🔢 mathematics

Spectral gap of Lee-Yang Hamiltonians

This paper proves that Lee-Yang Hamiltonians under a uniform Z-field possess a spectral gap of at least h/4 independent of system size and coupling strengths, a result derived from partition function zero-freeness that enables a polynomial-time quantum algorithm for computing their ground state energy.

Original authors: Chaithanya Rayudu, Jun Takahashi

Published 2026-07-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Chaithanya Rayudu, Jun Takahashi

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 a quantum computer as a giant, wobbly tower of blocks. Each block is a tiny magnet (a qubit) that can point up or down. The way these magnets push and pull on each other is described by a "Hamiltonian," which is just a fancy math recipe for the tower's energy. Usually, figuring out the lowest energy state (the ground state) of such a tower is a nightmare. It's like trying to find the single, perfect arrangement of blocks in a dark room where the rules change every second. For many of these towers, we don't even know if a "gap" exists—a safety buffer of energy that keeps the tower stable and prevents it from collapsing into chaos.

But in this paper, authors Chaithanya Rayudu and Jun Takahashi found a special class of these quantum towers, which they call Lee-Yang Hamiltonians. They proved something incredibly strong: if you add a uniform magnetic field (a gentle, steady wind blowing on all the blocks) to these specific towers, the system becomes rock-solid.

The Big Discovery: The "Safety Buffer"
The main finding is that for these Lee-Yang towers, no matter how big the tower is or how strong the magnets inside are, adding a field of strength $h$ guarantees a spectral gap of at least $h/4$ .

Think of the spectral gap as a moat around a castle. If the moat is wide enough (at least $h/4$ ), the castle (the ground state) is safe from invaders (thermal noise or small errors). The authors proved that this moat exists and is wide enough to keep the system stable, even if the castle is huge. This is a big deal because, for most quantum systems, proving such a gap exists is notoriously difficult—so hard that some famous guesses about it have remained unsolved for 40 years.

How They Did It: The "Zero-Free" Magic Trick
How did they prove this? They didn't just guess or run a simulation; they used a rigorous mathematical proof based on a classic idea called the Lee-Yang theorem.

Imagine the tower's behavior as a complex map with many paths. Usually, this map has "holes" or "zeros" where the math breaks down, making it impossible to predict what happens. The Lee-Yang theorem says that for these special Hamiltonians, if you look at the map in a specific way (the complex magnetic field plane), the zeros of the partition function are confined to the imaginary axis. This means that in the region surrounding the axis—specifically the unit polydisc—the map is completely "zero-free." There are no holes in this zone.

The authors used this "zero-free" property like a magic lens. They showed that because there are no holes in this specific region, the connections between different parts of the tower (called correlations) must fade away very quickly as you move through "imaginary time" (a mathematical way of looking at the system's history). They proved that these connections decay exponentially, like a sound fading out in a large hall. Because the connections die out so fast, the system must have that wide safety gap ( $h/4$ ) we talked about.

What This Means for Computers
This isn't just a math puzzle; it's a roadmap for building better quantum computers.

The Algorithm: The authors showed that because of this guaranteed gap, you can use a technique called "adiabatic quantum computation" to find the ground state energy of these systems efficiently. Imagine slowly turning a dial to guide the tower from a simple, easy-to-build state into the complex state you want. Because the gap is always there (at least $h/4$ ), the tower won't stumble or fall over during the journey.
The Speed: This means a quantum computer can solve the "ground state energy problem" for these Lee-Yang Hamiltonians in polynomial time. In plain English: it's fast. It's not an impossible task that takes forever; it's a task a computer can finish in a reasonable amount of time.
The "Sign Problem" Win: Many quantum simulations get stuck because of something called the "sign problem," which makes calculations explode in complexity. The authors note that while some of these Hamiltonians are easy for classical computers (because they don't have the sign problem), others are hard for classical computers but easy for quantum computers. This proves that quantum computers have a real advantage for a specific, important class of problems that classical computers might struggle with.

What They Didn't Claim
It's important to know what this paper doesn't say.

They didn't say all quantum systems are easy. This only works for the specific "Lee-Yang" class.
They didn't say they built a physical machine. This is a theoretical proof.
They didn't claim to solve the "Haldane conjecture" (a famous unsolved problem about integer spin chains) or prove that every Hamiltonian has a gap. They only proved it for this specific, well-defined family.

The Bottom Line
Rayudu and Takahashi took a difficult, abstract problem about quantum stability and solved it with a clever mathematical trick. They proved that for a broad class of quantum systems, a simple magnetic field creates a guaranteed safety buffer ( $h/4$ ) that keeps the system stable and makes it easy for a quantum computer to find the lowest energy state. It's a solid, proven step forward in understanding how to tame the wild world of quantum mechanics.

Technical Summary: Spectral Gap of Lee-Yang Hamiltonians

Problem Statement
The paper addresses the fundamental challenge of rigorously proving the existence of a spectral gap for many-body quantum Hamiltonians. While the spectral gap is crucial for determining low-temperature physical properties (such as exponential decay of correlations and phase stability) and enabling efficient adiabatic quantum computation, proving its existence is generally difficult and often undecidable for translation-invariant systems. Specifically, the authors focus on a broad class of Hamiltonians known as Lee-Yang Hamiltonians, which satisfy the Lee-Yang circle theorem (ensuring partition function zeros lie on the imaginary axis). The central question is whether these Hamiltonians, when subjected to a uniform magnetic field $h$ , possess a spectral gap that is uniform in system size and coupling strengths.

Methodology
The authors employ a multi-step analytical approach connecting the Lee-Yang theorem to correlation decay and spectral gaps:

Lee-Yang Zero-Freeness: The proof begins with the Lee-Yang circle theorem applied to the Trotterized partition function $Z_\epsilon$ . The authors utilize a version of the theorem that retains all variables from all Trotter layers, establishing that $Z_\epsilon$ is zero-free on the open unit polydisc. This implies the analyticity of $\log Z_\epsilon$ in this region.
Correlation Decay via Analyticity: The core technical innovation involves proving the exponential decay of imaginary-time correlations for products of $Z$ $Z$ -operators ( $Z_S = \prod_{i \in S} Z_i$ $Z_{S} = \prod_{i \in S} Z_{i}$ ).
- The authors express the Trotterized correlation function $f_{S,\epsilon}(k)$ in terms of the partition function evaluated at reflected points in the complex plane.
- They utilize a support lemma for the Taylor coefficients of $\log Z_\epsilon$ , demonstrating that non-zero coefficients correspond to "connected clusters" in the imaginary-time direction. This implies that monomials involving variables from widely separated layers must have high degrees.
- Combining this support property with the analyticity of $\log Z_\epsilon$ yields an exponential decay bound for the Trotterized correlator.
Continuum Limit and Error Control: To extend the result from the Trotterized system to the exact continuum limit, the authors introduce two critical steps to handle the $(\epsilon h)^{-1}$ $(ϵ h)^{- 1}$ prefactor that typically diverges as the Trotter step $\epsilon \to 0$ $ϵ \to 0$ :
- An additive error bound between the Trotterized and exact correlation functions.
- An adaptive choice of the Trotter step size $\epsilon$ , which shrinks exponentially with the time separation in the correlator. This ensures the prefactor growth is offset by the exponential decay, preserving the decay rate in the exact limit.
Spectral Gap Derivation: Finally, the authors relate the decay of the two-point correlation function at imaginary time $t = \beta/2$ to the spectral properties of the Hamiltonian. By taking the limit $\beta \to \infty$ , they show that the exponential decay of correlations implies a non-degenerate ground state and a lower bound on the spectral gap.

Key Contributions and Results

Uniform Spectral Gap: The main result (Theorem 11) proves that for any Lee-Yang Hamiltonian $H$ on $n$ qubits, the ground state of the field-perturbed Hamiltonian $H_h = H - h \sum_i Z_i$ is non-degenerate. Furthermore, the spectral gap above the ground state is at least $h/4$ . Crucially, this bound is independent of the system size $n$ and the coupling strengths of the Hamiltonian.
Correlation Decay: The paper establishes that connected correlators of diagonal operators decay exponentially in imaginary time with a rate proportional to $h/2$ (Theorem 10), specifically $|f_S(t)| \leq K e^{-\frac{h}{2} \min(t, \beta-t)}$ .
Algorithmic Implications: The existence of this uniform gap implies that the ground state of Lee-Yang Hamiltonians can be prepared efficiently via adiabatic evolution. Consequently, the authors place the ground-state energy problem for these Hamiltonians (including the bipartite quantum Max-Cut problem and the EPR Hamiltonian) in the complexity class BQP (Bounded-error Quantum Polynomial time). This improves upon previous StoqMA upper bounds.
Quantum Advantage Candidates: The result specifically addresses "phase-shifted EPR Hamiltonians," a family previously proposed as candidates for quantum advantage in ground-state energy estimation. The paper demonstrates that these specific Hamiltonians can be efficiently simulated on a quantum computer, despite the lack of known classical algorithms or sign-problem-free Monte Carlo methods for them.

Significance and Claims
The paper claims to provide a rigorous proof of a uniform spectral gap for a broad class of interacting quantum systems, a result that is notoriously difficult to obtain. By linking the Lee-Yang theorem (a property of partition function zeros) directly to the spectral gap via correlation decay, the work offers a new pathway for analyzing quantum many-body systems.

The authors emphasize that their result resolves an open question regarding the complexity of the bipartite quantum Max-Cut problem, placing it in BQP. They note that while the Lee-Yang class includes Hamiltonians amenable to sign-problem-free simulations (suggesting potential containment in BPP), it also contains families (like the phase-shifted EPR Hamiltonians) where no such classical simplification is known. For these cases, the paper provides the first evidence that efficient quantum algorithms exist for ground state preparation and energy estimation, contrasting with the current lack of candidate classical algorithms.

The work relies on the zero-freeness of the partition function as a tool to prove exponential decay, extending classical results by Penrose, Lebowitz, and others to the quantum regime with imaginary-time evolution. The authors explicitly state that their proof does not rely on the continuum limit of the Trotterization in the traditional sense but rather uses the discrete structure to derive bounds that survive the limit through adaptive step sizing.

Technical Summary: Spectral Gap of Lee-Yang Hamiltonians

More like this