A constructive commutative quantum Lovasz Local Lemma, and beyond

The recently proven Quantum Lovasz Local Lemma generalises the well-known Lovasz Local Lemma. It states that, if a collection of subspace constraints are "weakly dependent", there necessarily exists a state satisfying all constraints. It implies e.g. that certain instances of the kQSAT quantum satisfiability problem are necessarily satisfiable, or that many-body systems with "not too many" interactions are always frustration-free. However, the QLLL only asserts existence; it says nothing about how to find the state. Inspired by Moser's breakthrough classical results, we present a constructive version of the QLLL in the setting of commuting constraints, proving that a simple quantum algorithm converges efficiently to the required state. In fact, we provide two different proofs, one using a novel quantum coupling argument, the other a more explicit combinatorial analysis. Both proofs are independent of the QLLL. So these results also provide independent, constructive proofs of the commutative QLLL itself, but strengthen it significantly by giving an efficient algorithm for finding the state whose existence is asserted by the QLLL. We give an application of the constructive commutative QLLL to convergence of CP maps. We also extend these results to the non-commutative setting. However, our proof of the general constructive QLLL relies on a conjecture which we are only able to prove in special cases.Comment: 43 pages, 2 conjectures, no figures; unresolved gap in the proof; see arXiv:1311.6474 or arXiv:1310.7766 for correct proofs of the symmetric cas

Universal Quantum Hamiltonians

Quantum many-body systems exhibit an extremely diverse range of phases and physical phenomena. Here, we prove that the entire physics of any other quantum many-body system is replicated in certain simple, "universal" spin-lattice models. We first characterise precisely what it means for one quantum many-body system to replicate the entire physics of another. We then show that certain very simple spin-lattice models are universal in this very strong sense. Examples include the Heisenberg and XY models on a 2D square lattice (with non-uniform coupling strengths). We go on to fully classify all two-qubit interactions, determining which are universal and which can only simulate more restricted classes of models. Our results put the practical field of analogue Hamiltonian simulation on a rigorous footing and take a significant step towards justifying why error correction may not be required for this application of quantum information technology.Comment: 78 pages, 9 figures, 44 theorems etc. v2: Trivial fixes. v3: updated and simplified proof of Thm. 9; 82 pages, 47 theorems etc. v3: Small fix in proof of time-evolution lemma (this fix not in published version

Dissipative ground state preparation and the Dissipative Quantum Eigensolver

For any local Hamiltonian H, I construct a local CPT map and stopping condition which converges to the ground state subspace of H. Like any ground state preparation algorithm, this algorithm necessarily has exponential run-time in general (otherwise BQP=QMA), even for gapped, frustration-free Hamiltonians (otherwise BQP is in NP). However, this dissipative quantum eigensolver has a number of interesting characteristics, which give advantages over previous ground state preparation algorithms. - The entire algorithm consists simply of iterating the same set of local measurements repeatedly. - The expected overlap with the ground state subspace increases monotonically with the length of time this process is allowed to run. - It converges to the ground state subspace unconditionally, without any assumptions on or prior information about the Hamiltonian. - The algorithm does not require any variational optimisation over parameters. - It is often able to find the ground state in low circuit depth in practice. - It has a simple implementation on certain types of quantum hardware, in particular photonic quantum computers. - The process is immune to errors in the initial state. - It is inherently error- and noise-resilient, i.e. to errors during execution of the algorithm and also to faulty implementation of the algorithm itself, without incurring any computational overhead: the overlap of the output with the ground state subspace degrades smoothly with the error rate, independent of the algorithm's run-time. I give rigorous proofs of the above claims, and benchmark the algorithm on some concrete examples numerically.Comment: 58 pages, 6 tables+figures, 58 theorems etc. v2: Small generalisations and clarifications of results; 63 pages, 5 tables+figures, 62 theorems et

Simple universal models capture all classical spin physics

Spin models are used in many studies of complex systems---be it condensed matter physics, neural networks, or economics---as they exhibit rich macroscopic behaviour despite their microscopic simplicity. Here we prove that all the physics of every classical spin model is reproduced in the low-energy sector of certain `universal models'. This means that (i) the low energy spectrum of the universal model reproduces the entire spectrum of the original model to any desired precision, (ii) the corresponding spin configurations of the original model are also reproduced in the universal model, (iii) the partition function is approximated to any desired precision, and (iv) the overhead in terms of number of spins and interactions is at most polynomial. This holds for classical models with discrete or continuous degrees of freedom. We prove necessary and sufficient conditions for a spin model to be universal, and show that one of the simplest and most widely studied spin models, the 2D Ising model with fields, is universal.Comment: v1: 4 pages with 2 figures (main text) + 4 pages with 3 figures (supplementary info). v2: 12 pages with 3 figures (main text) + 35 pages with 6 figures (supplementary info) (all single column). v2 contains new results and major revisions (results for spin models with continuous degrees of freedom, explicit constructions, examples...). Close to published version. v3: minor typo correcte

Non-secret correlations can be used to distribute secrecy

A counter-intuitive result in entanglement theory was shown in [PRL 91 037902 (2003)], namely that entanglement can be distributed by sending a separable state through a quantum channel. In this work, following an analogy between the entanglement and secret key distillation scenarios, we derive its classical analog: secrecy can be distributed by sending non-secret correlations through a private channel. This strengthens the close relation between entanglement and secrecy.Comment: 4 page