⚛️ quantum physics

Non-commutative optimization problems with differential constraints

This paper introduces a method to transform non-commutative polynomial optimization problems with differential constraints into standard forms solvable by a complete hierarchy of semidefinite programming relaxations, demonstrating its effectiveness in approximating local observable averages in quantum spin systems under Hamiltonian evolution even in the thermodynamic limit.

Original authors: Mateus Araújo, Andrew J. P. Garner, Miguel Navascues

Published 2026-07-17

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CC BY 4.0

Original authors: Mateus Araújo, Andrew J. P. Garner, Miguel Navascues

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the future of a complex machine, like a giant clockwork toy with millions of moving gears. In the world of quantum physics, these "gears" are particles, and their movements are governed by strict rules called differential equations. These equations describe how things change over time, like a recipe for how a quantum system evolves from one moment to the next. However, there's a catch: these particles don't play nice with each other. If you swap the order in which you look at two particles, the result changes. This is called "non-commutativity," and it makes predicting their behavior incredibly hard.

For decades, scientists have had a powerful toolbox called "Non-Commutative Polynomial Optimization" (NPO) to solve problems about these quantum gears. Think of NPO as a super-smart calculator that can figure out the best possible outcome for a system, provided the rules are written as simple algebraic equations (like $x^2 + y^2 = 1$ ). But there's a big hole in this toolbox: it can't handle the "time" part. It can't easily process the differential equations that describe how the system changes over time. This means that for many real-world scenarios—like watching a quantum system evolve after a sudden jolt, or "quench"—scientists were stuck. They couldn't use their best tools to predict what would happen next, leaving them guessing about the future of quantum materials.

This paper introduces a clever new trick to fix that gap. The authors, Mateus Araújo, Andrew J. P. Garner, and Miguel Navascués, propose a method to translate these tricky "time-evolution" problems into the language that the NPO toolbox already understands. They call this new approach "Differential Non-Commutative Polynomial Optimization" (DNPO).

Here is the magic trick they discovered: Imagine you have a movie of a quantum system playing from start to finish. Instead of trying to solve the whole movie at once, the authors suggest turning the entire timeline into a single, giant, static snapshot. They do this by treating time not as a flowing river, but as just another variable in the equation, like a coordinate on a map. By doing this, they can rewrite the rules of the movie (the differential equations) as a set of static algebraic constraints. Suddenly, a problem that was impossible for their old tools becomes a standard NPO problem.

Once they have translated the problem, they can use a "hierarchy" of increasingly powerful computer algorithms (called Semidefinite Programming, or SDP) to solve it. Think of this hierarchy like a series of zoom lenses. The first lens gives a rough, blurry picture of the answer. The next lens zooms in a bit more, giving a sharper image. With each step up the hierarchy, the answer gets more precise. The paper proves that if the system's energy and size are bounded (which is true for almost all physical systems we care about), this process will eventually converge to the exact, perfect answer.

The authors tested this idea on some tough scenarios. First, they used it to predict the behavior of quantum systems after a "quench"—a sudden change in the system's energy, like flipping a switch. They found that even with just a few steps up their "zoom lens" hierarchy, they could get incredibly accurate predictions for how local parts of the system would behave, even for systems with dozens of particles. They also showed that this method works for systems that are theoretically infinite in size, like an endless chain of atoms, provided the system looks the same everywhere (a property called translation invariance).

In their simulations, they managed to calculate the future state of a chain of 25 quantum spins with high precision, and even touched on systems with infinitely many spins. The results were striking: the upper and lower bounds they calculated were so close together that the answer was effectively known. This means they didn't just guess; they provided a rigorous, mathematically proven range that the true answer must fall within.

The paper also tackled a different kind of puzzle: "quantum time series." Imagine you have a quantum system and you measure it at a few specific times, but you want to know what it was doing in between (interpolation) or what it will do later (extrapolation). Without this new method, this is a nightmare because the system's evolution is governed by those hard-to-handle differential equations. The authors showed that their DNPO approach could solve this, providing tight bounds on the system's behavior at any time, effectively filling in the gaps in the data with mathematical certainty.

While the paper doesn't claim to have solved every problem in quantum physics, it offers a complete and reliable roadmap for a huge class of them. It turns a problem that was previously considered too difficult for standard optimization tools into one that can be systematically solved. The authors suggest that this method could be a game-changer for studying how materials behave after sudden changes, a task that current approximation methods often struggle with. By bridging the gap between time-dependent dynamics and static optimization, they've given physicists a new way to peek into the future of the quantum world, one step at a time.

Technical Summary: Non-commutative optimization problems with differential constraints

Problem Definition
Non-commutative polynomial optimization (NPO) problems seek to minimize the state average of a polynomial of operator variables subject to polynomial constraints (positive semidefinite operators) over all possible Hilbert spaces, operators, and states. While NPO theory provides a complete hierarchy of semidefinite programming (SDP) relaxations for many quantum physics problems (e.g., Bell inequalities, ground state energies), it cannot inherently handle differential constraints. Specifically, the time evolution of quantum systems (open or closed) is modeled by differential equations on non-commuting variables, which cannot be expressed as static polynomial inequalities. This limitation prevents the formulation of tasks such as predicting the future behavior of many-body systems or mitigating readout errors in quantum devices within the standard NPO framework, except in highly restricted scenarios.

This paper introduces Differential Non-Commutative Polynomial Optimization (DNPO) problems. A DNPO involves minimizing an objective function over non-commuting variables where a subset of variables, indexed by a continuous parameter $t \in [0, 1]$ , satisfies a system of ordinary differential equations (ODEs). The optimization is performed over all Hilbert spaces, operators, and normalized states satisfying these differential and algebraic constraints.

Methodology
The core contribution is a method to relax any DNPO problem into an auxiliary standard NPO problem, thereby enabling the use of existing SDP hierarchies. The derivation proceeds as follows:

Operator Extension: The authors map the time-dependent operators $X(t)$ and the time parameter $t$ into an extended Hilbert space $\tilde{H} = H \otimes L^2[0, 1]$ . They define new operators $\tilde{X}$ and $\tilde{T}$ (where $\tilde{T}$ acts as the multiplication operator by $t$ ) such that the integral constraints and differential equations of the original problem translate into algebraic constraints on these new operators.
Differentiation Map: A linear differentiation map $D$ is defined on the abstract algebra of non-commutative polynomials. This map encodes the differential equation $dX_i/dt = g_i(t, X, Y)$ via the relation $D(h)(\tilde{T}, \tilde{X}, \tilde{Y}) = \int dt \frac{d}{dt}h(t, X(t), Y) \otimes |t\rangle\langle t|$ .
Fundamental Theorem of Calculus: The differential constraint is enforced by requiring that the expectation value of the differentiated polynomial equals the difference of the boundary values: $\omega(D(h)) = \omega(h(1)) - \omega(h(0))$ .
SDP Hierarchy Construction: The resulting NPO problem (denoted as $Q$ ) is relaxed into a hierarchy of SDPs by optimizing over linear functionals $\tilde{\omega}$ on the polynomial algebra, subject to moment matrix and localizing matrix positivity constraints.
Convergence Conditions: The authors prove that if the resulting NPO problem satisfies the Archimedean condition (essentially ensuring that the feasible operators are bounded), the hierarchy of SDP relaxations is complete. That is, the sequence of SDP lower/upper bounds converges to the exact solution of the original DNPO problem.
Markovian and Thermodynamic Extensions:
- For Markovian DNPOs (where variables at different times do not mix in constraints), a sparse relaxation is proposed using three distinct linear functionals ( $\tilde{\omega}_{[0,1]}$ , $\tilde{b}_0$ , $\tilde{b}_1$ ) to reduce computational cost.
- For thermodynamic limits (infinite systems), the method is adapted to enforce translation invariance (TI) on the initial state and Hamiltonian. A specific hierarchy is constructed for 1D chains, and the authors prove that the limit of this hierarchy corresponds to a valid physical solution in the thermodynamic limit.

Key Contributions

Formulation of DNPO: The paper formally defines DNPO problems, bridging the gap between non-commutative optimization and differential equations in quantum mechanics.
Complete Relaxation: It provides a constructive proof that any DNPO with bounded operators can be relaxed into an NPO, allowing the application of the standard NPO SDP hierarchy.
Convergence Proof: The authors establish that under the Archimedean condition, the SDP hierarchy converges to the exact solution of the DNPO.
Scalable Algorithms: The paper introduces sparse and translation-invariant variants of the hierarchy, making the method applicable to large finite systems and infinite thermodynamic limits.

Results and Numerical Tests
The authors validate the method through several numerical examples:

Academic Example: A simple non-commuting optimization problem is reformulated as a DNPO. The SDP hierarchy recovers the exact solution ( $e^{-1/4}$ ) at level $k=3$ .
Quantum Interpolation/Extrapolation: The method is applied to estimate the value of an observable at an unmeasured time $\tau$ given measurements at discrete times $\tau_k$ . The hierarchy provides tight bounds that become exact at the sample points and accurately bound the trajectory in between.
Quenched Spin Systems:
- Finite Systems: The method models the time evolution (quench) of a local observable in a 1D spin chain (Heisenberg Hamiltonian) starting from a product state. For a system of $n=18$ spins, level-2 relaxations provide upper and lower bounds differing by only $0.0012$ at $\tau=1$ .
- Thermodynamic Limit: The method is applied to an infinite 1D chain with translation-invariant initial states. Even with low hierarchy levels (e.g., $k_\sigma=2, k_t=2$ ) and small system windows ( $n=7$ ), the bounds are indistinguishable up to $\tau \approx 0.7$ . Increasing the hierarchy levels significantly improves predictability for longer times.

Significance and Claims
The paper claims that this work provides a complete hierarchy of tractable relaxations for characterizing the time-evolved local observables of condensed matter systems, applicable to both finite and infinite (thermodynamic) systems.

It solves the problem of extrapolating and interpolating quantum time series, a task for which no general algorithm previously existed.
It extends the 2-Reduced Density Matrix Theory (2-RDMT) toolbox to include quench dynamics, allowing for the modeling of ground states, Gibbs states, and steady states of Lindbladians under time evolution.
The authors emphasize that even low levels of the hierarchy yield very accurate approximations for reasonably long evolution times, even in the thermodynamic limit. This suggests the method is a viable alternative to tensor network states for studying quenched 2D models where tensor networks often fail.
The approach is general and can be extended to higher local dimensions (e.g., qudits) and higher spatial dimensions (e.g., 2D lattices) by defining appropriate translation operators and monomial sets.

Technical Summary: Non-commutative optimization problems with differential constraints

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