Nonstabilizerness Mpemba Effects

The Big Idea: The "Hot Water" of Quantum Magic

Imagine you are trying to bake the most complex, delicious cake possible (this represents a Quantum State that is powerful enough for advanced computing). To get there, you need to mix in a special, rare ingredient called "Magic" (scientists call this nonstabilizerness). Without this "Magic," your cake is just a plain sponge; with it, you can do anything.

Usually, you'd think the fastest way to get a rich, complex cake is to start with a batter that already has a lot of flavor. But this paper discovers a strange, counterintuitive rule: Sometimes, starting with a very plain, simple batter actually gets you to the complex cake faster than starting with a slightly more flavorful one.

This is called the Mpemba Effect. You might know it from the old saying that "hot water freezes faster than cold water." In this quantum world, "hot" means "less magical" and "cold" means "more magical." The paper proves that a "less magical" state can sometimes evolve into a "more magical" state faster than a "more magical" starting point.

The Experiment: A Quantum Kitchen

The researchers set up a digital "kitchen" using a Random Circuit. Think of this as a machine that shuffles and mixes quantum bits (qubits) according to strict rules.

The Rule: The machine must conserve a specific "charge" (like keeping the total number of eggs in the bowl constant, even if you move them around).
The Ingredients: They started with three different types of "tilted" batters:
1. Tilted Ferromagnetic: All the ingredients are lined up in a row, then tilted slightly.
2. Tilted Néel: The ingredients are arranged in a checkerboard pattern, then tilted.
3. Tilted Domain-Wall: Half the ingredients are on one side, half on the other, then tilted.

The Surprise Findings

The researchers measured how much "Magic" was generated over time. Here is what they found:

1. The "Hot Water" Wins (The Mpemba Effect)
When they used the Tilted Ferromagnetic batter, they saw the effect clearly.

Scenario A: Start with a batter tilted slightly (very simple, low Magic).
Scenario B: Start with a batter tilted more (slightly more complex, higher Magic).
Result: Even though Scenario B started with more Magic, Scenario A caught up and eventually surpassed it. The "simpler" start generated the necessary complexity faster.

2. It's Not Just About the Ingredients (The Charge)
You might think this happens because the "simpler" batter had a different number of eggs (charge). But the researchers found something deeper.

They compared the Tilted Néel and Tilted Domain-Wall batters.
These two had the exact same number of eggs (identical charge distribution) and the exact same starting flavor (identical initial Magic).
The Twist: The Domain-Wall batter showed the Mpemba effect (the simple one won), but the Néel batter did not (the one with more initial flavor stayed ahead).

3. The Secret: How the Ingredients are Arranged
Why did the Domain-Wall win but the Néel lose?

The Néel batter was mixed evenly throughout. The machine could stir it everywhere at once, so the speed of mixing depended mostly on the total number of eggs.
The Domain-Wall batter had a specific structure: a wall separating two groups. The machine had to slowly "eat away" at this wall to mix everything.
The Lesson: It's not just how much of a resource you have, but how it is arranged in space. The specific shape of the starting state changes how fast the "Magic" spreads.

Does This Only Happen in Digital Simulations?

The researchers wanted to know if this was just a trick of their computer simulation (random circuits) or a real law of physics.

They tested it on a non-integrable Hamiltonian (a more realistic, messy physical system, like a real chain of magnets).
Result: The effect still happened! In fact, for some states, the effect only appeared in the real physical system and not in the simulation.
This proves that the "Mpemba Effect for Magic" is a fundamental feature of quantum physics, not just a quirk of a specific computer model.

The Takeaway

This paper tells us that when preparing complex quantum states, the starting point isn't everything.

You don't always need to start with the "closest" or "most advanced" state to get to your goal.
Sometimes, a simpler, more structured starting point allows the system to "run faster" toward the goal.
The secret lies in the spatial arrangement of the initial state, not just the total amount of resources or charges it holds.

In short: In the quantum world, taking a "step back" (starting with less magic) can sometimes be the fastest way to take a giant leap forward.

Technical Summary: Nonstabilizerness Mpemba Effects

Problem Statement
Quantum magic (nonstabilizerness) is a critical resource for universal quantum computation, distinguishing states that can be efficiently classically simulated (stabilizer states) from those that cannot. A central question in quantum state preparation is how to generate these nonstabilizer resources most efficiently. While the Mpemba effect—the counterintuitive phenomenon where a system starting further from equilibrium relaxes faster than one starting closer—has been explored in classical thermodynamics and various quantum contexts (e.g., symmetry restoration, open system relaxation), its existence in the dynamical generation of quantum magic remained an open question. Prior work in random circuit models found no Mpemba behavior for magic, suggesting that conservation laws might be required to alter the dynamical landscape. This paper investigates whether a state with lower initial magic can generate magic faster than a state with higher initial magic under symmetry-constrained dynamics, and identifies the microscopic features governing such behavior.

Methodology
The authors study the dynamics of the second stabilizer Rényi entropy ( $M_2$ ), a monotone quantifying nonstabilizerness, in two distinct settings:

U(1)-Symmetric Random Circuits: A brickwork circuit of nearest-neighbor two-qubit gates conserving total U(1) charge ( $Q = \sum n_j$ ). The gates are block-diagonal in the charge basis, with Haar-random unitaries acting within the charge- $q$ subspaces.
Nonintegrable Hamiltonian Dynamics: Evolution under a mixed-field Ising model (MFIM) with no U(1) symmetry, serving as a control to test the necessity of Abelian symmetry.

The study utilizes three families of "tilted" product states as initial conditions, generated by applying uniform Pauli-Y rotations to ferromagnetic, Néel, and domain-wall configurations. These families are parameterized by a tilt angle $\theta$ . Crucially, the authors compare states with identical initial $M_2$ and identical conserved charge distributions ( $p(q)$ ) but different spatial structures within charge sectors (specifically, tilted Néel vs. tilted domain-wall states).

Key Contributions and Results

Inverse Mpemba Effect in Magic Generation: The authors demonstrate an "inverse Mpemba effect" in the generation of magic. Specifically, for tilted ferromagnetic states (TFS) and tilted domain-wall states (TDWS), states with lower initial magic (larger $\theta$ in the range $[\pi/4, \pi/2]$ ) cross and eventually exceed the magic of states with higher initial magic (smaller $\theta$ ) at late times. This occurs because the state with lower initial magic occupies charge sectors with larger Hilbert space dimensions (closer to half-filling), facilitating faster thermalization and magic generation.
Decoupling of Charge Distribution and Dynamics: A pivotal finding is that the conserved charge distribution $p(q)$ $p (q)$ alone does not determine the presence of the Mpemba effect. The authors show that tilted Néel states (TNS) and tilted domain-wall states (TDWS) share identical charge distributions $p(q)$ $p (q)$ and identical initial $M_2$ $M_{2}$ . However, only the TDWS family exhibits the Mpemba effect, while TNS does not.
- Mechanism: The difference arises from the spatial structure within charge sectors. TNS states thermalize rapidly regardless of $\theta$ because their local structure is already highly entangled. In contrast, TDWS states contain extended locally ferromagnetic regions where U(1)-symmetric gates act trivially or weakly. This creates a slow initial relaxation timescale (scaling with system size) for small $\theta$ , allowing the "hotter" (lower initial magic) state to catch up and overtake the "colder" state.
Generality Beyond Random Circuits and Abelian Symmetry: The phenomenon is not restricted to random circuits or Abelian symmetries.
- In SU(2)-symmetric circuits, similar effects are observed (detailed in Supplemental Material).
- In the nonintegrable MFIM Hamiltonian, the Mpemba effect reappears for the tilted Néel family (which showed no effect in the U(1) circuit). Here, energy (rather than U(1) charge) acts as the conserved quantity. States with larger $\theta$ have energies closer to the spectrum center (infinite temperature), leading to faster relaxation and the crossing of magic trajectories.
Analytical Predictions for Late-Time Magic: The authors derive analytical expressions for the late-time $M_2$ in the U(1)-symmetric circuit. They show that the steady-state magic is extensively constrained by the initial charge-sector distribution, deviating systematically from the symmetry-free Haar prediction. For tilted ferromagnetic states, late-time magic increases monotonically with $\theta$ ; for tilted Néel and domain-wall states, it becomes nearly independent of $\theta$ once $\theta \gtrsim \pi/8$ .

Significance
The paper establishes quantum magic as a distinct arena for Mpemba physics. It refutes the notion that the Mpemba effect in resource generation is solely determined by the distribution of conserved quantities (like charge or energy). Instead, the authors demonstrate that the spatial structure of the initial state within each conserved sector is a critical control parameter.

The findings imply that the fastest route to a target many-body resource state does not necessarily originate from the initial state that appears "closest" in terms of resource content or conserved charge distribution. This insight offers a new guiding principle for the efficient preparation of resource states relevant to measurement-based quantum computation and fault-tolerant protocols, suggesting that engineering specific spatial correlations in initial states can accelerate the generation of nonstabilizer resources. The work complements recent studies on symmetry-constrained steady-state magic and opens avenues for exploring Mpemba-like acceleration in other quantum resources such as coherence and imaginarity.