{"_id":"50272","language":[{"iso":"eng"}],"publication":"arXiv:2401.02368","author":[{"last_name":"Rudolph","full_name":"Rudolph, Dorian","first_name":"Dorian"},{"last_name":"Gharibian","full_name":"Gharibian, Sevag","orcid":"0000-0002-9992-3379","id":"71541","first_name":"Sevag"},{"last_name":"Nagaj","full_name":"Nagaj, Daniel","first_name":"Daniel"}],"year":"2024","title":"Quantum 2-SAT on low dimensional systems is $\\mathsf{QMA}_1$-complete: Direct embeddings and black-box simulation","user_id":"71541","abstract":[{"text":"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.","lang":"eng"}],"type":"preprint","status":"public","date_created":"2024-01-07T20:09:13Z","external_id":{"arxiv":["2401.02368"]},"date_updated":"2024-01-07T20:10:13Z","citation":{"chicago":"Rudolph, Dorian, Sevag Gharibian, and Daniel Nagaj. “Quantum 2-SAT on Low Dimensional Systems Is $\\mathsf{QMA}_1$-Complete: Direct Embeddings and Black-Box Simulation.” *ArXiv:2401.02368*, 2024.","short":"D. Rudolph, S. Gharibian, D. Nagaj, ArXiv:2401.02368 (2024).","ama":"Rudolph D, Gharibian S, Nagaj D. Quantum 2-SAT on low dimensional systems is $\\mathsf{QMA}_1$-complete: Direct embeddings and black-box simulation. *arXiv:240102368*. Published online 2024.","mla":"Rudolph, Dorian, et al. “Quantum 2-SAT on Low Dimensional Systems Is $\\mathsf{QMA}_1$-Complete: Direct Embeddings and Black-Box Simulation.” *ArXiv:2401.02368*, 2024.","apa":"Rudolph, D., Gharibian, S., & Nagaj, D. (2024). Quantum 2-SAT on low dimensional systems is $\\mathsf{QMA}_1$-complete: Direct embeddings and black-box simulation. In *arXiv:2401.02368*.","bibtex":"@article{Rudolph_Gharibian_Nagaj_2024, title={Quantum 2-SAT on low dimensional systems is $\\mathsf{QMA}_1$-complete: Direct embeddings and black-box simulation}, journal={arXiv:2401.02368}, author={Rudolph, Dorian and Gharibian, Sevag and Nagaj, Daniel}, year={2024} }","ieee":"D. Rudolph, S. Gharibian, and D. Nagaj, “Quantum 2-SAT on low dimensional systems is $\\mathsf{QMA}_1$-complete: Direct embeddings and black-box simulation,” *arXiv:2401.02368*. 2024."}}