Optimal Fermionic Joint Measurements for Estimating Non-Commuting Majorana Observables

TL;DR

This work develops fermionic joint measurements for non-commuting Majorana observables, linking incompatibility robustness to SYK-model spectra and providing asymptotically optimal noise scaling for degree-$2k$ observables (up to $k\le 5$). A constructive scheme uses fermionic Gaussian unitaries and random Majorana monomials to implement a parent POVM, with detailed Turán-graph–based designs for $k=1$ and randomized partitions for higher $k$. The framework enables efficient estimation of $k$-body marginals and Hamiltonians, with sample complexities matching fermionic shadows up to constants and offering advantages in the number of unitaries required. The results illuminate deep connections between measurement incompatibility, graph/design theory, and SYK physics, and point to practical pathways for quantum chemistry applications via fermionic joint measurements and randomized processing.

Abstract

An important class of fermionic observables, relevant in tasks such as fermionic partial tomography and estimating energy levels of chemical Hamiltonians, are the binary measurements obtained from the product of anti-commuting Majorana operators. In this work, we investigate efficient estimation strategies of these observables based on a joint measurement which, after classical post-processing, yields all sufficiently unsharp (noisy) Majorana observables of even-degree. By exploiting the symmetry properties of the Majorana observables, as described by the braid group, we show that the incompatibility robustness, i.e., the minimal classical noise necessary for joint measurability, relates to the spectral properties of the Sachdev-Ye-Kitaev (SYK) model. In particular, we show that for an $n$ mode fermionic system, the incompatibility robustness of all degree-$2k$ Majorana observables satisfies $Θ(n^{-k/2})$ for $k\leq 5$. Furthermore, we present a joint measurement scheme achieving the asymptotically optimal noise, implemented by a small number of fermionic Gaussian unitaries and sampling from the set of all Majorana monomials. Our joint measurement, which can be performed via a randomization over projective measurements, provides rigorous performance guarantees for estimating fermionic observables comparable with fermionic classical shadows.

Figures (8)

• Figure 1: Circuit implementation of the joint measurement $\mathsf{G}^O$, defined in Eq. (\ref{['eq:jm_gen']}), on an $n$-mode fermionic state. The state evolves under the action of a Majorana monomial $\gamma_X$, sampled uniformly at random from the set of all Majorana monomials, followed by a fermionic Gaussian unitary $U_O$. The $n$ measurements $\{\gamma_R\,|\,R\in\mathcal{D}_2\}$ are performed on the $n$ modes, with outcomes $q_{R}\in\{\pm 1\}$. In Sec. \ref{['sec:nearly_optimal']} we specify a choice of $U_O$ which optimizes the accuracy of the joint measurement. In the qubit setting---under the Jordan-Wigner transformation---the same circuit is implemented by performing a random Pauli operator $P_i\in\{\mathbbm{1},X,Y,Z\}$ on each of the $n$ qubits, followed by a matchgate circuit and the computational basis measurement.
• Figure 2: (Left.) A perfect matching $\mathcal{M}$ of the Turán graph $T(12,4)$, with 12 vertices partitioned into 4 (coloured) subsets $Y_1,\ldots, Y_4$, each containing 3 vertices. The sparse arrangement ensures that, given any two partition subsets, only one edge in the perfect matching connects them. (Right.) A second Turán graph is obtained from the first via a permutation $\sigma$ of the vertices, exchanging $v_1\leftrightarrow v_2$ if and only if $\{v_1,v_2\}\in\mathcal{M}$. Consequently, each partition subset $Y_1^{\sigma}\ldots, Y_4^{\sigma}$, has no identically coloured vertices.
• Figure 3: Pictorial representation, for $n=6$, of the orthogonal matrices $O^{(1)}$ (left) and $O^{(2)}$ (right) defining the POVMs $\mathsf{G}^{(1)}$ and $\mathsf{G}^{(2)}$, respectively. The block diagonal matrix $D$ consists of four $3\times3$ lower-flat orthogonal matrices $F^{(i)}$, each assigned a distinct colour, e.g., the elements of $F^{(1)}$ are coloured red. Uncoloured entries represent zeros. The matrix $O^{(1)}$ is obtained by a permutation of the rows of $D$, as defined by a perfect matching of the Turán graph $T(12,4)$. The matrix $O^{(2)}$ is obtained by a permutation of the columns of $O^{(1)}$, as defined by a mapping between the vertices of two Turán graphs (see Fig. \ref{['fig:perfect_matching']}).
• Figure 4: (Left.) An example, for $n=15$, of a perfect matching $\mathcal{M}$ of the Turán graph $T(2n,\ell+1)$, with $\ell=5$. The vertices (representing single Majorana operators) are partitioned into $6$ disjoint subsets $Y_i$ of cardinality $5$. The six partition subsets are assigned distinct colors and the sparsely arranged perfect matching (see Def. \ref{['def:sparse']}) ensures the existence of an edge between every two distinct colors. (Right.) A Turán graph $T(30,6)$ obtained by permuting vertices connected by an edge in the original Turán graph, i.e., $v_1\leftrightarrow v_2$ if $\{v_1,v_2\}\in\mathcal{M}$. This ensures each partition subset $Y_i^{\sigma}$ contains no identically colored vertices.
• Figure 5: A perfect matching of the graph $T(2n,\ell+1)$, with its $2n$ vertices partitioned into disjoint subsets $Y_i$, $i=1,\ldots, \ell+1$, each of cardinality $\ell$. First, each vertex in $Y_1$ is connected to a vertex of a different subset via the edges $(1,\ell+1), (2,2\ell+1),\ldots,(\ell,\ell^2+1)$, ensuring that $Y_1$ is connected to every other partition subset. This strategy repeats for the remaining subsets, i.e., vertices in $Y_2$ are paired to vertices of different subsets by the edges $(\ell+2,2\ell+2),(\ell+3,3\ell+2),\ldots,(2\ell,\ell^2+2)$. This continues to the penultimate subset $Y_{\ell}$ that has only one vertex left unpaired, and is joined to the remaining vertex of $Y_{\ell+1}$ by the edge $(\ell^2,\ell(\ell+1))$. We note that the perfect matching is used to define the permutation $\pi$ in Eq. (\ref{['A:perm']}), required to construct $O^{(1)}\in O(2n)$ (cf. Eq. (\ref{['eq:matrixQ']})). The permutation is defined such that each edge is mapped to a unique element in $\mathcal{D}_2$, e.g., $(1,\ell+1)\rightarrow (1,2)$.

Theorems & Definitions (41)

• Proposition 1
• Proposition 2
• Theorem 1
• Lemma 1
• Theorem 2
• Lemma 2
• Lemma 3
• Proposition 3
• Proposition 4
• Conjecture 1