---
res:
bibo_abstract:
- "Despite the fundamental role the Quantum Satisfiability (QSAT) problem has\r\nplayed
in quantum complexity theory, a central question remains open: At which\r\nlocal
dimension does the complexity of QSAT transition from \"easy\" to \"hard\"?\r\nHere,
we study QSAT with each constraint acting on a $k$-dimensional and\r\n$l$-dimensional
qudit pair, denoted $(k,l)$-QSAT. Our first main result shows\r\nthat, surprisingly,
QSAT on qubits can remain $\\mathsf{QMA}_1$-hard, in that\r\n$(2,5)$-QSAT is $\\mathsf{QMA}_1$-complete.
In contrast, $2$-SAT on qubits is\r\nwell-known to be poly-time solvable [Bravyi,
2006]. Our second main result\r\nproves that $(3,d)$-QSAT on the 1D line with
$d\\in O(1)$ is also\r\n$\\mathsf{QMA}_1$-hard. Finally, we initiate the study
of 1D $(2,d)$-QSAT by\r\ngiving a frustration-free 1D Hamiltonian with a unique,
entangled ground state.\r\n Our first result uses a direct embedding, combining
a novel clock\r\nconstruction with the 2D circuit-to-Hamiltonian construction
of [Gosset, Nagaj,\r\n2013]. Of note is a new simplified and analytic proof for
the latter (as\r\nopposed to a partially numeric proof in [GN13]). This exploits
Unitary Labelled\r\nGraphs [Bausch, Cubitt, Ozols, 2017] together with a new \"Nullspace
Connection\r\nLemma\", allowing us to break low energy analyses into small patches
of\r\nprojectors, and to improve the soundness analysis of [GN13] from\r\n$\\Omega(1/T^6)$
to $\\Omega(1/T^2)$, for $T$ the number of gates. Our second\r\nresult goes via
black-box reduction: Given an arbitrary 1D Hamiltonian $H$ on\r\n$d'$-dimensional
qudits, we show how to embed it into an effective null-space\r\nof a 1D $(3,d)$-QSAT
instance, for $d\\in O(1)$. Our approach may be viewed as a\r\nweaker notion of
\"simulation\" (\\`a la [Bravyi, Hastings 2017], [Cubitt,\r\nMontanaro, Piddock
2018]). As far as we are aware, this gives the first\r\n\"black-box simulation\"-based
$\\mathsf{QMA}_1$-hardness result, i.e. for\r\nfrustration-free Hamiltonians.@eng"
bibo_authorlist:
- foaf_Person:
foaf_givenName: Dorian
foaf_name: Rudolph, Dorian
foaf_surname: Rudolph
- foaf_Person:
foaf_givenName: Sevag
foaf_name: Gharibian, Sevag
foaf_surname: Gharibian
foaf_workInfoHomepage: http://www.librecat.org/personId=71541
orcid: 0000-0002-9992-3379
- foaf_Person:
foaf_givenName: Daniel
foaf_name: Nagaj, Daniel
foaf_surname: Nagaj
dct_date: 2024^xs_gYear
dct_language: eng
dct_title: 'Quantum 2-SAT on low dimensional systems is $\mathsf{QMA}_1$-complete: Direct
embeddings and black-box simulation@'
...