No quantum solutions to linear constraint systems from monomial measurement-based quantum computation in odd prime dimension

TL;DR

This work addresses whether state-dependent contextuality in MBQC with qudits of dimension $d>2$ can yield quantum solutions to a linear constraint system (LCS) naturally associated to the MBQC. By defining a general mapping from deterministic, non-adaptive $ ext{Z}_d$-MBQC to an LCS $A(M)x=b(o)mod d$ and focusing on monomial MBQC within the normaliser $N_{SU(p)}(p)$ for odd primes $p$, the authors prove a no-go theorem: any quantum solution to the associated LCS within this extended group must be classical, implying no quantum lift beyond classical solvability. This result extends prior no-quantum-solution findings from Pauli-restricted settings to a broader monomial/normaliser regime, and it sharpens the distinction between state-dependent contextuality in MBQC and state-independent LCS contextuality, particularly highlighting the need for state-independent (Kochen–Specker-type) ideas to realize genuine quantum LCS solutions in odd prime dimensions. The work thus constrains how contextual MBQC can translate into quantum advantage via LCS solutions in finite odd prime dimensions, guiding future exploration toward intrinsically state-independent contextuality frameworks.

Abstract

We combine the study of resources in measurement-based quantum computation (MBQC) with that of quantum solutions to linear constraint systems (LCS). Contextuality of the input state in MBQC has been identified as a key resource for quantum advantage, and in a stronger form, underlies algebraic relations between (measurement) operators which obey classically unsatisfiable (linear) constraints. Here, we compare these two perspectives on contextuality, and study to what extent they are related. More precisely, we associate a LCS to certain MBQC which exhibit strong forms of state-dependent contextuality, and ask if the measurement operators in such MBQC give rise to state-independent contextuality in the form of quantum solutions of its associated LCS. Our main result rules out such quantum solutions for a large class of MBQC. This both sharpens the distinction between state-dependent and state-independent forms of contextuality, and further generalises results on the non-existence of quantum solutions to LCS in finite odd (prime) dimension.

Figures (3)

• Figure 1: Mermin-Peres square.
• Figure 2: Mermin's star.
• Figure 4: Two types of intersections between conjugate Heisenberg-Weyl subgroups $H\cong H(\xi,\sigma),H(\xi',\sigma')\leq N$ (see Lm. \ref{['lem: comm_tildeK']}), where $\xi,\xi'\in T$ and $\sigma,\sigma'\in S_p$ (with notation as in proof of Thm. \ref{['thm: FCO-2']}). If $(\xi',\sigma')\neq(\xi\chi,\sigma)^a$ for all $a\in\mathbb Z_p$ and $(\chi,\mathrm{id})\in\tau\cdot H\cap T_{(p)}\cong\{\omega^cZ^b\mid b,c\in\mathbb Z_p\}$ (with $Z$ the Pauli-Z operator in $H$) and $\tau=\phi^N_{S_p}(\xi,\sigma)$, the intersection is the center (left); if instead $(\xi',\sigma')\neq(\xi\chi,\sigma)^a$ (for some $a\in\mathbb Z_p$ and $\chi\in\tau\cdot H\cap T_{(p)}$ with $\tau=\phi^N_{S_p}(\xi,\sigma)$), the intersection additionally contains the subgroup $\tau\cdot\langle Z\rangle$, conjugated by $\tau$ (right).

Theorems & Definitions (69)

• Theorem 1
• proof
• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Lemma 1
• proof
• Lemma 2
• proof