[Paper Review] On the uncomputability of the spectral gap

Seth Lloyd|arXiv (Cornell University)|Feb 18, 2016

Quantum Computing Algorithms and Architecture2 references4 citations

TL;DR

This paper reviews the uncomputability of the spectral gap in quantum Hamiltonian systems capable of universal computation, demonstrating that the gap is undecidable due to the spectral distinction between halting (discrete, gapped) and non-halting (continuous, gapless) computations. The result arises from mapping the halting problem to the spectral properties of unitary and Hamiltonian models, with the Feynman Hamiltonian serving as a key example where the gap's computability depends on the undecidability of the halting problem.

ABSTRACT

This paper reviews the 1994 proof that the spectral gap of Hamiltonian quantum systems capable of universal computation is uncomputable.

Motivation & Objective

• To re-interpret and restate the 1994 proof of spectral gap uncomputability in modern quantum information language.
• To clarify the fundamental mechanism linking the undecidability of the halting problem to the spectral properties of quantum systems.
• To contrast the original models (Benioff, Deutsch, Feynman, Margolus) with the more recent planar, physics-like systems of Cubitt et al.
• To emphasize that the uncomputability of the spectral gap is not a new discovery but was established earlier in foundational quantum computation models.

Proposed method

• Modeling quantum computation via unitary evolution with a clock register that tracks time steps, using states $|\ell\rangle$ to represent computational time.
• Defining the unitary evolution operator $U = \sum_{\ell} U_{\ell} \otimes |\ell+1\rangle\langle\ell|$, with $U_{-\ell} = U_{\ell}^\dagger$ to allow backward evolution.
• Introducing a sign qubit to make halting computations cyclic, ensuring eigenstates exist for finite computations.
• Transforming the unitary evolution into a Hamiltonian via $H = U + U^\dagger$, which governs continuous-time dynamics.
• Analyzing the spectrum of $H$ by deriving its eigenvalues as $2\cos(2\pi k/m)$ for halting cases and $2\cos(2\pi a)$ for non-halting cases.
• Establishing that the spectral gap's existence depends on whether the computation halts, which is undecidable by the halting problem.

Experimental results

Research questions

• RQ1 Can the spectral gap of a Hamiltonian system capable of universal quantum computation be computed algorithmically?
• RQ2 How does the halting behavior of a quantum computation influence the spectral properties of its associated Hamiltonian?
• RQ3 What is the relationship between the undecidability of the halting problem and the spectral gap in quantum systems?
• RQ4 Why is the spectral gap uncomputable in the Feynman Hamiltonian model despite its physical simplicity?
• RQ5 To what extent does the uncomputability of the gap depend on the model's structure, such as the presence of a clock register?

Key findings

• The spectral gap of a Hamiltonian system capable of universal quantum computation is uncomputable because it depends on the undecidability of the halting problem.
• For halting computations, the spectrum of the unitary evolution operator $U$ is discrete with eigenvalues $e^{2\pi i k/m}$, leading to a finite spectral gap in the corresponding Hamiltonian $H = U + U^\dagger$.
• For non-halting computations, the spectrum of $U$ becomes continuous with eigenvalues $e^{2\pi i a}$ for $a \in [0,1)$, resulting in a gapless spectrum for $H$.
• The eigenstates of $H$ are identical to those of $U$, taking the form of uniform superpositions $|k,b_0\rangle = \frac{1}{\sqrt{m}} \sum_{\ell=0}^{m-1} e^{-2\pi i k\ell/m} U^\ell |b_0\rangle |0\rangle$ for halting cases and plane waves $|a,b_0\rangle = \sum_{\ell=-\infty}^{\infty} e^{-2\pi i a\ell} U^\ell |b_0\rangle |0\rangle$ for non-halting ones.
• The spectral gap of the Feynman Hamiltonian is uncomputable because determining whether the spectrum is discrete or continuous is equivalent to solving the halting problem.
• The result establishes a foundational link between quantum computation, undecidability, and spectral theory, showing that the gap cannot be determined algorithmically even in principle.