 4-designs
• Remark 1: Purity estimation with unitary 4-designs
• Lemma 2: Dichotomic decomposition
• Theorem 1: Improved scaling in estimating dichotomic observables with block-diagonal unitary 4-designs
• Lemma 3: Dichotomic decomposition of general observables
• proof
• Theorem 2: Estimating general observables with block-diagonal unitary 4-designs
• Lemma 4: Estimating traceless dichotomic observables with block-diagonal Clifford gates
• Theorem 3: Estimating observables via Pauli sampling and block-diagonal Clifford gates
• Corollary 1: Performance guarantee for local observables and Hamiltonians