Optimal Overlapping Tomography

TL;DR

This work presents protocols for overlapping tomography that are optimal with respect to the number of measurement settings, and proves that using general projective measurements, all $k$-body marginals can be reconstructed with only $3^k$ settings, independently of the system size.

Abstract

Characterising large-scale quantum systems is central to fundamental physics and essential for applications of quantum technologies. While a full characterisation requires exponentially increasing resources, focusing on application-relevant information can often lead to significantly simplified analysis. Overlapping tomography is such a scheme, allowing one to obtain all the information contained in specific subsystems of multiparticle quantum systems in an efficient manner, but the ultimate limits of this approach remain elusive. We present protocols for overlapping tomography that are optimal with respect to the number of measurement settings. First, by providing algorithmic approaches based on graph theory we find the minimal number of Pauli settings, relating overlapping tomography to the problem of covering arrays in combinatorics. This significantly reduces the number of measurement settings, showing for instance that two-body overlapping tomography of nearest neighbours in qubit systems with planar topologies can always be performed with nine Pauli settings. Second, we prove that using general projective measurements, all $k$-body marginals can be reconstructed with only $3^k$ settings, independently of the system size. Finally, we demonstrate the practical applicability of our methods in a six-photon experiment. Our results will find applications in learning noise and interaction patterns in quantum computers as well as characterising fermionic systems in quantum chemistry.

Figures (14)

• Figure 1: (a) Measurement graph $K_{3,3}$ for three qubits. Each qubit is represented by three vertices corresponding to the Pauli operators $X$, $Y$, and $Z$. Edges represent two-body Pauli operators, and cliques (triangles) correspond to three-qubit Pauli settings. (b) A minimal edge clique covering of $K_{3,3}$. Each colour groups edges into a triangle, minimising the total number of triangles needed to cover all edges. (c) Three-qubit Pauli settings derived from the clique covering, ensuring all two-body Pauli operators for each qubit pair are included. Colours match the triangles in (b).
• Figure 2: Number of measurement settings needed for two-body overlapping tomography of $n$ qubits ($k=2$). Blue circles: minimal number of Pauli settings possible, corresponding to covering array numbers kokkala2020structure. Red squares: minimal number of projective settings, $3^2 = 9$. Orange triangles: number of Pauli settings needed in the construction of Refs. cotler2020quantumbonetmonroig2020nearly. Green pentagons: number of Pauli settings needed in the construction of Ref. garciaperez2020pairwise.
• Figure 3: Experimental setup for generating and detecting six-photon Dicke state with three excitations. Ultraviolet pulses (390 nm, 80 MHz repetition rate, 300 mW average power) are focused onto a type-II beamlike $\beta$-barium borate (BBO) crystal using a lens ($f_1=75$mm) to generate three photon pairs photons simultaneously in the third-order spontaneous parametric down-conversion (SPDC) emissions. The photons are recollimated with two lenses ($f_2=100$ mm), spectrally and spatially filtered by interference filters (IFs) ($\Delta\lambda=10$nm) and single-mode fibres, and merged into a single path by a polarising beam splitter (PBS). Hong-Ou-Mandel-type interference visibility is $0.92$. The six indistinguishable photons are distributed into six output modes using three 50:50 and two 67:33 beam splitters (BSs), achieving a maximal success probability of $5/324$ for detecting one photon in each measurement part (MP). Arbitrary polarisation analysis in each output mode is conducted by the MP, composed of a quarter-wave plate (QWP), a half-wave plate (HWP), a PBS, and two single-photon avalanche detectors (SPADs).
• Figure 4: Two-qubit reduced states from overlapping tomography of a six-qubit Dicke state. (a) Results for the optimal Pauli scheme (12 settings): the real and imaginary parts of the marginal $\varrho_{2,3}^P$ are shown (see Appendix \ref{['app-sec:exp']} of SM *[See Supplemental Material at ] [ for the appendiceswhich includes Refs. [42--49].] supp for other marginals), along with the two-qubit mixed state fidelities between experimental states $\varrho_{i_1, i_2}^P$ and ideal marginals $\varrho_{i_1, i_2}$ for each pair. (b) Similar analysis for the minimal non-Pauli scheme (nine settings). In both cases, fidelity error bars indicate $\pm1 \sigma$ uncertainties estimated via Monte Carlo sampling with Poissonian photon statistics.
• Figure 5: (a) Connectivity graph $G$ of three qubits. The set of edges $\{\{1,2\}, \{2,3\}\}$ represents the two-qubit marginals that are desired, i.e., we aim to reconstruct the marginals $\varrho_{12}$ and $\varrho_{23}$ of a three-qubit state $\varrho$. (b) Covering graph $G^{\times 3}$ of $G$. Each edge represents a two-body Pauli operator that is needed to recover the two-qubit marginals. For instance, the expectation value of $X_1Y_2$ is required and therefore the edge connecting the vertices $X_1$ and $Y_2$ is drawn in the covering graph. On the other hand, the expectation value of $X_1Z_3$ is not needed, and the vertices $X_1$ and $Z_3$ are not connected. (c) Measurement graph $K_{3,3}$. Each triangle represent a possible three-qubit Pauli setting. As an example, the vertices $Y_1$, $X_2$, and $X_3$ are pairwise connected and represent the measurement setting $Y_1X_2X_3$. From the measurement data of $Y_1X_2X_3$, it is possible to obtain the expectation values of $Y_1X_2$, $Y_1X_3$, and $X_2X_3$. We therefore say that the three-qubit Pauli setting $Y_1X_2X_3$ covers $Y_1X_2$, $Y_1X_3$, and $X_2X_3$, inspired by the fact that the triangle connecting the vertices $Y_1$, $X_2$, and $X_3$ covers the edges $Y_1X_2$, $Y_1X_3$, and $X_2X_3$.

Theorems & Definitions (4)

• Lemma 1
• proof
• Theorem 2
• proof