Wandering range of robust quantum symmetries

This paper introduces the concept of the wandering range to quantify how robust quantum symmetries deviate under perturbations, identifying conditions for linear scaling and deriving explicit nonperturbative bounds on this deviation.

The Gist

Imagine you are trying to keep a spinning top perfectly balanced on a table. In the ideal world, if you spin it just right, it stays upright forever. In physics, this "perfect balance" is called a symmetry. It's a rule that says, "No matter how much time passes, this part of the system stays exactly the same."

But in the real world, nothing is perfect. There are bumps on the table, tiny gusts of wind, or vibrations. In physics, these are called perturbations. When you add a little bit of "noise" (a perturbation) to your spinning top, does it stay balanced?

• Fragile Symmetries: Some tops wobble wildly and fall over quickly. Their balance is destroyed by even the tiniest nudge.
• Robust Symmetries: Other tops are like acrobats. Even with a little wind, they might wobble a tiny bit, but they keep spinning upright for a long time. They are "robust."

This paper is about measuring how much those "robust" tops wobble when you push them. The authors call this measurement the "Wandering Range."

The Big Question: How much does it wobble?

The authors wanted to know: If I push the system with a force of size $\epsilon$ (epsilon), how far does the symmetry "wander" from its perfect spot?

In the past, we knew that for small systems (like a toy with only a few gears), the wobble is directly proportional to the push. If you push twice as hard, it wobbles twice as much. This is called linear scaling.

However, for huge, complex systems (infinite-dimensional quantum systems, like a real atom or a superconductor), things get tricky. The authors discovered that for some states, the wobble doesn't behave nicely. It might be very slow to grow, or it might depend heavily on where you look in the system.

The Main Discoveries

The paper finds the "Golden Rules" for when the wobble behaves nicely (linearly) again.

1. The "Simple" Cases (The Easy Wins)

The authors prove that if you look at the system in two specific ways, the wobble is perfectly predictable:

• The "Pure" States: If you look at the system using only its fundamental building blocks (the eigenvectors), the wobble is small and linear.
• The "Finite" Symmetries: If the symmetry only affects a limited, finite part of the system (like a small group of gears), the wobble is also linear.

Analogy: Imagine a massive orchestra. If you ask a single violinist to play slightly off-key (a small perturbation), the whole orchestra might sound a bit messy. But if you ask just the first chair violinist to adjust, or if you only listen to the woodwinds section, the "messiness" is very easy to calculate and stays proportional to the mistake.

2. The "Super-Robust" Symmetries (The Heavy Hitters)

The most exciting part of the paper deals with Completely Robust Symmetries. These are symmetries that are so strong they survive any kind of bounded noise.

The authors developed a new mathematical tool (a fancy version of a "Schrieffer-Wolff transformation," which sounds like a wizard's spell) to prove that for these super-robust symmetries, the wobble is always linear, no matter how big the system is.

They even gave a formula for the maximum wobble:
$$\text{Wobble} \approx \frac{\text{Push Strength}}{\text{Gap Size}}$$

The Analogy of the "Gap":
Imagine the energy levels of the system are like rungs on a ladder. The "spectral gap" is the distance between the rungs.

• If the rungs are far apart (a large gap), the system is stiff. A push won't make it wobble much.
• If the rungs are very close together (a small gap), the system is floppy. The same push will make it wobble a lot.

The paper proves that as long as there is a minimum distance between the rungs (a non-vanishing gap), you can predict exactly how much the symmetry will wander.

How They Did It: The "Quantum KAM" Machine

To prove this, the authors used a technique called KAM iteration (named after Kolmogorov, Arnold, and Moser).

The Metaphor:
Imagine you are trying to straighten a crooked picture frame on a wall.

1. You push it a little bit. It's still crooked.
2. You push it again, but this time you adjust your angle based on the first mistake.
3. You keep doing this, getting closer and closer to perfect.

The authors used a mathematical version of this "iterative push." They built a new, imaginary Hamiltonian (a "perfect" version of the system) that commutes with the original one. This imaginary system acts as a "shadow" that mimics the real, messy system but is perfectly organized. By comparing the real system to this perfect shadow, they could calculate the exact error.

They also used a sequence of numbers called Catalan numbers (famous in combinatorics) to prove that their "iterative push" wouldn't explode into chaos but would actually converge to a stable answer.

Why Does This Matter?

This isn't just abstract math. It's crucial for Quantum Computing and Quantum Simulation.

• The Problem: We want to build quantum computers to simulate complex molecules or materials. But real quantum computers are noisy. They have errors.
• The Hope: If we can design a quantum system with "Robust Symmetries," we can protect our information from these errors.
• The Takeaway: This paper tells engineers exactly how much error they can tolerate before their "protection" (symmetry) breaks down. It gives them a safety margin: "If your noise is this small, and your energy gaps are this big, your quantum memory will stay stable."

Summary in One Sentence

This paper provides a precise mathematical ruler to measure how much "perfect order" (symmetry) in a quantum system gets messed up by real-world noise, proving that for the most stable systems, the messiness grows in a perfectly predictable, linear way.

Technical Summary

1. Problem Statement

The paper addresses the stability of quantum symmetries under Hamiltonian perturbations. In quantum mechanics, a symmetry $S$ is an operator commuting with the Hamiltonian $H$ ($[S, H]=0$), implying it is conserved during time evolution ($e^{itH}Se^{-itH} = S$). However, in realistic scenarios (e.g., analog quantum simulation), the Hamiltonian is subject to perturbations $H(\varepsilon) = H + \varepsilon V$. Under such perturbations, symmetries are generally no longer conserved.

The authors distinguish between:

• Fragile Symmetries: Deviations from the initial symmetry value accumulate over time, leading to significant errors.
• Robust Symmetries: The deviation remains small as the perturbation strength $\varepsilon \to 0$.

The Core Question: While it is known that robust symmetries converge to their unperturbed values as $\varepsilon \to 0$, the rate of this convergence is not guaranteed to be linear ($O(\varepsilon)$). In infinite-dimensional Hilbert spaces, the "wandering range" (the maximal deviation over time) can scale as $O(\varepsilon^\gamma)$ with $\gamma$ arbitrarily close to 0, or even fail to converge uniformly in the operator norm. The paper aims to:

1. Identify conditions under which the wandering range scales linearly with $\varepsilon$ ($O(\varepsilon)$).
2. Derive explicit, non-perturbative bounds for this deviation, generalizing finite-dimensional results to infinite-dimensional systems.

2. Methodology

The authors employ a combination of spectral theory, perturbation theory, and a quantum adaptation of the Kolmogorov-Arnold-Moser (KAM) iteration scheme.

• Definitions:

  • Wandering Range: Defined as $\delta_{H(\varepsilon)}(S, \psi) = \sup_{t \in \mathbb{R}} \| (e^{itH(\varepsilon)} S e^{-itH(\varepsilon)} - S) \psi \|$.
  • Admissible Perturbations: Perturbations that preserve the pure point spectrum structure and allow for a continuous unitary transformation relating the perturbed and unperturbed eigenprojections.
  • Completely Robust Symmetries: Symmetries belonging to the bicommutant of $H$ ($S \in \{H\}''$), meaning they commute with $H$ and all other symmetries of $H$.

• Technical Framework:

  • Schrieffer-Wolff Transformation: The core of the proof involves constructing a unitary transformation $W(\varepsilon) = e^{iK(\varepsilon)}$ that block-diagonalizes the perturbed Hamiltonian. This transforms $H(\varepsilon)$ into an "eternal block-diagonal approximation" $H + \varepsilon \hat{V}(\varepsilon)$, where $\hat{V}(\varepsilon)$ commutes with the original $H$.
  • Homological Equation: The construction relies on solving the commutator equation $i[X, H] = \{B\}$ (where $\{B\}$ is the off-diagonal part of an operator $B$). The authors extend the solution of this equation to unbounded Hamiltonians with a non-vanishing spectral gap.
  • KAM Iteration: The unitary transformation and the effective Hamiltonian are constructed as formal power series in $\varepsilon$. The coefficients are determined recursively order-by-order.
  • Convergence Analysis: To prove the convergence of the formal series, the authors utilize the combinatorial properties of Catalan numbers. They establish that the norms of the series coefficients are bounded by Catalan numbers, allowing them to prove convergence within a specific radius of $\varepsilon$ determined by the spectral gap and perturbation strength.

3. Key Contributions and Results

A. Linear Scaling for Specific Cases (Section 2)

The authors prove that the wandering range is of order $O(\varepsilon)$ under two specific conditions:

1. Dense Subspace: For states $\psi$ belonging to the linear span of the eigenvectors of the unperturbed Hamiltonian (Theorem 2.1).
2. Finite-Rank Symmetries: If the symmetry operator $S$ has finite rank, the convergence is uniform in the operator norm (Theorem 2.1).

• Result: For linear perturbations $H(\varepsilon) = H + \varepsilon V$ where $H$ has a compact resolvent, the wandering range scales linearly with $\varepsilon$ for these cases.

B. Uniform Bounds for Completely Robust Symmetries (Section 3)

The paper's most significant contribution is the derivation of a uniform bound for completely robust symmetries (those in the bicommutant $\{H\}''$) under arbitrary bounded perturbations (Theorem 3.1).

• Theorem 3.1: If $H$ has a pure point spectrum with a non-vanishing minimal spectral gap $\eta$, and the perturbation is uniformly bounded by $v$, then for sufficiently small $\varepsilon$:
  $$\sup_{t \in \mathbb{R}} \| e^{itH(\varepsilon)} S e^{-itH(\varepsilon)} - S \| \leq \beta \frac{v}{\eta} \|S\| \varepsilon$$
  where $\beta \approx 15.3$ is a constant derived from the convergence analysis.
• Significance: This result generalizes finite-dimensional bounds to infinite-dimensional unbounded Hamiltonians and removes the dependence on the number of distinct eigenvalues (which is infinite in these systems).

C. Eternal Block-Diagonal Approximation (Theorem 3.2)

The authors construct a new Hamiltonian $H + \varepsilon \hat{V}(\varepsilon)$ that:

1. Commutes with the original $H$ (and thus with any completely robust symmetry).
2. Generates a dynamics that approximates the true perturbed dynamics uniformly in time.
   This "eternal block-diagonal approximation" is the mechanism that guarantees the linear scaling of the wandering range.

D. Convergence via Catalan Numbers (Section 4)

The proof of the existence of the transformation $W(\varepsilon)$ relies on a rigorous convergence analysis of the KAM series.

• The authors show that the norms of the series coefficients are bounded by Catalan numbers ($d_s$).
• They prove that the series converges for $\varepsilon \leq \frac{\eta}{v\rho}$, where $\rho \approx 34.8$. This provides a concrete, non-perturbative regime where the linear scaling holds.

4. Significance and Applications

• Theoretical Advancement: The paper bridges the gap between finite-dimensional quantum mechanics (where linear scaling is often assumed) and infinite-dimensional systems. It rigorously establishes that linear scaling is not automatic in infinite dimensions but holds under specific spectral and symmetry conditions.
• Robustness in Quantum Simulation: The results are crucial for analog quantum simulation. They provide a theoretical guarantee that if a target system possesses a "completely robust" symmetry (like total spin or particle number in certain models), the simulation will preserve this symmetry with an error that scales linearly with experimental imperfections, provided the spectral gap is non-zero.
• Physical Models: The authors apply their results to a Josephson junction in an inductive loop. They demonstrate how to identify the unperturbed Hamiltonian (harmonic oscillator) and the perturbation (cosine potential) to verify the conditions for the KAM iteration, providing concrete criteria for when the symmetry protection holds.
• Limitations and Future Work: The method requires a non-vanishing minimal spectral gap. The authors note that this excludes systems with continuous bands (solid-state physics) or systems with vanishing gaps (e.g., the hydrogen atom), identifying these as directions for future research.

Summary

This paper provides a rigorous mathematical framework for understanding the stability of quantum symmetries. By combining spectral theory with a quantum KAM iteration scheme, the authors prove that for Hamiltonians with a spectral gap, the deviation of robust symmetries under perturbation is linear in the perturbation strength. They provide explicit, non-perturbative bounds that depend only on the perturbation strength and the spectral gap, offering a powerful tool for analyzing error resilience in quantum systems.