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

• URL: http://arxiv.org/abs/2302.11578v3
• Date: Mon, 10 Jun 2024 09:47:53 GMT
• Title: Guidable Local Hamiltonian Problems with Implications to Heuristic Ansätze State Preparation and the Quantum PCP Conjecture
• Authors: Jordi Weggemans, Marten Folkertsma, Chris Cade,
• Abstract summary: We study 'Merlinized' versions of the recently defined Guided Local Hamiltonian problem. 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 show that guidable local Hamiltonian problems for both classes of guiding states are $mathsfQCMA$-complete in the inverse-polynomial precision setting.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 Ans\"atze selected by classical heuristics are just as powerful as quantum Ans\"atze 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 $\mathsf{MA}$.

Related papers

• Optimizing random local Hamiltonians by dissipation [44.99833362998488]
  We prove that a simplified quantum Gibbs sampling algorithm achieves a $Omega(frac1k)$-fraction approximation of the optimum. Our results suggest that finding low-energy states for sparsified (quasi)local spin and fermionic models is quantumly easy but classically nontrivial.
  arXiv  Detail & Related papers  (2024-11-04T20:21:16Z)
• Quasi-quantum states and the quasi-quantum PCP theorem [0.21485350418225244]
  We show that solving the $k$-local Hamiltonian over the quasi-quantum states is equivalent to optimizing a distribution of assignment over a classical $k$-local CSP. Our main result is a PCP theorem for the $k$-local Hamiltonian over the quasi-quantum states in the form of a hardness-of-approximation result.
  arXiv  Detail & Related papers  (2024-10-17T13:43:18Z)
• Phase-space gaussian ensemble quantum camouflage [0.0]
  We extend the phase-space description of the Weyl-Wigner quantum mechanics to a subset of non-linear Hamiltonians in position and momentum. For gaussian statistical ensembles, the exact phase-space profile of the quantum fluctuations over the classical trajectories are found.
  arXiv  Detail & Related papers  (2024-09-24T18:14:07Z)
• Quantum PCPs: on Adaptivity, Multiple Provers and Reductions to Local Hamiltonians [0.0]
  We show that non-adaptive quantum PCPs can simulate adaptive quantum PCPs when the number of proof queries is constant. We also show that there exists (quantum) oracles relative to which certain quantum PCP statements are false.
  arXiv  Detail & Related papers  (2024-03-07T19:00:06Z)
• The Power of Unentangled Quantum Proofs with Non-negative Amplitudes [55.90795112399611]
  We study the power of unentangled quantum proofs with non-negative amplitudes, a class which we denote $textQMA+(2)$. In particular, we design global protocols for small set expansion, unique games, and PCP verification. We show that QMA(2) is equal to $textQMA+(2)$ provided the gap of the latter is a sufficiently large constant.
  arXiv  Detail & Related papers  (2024-02-29T01:35:46Z)
• Sparse random Hamiltonians are quantumly easy [105.6788971265845]
  A candidate application for quantum computers is to simulate the low-temperature properties of quantum systems. This paper shows that, for most random Hamiltonians, the maximally mixed state is a sufficiently good trial state. Phase estimation efficiently prepares states with energy arbitrarily close to the ground energy.
  arXiv  Detail & Related papers  (2023-02-07T10:57:36Z)
• Optimizing local Hamiltonians for the best metrological performance [0.0]
  We discuss efficient methods to optimize the metrological performance over local Hamiltonians in a bipartite quantum system. We present the quantum Fisher information in a bilinear form and maximize it by iterating a see-saw. We consider a number of other problems in quantum information theory that can be solved in a similar manner.
  arXiv  Detail & Related papers  (2022-06-06T18:01:03Z)
• Theory of Quantum Generative Learning Models with Maximum Mean Discrepancy [67.02951777522547]
  We study learnability of quantum circuit Born machines (QCBMs) and quantum generative adversarial networks (QGANs) We first analyze the generalization ability of QCBMs and identify their superiorities when the quantum devices can directly access the target distribution. Next, we prove how the generalization error bound of QGANs depends on the employed Ansatz, the number of qudits, and input states.
  arXiv  Detail & Related papers  (2022-05-10T08:05:59Z)
• Efficient Bipartite Entanglement Detection Scheme with a Quantum Adversarial Solver [89.80359585967642]
  Proposal reformulates the bipartite entanglement detection as a two-player zero-sum game completed by parameterized quantum circuits. We experimentally implement our protocol on a linear optical network and exhibit its effectiveness to accomplish the bipartite entanglement detection for 5-qubit quantum pure states and 2-qubit quantum mixed states.
  arXiv  Detail & Related papers  (2022-03-15T09:46:45Z)
• Average-case Speedup for Product Formulas [69.68937033275746]
  Product formulas, or Trotterization, are the oldest and still remain an appealing method to simulate quantum systems. We prove that the Trotter error exhibits a qualitatively better scaling for the vast majority of input states. Our results open doors to the study of quantum algorithms in the average case.
  arXiv  Detail & Related papers  (2021-11-09T18:49:48Z)
• Variational quantum eigensolvers for sparse Hamiltonians [0.0]
  Hybrid quantum-classical variational algorithms such as the variational quantum eigensolver (VQE) are promising applications for noisy, intermediate-scale quantum computers. We extend VQE to general sparse Hamiltonians.
  arXiv  Detail & Related papers  (2020-12-13T22:36:51Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.