Undecidability of the Spectral Gap in One Dimension (1810.01858v2)

Abstract: The spectral gap problem - determining whether the energy spectrum of a system has an energy gap above ground state, or if there is a continuous range of low-energy excitations - pervades quantum many-body physics. Recently, this important problem was shown to be undecidable for quantum spin systems in two (or more) spatial dimensions: there exists no algorithm that determines in general whether a system is gapped or gapless, a result which has many unexpected consequences for the physics of such systems. However, there are many indications that one dimensional spin systems are simpler than their higher-dimensional counterparts: for example, they cannot have thermal phase transitions or topological order, and there exist highly-effective numerical algorithms such as DMRG - and even provably polynomial-time ones - for gapped 1D systems, exploiting the fact that such systems obey an entropy area-law. Furthermore, the spectral gap undecidability construction crucially relied on aperiodic tilings, which are not possible in 1D. So does the spectral gap problem become decidable in 1D? In this paper we prove this is not the case, by constructing a family of 1D spin chains with translationally-invariant nearest neighbour interactions for which no algorithm can determine the presence of a spectral gap. This not only proves that the spectral gap of 1D systems is just as intractable as in higher dimensions, but also predicts the existence of qualitatively new types of complex physics in 1D spin chains. In particular, it implies there are 1D systems with constant spectral gap and non-degenerate classical ground state for all systems sizes up to an uncomputably large size, whereupon they switch to a gapless behaviour with dense spectrum.

• The paper demonstrates that determining a non-zero spectral gap in 1D quantum spin chains is undecidable by reducing the problem to the Halting Problem.
• It introduces a novel Marker Hamiltonian combined with an augmented quantum Turing machine to simulate quantum phase estimation within a one-dimensional framework.
• The findings imply that even efficient numerical methods may fail to predict long-range behavior, as the ground state energy density in 1D systems is fundamentally uncomputable.

The paper "Undecidability of the Spectral Gap in One Dimension" (1810.01858) addresses a fundamental question in quantum many-body physics: whether it is possible to algorithmically determine if a quantum system has a non-zero energy gap above its ground state in the thermodynamic limit. The spectral gap is a crucial property determining many physical characteristics, including the decay of correlations, the stability of topological phases, and the applicability of certain efficient simulation algorithms.

Previously, it was shown that this problem is undecidable for quantum spin systems in two or more spatial dimensions (Ota et al., 2015). This result relied heavily on embedding classical aperiodic tiling problems, which are known to be undecidable in 2D, into the Hamiltonian's ground state properties. One-dimensional systems, however, are often considered simpler than their higher-dimensional counterparts. They cannot host thermal phase transitions or topological order, and powerful numerical methods like DMRG exist and are even provably efficient for gapped 1D systems due to the area law for entanglement entropy. Furthermore, aperiodic tilings, key to the 2D proof, are not possible in 1D in the same way (classical 1D tiling is decidable). These factors led to the expectation that the spectral gap problem might be decidable in 1D.

This paper proves that, contrary to these expectations, the spectral gap is also undecidable for 1D quantum spin chains. The authors construct a family of translationally-invariant nearest-neighbor Hamiltonians on a chain of qudits (quantum systems with local dimension $d$) for which determining the presence of a spectral gap is equivalent to solving the Halting Problem for a universal Turing machine (UTM), which is known to be undecidable.

The core challenge in 1D is the absence of complex, fractal-like classical tiling structures that can naturally embed computations across all length scales, as used in the 2D proof. The paper overcomes this by introducing a novel construction involving a "Marker Hamiltonian" and an embedded Quantum Turing Machine (QTM).

The construction involves combining several components:

1. Marker Hamiltonian ($H'$): This local Hamiltonian, acting on the spin chain, creates a structure analogous to the 2D tiling. It introduces special spin states called "markers" that can partition the chain into segments. The Hamiltonian provides an energy bonus (negative energy contribution) to states where segments are bounded by markers. Crucially, this bonus is designed to decrease rapidly (e.g., doubly exponentially) as the length of a segment increases. This encourages the formation of shorter segments to minimize energy, provided other parts of the Hamiltonian don't penalize this.
2. Augmented Phase Estimation QTM: A QTM is designed to perform quantum phase estimation (QPE) on a gate that encodes an input parameter $\phi$ (derived from an integer input $\iota$) for a classical UTM. This QTM runs on the "tape" provided by the segments created by the Marker Hamiltonian. A key augmentation is added to detect if the QPE successfully extracts the full binary expansion of $\phi$ or if it is truncated due to insufficient tape length (segment length). If the QPE is truncated, a large energy penalty is inflicted on the corresponding history state.
3. Encoding the Halting Problem: The QTM (performing QPE and feeding the result to a universal classical UTM) is embedded into a history state Hamiltonian ($H_C$). This Hamiltonian's ground state represents a superposition over the computation's history. Unlike the 2D case, this construction penalizes the QTM/UTM computation if it fails to halt within the available tape segment. Failing to halt can mean either entering a non-terminating loop (detected by the clock running out of space) or running out of tape space before completing the computation. A local penalty term $P$ is added to $H_C$ to inflict this energy cost.
4. Combining Components: The total Hamiltonian $H$ is constructed by summing the Marker Hamiltonian (scaled by a small factor $\mu$) and the history state Hamiltonian with penalties: $H = \mu H' + H_C$. The Marker Hamiltonian provides a bonus for short segments, while $H_C$ penalizes computations that don't halt within those segments.
   • If the UTM does not halt on input $\iota$, any segment, regardless of length, will result in the QTM/UTM computation eventually failing to halt (either running out of tape or looping), incurring a penalty. This penalty term dominates the bonus from the Marker Hamiltonian, resulting in a net positive ground state energy contribution from each segment. The lowest energy configuration corresponds to the longest possible segment (the whole chain), and the ground state energy $\lambda_{min}(H_N) \ge 0$ for all $N$.
   • If the UTM does halt on input $\iota$ after a certain time $T_{halt}$ and using $w_{halt}$ tape, there exists a minimum segment length ($w_{halt}$) above which the computation will successfully halt and avoid the penalty. Since the Marker Hamiltonian gives a bonus that decreases with segment length, the minimum energy configuration is achieved by partitioning the chain into as many segments of length $w_{halt}$ as possible. Each such segment contributes a finite, negative energy bonus ($-\Omega(1/4^{f(w_{halt})})$). As the system size $N$ increases, the total ground state energy diverges to $-\infty$ proportional to $-\lfloor N/w_{halt} \rfloor$.
5. Final Spectral Gap Construction: This combined Hamiltonian $H$ (with its spectrum shifted up such that $\lambda_{min}(H_N) \ge 1$ if the TM doesn't halt, and $\lambda_{min}(H_N) \to -\infty$ if it halts) is then combined with a trivial gapped Hamiltonian ($trivial$, with ground state energy 0 and gap $\ge 1$) and a Hamiltonian with a dense spectrum ($dense$, in $[0, \infty)$). A guard Hamiltonian is added to prevent low-energy states from being superpositions across the Hilbert spaces of $H$ and $trivial$. The resulting total Hamiltonian $H_{tot}$ is defined on a larger Hilbert space and has the property that its low-energy spectrum is determined by the minimum of the ground state energies of $H$ (plus $dense$) and $trivial$.
   • If the UTM does not halt, $\lambda_{min}(H_N) \ge 1$. The lowest energy state is the ground state of $trivial$ at energy 0, and there is a spectral gap of at least 1.
   • If the UTM halts, $\lambda_{min}(H_N) \to -\infty$. The dense spectrum of $dense$ is effectively pulled down by $H$, resulting in a dense spectrum near the ground state, and the system is gapless.

Since the mapping from the integer input $\iota$ to the Halting Problem outcome is a reduction, and the Halting Problem is undecidable, determining whether the constructed Hamiltonian is gapped or gapless based on $\iota$ is also undecidable. The authors provide explicit forms for the local interactions, showing they are translationally invariant, nearest-neighbor, and depend on $\iota$ only through numerical factors involving its binary expansion. The result holds even for arbitrarily small perturbations of a classical Hamiltonian, highlighting the instability of the gapped phase to quantum effects.

The practical implications are significant. It means there can be no general algorithm to predict the thermodynamic phase of such systems based on their microscopic description. Furthermore, it suggests that numerical methods, even those that are provably efficient for gapped 1D systems (like DMRG), might fail to predict the correct long-range behavior or entanglement properties for all 1D systems, even if they appear gapped for finite, albeit uncomputably large, system sizes. The construction also implies that the ground state energy density of 1D spin chains is generally uncomputable.

The paper discusses extensions, including a limited result for periodic boundary conditions and the possibility of making the $\iota$-dependence reside solely in the one-local transverse field terms. It leaves open the question of whether a dimension threshold exists below which the spectral gap problem becomes decidable, noting that the presented construction uses a large local dimension.