Guidable Local Hamiltonian Problems with Implications to Heuristic Ansätze State Preparation and the Quantum PCP Conjecture

TL;DR

This work studies guidable local Hamiltonian problems, where a guiding state is promised to exist, and analyzes how classical Ansätze compare to quantumly prepared ones under quantum phase estimation. It establishes $\mathsf{QCMA}$-completeness for two guidable LH variants with $k=2$ and inverse-polynomial precision, while showing $\mathsf{NP}$/$\mathsf{NqP}$ containment in constant-precision regimes for classically evaluatable guiding states; it further connects to $\mathsf{QCPCP}$ and proves a nontrivial upper bound $\mathsf{QCPCP}[O(1)]\subseteq \mathsf{BQP}^{\mathsf{NP}[1]}$, revealing how classical and quantum witness strategies interact in this setting. The paper then articulates three implications for the quantum PCP conjecture: a no-go result against classical reductions from $\mathsf{QPCP}$ to constant-gap LH, no-go constraints on fidelity-preserving gap amplifications, and proposed stronger NLTS-type conjectures (NLCES) to guide future explorations; it additionally extends several results to the $\mathsf{MA}$ class. Together, these results clarify when classical Ansätze can rival quantum heuristics in ground-state energy estimation, illuminate the feasibility of quantum-classical PCP frameworks, and sharpen the theoretical landscape surrounding the quantum PCP program.

Abstract

We study 'Merlinized' versions of the recently defined Guided Local Hamiltonian problem, which we call 'Guidable Local Hamiltonian' problems. Unlike their guided counterparts, these problems do not have a guiding state provided as a part of the input, but merely come with the promise that one exists. We consider in particular two classes of guiding states: those that can be prepared efficiently by a quantum circuit; and those belonging to a class of quantum states we call classically evaluatable, for which it is possible to efficiently compute expectation values of local observables classically. We show that guidable local Hamiltonian problems for both classes of guiding states are $\mathsf{QCMA}$-complete in the inverse-polynomial precision setting, but lie within $\mathsf{NP}$ (or $\mathsf{NqP}$) in the constant precision regime when the guiding state is classically evaluatable. Our completeness results show that, from a complexity-theoretic perspective, classical Ansätze selected by classical heuristics are just as powerful as quantum Ansätze prepared by quantum heuristics, as long as one has access to quantum phase estimation. In relation to the quantum PCP conjecture, we (i) define a complexity class capturing quantum-classical probabilistically checkable proof systems and show that it is contained in $\mathsf{BQP}^{\mathsf{NP}[1]}$ for constant proof queries; (ii) give a no-go result on 'dequantizing' the known quantum reduction which maps a $\mathsf{QPCP}$-verification circuit to a local Hamiltonian with constant promise gap; (iii) give several no-go results for the existence of quantum gap amplification procedures that preserve certain ground state properties; and (iv) propose two conjectures that can be viewed as stronger versions of the NLTS theorem. Finally, we show that many of our results can be directly modified to obtain similar results for the class $\mathsf{MA}$.

Figures (6)

• Figure 1: Complexity characterization of $\mathsf{CGaLH}^{*}(k,\delta,\zeta)$ over parameter regime $\delta$ and $\zeta$, for $k = \mathcal{O}(1)$. Any classification indicates completeness for the respective complexity class, except for $\mathsf{NqP}$, for which we only know containment (indicated by the $'\dagger'$). Here completeness for certain parameter combinations means that for all functions of the indicated form, the problem is contained in the complexity class, and for a subset of these functions the problem is also hard. The results for $\mathsf{QPCP}[\mathcal{O}(1)]$ and $\mathsf{QMA}$ follow directly from Aharonov2008The and Kitaev2002ClassicalAQ.
• Figure 2: Visualization of the (conjectured) relations between classes of quantum states considered in this work, given a Hilbert space of a fixed dimension. For MPS, we only consider states with polynomially-bounded bond and local dimension. We take $\xi \leq \epsilon/8\leq 1/3$, such that by Theorem \ref{['thm:sampl_vs_ces']} we have that (i) all $\xi$-samplable states are also $\epsilon$-classically evaluatable and (ii) constant-depth and IQP circuits are not $\xi$-samplable. One also expects that there are quantum states (which can be prepared by a polynomial time quantum circuit) which are neither classically evaluatable nor samplable, or else $\mathsf{QMA}$ ($\mathsf{QCMA}$) would be in $\mathsf{NP}$ or $\mathsf{MA}$, respectively.
• Figure 3: Illustration of the key ideas to construct the desired witness distribution in the yes-case in the first part of the reduction. The blue lines are witnesses, for which their position with respect to the $y$-axis represents the corresponding acceptance probabilities. The dark red lines represent the completeness and soundness parameters. $a) \rightarrow b)$ represents the randomized reduction from a $\mathsf{QCMA}$-problem to a $\mathsf{UQCMA}$ one, $b) \rightarrow c)$ the error reduction and finally $d) \rightarrow e)$ The spectra of $H_\text{yes}$ and $H_\text{no}$ when $x\in A_\text{yes}$. $H_\text{yes}$ follows from the circuit-to-Hamiltonian mapping with the small penalty resulting in a Hamiltonian with fine control over its low-energy subspace, allowing one to ensure that its ground state is unique and can be made exponentially close to the history state corresponding to the unique accepting witness. The light blue shaded area represents the fact that we do not know the exact energy values corresponding to non-accepting witnesses, except for the fact that they are separated from $\lambda_0(H_\text{yes})$ by at least $\gamma(H_\text{yes}) = \Omega(1/\tilde{T}^6)$ for our choice of $\epsilon$. $H_\text{no}$ is chosen such that its ground state energy lies exactly in the gap of $H_\text{yes}$ in the sc Yes-case. Observe that if one was able to show that $\mathsf{QMA} \subseteq_r \mathsf{UQMA}$, one could use the same proof construction to show $\mathsf{QMA}$-hardness of inverse-poly-gapped Hamiltonians, for which we only yet know that they are $\mathsf{QCMA}$-hard. $\mathsf{QCMA}$-hardness for inverse-poly-gapped Hamiltonians was already shown in Aharonov2022pursuitof (in fact they even show it for 1D Hamiltonians), and rediscovered in this work.
• Figure 4: Illustration of the approximate low-energy projector $\Pi_\alpha$ in both the yes-case with $\alpha = \frac{a+b}{2}$. The orange crosses correspond to the energy values, and the attached shaded lines indicate the fidelity of the guiding state with the space spanned by all eigenstates $\ket{\psi_l}$ of $H$ that have energy at most $\lambda_i$. The polynomial approximation of the shifted sign function is displayed as $Q_\alpha(x)$, and the $\epsilon'$-error approximation regimes are indicated with the blue-shaded areas. In the red regime we do not have tight bounds on the error, except that the function values are in $[0,1]$. For small enough $\epsilon$, in the yes-case the contribution of the ground state to the value of $\norm{\tilde{\Pi}_\alpha \ket{u}}^2$ should be larger than that computed in the no-case due to contributions of higher energy values, as a result from an inexact implementation of the low-energy projector. In the no-case, all energy values will be larger than $b$.
• Figure 5: Known inclusions between some complexity classes and our proposed class $\mathsf{QCPCP}[q]$, with $q= \mathcal{O}(1)$. A line drawn from complexity class $A$ to another class $B$, where $B$ is placed above $A$ means that $A \subseteq B$. Note that the complexity of our proposed class $\mathsf{QCPCP}[\mathcal{O}(1)]$ is non-trivial, as it contains both $\mathsf{NP}$ and $\mathsf{BQP}$ for which it is believed that both $\mathsf{BQP} \not\subset \mathsf{NP}$ and $\mathsf{BQP} \not\supset \mathsf{NP}$.

Theorems & Definitions (103)

• Definition 1.1 (Informal): Classically evaluatable and quantumly preparable states, from Definition \ref{['def:cds']}
• Definition 1.2 (Informal): Guidable Local Hamiltonian problems, from Definition \ref{['def:GaLH']}
• Theorem 1.1 (Informal): Complexity of guidable local Hamiltonian problems, from Corollary \ref{['cor:CGaLHstar']} and Theorem \ref{['thm:QGaLH']}
• Corollary 1.1: Classical versus quantum state preparation
• Theorem 1.2 (Informal): Classical containment of the classically guidable local Hamiltonian problem, from Theorem \ref{['thm:cp_NP_NqP']}.
• Definition 1.3 (Informal): Quantum-classical PCP, from Definition \ref{['def:QCPCP']}
• Theorem 1.3 (Informal): Upper bound on $\mathsf{QCPCP}$, from Theorem \ref{['thm:QCPCP_NP_qr']}
• Theorem 1.4 (Informal): No-go for classical polynomial-time reductions, from Theorem \ref{['thm:clas_red_no_go']}
• Theorem 1.5 (Informal): No-go results for Hamiltonian gap amplification, from Theorem \ref{['thm:QCPCP_nogos']}
• Conjecture 1.1 (Informal): NLCES conjecture, from Conjecture \ref{['conj:NLCES']}