Extracting conserved operators from a projected entangled pair state

This paper presents a method to extract geometrically local conserved operators, including parent Hamiltonians, from infinite projected entangled pair states (iPEPS) by analyzing the quantum geometry of parameter-deformed manifolds via static structure factors, successfully identifying both frustration-free and non-frustration-free Hamiltonians with improved locality for states like the short-range RVB and deformed toric code.

The Gist

Imagine you are a detective trying to figure out the rules of a complex game, but you don't have the rulebook. All you have is a single, perfect snapshot of the game in progress. You see the pieces arranged in a specific, intricate pattern. Your goal? To reverse-engineer the rules (the "Hamiltonian") that created this pattern.

This is exactly the problem physicists face with Quantum Matter. They have powerful computer models (called iPEPS) that describe how quantum particles behave, but they often don't know the exact "laws of physics" (the Hamiltonian) that govern those particles. Usually, figuring out the rules from the snapshot is incredibly hard, like trying to guess the recipe of a cake just by looking at a single crumb.

This paper introduces a new, clever detective tool to solve this mystery. Here is how it works, using some everyday analogies:

1. The Problem: The "Black Box" of Quantum Physics

Think of a quantum system as a giant, invisible machine.

• The Input: We have a "snapshot" of the machine's state (the iPEPS).
• The Goal: We want to find the "control panel" (the Hamiltonian) that keeps the machine running in this specific state.
• The Difficulty: In the quantum world, the machine is so complex that standard methods often fail or require impossible amounts of computing power. It's like trying to figure out how a symphony orchestra plays a song just by listening to one second of it, without knowing the sheet music.

2. The Solution: The "Ripple Test"

The authors' method is like testing a still pond to find out what's hidden underneath.

• The Setup: Imagine the quantum state is a calm, flat pond.
• The Test: They gently poke the pond with a stick (mathematically, they "deform" the state slightly with a parameter).
• The Observation: They watch how the water ripples.
  • If they poke the pond in a random direction, big ripples appear. This means the state is sensitive to that change.
  • The Magic: If they poke the pond in a special direction and no ripples appear at all, they have found a "conserved quantity." In physics, if a system doesn't change when you poke it, it means there is a hidden rule (a conserved operator) protecting it.

3. The "Fingerprint" of the Rules

The paper uses a mathematical tool called Static Structure Factors. Think of this as a "fingerprint scanner" for the quantum state.

• Normally, scientists look at how particles talk to each other (correlations).
• This new method looks at how the entire pattern reacts to a poke.
• If the "fingerprint" shows zero reaction (zero "fidelity susceptibility") for a specific type of poke, that poke corresponds to a Conserved Operator.
• If that operator is the "Hamiltonian" (the energy rule), then the state is a stable solution to that rule.

4. What Did They Discover?

The authors tested their "Ripple Test" on several famous quantum puzzles:

• The AKLT State (The Perfect Crystal): They looked at a state known to be perfectly ordered. Their method successfully found the exact rules that create it, proving their detective tool works.
• The XX Model (The Chaotic Magnet): They looked at a messy, disordered state. Usually, finding the rules here is a nightmare. But their method found the rules anyway, even though the state wasn't perfectly "frustration-free" (a fancy way of saying the rules were messy and conflicting).
• The RVB State (The Quantum Soup): This is a state thought to be related to high-temperature superconductors (the holy grail of energy). It's incredibly complex. Their method found a simplified set of rules (a 4-site Hamiltonian) that keeps this "soup" stable. This is a big deal because previous methods required rules involving 8 or more sites, which are too complicated to build in a lab. Their method found a simpler, more practical recipe.
• The "Quantum Scars" (The Ghosts in the Machine): They found a set of rules where a specific state isn't the ground state (the lowest energy), but an excited state that refuses to decay. It's like a ghost that haunts the machine, staying in the middle of the energy spectrum without fading away. This is a phenomenon called Quantum Many-Body Scars, and finding the rules that create them is a major breakthrough.

5. Why Does This Matter?

• For Scientists: It's a new way to "learn" the laws of physics directly from data, without needing to guess the model first.
• For Engineers: The rules they found are often simpler (more "local") than previous methods. This means it's easier to build these quantum states in real laboratories using quantum computers.
• For the Future: It opens the door to discovering new materials and exotic quantum states that we didn't even know existed, simply by analyzing the "ripples" in their quantum snapshots.

In a nutshell: The authors built a mathematical "poke-test" that allows us to reverse-engineer the laws of quantum physics from a single snapshot of a quantum state. It's like looking at a perfectly arranged pile of sand and deducing the exact wind patterns that created it, even if the wind was chaotic.

Technical Summary

1. Problem Statement

The paper addresses the inverse problem of Hamiltonian learning: given a quantum many-body state (specifically a tensor network state like a Projected Entangled Pair State, PEPS), how can one reconstruct the underlying Hamiltonian or other conserved operators for which this state is an eigenstate?

• Context: While efficient algorithms exist for 1D Matrix Product States (MPS), extending this to 2D PEPS is significantly harder due to the #P-complete nature of exact PEPS contraction.
• Challenge: Practical 2D calculations rely on approximate contraction schemes (e.g., Corner Transfer Matrix Renormalization Group, CTMRG), which introduce errors in correlation functions. It was previously unclear whether one could reliably extract conserved operators from these approximate states, especially for non-frustration-free Hamiltonians or critical (gapless) systems.
• Goal: To develop a method that extracts geometrically $k$-local conserved operators (including Hamiltonians) from a given infinite PEPS (iPEPS), even when the state is only an approximate eigenstate or obtained via variational optimization.

2. Methodology

The authors propose a method based on the static structure factors of multi-site operators. The core logic is that if a state $|\Psi\rangle$ is an eigenstate of a conserved operator $\hat{H}$, the quantum fluctuations of $\hat{H}$ in that state must vanish (or be minimal for approximate eigenstates).

A. Theoretical Framework

1. Static Structure Factor Matrix ($S$):
   The method defines a matrix $S^{\alpha, \beta}(q)$ representing the Fourier-transformed connected correlation functions of a basis set of geometrically $k$-local Hermitian operators $\{\hat{o}^\alpha_x\}$:
   $$S^{\alpha, \beta}(q) = \sum_x e^{-iq\cdot x} \left( \frac{\langle \Psi | \hat{o}^\alpha_x \hat{o}^\beta_0 | \Psi \rangle}{\langle \Psi | \Psi \rangle} - \frac{\langle \Psi | \hat{o}^\alpha_x | \Psi \rangle \langle \Psi | \hat{o}^\beta_0 | \Psi \rangle}{\langle \Psi | \Psi \rangle^2} \right)$$
2. Kernel Extraction:
   Conserved operators $\hat{H} = \sum_x e^{iq\cdot x} \hat{h}_x$ correspond to the kernel (null space) of the symmetric part of $S$. If $\hat{H}|\Psi\rangle = \lambda|\Psi\rangle$, then the eigenvalue of $S$ associated with the vectorized coefficients of $\hat{H}$ is zero (representing zero variance/fluctuation).
3. Quantum Geometry Connection:
   The paper establishes that the static structure factor matrix is proportional to the quantum metric tensor (fidelity susceptibility) of the manifold of states deformed by the operators. Conserved operators correspond to directions of vanishing fidelity susceptibility.

B. Computational Implementation: Generating Function Method

To evaluate $S$ for multi-site operators efficiently in 2D, the authors utilize and generalize the generating function method:

1. Generating Function Definition:
   Instead of summing infinite series directly, they define a generating function state:
   $$\langle G_\alpha(\mu, q) | = \langle \Psi | \prod_x (1 + \mu e^{-iq\cdot x} \hat{o}^\alpha_x)$$
   where $\mu$ is a scalar parameter.
2. Differentiation:
   The static structure factor is obtained by differentiating the expectation value with respect to $\mu$ at $\mu=0$:
   $$S^{\alpha, \beta}(q) = \left. \frac{d}{d\mu} \frac{\langle G_\alpha(\mu, q) | \hat{o}^\beta_0 | \Psi \rangle}{\langle G_\alpha(\mu, q) | \Psi \rangle} \right|_{\mu=0}$$
3. Tensor Network Construction:
   • The product term creates an infinite Projected Entangled Pair Operator (iPEPO).
   • The authors show that for multi-site operators, this iPEPO can be constructed with a manageable bond dimension $D'$.
   • Differentiation Strategy: They found that Automatic Differentiation (AD) is unstable for multi-site operators due to zero singular values in the iPEPO tensors at $\mu=0$. Instead, they employ a central 5-point finite difference formula to achieve high precision.
4. Handling Approximations:
   The method uses CTMRG to contract the resulting tensor networks. The authors demonstrate that despite CTMRG errors (controlled by environment bond dimension $\chi$), the lowest eigenvalues of $S$ remain robust enough to identify conserved operators.

3. Key Contributions

• Generalization to 2D: Successfully extends Hamiltonian learning from 1D MPS to 2D iPEPS, overcoming the complexity of multi-site operator evaluation.
• Robustness to Approximation: Proves that conserved operators can be extracted even when the input state is a variational approximation or when the contraction scheme (CTMRG) introduces errors.
• Beyond Frustration-Free: Unlike standard parent Hamiltonian constructions which often yield frustration-free Hamiltonians (where the state is the ground state), this method can identify non-frustration-free operators where the state is an excited eigenstate.
• Improved Locality: The extracted Hamiltonians often have better locality (fewer sites involved) compared to standard constructions.

4. Key Results and Benchmarks

The authors validate the method on several distinct physical systems:

A. 2D AKLT State (Exact Benchmark)

• System: Spin-2 Affleck-Kennedy-Lieb-Tasaki state on a square lattice.
• Result: The method successfully recovered the exact SO(3) symmetry generators (1-site) and identified 81 non-trivial frustration-free 2-site parent Hamiltonians (projectors onto the fusion channel 4).
• Significance: Confirms the method works perfectly for exact PEPS with finite correlation length.

B. 2D Quantum XX Model (Variational Benchmark)

• System: Spin-1/2 XX model in a staggered Z-field, optimized via variational iPEPS.
• Result:
  • Recovered the U(1) symmetry generator ($\sum Z_i$) even in the ferromagnetic phase where symmetry is spontaneously broken.
  • Extracted the non-frustration-free parent Hamiltonian $H(J)$ with high precision (matching coefficients to the 4th decimal place) despite the input being a variational approximation.
• Significance: Demonstrates applicability to variational states and systems with spontaneous symmetry breaking.

C. Short-Range RVB State (Critical System)

• System: Resonating Valence Bond (RVB) state on a square lattice (gapless, divergent correlation length).
• Result: Extracted an approximate 4-site-plaquette local Hamiltonian (Eq. 6) containing Heisenberg terms and four-spin interactions.
  • The RVB state is shown to be a good approximate eigenstate (low variance).
  • The energy expectation value of the RVB state in this Hamiltonian is nearly identical to the ground state energy of the Hamiltonian itself.
• Significance: Shows the method works for critical systems with divergent correlation lengths, finding a local parent Hamiltonian that is more efficient than standard 8-site constructions.

D. Deformed Ising/Toric Code (Quantum Many-Body Scars)

• System: A family of deformed Ising wavefunctions $|\Psi_{Ising}(\beta)\rangle$ and their dual deformed Toric Code states.
• Result: Found a Hamiltonian for which these states are excited eigenstates with zero energy (lying in the middle of the spectrum), rather than ground states.
• Significance: This identifies a potential realization of Quantum Many-Body Scars (QMBS), where low-entanglement states exist as excited states in non-integrable models. The Hamiltonian commutes with the deformation parameter, making the entire family of states eigenstates.

5. Significance and Outlook

• Data-Driven Physics: Provides a systematic, automated way to discover Hamiltonians and symmetries directly from wavefunction ansatzes or experimental data (via structure factors), reducing reliance on physical intuition.
• Scalability: The computational cost scales polynomially with the bond dimension, making it feasible for large systems.
• Future Directions:
  • Applying the method to extract emergent continuous symmetries at deconfined quantum critical points.
  • Generalizing to extract Hamiltonians from PEPOs representing Gibbs states at finite temperatures.
  • Tightening the theoretical bounds on learning complexity for tensor network states.

In conclusion, this work establishes a robust framework for "reverse-engineering" quantum dynamics from static tensor network states, bridging the gap between variational wavefunction ansatzes and the physical Hamiltonians that govern them, even in the presence of approximation errors and criticality.