Hamiltonian Locality in Quantum Systems

• Hamiltonian locality is defined as the sum of interactions acting nontrivially on small subsystems, characterized by bounded support and rapid decay with distance.
• Dynamical constraints such as Lieb–Robinson bounds result in effective light cones, establishing causal limits on the propagation of information in quantum systems.
• Locality influences computational complexity and Hamiltonian learning, with k-local and quasi-local structures underpinning challenges and efficiencies in quantum modeling.

A Hamiltonian is said to be local if it is composed as a sum of terms, each of which acts nontrivially only on small subsystems, typically of bounded cardinality, or in a preferred basis, with interactions between components decaying rapidly with "distance." The notion of locality is central in quantum many-body physics, quantum information science, and computational complexity theory, underpinning both physical phenomena (such as causality and propagation bounds) and computational tractability. However, “locality” is a basis-dependent structure, and recent advances have clarified the subtleties of basis-agnostic definitions, the emergence of locality from spectra, operational property testing, implications for simulation complexity, and state-dependent locality in exotic models.

1. Formal Definitions of Locality

Tensor-product and geometric locality:

The traditional setting assumes a Hilbert space structured as a tensor product $\mathcal{H} = \bigotimes_{i=1}^n \mathcal{H}_i$. A Hamiltonian $H$ is $k$-local if it can be written as

$H = \sum_j H_j$

where each $H_j$ acts nontrivially on at most $k$ (not necessarily contiguous) subsystems $i_1, ..., i_k$ and as the identity elsewhere (0808.2117, Cubitt et al., 2013). For instance, a conventional two-dimensional spin model features

$H = \sum_{\langle i,j \rangle} h_{ij}$

with $h_{ij}$ coupling at most two spins. "Geometric" locality further restricts supports so that each $h_{ij}$ acts on nearby or adjacent sites per a prescribed interaction graph (e.g., a lattice).

Pauli-basis and matrix-representation locality:

Given a Pauli basis, an $n$-qubit operator $H$ can always be expanded as

$$H = \sum_P \alpha_P P, \qquad P \in \{I,X,Y,Z\}^{\otimes n}$$

with $|P|$ the weight (number of non-identity tensor factors). $H$ is $k$-local if $\alpha_P \neq 0 \implies |P| \leq k$ (Kallaugher et al., 10 May 2025, Bluhm et al., 5 Mar 2024).

Basis-agnostic (block-matrix) locality:

A generalized definition fixes an orthonormal basis $\{|i\rangle\}$ and partitions $H$ into "blocks" $H_\mathcal{Z}$ (submatrices indexed by finite sets $\mathcal{Z}$ of basis labels). $H$ is $\mu$-local in this basis if there exists $\mu \geq 0$ such that for every finite block $\mathcal{P}$,

$$\sum_{\mathcal{Z}: \mathcal{Z} \cap \mathcal{P} \neq \emptyset} |\mathcal{Z}| \| H_\mathcal{Z} \| e^{\mu\, \mathrm{diam}(\mathcal{Z})} \leq |\mathcal{P}|\, a_\mu$$

for some $a_\mu \geq 0$ (Koochakie et al., 2013). This covers locality in non-tensor bases (e.g., energy eigenbasis).

2. Lieb–Robinson Bounds and Dynamical Locality

Locality of the Hamiltonian constrains the causal structure of quantum evolution via Lieb–Robinson bounds. For a $\mu$-local Hamiltonian in a fixed basis, the commutator of evolved local observables decays exponentially in the separation of supports: $$\| [A^t, B] \| \leq 2 \min(|\mathcal{A}|, |\mathcal{B}|) \|A\| \|B\| e^{-\mu d(\mathcal{A}, \mathcal{B})} (e^{\langle a_\mu \rangle_t |t|} - 1)$$ where $A$ and $B$ are supported on disjoint blocks $\mathcal{A}$ and $\mathcal{B}$, and $d(\mathcal{A}, \mathcal{B})$ is their distance (Koochakie et al., 2013). This determines an effective light cone for information propagation with a maximally allowed "Lieb–Robinson velocity"

$v_{LR} = \langle a_\mu \rangle_t / \mu$

Even without a strict tensor-product decomposition, basis locality implies exponential suppression of operator spread outside a block.

Applications include:

• Exponential decay of off-diagonal propagator amplitudes,
• Fundamental limits on correlation propagation speed,
• Adiabatic evolution: small LR speed in the instantaneous energy basis yields adiabaticity conditions, with minimal run time for adiabatic quantum computing set by $v_{LR}$ and the spectral gap (Koochakie et al., 2013).

3. Locality and Computational Complexity

The $k$-local Hamiltonian problem—deciding whether a $k$-local $H$ has ground energy below $a$ or above $b$—forms the quantum analogue of CNF-SAT. The computational complexity is sharply dictated by $k$ and related structural constraints:

• For $k = 1$, the problem is in P (efficiently solvable).
• For $k = 2$, the classification (on qubits) is:
  • Diagonal in some basis: NP-complete (classical Ising),
  • Stoquastic (sign problem-free): StoqMA-complete,
  • Otherwise: QMA-complete (e.g. Heisenberg, XY) (Cubitt et al., 2013).
• For $k \geq 3$, general $k$-local Hamiltonians are QMA-complete (0808.2117, Hallgren et al., 2013).
• With succinctly described ground states, the $k=3$ (and stoquastic $k=4$) local Hamiltonian problem is MA-complete rather than QMA-complete (Waite et al., 30 Sep 2025).

For commuting local Hamiltonians, the complexity can dramatically shift. In 2-local commuting cases, the problem is in NP, and no topological order appears for $k \leq 3$ on qubits or $k=3$ on qutrits with nearly Euclidean interaction graphs; topological order only emerges for $k \geq 4$ or $d \geq 4$ (Aharonov et al., 2011, 1803.02213).

4. Testing, Learning, and Emergence of Locality

Locality testing:

Given oracle access to $U(t) = e^{-iHt}$, one may ask whether $H$ is $k$-local. If the distance is measured in the operator norm, distinguishing $k$-locality generically requires exponentially many queries in $n$, both in incoherent and coherent models; this is as hard as tomography. However, in the average-case Frobenius norm, randomized measurement protocols can test $k$-locality with polynomial resources in $1/\varepsilon$ (Bluhm et al., 5 Mar 2024). Recent advances provide algorithms with Heisenberg-limited evolution time complexity, and tight lower bounds (Kallaugher et al., 10 May 2025).

Hamiltonian learning:

While locality testing (in average-case norms) is efficient, learning an arbitrary local Hamiltonian to nontrivial accuracy remains exponentially hard (even with locality constraints), creating an exponential separation between testing and learning (Bluhm et al., 5 Mar 2024). For truly local (e.g., $k$-local) models, leveraging only local measurements suffices for unique and robust recovery of the Hamiltonian in each finite region, with sample and computational complexity scaling polynomially in region size (Bairey et al., 2018).

Emergence from spectra and chaos:

Even without a predefined tensor structure, locality can often be inferred, or even "emerges," from the spectrum. Generically, the local tensor factorization of a Hamiltonian is uniquely determined by its energy spectrum, apart from measure-zero cases supporting dualities (e.g., the Ising/Kramers–Wannier transformation) (Cotler et al., 2017). For random matrices (GOE/GUE), there always exists a basis in which the Hamiltonian is approximately 2-local up to exponentially small errors; this effect provides a mechanism for the dynamical emergence of locality from chaos in high-dimensional nonlocal models (Loizeau et al., 2023).

5. Variants and Subtleties: Quasi-locality, State-Dependence, and Special Models

Quasi-local and state-dependent locality:

In field-theoretic or deformed models, such as $T \overline T$-deformed CFTs or negativity Hamiltonians coding the entanglement structure of mixed states, the Hamiltonian may acquire a "quasi-local" structure: being local integrals up to mild nonlocal corrections (such as terms coupling only mirrored points across a boundary) (Monten et al., 9 Nov 2024, Murciano et al., 2022). All finite-order perturbative corrections in $T\overline T$ preserve such quasi-locality.

Relatively local Hamiltonians:

Certain models (motivated by background-independent quantum gravity) have Hamiltonians that are nonlocal in their bare form but "inherit" a local interaction structure from the entanglement pattern of the state: dynamics and geometry are emergent and state-dependent, with coordinate velocities of entanglement growth and operator spread arbitrarily small for nearly unentangled initial states (Lee, 2018). In these cases, locality is not an operator property but a property of the operator-state pair.

6. Practical and Physical Consequences

• Quantum battery models: The maximal charging power achievable by quantum batteries is sharply bounded by the locality of both the battery and charger Hamiltonians, together with the per-site energy capacity. Interactions extending over $k$ sites (for the charger) and $q$ sites (for the battery) combine multiplicatively to enhance the bound, but only subject to $g$-extensivity: the limitation that each site can only store/buffer order-one energy (Sarkar et al., 21 Jan 2025).
• Quantum computation: Geometric locality sets minimal constraints in Hamiltonian-based quantum computation. Universal Hamiltonian quantum computers can be constructed with $k=2, d=8$ (non-translationally invariant), $k=3, d=5$ (with nontrivial gadgetry), and so on, mapping out a trade-off between interaction locality and on-site dimension (Wei et al., 2015).
• Entanglement Hamiltonians and negativity: The operator content governing the spectrum of reduced states (and hence correlations, negativity, etc.) is regulated by the locality properties in the underlying Hamiltonian, with corrections (e.g., quasi-locality) controlling the deviation from area-law behavior (Murciano et al., 2022).

7. Summary Table: Key Locality Concepts

Concept	Locality Structure	Reference Example Papers
$k$-local (tensor-product)	Terms support size ≤ $k$	(Cubitt et al., 2013, 0808.2117)
$\mu$-local (block basis)	Exponential decay in chosen basis	(Koochakie et al., 2013)
Quasi-local	Local plus mild, structure-constrained nonlocal terms	(Monten et al., 9 Nov 2024, Murciano et al., 2022)
State-dependent locality	Operator's effective locality set by state	(Lee, 2018)
Emergent locality (spectral)	Unique tensor structure fixed by spectrum	(Cotler et al., 2017, Loizeau et al., 2023)

These results reveal that Hamiltonian locality is a multi-faceted, representation-dependent, and operationally testable property fundamental to physical theory, computational complexity, and the structure of quantum many-body dynamics. Robust consequences—causal bounds, computational intractability of ground state energy, tractability of learning and property testing, and energy transfer bounds—are all regulated, directly or indirectly, by the locality principle instantiated within the chosen or emergent basis.