Undecidability of the Spectral Gap (short version) (1502.04135v3)

Abstract: The spectral gap - the energy difference between the ground state and first excited state - is central to quantum many-body physics. Many challenging open problems, such as the Haldane conjecture, existence of gapped topological spin liquid phases, or the Yang-Mills gap conjecture, concern spectral gaps. These and other problems are particular cases of the general spectral gap problem: given a quantum many-body Hamiltonian, is it gapped or gapless? Here we prove that this is an undecidable problem. We construct families of quantum spin systems on a 2D lattice with translationally-invariant, nearest-neighbour interactions for which the spectral gap problem is undecidable. This result extends to undecidability of other low energy properties, such as existence of algebraically decaying ground-state correlations. The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phase-estimation algorithm followed by a universal Turing Machine. The spectral gap depends on the outcome of the corresponding Halting Problem. Our result implies that there exists no algorithm to determine whether an arbitrary model is gapped or gapless. It also implies that there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics.

• The paper establishes the undecidability of determining the spectral gap in 2D quantum many-body Hamiltonians by linking it to the Halting Problem.
• It constructs specific Hamiltonians using quantum phase estimation, quantum Turing machines, and aperiodic tilings to encode computation into the ground state energy.
• The findings highlight fundamental limits for algorithmic and numerical methods in simulating quantum systems, revealing new theoretical boundaries in condensed matter physics.

The paper "Undecidability of the Spectral Gap" (1502.04135) proves that determining whether a given quantum many-body Hamiltonian on a 2D lattice has a spectral gap above the ground state in the thermodynamic limit is an undecidable problem. This means there cannot exist a general algorithm that takes a description of the local interactions as input and outputs whether the system is gapped or gapless. Furthermore, for certain specific Hamiltonians, the presence or absence of a gap may be independent of the standard axioms of mathematics.

The spectral gap is a fundamental property in condensed matter physics, distinguishing gapped systems (with non-critical behavior, like short-range correlations and massive excitations) from gapless systems (exhibiting criticality, long-range correlations, and continuous spectrum above the ground state). Many significant open problems in physics, such as the Haldane conjecture for 1D integer spin chains, the existence of gapped spin liquid phases, and the Yang-Mills mass gap problem, are specific instances of this general question.

The proof of undecidability relies on relating the spectral gap problem to the Halting Problem for a Universal Turing Machine (UTM), which is known to be undecidable. The core idea is to construct a family of Hamiltonians such that the presence of a spectral gap is directly equivalent to whether a specific UTM halts on a particular input.

The construction proceeds in several steps:

1. Relating Spectral Gap to Ground State Energy Density: The authors show how to construct a Hamiltonian $H(\varphi)$ whose spectral gap behavior is determined by the ground state energy density of another Hamiltonian $H_u(\varphi)$. They use a construction involving a local Hilbert space which is a tensor product of Hilbert spaces for $H_u$, a known gapless Hamiltonian $H_d$ with zero ground state energy, and an auxiliary state $\ket{0}$. The local interaction $h(\varphi)^{(i,j)}$ between sites $i$ and $j$ is roughly structured as:

   $h(\varphi)^{(i,j)} \approx \proj{0}^{(i)} \otimes (-\proj{0})^{(j)} + h_{u}^{(i,j)}(\varphi)\otimes \id_d^{(i,j)} + \id_{u}^{(i,j)}\otimes h_d^{(i,j)}$

   where $\id_u, \id_d$ are identity operators on the respective Hilbert spaces. The spectrum of the resulting Hamiltonian $H(\varphi)$ is shown to contain $$\{0\} \cup \{\text{spectrum of } H_u(\varphi) + \text{spectrum of } H_d\} \cup S$$, where $S \ge 1$. Since $H_d$ is gapless with zero ground state energy, if $H_u(\varphi)$ has a strictly positive ground state energy density (meaning its ground state energy $\lambda_0(H_u)$ diverges to $+\infty$ with system size $L$), then $H(\varphi)$ will have a gap $\ge 1$ above its lowest eigenvalue (which is fixed at 0 by the $\proj{0}$ terms). If $H_u(\varphi)$ has a ground state energy density that tends to 0 from below ($\lambda_0(H_u) \le 0$ in the thermodynamic limit), then the spectrum of $H_u(\varphi) + H_d$ will extend down to or below zero, and combined with the continuous spectrum of $H_d$, $H(\varphi)$ will be gapless.

2. Encoding Computation in Ground States: The next step is to construct a Hamiltonian $H_u(\varphi)$ whose ground state energy density depends on the output of a computation, specifically, whether a UTM halts. This is achieved by building on the concept of encoding a computation history in the ground state of a Hamiltonian, pioneered by Feynman and developed by Kitaev and others. For a Quantum Turing Machine (QTM), a 1D translationally-invariant Hamiltonian can be constructed whose ground state is a superposition of the computation steps: $$\frac{1}{\sqrt{T}}\sum_{t=0}^{T-1}\ket{t}\ket{\psi_t}$$, where $\ket{t}$ encodes time and $\ket{\psi_t}$ is the state of the QTM at time $t$.
3. Encoding the Input via Quantum Phase Estimation: To encode an arbitrary input $n$ for the UTM into the fixed parameters of a translationally-invariant Hamiltonian, the authors utilize Quantum Phase Estimation (QPE). A QTM $P_n$ is designed that takes an initial state encoding a number $N=2^x-1 \geq n$ and performs QPE on a unitary operator $U$ whose phase $\varphi(n)$ is determined by the binary digits of $n$. If $x$ is large enough, QPE outputs the binary representation of $n$ exactly. $P_n$ then writes $n$ onto the tape of a UTM. The entire computation of $P_n$ followed by the UTM is encoded in a history state Hamiltonian. The phase $e^{2\pi i \varphi(n)}$ and other terms related to the QPE become matrix elements in the local Hamiltonian. An on-site term penalizing the halting state is added to the quantum layer. In a simple 1D history state Hamiltonian, this penalty energy per site tends to zero with system size, resulting in zero energy density whether the machine halts or not.
4. Utilizing 2D Aperiodic Tilings: To make the ground state energy density non-zero only if the UTM halts, the second spatial dimension is used in conjunction with aperiodic Wang tilings, specifically the Robinson tiling. The Robinson tiling enforces a recursive pattern of squares of sizes $4^k$ across the 2D lattice. A 2D Hamiltonian is constructed with two layers: a classical layer enforcing the Robinson tiling and a quantum layer. The quantum layer effectively places 1D QTM history state Hamiltonians along the edges ("segments") of the squares in the tiling. Each segment runs the same QTM (encoding $P_n$ followed by the UTM on input $n$). The size of the square determines the effective tape length/runtime available to the QTM on that segment.
   • If the UTM does not halt on input $n$, QTMs on segments larger than the size needed for $n$ will not halt and contribute zero energy to the quantum layer. Segments smaller than this size contribute some fixed energy. A negative on-site bias $-\alpha(n)\id$ is added to cancel these smaller contributions, making the overall ground state energy density tend to 0 from below.
   • If the UTM halts on input $n$, QTMs on all segments large enough to simulate the halting computation will halt, contributing a small, positive energy penalty per segment. The Robinson tiling ensures there are quadratically many such segments with increasing system size. These positive contributions sum up, leading to a strictly positive ground state energy density that diverges with system size. The robustness of the tiling ensures this behavior cannot be avoided by introducing defects.

This construction results in a family of 2D translationally-invariant, nearest-neighbor Hamiltonians $H(n)$ (indexed by the input $n$ to the UTM) such that $H(n)$ is gapped if and only if the UTM halts on input $n$. Since the Halting Problem is undecidable, the spectral gap problem for this family of Hamiltonians is also undecidable.

Practical Implications and Considerations:

• Limits of Algorithms: The result implies that no single algorithm can solve the spectral gap problem for all 2D lattice Hamiltonians of this type. While specific models (like Ising, AKLT, some Heisenberg chains) might be solvable, there is no general procedure.
• Limits of Numerical Methods: Standard numerical techniques like finite-size scaling, which extrapolate thermodynamic limit behavior from finite systems, are fundamentally limited for the constructed models. The size at which a system transitions from appearing gapless to gapped (or vice versa) can be arbitrarily large and uncomputable. This means simulations on very large but finite lattices might show gapless behavior, while the true infinite-system limit is gapped, or vice versa, with the switch happening at a system size beyond computational reach.
• New Physical Phenomena: The models suggest the possibility of a new type of "phase transition" driven purely by system size, occurring at uncomputably large scales. They also point to extreme instability where minute changes in parameters could induce uncomputably many gapped/gapless transitions.
• Model Specificity: The constructed Hamiltonians are highly artificial. They require specific interactions and potentially large local Hilbert spaces. It is an open question whether this undecidability extends to more "natural" or simpler Hamiltonians (e.g., with small local Hilbert space dimension).
• Dimensionality: The proof relies on the 2D nature of the tiling. Undecidability of the spectral gap in 1D remains an open problem, although many related problems (like the 1D Local Hamiltonian problem) are known to be QMA-complete.
• Implementation Complexity: Even if one could construct such a Hamiltonian in a lab, determining its gap experimentally would likely involve measuring properties of very large systems, facing the same uncomputability issues encountered in numerical simulation. The state preparation and measurement for the complex ground states encoding computation histories would be extremely challenging.

In summary, this paper delivers a profound theoretical result about the fundamental limits of determining spectral properties in quantum many-body systems, highlighting deep connections between computational theory and condensed matter physics. While the constructed models are theoretical, they demonstrate limitations inherent in any general approach to the spectral gap problem, impacting both theoretical analysis and practical simulation strategies.