77 research outputs found

### 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

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