18 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
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
Quantum Side Information: Uncertainty Relations, Extractors, Channel Simulations
In the first part of this thesis, we discuss the algebraic approach to
classical and quantum physics and develop information theoretic concepts within
this setup.
In the second part, we discuss the uncertainty principle in quantum
mechanics. The principle states that even if we have full classical information
about the state of a quantum system, it is impossible to deterministically
predict the outcomes of all possible measurements. In comparison, the
perspective of a quantum observer allows to have quantum information about the
state of a quantum system. This then leads to an interplay between uncertainty
and quantum correlations. We provide an information theoretic analysis by
discussing entropic uncertainty relations with quantum side information.
In the third part, we discuss the concept of randomness extractors. Classical
and quantum randomness are an essential resource in information theory,
cryptography, and computation. However, most sources of randomness exhibit only
weak forms of unpredictability, and the goal of randomness extraction is to
convert such weak randomness into (almost) perfect randomness. We discuss
various constructions for classical and quantum randomness extractors, and we
examine especially the performance of these constructions relative to an
observer with quantum side information.
In the fourth part, we discuss channel simulations. Shannon's noisy channel
theorem can be understood as the use of a noisy channel to simulate a noiseless
one. Channel simulations as we want to consider them here are about the reverse
problem: simulating noisy channels from noiseless ones. Starting from the
purely classical case (the classical reverse Shannon theorem), we develop
various kinds of quantum channel simulation results. We achieve this by using
classical and quantum randomness extractors that also work with respect to
quantum side information.Comment: PhD thesis, ETH Zurich. 214 pages, 13 figures, 1 table. Chapter 2 is
based on arXiv:1107.5460 and arXiv:1308.4527 . Section 3.1 is based on
arXiv:1302.5902 and Section 3.2 is a preliminary version of arXiv:1308.4527
(you better read arXiv:1308.4527). Chapter 4 is (partly) based on
arXiv:1012.6044 and arXiv:1111.2026 . Chapter 5 is based on arXiv:0912.3805,
arXiv:1108.5357 and arXiv:1301.159
Multipartite Quantum States and their Marginals
Subsystems of composite quantum systems are described by reduced density
matrices, or quantum marginals. Important physical properties often do not
depend on the whole wave function but rather only on the marginals. Not every
collection of reduced density matrices can arise as the marginals of a quantum
state. Instead, there are profound compatibility conditions -- such as Pauli's
exclusion principle or the monogamy of quantum entanglement -- which
fundamentally influence the physics of many-body quantum systems and the
structure of quantum information. The aim of this thesis is a systematic and
rigorous study of the general relation between multipartite quantum states,
i.e., states of quantum systems that are composed of several subsystems, and
their marginals. In the first part, we focus on the one-body marginals of
multipartite quantum states; in the second part, we study general quantum
marginals from the perspective of entropy.Comment: PhD thesis, ETH Zurich. The first part contains material from
arXiv:1208.0365, arXiv:1204.0741, and arXiv:1204.4379. The second part is
based on arXiv:1302.6990 and arXiv:1210.046
Symmetry reduction in convex optimization with applications in combinatorics
This dissertation explores different approaches to and applications of symmetry reduction in convex optimization. Using tools from semidefinite programming, representation theory and algebraic combinatorics, hard combinatorial problems are solved or bounded. The first chapters consider the Jordan reduction method, extend the method to optimization over the doubly nonnegative cone, and apply it to quadratic assignment problems and energy minimization on a discrete torus. The following chapter uses symmetry reduction as a proving tool, to approach a problem from queuing theory with redundancy scheduling. The final chapters propose generalizations and reductions of flag algebras, a powerful tool for problems coming from extremal combinatorics