Quantum SAT Problems with Finite Sets of Projectors are Complete for a Plethora of Classes

• URL: http://arxiv.org/abs/2506.07244v1
• Date: Sun, 08 Jun 2025 17:59:55 GMT
• Title: Quantum SAT Problems with Finite Sets of Projectors are Complete for a Plethora of Classes
• Authors: Ricardo Rivera Cardoso, Alex Meiburg, Daniel Nagaj,
• Abstract summary: We demonstrate new qubit variants of the Quantum Satisfiability (QSAT) problem.<n>We first prove there are qudit QSAT problems that are complete for $mathsfBQP_1$, $mathsfcoRP$, $mathsfQCMA$.<n>We then show that any QCSP can be reduced to a problem in qubits while maintaining the same complexity.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Previously, all known variants of the Quantum Satisfiability (QSAT) problem, i.e. deciding whether a $k$-local ($k$-body) Hamiltonian is frustration-free, could be classified as being either in $\mathsf{P}$; or complete for $\mathsf{NP}$, $\mathsf{MA}$, or $\mathsf{QMA_1}$. Here, we demonstrate new qubit variants of this problem that are complete for $\mathsf{BQP_1}$, $\mathsf{coRP}$, $\mathsf{QCMA}$, $\mathsf{PI(coRP,NP)}$, $\mathsf{PI(BQP_1,NP)}$, $\mathsf{PI(BQP_1,MA)}$, $\mathsf{SoPU(coRP,NP)}$, $\mathsf{SoPU(BQP_1,NP)}$, and $\mathsf{SoPU(BQP_1,MA)}$. Our result implies that a complete classification of quantum constraint satisfaction problems (QCSPs), analogous to Schaefer's dichotomy theorem for classical CSPs, must either include these 13 classes, or otherwise show that some are equal. Additionally, our result showcases two new types of QSAT problems that can be decided efficiently, as well as the first nontrivial $\mathsf{BQP_1}$-complete problem. We first prove there are qudit QSAT problems that are complete for $\mathsf{BQP_1}$, $\mathsf{coRP}$, and $\mathsf{QCMA}$ by re-defining elements of the circuit-to-Hamiltonian transformation. We then show that any QCSP can be reduced to a problem in qubits while maintaining the same complexity - something believed not to be possible classically. The remaining six problems are obtained by considering "sums" and "products" of the first seven QSAT problems. Before this work, the QSAT problems generated in this way resulted in complete problems for $\mathsf{PI}$ and $\mathsf{SoPU}$ classes that were trivially equal to other known classes. We thus commence the study of these new and seemingly nontrivial classes. While [Meiburg, 2021] first sought to prove completeness for the first three classes, we note that his constructions are flawed. Here, we rework them and obtain improvements on the required qudit dimensionality.

Related papers

• Towards a universal gateset for $\mathsf{QMA}_1$ [0.0]
  We prove that for all gatesets in the cyclotomic field $mathbbQ(zeta_2k),zeta_2k=e2pi i/2k$, $G_2k$ is universal for all gatesets in $mathbbQ(zeta_2k),zeta_2k=e2pi i/2k$.<n>We also prove that the Gapped Clique Homology problem defined by King
  arXiv  Detail & Related papers  (2024-11-04T23:39:27Z)
• Quantum Sabotage Complexity [0.7812210699650152]
  We show $mathsfQ(f_mathsfsab)$, the quantum query complexity of $f_mathsfsab$. We show that when $f$ is the Indexing function, $mathsfQ(f_mathsfsab)=Theta(sqrtmathsfsab)$, ruling out the possibility that $mathsfQ(f_mathsfsab)=Theta(sqrtmathsf
  arXiv  Detail & Related papers  (2024-08-22T17:57:58Z)
• Provably learning a multi-head attention layer [55.2904547651831]
  Multi-head attention layer is one of the key components of the transformer architecture that sets it apart from traditional feed-forward models. In this work, we initiate the study of provably learning a multi-head attention layer from random examples. We prove computational lower bounds showing that in the worst case, exponential dependence on $m$ is unavoidable.
  arXiv  Detail & Related papers  (2024-02-06T15:39:09Z)
• Quantum Polynomial Hierarchies: Karp-Lipton, error reduction, and lower bounds [1.3927943269211591]
  This work studies three quantum-verifier based generalizations of $mathsfPH$. We first resolve several open problems from [GSSSY22], including a collapse theorem and a Karp-Lipton theorem for $mathsfQCPH$. We show one-sided error reduction for $mathsfpureQPH$, as well as the first bounds relating these quantum variants of $mathsfPH$.
  arXiv  Detail & Related papers  (2024-01-03T09:12:25Z)
• The Entangled Quantum Polynomial Hierarchy Collapses [0.0]
  We show that the entangled quantum quantum hierarchy $mathsfQEPH$ collapses to its second level. We also show that only quantum superposition (not classical probability) increases the computational power of these hierarchies.
  arXiv  Detail & Related papers  (2024-01-02T22:25:56Z)
• Quantum Depth in the Random Oracle Model [57.663890114335736]
  We give a comprehensive characterization of the computational power of shallow quantum circuits combined with classical computation. For some problems, the ability to perform adaptive measurements in a single shallow quantum circuit is more useful than the ability to perform many shallow quantum circuits without adaptive measurements.
  arXiv  Detail & Related papers  (2022-10-12T17:54:02Z)
• Cryptographic Hardness of Learning Halfspaces with Massart Noise [59.8587499110224]
  We study the complexity of PAC learning halfspaces in the presence of Massart noise. We show that no-time Massart halfspace learners can achieve error better than $Omega(eta)$, even if the optimal 0-1 error is small.
  arXiv  Detail & Related papers  (2022-07-28T17:50:53Z)
• Learning a Single Neuron with Adversarial Label Noise via Gradient Descent [50.659479930171585]
  We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations. The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
  arXiv  Detail & Related papers  (2022-06-17T17:55:43Z)
• Low-degree learning and the metric entropy of polynomials [44.99833362998488]
  We prove that any (deterministic or randomized) algorithm which learns $mathscrF_nd$ with $L$-accuracy $varepsilon$ requires at least $Omega(sqrtvarepsilon)2dlog n leq log mathsfM(mathscrF_n,d,|cdot|_L,varepsilon) satisfies the two-sided estimate $$c (1-varepsilon)2dlog
  arXiv  Detail & Related papers  (2022-03-17T23:52:08Z)
• Threshold Phenomena in Learning Halfspaces with Massart Noise [56.01192577666607]
  We study the problem of PAC learning halfspaces on $mathbbRd$ with Massart noise under Gaussian marginals. Our results qualitatively characterize the complexity of learning halfspaces in the Massart model.
  arXiv  Detail & Related papers  (2021-08-19T16:16:48Z)
• Hardness of Learning Halfspaces with Massart Noise [56.98280399449707]
  We study the complexity of PAC learning halfspaces in the presence of Massart (bounded) noise. We show that there an exponential gap between the information-theoretically optimal error and the best error that can be achieved by a SQ algorithm.
  arXiv  Detail & Related papers  (2020-12-17T16:43:11Z)
• StoqMA meets distribution testing [0.0]
  We provide a novel connection between $mathsfStoqMA$ and distribution testing via reversible circuits. We show that both variants of $mathsfStoqMA$ that without any ancillary random bit and with perfect soundness are contained in $mathsfNP$. Our results make a step towards collapsing the hierarchy $mathsfMA subseteq mathsfStoqMA subseteq mathsfSBP$ [BBT06], in which all classes are contained in $
  arXiv  Detail & Related papers  (2020-11-11T12:30:42Z)

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.