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

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

The Complexity of Relating Quantum Channels to Master Equations

Completely positive, trace preserving (CPT) maps and Lindblad master equations are both widely used to describe the dynamics of open quantum systems. The connection between these two descriptions is a classic topic in mathematical physics. One direction was solved by the now famous result due to Lindblad, Kossakowski Gorini and Sudarshan, who gave a complete characterisation of the master equations that generate completely positive semi-groups. However, the other direction has remained open: given a CPT map, is there a Lindblad master equation that generates it (and if so, can we find it's form)? This is sometimes known as the Markovianity problem. Physically, it is asking how one can deduce underlying physical processes from experimental observations. We give a complexity theoretic answer to this problem: it is NP-hard. We also give an explicit algorithm that reduces the problem to integer semi-definite programming, a well-known NP problem. Together, these results imply that resolving the question of which CPT maps can be generated by master equations is tantamount to solving P=NP: any efficiently computable criterion for Markovianity would imply P=NP; whereas a proof that P=NP would imply that our algorithm already gives an efficiently computable criterion. Thus, unless P does equal NP, there cannot exist any simple criterion for determining when a CPT map has a master equation description. However, we also show that if the system dimension is fixed (relevant for current quantum process tomography experiments), then our algorithm scales efficiently in the required precision, allowing an underlying Lindblad master equation to be determined efficiently from even a single snapshot in this case. Our work also leads to similar complexity-theoretic answers to a related long-standing open problem in probability theory.Comment: V1: 43 pages, single column, 8 figures. V2: titled changed; added proof-overview and accompanying figure; 50 pages, single column, 9 figure

Entanglement subvolume law for 2D frustration-free spin systems

Let $H$ be a frustration-free Hamiltonian describing a 2D grid of qudits with local interactions, a unique ground state, and local spectral gap lower bounded by a positive constant. For any bipartition defined by a vertical cut of length $L$ running from top to bottom of the grid, we prove that the corresponding entanglement entropy of the ground state of $H$ is upper bounded by $\tilde{O}(L^{5/3})$. For the special case of a 1D chain, our result provides a new area law which improves upon prior work, in terms of the scaling with qudit dimension and spectral gap. In addition, for any bipartition of the grid into a rectangular region $A$ and its complement, we show that the entanglement entropy is upper bounded as $\tilde{O}(|\partial A|^{5/3})$ where $\partial A$ is the boundary of $A$. This represents the first subvolume bound on entanglement in frustration-free 2D systems. In contrast with previous work, our bounds depend on the local (rather than global) spectral gap of the Hamiltonian. We prove our results using a known method which bounds the entanglement entropy of the ground state in terms of certain properties of an approximate ground state projector (AGSP). To this end, we construct a new AGSP which is based on a robust polynomial approximation of the AND function and we show that it achieves an improved trade-off between approximation error and entanglement

Zero-error channel capacity and simulation assisted by non-local correlations

Shannon's theory of zero-error communication is re-examined in the broader setting of using one classical channel to simulate another exactly, and in the presence of various resources that are all classes of non-signalling correlations: Shared randomness, shared entanglement and arbitrary non-signalling correlations. Specifically, when the channel being simulated is noiseless, this reduces to the zero-error capacity of the channel, assisted by the various classes of non-signalling correlations. When the resource channel is noiseless, it results in the "reverse" problem of simulating a noisy channel exactly by a noiseless one, assisted by correlations. In both cases, 'one-shot' separations between the power of the different assisting correlations are exhibited. The most striking result of this kind is that entanglement can assist in zero-error communication, in stark contrast to the standard setting of communicaton with asymptotically vanishing error in which entanglement does not help at all. In the asymptotic case, shared randomness is shown to be just as powerful as arbitrary non-signalling correlations for noisy channel simulation, which is not true for the asymptotic zero-error capacities. For assistance by arbitrary non-signalling correlations, linear programming formulas for capacity and simulation are derived, the former being equal (for channels with non-zero unassisted capacity) to the feedback-assisted zero-error capacity originally derived by Shannon to upper bound the unassisted zero-error capacity. Finally, a kind of reversibility between non-signalling-assisted capacity and simulation is observed, mirroring the famous "reverse Shannon theorem".Comment: 18 pages, 1 figure. Small changes to text in v2. Removed an unnecessarily strong requirement in the premise of Theorem 1