Certified bounds on optimization problems in quantum theory

TL;DR

<3-5 sentence high-level summary> This paper tackles certifiability in non-commutative polynomial optimization (NPO) arising in quantum information and physics by turning floating-point semidefinite programming (SDP) relaxations into exact rational certificates. It introduces a three-step post-processing pipeline—rounding, Frobenius-optimal projection, and lifting—to transform numerical SOS/SOHS data into rigorous bounds, and extends this framework to sparsity- and symmetry-adapted hierarchies. The authors demonstrate the approach on maximal Bell-inequality violations and ground-state observables in many-body systems, showing that certified bounds are reliable even when numerical bounds fail, and they analyze the trade-offs in dense, sparse, and symmetry-adapted settings. This work advances the use of SDP-based relaxations as robust, provable tools for quantum certification and device-independent analysis.

Abstract

Semidefinite relaxations of polynomial optimization have become a central tool for addressing the non-convex optimization problems over non-commutative operators that are ubiquitous in quantum information theory and, more in general, quantum physics. Yet, as these global relaxation methods rely on floating-point methods, the bounds issued by the semidefinite solver can - and often do - exceed the global optimum, undermining their certifiability. To counter this issue, we introduce a rigorous framework for extracting exact rational bounds on non-commutative optimization problems from numerical data, and apply it to several paradigmatic problems in quantum information theory. An extension to sparsity and symmetry-adapted semidefinite relaxations is also provided and compared to the general dense scheme. Our results establish rational post-processing as a practical route to reliable certification, pushing semidefinite optimization toward a certifiable standard for quantum information science.

Figures (11)

• Figure 1: The geometric picture of our procedure. (1) $G_0$ is rounded to a rational $\tilde{G}_0$. (2) $\tilde{G}_0$ is projected onto $\mathcal{L}$, possibly leaving the PSD cone. (3) $\mathcal{P}(\tilde{G}_0)$ is lifted back into the PSD cone, certifying $\lambda_d^{\mathrm{rat}}$ as an upper bound.
• Figure 2: $(a)$ The correlation graph $C$ for $f(\underline{X}) = X_1X_5 + X_1 X_2 + X_2 X_3 + X_3 X_4 + X_4 X_1$. $(b)$ A minimal chordal extension of $C$. $(c)$ Another chordal extension of $C$.
• Figure 3: Numerical results for the certification scheme on Bell inequalities A2-A89
• Figure 4: Numerical bounds $\lambda_d$ and rationalized bounds $\lambda_d^{\mathrm{rat}}$ relative to exact maximal violations $\lambda_{\max}$ for $\mathcal{B}_2$ (a) and $\mathcal{B}_3$ (b).
• Figure 5: Difference in bound losses $\delta_d - \delta_d^{\mathrm{sp}}$ between dense and sparse certification schemes for Bell inequalities $A_2$–$A_{89}$ at relaxation orders $d=1$ (a), $d=2$ (b), and $d=3$ (c).

Theorems & Definitions (16)

• Theorem 1: Helton–McCullough Positivstellensatz helton_positivstellensatz_2004
• Lemma 2: Frobenius optimal Gram projection
• proof : Proof of Lemma \ref{['thm:frobopt']}
• Corollary 3
• proof : Proof of Corollary \ref{['thm:binomial_proj']}
• Theorem 4: Certified bounds to NPOs
• Lemma 5: Constant SOHS decomposition
• proof : Proof of Theorem \ref{['thm:theorem2']}
• Theorem 6: Certified tightening for interior pre-certificates
• Lemma 7: Frobenius optimal sparse Gram projection