Physically-Motivated Guiding States for Local Hamiltonians

TL;DR

This work analyzes the Guided Local Hamiltonian problem through physically-motivated guiding states, linking ground-state energy estimation to computational complexity. By proving $\textbf{BQP}$ containment for several efficiently preparable state families (SCSS, SCESS, fixed-weight, MPS, Gaussian) and establishing $\textbf{BQP}$-hardness for 2-local Hamiltonians with these states, it delineates when quantum advantage persists versus when classical heuristics remain viable via dequantisation. A central contribution is the Goldilocks zone, where guiding states are efficiently described, prepareable, and sample-query accessible, enabling meaningful quantum-vs-classical comparisons. The results extend previous hardness and containment results to more physically relevant settings (Quantum Chemistry, Hamiltonian complexity) and clarify the structure of reductions needed to preserve hardness. Overall, the paper advances the complexity landscape for ground-state estimation, guiding experimental relevance and highlighting boundaries of quantum advantage for guided quantum simulations.

Abstract

This work characterises families of guiding states for the Guided Local Hamiltonian problem, revealing new connections between physical constraints and computational complexity. Focusing on states motivated by Quantum Chemistry and Hamiltonian Complexity, we extend prior BQP-hardness results beyond semi-classical subset states. We demonstrate that broader state families preserve hardness, while maintaining classical tractability under practical parameter regimes. Crucially, we provide a constructive proof of BQP containment for the canonical problem, showing the problem is BQP-complete when provided with a polynomial-size classical description of the guiding state. Our results show quantum advantage persists for physically meaningful state classes, and classical methods remain viable when guiding states admit appropriate descriptions. We identify a Goldilocks zone of guiding states that are efficiently preparable, succinctly described, and sample-query accessible, allowing for a meaningful comparison between quantum and classical approaches. Our work furthers the complexity landscape for ground state estimation problems, presenting steps toward experimentally relevant settings while clarifying the boundaries of quantum advantage.

Figures (11)

• Figure 1: Summary of the steps taken to prove BQP - hardness for the Guided Local Hamiltonian problem. We reduce from arbitrary BQP circuits to a local Hamiltonian $H_{\mu}$, defined for a specific clock register encoding $\mu$. Next, we construct a guiding state $\xi$ that is guaranteed to have overlap with the ground state $\eta_\nu$ of the Hamiltonian $H_{\mu}$. Then we perform a perturbative analysis to show that the ground state of the perturbed Hamiltonian $\hat{H}_{\mu}$ is close to the guiding state $\xi$. Further gadget reductions are employed to reduce the locality to $2$, concluding the proof of BQP - hardness.
• Figure 2: The overlap between the different state types producing the Goldilocks zone. The upper region represents those states that recover the quantum result of this work. The lower region represents those states that recover the classical result of Ref. GLG22. States lying in the intersection (dashed) are those that can be used to prove both results, under the right conditions.
• Figure 3: The Goldilocks zone --- the outer limit of guiding state that recovers both the classical and quantum results for the Guided Local Hamiltonian problem. The conjectured relationship between the physically-motivated states and the semi-classical subset states for which we have proven BQP completeness.
• Figure 4: An example of part of the unitary operator $U_c$ that implements the cycle increment $x_i \mapsto x_{i+1}$. The first multi-controlled $X$ gate is applied to the workspace qubits, controlled by the $n$-bit string $x_i$, with the target qubit as the ancilla qubit. Notice that the ancilla qubit is only flipped if the workspace qubits are in the state $|x_i\rangle$. The second gate is a controlled-$V_i$ operator that applies the Gray code rotation $V_i$ to the workspace qubits, conditioned on the ancilla qubit being $|1\rangle$.
• Figure 5: A schematic of the Feynman-Kitaev construction showing the expected evolution of unitaries $U_1,\dots,U_4$ controlled by the clock register. Each gate is coupled to a timestamp encoded in binary and denoted using $\mu(\boldsymbol{t})$.

Theorems & Definitions (66)

• Definition 1: Polynomial-time Generated Quantum Circuit
• Definition 2: BQP
• Definition 3: [State Type] Guided Local Hamiltonian problem
• Lemma 1: QPE LinNotes22
• Lemma 2
• Lemma 3
• Theorem 1
• Corollary 1
• Theorem 2
• Corollary 2