Quantum 2-SAT on low dimensional systems is $\mathsf{QMA}_1$-complete: Direct embeddings and black-box simulation
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914630276743168 |
|---|---|
| author | Rudolph, Dorian Gharibian, Sevag Nagaj, Daniel |
| author_facet | Rudolph, Dorian Gharibian, Sevag Nagaj, Daniel |
| contents | Despite the fundamental role the Quantum Satisfiability (QSAT) problem has played in quantum complexity theory, a central question remains open: At which local dimension does the complexity of QSAT transition from "easy" to "hard"? Here, we study QSAT with each constraint acting on a $k$-dimensional and $l$-dimensional qudit pair, denoted $(k,l)$-QSAT. Our first main result shows that, surprisingly, QSAT on qubits can remain $\mathsf{QMA}_1$-hard, in that $(2,5)$-QSAT is $\mathsf{QMA}_1$-complete. In contrast, $2$-SAT on qubits is well-known to be poly-time solvable [Bravyi, 2006]. Our second main result proves that $(3,d)$-QSAT on the 1D line with $d\in O(1)$ is also $\mathsf{QMA}_1$-hard. Finally, we initiate the study of 1D $(2,d)$-QSAT by giving a frustration-free 1D Hamiltonian with a unique, entangled ground state.
Our first result uses a direct embedding, combining a novel clock construction with the 2D circuit-to-Hamiltonian construction of [Gosset, Nagaj, 2013]. Of note is a new simplified and analytic proof for the latter (as opposed to a partially numeric proof in [GN13]). This exploits Unitary Labelled Graphs [Bausch, Cubitt, Ozols, 2017] together with a new "Nullspace Connection Lemma", allowing us to break low energy analyses into small patches of projectors, and to improve the soundness analysis of [GN13] from $Ω(1/T^6)$ to $Ω(1/T^2)$, for $T$ the number of gates. Our second result goes via black-box reduction: Given an arbitrary 1D Hamiltonian $H$ on $d'$-dimensional qudits, we show how to embed it into an effective null-space of a 1D $(3,d)$-QSAT instance, for $d\in O(1)$. Our approach may be viewed as a weaker notion of "simulation" (à la [Bravyi, Hastings 2017], [Cubitt, Montanaro, Piddock 2018]). As far as we are aware, this gives the first "black-box simulation"-based $\mathsf{QMA}_1$-hardness result, i.e. for frustration-free Hamiltonians. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02368 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantum 2-SAT on low dimensional systems is $\mathsf{QMA}_1$-complete: Direct embeddings and black-box simulation Rudolph, Dorian Gharibian, Sevag Nagaj, Daniel Quantum Physics Computational Complexity Despite the fundamental role the Quantum Satisfiability (QSAT) problem has played in quantum complexity theory, a central question remains open: At which local dimension does the complexity of QSAT transition from "easy" to "hard"? Here, we study QSAT with each constraint acting on a $k$-dimensional and $l$-dimensional qudit pair, denoted $(k,l)$-QSAT. Our first main result shows that, surprisingly, QSAT on qubits can remain $\mathsf{QMA}_1$-hard, in that $(2,5)$-QSAT is $\mathsf{QMA}_1$-complete. In contrast, $2$-SAT on qubits is well-known to be poly-time solvable [Bravyi, 2006]. Our second main result proves that $(3,d)$-QSAT on the 1D line with $d\in O(1)$ is also $\mathsf{QMA}_1$-hard. Finally, we initiate the study of 1D $(2,d)$-QSAT by giving a frustration-free 1D Hamiltonian with a unique, entangled ground state. Our first result uses a direct embedding, combining a novel clock construction with the 2D circuit-to-Hamiltonian construction of [Gosset, Nagaj, 2013]. Of note is a new simplified and analytic proof for the latter (as opposed to a partially numeric proof in [GN13]). This exploits Unitary Labelled Graphs [Bausch, Cubitt, Ozols, 2017] together with a new "Nullspace Connection Lemma", allowing us to break low energy analyses into small patches of projectors, and to improve the soundness analysis of [GN13] from $Ω(1/T^6)$ to $Ω(1/T^2)$, for $T$ the number of gates. Our second result goes via black-box reduction: Given an arbitrary 1D Hamiltonian $H$ on $d'$-dimensional qudits, we show how to embed it into an effective null-space of a 1D $(3,d)$-QSAT instance, for $d\in O(1)$. Our approach may be viewed as a weaker notion of "simulation" (à la [Bravyi, Hastings 2017], [Cubitt, Montanaro, Piddock 2018]). As far as we are aware, this gives the first "black-box simulation"-based $\mathsf{QMA}_1$-hardness result, i.e. for frustration-free Hamiltonians. |
| title | Quantum 2-SAT on low dimensional systems is $\mathsf{QMA}_1$-complete: Direct embeddings and black-box simulation |
| topic | Quantum Physics Computational Complexity |
| url | https://arxiv.org/abs/2401.02368 |