Guided Local Hamiltonian Complexity

• The Guided Local Hamiltonian Problem is defined by supplementing a local Hamiltonian instance with a guiding state that guarantees nontrivial overlap with the target eigenstate.
• This framework shifts complexity from QMA-complete to BQP-complete, enabling precise energy estimation and illuminating quantum versus classical tractability.
• Algorithmic advances such as quantum phase estimation and randomized quantum imaginary-time evolution provide practical tools and inspire classical dequantization methods.

Searching arXiv for recent and foundational papers on the Guided Local Hamiltonian Problem and closely related variants. The Guided Local Hamiltonian Problem (GLH) is a guided variant of the Local Hamiltonian problem in which the input includes, in addition to a local Hamiltonian, a state promised to have nontrivial overlap with the target low-energy eigenstate. In the formulation emphasized by recent work, the standard $k$-local Hamiltonian problem asks for the ground-state energy $E_0$ of $H=\sum_{X\in S}\lambda_X h_X$ to within additive error, whereas GLH augments the instance by a guiding state $\lvert \psi_I\rangle$ satisfying $|\langle \psi_I \mid \psi_0\rangle| \ge \gamma$, where $\lvert \psi_0\rangle$ is the ground state and $\gamma$ is a known lower bound. This extra promise changes the complexity-theoretic character of the problem: the ordinary Local Hamiltonian problem is $QMA$-complete, while GLH is $BQP$-complete, making it a natural framework for studying where quantum advantage persists and where it can be eroded by additional structure or by classical dequantization (Zhang et al., 2024).

1. Formal definition and core promise structure

GLH was recently introduced by Gharibian and Le Gall as a variant of local Hamiltonian energy estimation in which a helpful state is supplied as part of the instance (Cade et al., 2022). In the standard local Hamiltonian promise problem, one is given a Hamiltonian

$H=\sum_{i=1}^m H_i$

or, in another common notation,

$E_0$0

with each term acting nontrivially on at most $E_0$1 qubits, and one must distinguish low from high ground energy under a promised additive gap. This problem is the quantum analogue of classical constraint satisfaction and is $E_0$2-complete (Zhang et al., 2024).

The defining feature of GLH is the guiding-state input. In the basic ground-state version, the instance includes a state $E_0$3 or $E_0$4 together with the promise that it has nontrivial overlap with the ground space, typically written as

$E_0$5

where $E_0$6 projects onto the ground space. The problem is then to decide whether the smallest eigenvalue is at most $E_0$7 or at least $E_0$8, with a promise gap such as $E_0$9 in one canonical formulation (Zhang et al., 2024).

A broader formulation, often denoted Guided Local Hamiltonian Low Energy, extends the target from the ground state to the $H=\sum_{X\in S}\lambda_X h_X$0-th eigenstate. In that setting, the promise becomes

$H=\sum_{X\in S}\lambda_X h_X$1

where $H=\sum_{X\in S}\lambda_X h_X$2 projects onto the eigenspace of the $H=\sum_{X\in S}\lambda_X h_X$3-th eigenvalue $H=\sum_{X\in S}\lambda_X h_X$4, and the task is to distinguish $H=\sum_{X\in S}\lambda_X h_X$5 from $H=\sum_{X\in S}\lambda_X h_X$6. This encompasses both the ground-state case $H=\sum_{X\in S}\lambda_X h_X$7 and guided excited-state energy estimation (Cade et al., 2022).

The model is tightly linked to restricted descriptions of the guiding state. Several papers focus on “semi-classical” states, including sparse subset states

$H=\sum_{X\in S}\lambda_X h_X$8

with $H=\sum_{X\in S}\lambda_X h_X$9, as well as semi-classical encoded states obtained by applying local isometries to such sparse superpositions. These representations matter because they determine whether the guide is only quantumly useful, also classically sampleable, or strong enough to support dequantized algorithms (Cade et al., 2022).

2. Complexity-theoretic position

The central complexity-theoretic fact is that the guiding promise does not trivialize local Hamiltonian estimation. Instead, it changes the problem class from $\lvert \psi_I\rangle$0-complete to $\lvert \psi_I\rangle$1-complete in the standard guided setting, thereby isolating a regime in which a quantum computer can use the guide efficiently but worst-case classical tractability remains nontrivial (Zhang et al., 2024).

Problem family	Additional structure	Complexity status
Local Hamiltonian	No guide	$\lvert \psi_I\rangle$2-complete (Zhang et al., 2024)
Guided Local Hamiltonian	Guiding state with promised overlap	$\lvert \psi_I\rangle$3-complete (Zhang et al., 2024)
Guided Local Hamiltonian Low Energy	Guide for the $\lvert \psi_I\rangle$4-th eigenstate	$\lvert \psi_I\rangle$5-hard; $\lvert \psi_I\rangle$6-complete in stated parameter regimes (Cade et al., 2022)
Guidable LH with classically evaluatable or quantumly preparable guide	Existence of guide, not supplied as input	$\lvert \psi_I\rangle$7-complete at inverse-polynomial precision (Weggemans et al., 2023)
LH with succinct ground state	Exact ground state has amplitude-computing classical circuit	$\lvert \psi_I\rangle$8-complete (Jiang, 2023)
Stoquastic GLH	Stoquastic $\lvert \psi_I\rangle$9 and succinct guiding state	promise $|\langle \psi_I \mid \psi_0\rangle| \ge \gamma$0-hard (Waite, 30 Sep 2025)

Two refinements established in 2022 are especially important. First, $|\langle \psi_I \mid \psi_0\rangle| \ge \gamma$1-hardness persists already for $|\langle \psi_I \mid \psi_0\rangle| \ge \gamma$2-local Hamiltonians. Second, the hardness survives even when the guiding state has fidelity $|\langle \psi_I \mid \psi_0\rangle| \ge \gamma$3 with the target eigenstate, rather than merely fidelity near $|\langle \psi_I \mid \psi_0\rangle| \ge \gamma$4. The same work extends the hardness framework to excited-state estimation and to physically motivated families including non-2SLD $|\langle \psi_I \mid \psi_0\rangle| \ge \gamma$5-local Hamiltonians on a $|\langle \psi_I \mid \psi_0\rangle| \ge \gamma$6D square lattice, the antiferromagnetic Heisenberg model $|\langle \psi_I \mid \psi_0\rangle| \ge \gamma$7, and the antiferromagnetic $|\langle \psi_I \mid \psi_0\rangle| \ge \gamma$8 model $|\langle \psi_I \mid \psi_0\rangle| \ge \gamma$9 on a $\lvert \psi_0\rangle$0D triangular lattice (Cade et al., 2022).

A separate line of results shows that when the guide is no longer part of the input but is only promised to exist, the complexity shifts again. For guidable local Hamiltonian problems with classically evaluatable or efficiently quantum-preparable states, inverse-polynomial precision yields $\lvert \psi_0\rangle$1-completeness, whereas constant precision with classically evaluatable guides yields containment in $\lvert \psi_0\rangle$2 or $\lvert \psi_0\rangle$3, depending on the overlap regime (Weggemans et al., 2023). Conversely, when the promise is strengthened from mere overlap to an exact succinct classical description of a ground state, the problem drops to $\lvert \psi_0\rangle$4-complete (Jiang, 2023).

The stoquastic case does not collapse to a trivially classical regime. The guided stoquastic local Hamiltonian problem is shown to be promise $\lvert \psi_0\rangle$5-hard for $\lvert \psi_0\rangle$6-local, $\lvert \psi_0\rangle$7-local, and square-lattice stoquastic Hamiltonians, while a pinned stoquastic variant is $\lvert \psi_0\rangle$8-hard (Waite, 30 Sep 2025). This demonstrates that “stoquastic” and “guided” do not, by themselves, imply an easy classical complexity classification.

3. Quantum algorithmic frameworks

One constructive route to $\lvert \psi_0\rangle$9 containment is direct quantum phase estimation (QPE). If a guiding state can be prepared efficiently from its classical description and has inverse-polynomial overlap with the target low-energy state, QPE yields an $\gamma$0-additive energy estimate. In one formal lemma, for a guiding state $\gamma$1 satisfying $\gamma$2, QPE obtains an $\gamma$3-additive approximation to the ground-state energy with probability at least $\gamma$4, using $\gamma$5 repetitions and total cost

$\gamma$6

This argument underlies $\gamma$7 containment for several guiding-state families, including semi-classical subset states, semi-classical encoded subset states, fixed-weight states, matrix product states, and Gaussian states; Fendley states are excluded from that containment theorem because an efficient classical description and preparation procedure is not established there (Waite et al., 30 Sep 2025).

A second algorithmic framework is randomized quantum imaginary-time evolution (RQITE). Its central quantity is

$\gamma$8

As $\gamma$9 grows, the ground-state contribution becomes dominant. The energy-search procedure is organized through

$QMA$0

which behaves monotonically and drops near zero as $QMA$1 approaches the ground-state energy. A threshold $QMA$2 is then used to certify an $QMA$3-accurate estimate $QMA$4 of $QMA$5 (Zhang et al., 2024).

These two algorithmic viewpoints emphasize different resources. QPE exploits efficient state preparation plus overlap. RQITE exploits access to an imaginary-time partition-function-like observable. Both are compatible with the $QMA$6-complete status of GLH, but they expose different dequantization opportunities: QPE is naturally tied to state families with succinct preparation, whereas RQITE exposes analytic objects amenable to cluster expansion and analytic continuation (Waite et al., 30 Sep 2025).

4. Classical dequantization and accuracy regimes

A major development in 2024 was a dequantized classical algorithm for the RQITE approach that removes earlier restrictions and applies to more realistic Hamiltonians (Zhang et al., 2024). The key object is again $QMA$7, but the classical algorithm approximates it through a cluster expansion for $QMA$8. For sufficiently small $QMA$9, the exponential can be expanded in connected clusters of local Hamiltonian terms, with disconnected clusters factorizing. This yields efficient classical approximation when the guiding state is semi-classical in the sense

$BQP$0

where the $BQP$1 are product states accessible classically (Zhang et al., 2024).

The limited-accuracy regime is controlled by the threshold

$BQP$2

where $BQP$3 is the maximum degree of the interaction graph. The dequantized algorithm works when $BQP$4, which corresponds to the energy-accuracy threshold

$BQP$5

In that regime, the runtime takes the form

$BQP$6

with $BQP$7 the number of classical components in the guiding state and $BQP$8 the number of Hamiltonian terms (Zhang et al., 2024). The result shows that some apparent quantum advantage disappears once only coarse precision is demanded.

The more substantial advance concerns arbitrary constant accuracy. Rather than extending the cluster expansion directly to large $BQP$9, the dequantization uses analytic continuation. Under the stronger overlap condition

$H=\sum_{i=1}^m H_i$0

the partition function is proved to be zero-free in the right half-plane,

$H=\sum_{i=1}^m H_i$1

so $H=\sum_{i=1}^m H_i$2 is analytic there. A conformal map $H=\sum_{i=1}^m H_i$3 and complex Taylor expansion then extend the method to larger constant $H=\sum_{i=1}^m H_i$4, giving arbitrary constant precision under this stronger overlap assumption (Zhang et al., 2024).

A central practical consequence is the removal of the constant operator norm restriction $H=\sum_{i=1}^m H_i$5. Earlier dequantizations relied on such normalization, which is unrealistic when $H=\sum_{i=1}^m H_i$6 scales as $H=\sum_{i=1}^m H_i$7. The cluster-expansion and analytic-continuation framework works directly with the imaginary-time partition function and therefore applies to general local Hamiltonians with polynomially growing norm (Zhang et al., 2024). The paper is explicit, however, that this is not a general classical algorithm for all GLH instances; the classically simulable regime is identified for structured subclasses rather than for the worst case.

A closely related classical perspective appears in later work on physically motivated guiding states. There, dequantized QSVT-style arguments show classical tractability when the guide admits efficient sample-query access, the overlap is constant, and the precision regime is constant; the tractable families include semi-classical subset states, semi-classical encoded subset states, fixed-weight states, matrix product states, and Gaussian states (Waite et al., 30 Sep 2025).

5. Guiding-state families and specialized Hamiltonian classes

Much of the subsequent literature is concerned with determining which guiding states preserve hardness and which admit classical treatment. One 2025 classification isolates several physically motivated families: fixed-weight states, matrix product states, Gaussian states, and Fendley states. These families are motivated respectively by fixed excitation number sectors, low-entanglement $H=\sum_{i=1}^m H_i$8D many-body states, free-fermion or matchgate-solvable states, and generalized free-fermion-solvable states beyond Gaussian. The resulting picture is deliberately mixed: fixed-weight states, MPSs, Gaussian states, and Fendley states preserve $H=\sum_{i=1}^m H_i$9-hardness even for $E_0$00-local Hamiltonians, while $E_0$01 containment is proved for all but Fendley states because efficient classical preparation is not established there (Waite et al., 30 Sep 2025).

The same work introduces the “Goldilocks zone” for guiding states: the intersection of states that are efficiently preparable with succinct classical descriptions and states that admit efficient sample-query access. This is the regime in which rigorous quantum and classical comparisons are both meaningful. The stated message is that many physically motivated families fall in this overlap region, so the comparison between quantum advantage and dequantization is not limited to artificial sparse states (Waite et al., 30 Sep 2025).

On the Hamiltonian side, hardness is not confined to abstract circuit Hamiltonians. Guided hardness has been extended to $E_0$02-local Hamiltonians, non-2SLD families on a $E_0$03D square lattice, antiferromagnetic Heisenberg interactions, antiferromagnetic $E_0$04 interactions on a $E_0$05D triangular lattice, and excited-state energy estimation. These results show that the guiding promise does not eliminate hardness even for geometrically constrained or physically motivated interaction families (Cade et al., 2022).

The stoquastic setting gives a distinct specialization. Guided $E_0$06-local stoquastic local Hamiltonian is $E_0$07-hard for any $E_0$08, and the hardness survives locality reduction to guided $E_0$09-local stoquastic Hamiltonians and further to the square lattice. In the same paper, a guided pinned stoquastic $E_0$10-local Hamiltonian problem is shown to be $E_0$11-hard (Waite, 30 Sep 2025). This places stoquastic GLH in a complexity regime that is neither the generic $E_0$12 world nor obviously classically easy.

6. Related notions of “guidance” and broader significance

The term “guided” is used in several distinct senses across Hamiltonian complexity, and disambiguation matters. In the formal GLH problem, the guide is an explicit input state promised to overlap with the relevant eigenstate. In “guidable” local Hamiltonian problems, by contrast, the guide is not provided; only its existence is promised, and the state becomes a Merlin witness. For classically evaluatable or efficiently quantum-preparable guides, these guidable variants are $E_0$13-complete at inverse-polynomial precision, while the classically evaluatable constant-precision regime lies in $E_0$14 or $E_0$15 (Weggemans et al., 2023).

An even stronger notion is the local Hamiltonian problem with a succinct ground state. There the promise is that some exact ground state has a polynomial-size classical circuit computing amplitudes up to a global scale, and the resulting problem is $E_0$16-complete. This is strictly stronger than the standard overlap-based guiding promise and is verified using a protocol based on the fixed-node quantum Monte Carlo method and a continuous-time Markov chain construction (Jiang, 2023).

A different meaning of guidance appears in commuting local Hamiltonians. “Guided reductions” are reductions in which the prover provides a guide string telling the verifier how to construct a simpler commuting Hamiltonian. This framework yields NP containment for rank-1 commuting local Hamiltonians in $E_0$17D independent of local dimension and for a nontrivial family of rank-1 $E_0$18D commuting Hamiltonians with qudits on edges. Despite the terminological overlap, these guided reductions are not the same object as the guiding-state promise of GLH (Bostanci et al., 2024).

The broader significance of GLH is clearest when set against the hardness of the unguided problem. The standard Local Hamiltonian problem remains $E_0$19-complete even in highly constrained settings such as nearest-neighbor $E_0$20-local Hamiltonians on a line of $E_0$21-state qudits, showing that geometry alone does not eliminate worst-case hardness (Hallgren et al., 2013). It also supports sophisticated verification structures, such as a one-round multiprover interactive proof with five entangled provers, $E_0$22-bit classical questions, and constant-size quantum answers (Fitzsimons et al., 2014). Against that background, GLH identifies a narrower regime in which access to a helpful state changes the complexity from $E_0$23 to $E_0$24, yet still leaves substantial room for classical dequantization on structured instances.

A final source of confusion is the informal use of “guided” for variational algorithms. The Hamiltonian Quantum Approximate Optimization Algorithm uses classical side-information, such as a MaxCut partition, to choose a symmetry-breaking driver and initialize parameters, but that work explicitly states that it is not the Guided Local Hamiltonian Problem in the formal complexity-theoretic sense (Kannan et al., 2024). The formal GLH literature is distinguished by the explicit overlap promise, the decision-problem formulation, and the resulting fine-grained boundary between $E_0$25, $E_0$26, $E_0$27, $E_0$28, $E_0$29, $E_0$30, and dequantized classical regimes.