FRONT PAGE
PROJECT FINAL REPORT
Grant Agreement number: 233859
Project acronym: QUEVADIS
Project title: Quantum Engineering via Dissipation
Funding Scheme:
Period covered: from 1/6/2009 to 30/9/2012
Name of the scientific representative of the project's co-ordinator[1], Title and Organisation:
Prof. Frank Verstraete
Faculty of Physics
University of Vienna
Tel: +43 1 4277 51219, Mobile: +43 664 60277 51219
E-mail:
Project website address: www.quevadis.at
4.1 Final publishable summary report
This section must be of suitable quality to enable direct publication by the Commission and should preferably not exceed 40 pages. This report should address a wide audience, including the general public.
The publishable summary has to include 5 distinct parts described below:
· An executive summary (not exceeding 1 page).
· A summary description of project context and objectives (not exceeding 4 pages).
· A description of the main S&T results/foregrounds (not exceeding 25 pages),
· The potential impact (including the socio-economic impact and the wider societal implications of the project so far) and the main dissemination activities and exploitation of results (not exceeding 10 pages).
· The address of the project public website, if applicable as well as relevant contact details.
Furthermore, project logo, diagrams or photographs illustrating and promoting the work of the project (including videos, etc…), as well as the list of all beneficiaries with the corresponding contact names can be submitted without any restriction.
4.1.1 Executive summary.
The project QUEVADIS envisioned the study of dissipative processes as means of performing quantum information theoretic tasks. This point of view was certainly a paradigm shift, as dissipation had traditionally been perceived as the main enemy of quantum information.
Since the start of this project, this line of idea has proven to open up many novel research opportunities, both experimental and theoretical. A lot of groups around the world have started working on dissipation-based ideas for quantum information processing. The QUEVADIS (Quantum Engineering via Dissipation) project consisted of 5 work packages; all 16 milestones and deliverables have been delivered.
Furthermore, we are very happy that the project lead to many novel interesting research project that were not anticipated in the proposal, as should be for any theoretical project.
As a consequence, a large number of research papers has been published in leading physics journals. In particular, QUEVADIS lead to the publication of 1 paper in Nature, 2 in Nature Physics, and 18 in Physical Review Letters.
Highlights of the project include
1. The original paper identifying the possibility of using dissipation for quantum information theoretic tasks (Nat. Phys. 5, 633, 2009)
2. The uncovering of a relationship between quantum field theories and the description of quantum Markov chains (Phys. Rev. Lett 103, 080501, 2009)
3. A collaboration with experimentalists allowing for a quantum memory stabilized by engineered dissipation (Phys. Rev. Lett 107, 080503, 2011)
4. Setting up the mathematical framework for describing convergence rates of quantum Markov chains (J. Math. Phys. 51, 122201, 2010)
5. The construction of a quantum algorithm for simulating thermal states of generic quantum many-body Hamiltonians, as a quantum generalization of the ubiquitous Metropolis algorithm (Nature 471, 87, 2011)
6. The discovery of dissipative quantum phase transitions in central spin systems (Phys. Rev. A 86, 012116, 2012)
7. The construction of a dissipation based quantum algorithm for contracting tensor networks (Phys. Rev. Lett. 108, 110502, 2012)
4.1.2 Description of project context and objectives
The project fits within the very active worldwide effort of trying to harness the power of quantum mechanics for information theoretic tasks. A central premise of this project has been the fact that dissipation can be a useful resource for information theoretic tasks. Within this project, we have on the one hand been able to show that novel quantum algorithms can be constructed that exploit dissipation, and on the other hand that engineering dissipation can lead to long-lived quantum memories.
The project was divided into 5 workpackages; each was concerned with different aspects of dissipation:
1. Work package 1 aimed at constructing a mathematical theory of fixed points and convergence rates of quantum Markov chains. The milestones and deliverables were
a. Criteria for emergence of symmetries, coherence and other fixed point properties
b. Criteria which separate quantum from classical evolutions
c. Quantum counterparts of results for convergence of Markov chains
d. Criteria for the reachability of dissipative evolutions under coherent control.
All were delivered. It turns out that the study of quantum Markov chains form a very rich mathematical subject, with connections to many different branches in mathematics and statistics.
2. Work package 2 had as topic the study of dissipative quantum engineering protocols. The milestones and deliverables were
a. Estimate the speed of convergence for creating MPS, PEPS and/or quasi-free states
b. Construct quantum Metropolis type algorithm for simulating thermal quantum states
c. Quantify the computational complexity of finding ground states (in particular of quantum spin glasses) by means of the gap of the corresponding Liouvillian
Also here, all were delivered. A real breakthrough has been the construction of a quantum algorithm for simulating generic quantum many-body systems, i.e. a quantum version of the Metropolis algorithm.
3. The topic of research for work package 3 was dissipative quantum computing. This opened up the possibility of doing quantum computing or building quantum memories with a scheme that violates most of the DiVincenzo criteria for quantum computation. The milestones and deliverables were
a. Construction of a universal dissipative gate set
b. Study of the robustness of code subspaces when stabilized by noisy quantum dissipative processes
c. New quantum algorithms
All deliverables were delivered, and we obtained a much better understanding of dissipatively engineered quantum memories, and constructed novel quantum algorithms for contracting tensor networks.
4. The central question addressed in workpackage 4 is to identify novel effects or phenomena that arise due to dissipation. For this, we have identified novel non-equilibrium phase transitions, and developed a whole range of matrix product / tensor network methods for simulating strongly correlated quantum systems. The milestones and deliverables were
a. Characterize the phase diagram of non-equilibrium quantum models
b. Develop numerical renormalization group methods for simulating quantum dissipative processes
c. Develop a real-space renormalization group formalism for non-equilibrium quantum dissipative processes
All deliverables were delivered.
5. Workpackage 5 was of crucial importance as it provided the link between the theoretical work and experiments. A highlight was certainly the demonstration of dissipatively driven entanglement of two macroscopic ensembles. The milestones and deliverables were
a. Develop procedures for simulating quantum dissipative processes in ion traps
b. Develop procedures for simulating quantum dissipative processes in neutral atoms
c. Develop procedures for simulating quantum dissipative processes in atomic ensembles
All deliverables were delivered.
4.1.3 Description of the main S&T results/foregrounds
All the main results of the project have been published in high-quality physics journals. In this section, we will give a description of a representative selection of the published papers, arranged according to the different topics. Some papers are included several times as they were crucial for more than 1 WP; for those, the abstract is included only once.
1. WP1: Mathematical theory of fixed points and convergence rates
a. Criteria for emergence of symmetries, coherence and other fixed point properties
1. Characterizing symmetries in Projected Entangled Pair States
Authors: D. Pérez-García, M. Sanz, C.E. González-Guillén, M.M. Wolf, J.I. Cirac
Journal: New J. Phys. 12, 025010 (2010) [arXiv:0908.1674].
We show that two different tensors defining the same translational invariant injective projected entangled pair state (PEPS) in a square lattice must be the same up to a trivial gauge freedom. This allows us to characterize the existence of any local or spatial symmetry in the state. As an application of these results we prove that a SU(2) invariant PEPS with half-integer spin cannot be injective, which can be seen as a Lieb–Shultz–Mattis theorem in this context. We also give the natural generalization for U(1) symmetry in the spirit of Oshikawa–Yamanaka–Affleck, and show that a PEPS with Wilson loops cannot be injective.
2. PEPS as ground states: degeneracy and topology
Authors: N. Schuch, J.I. Cirac, D. Pérez-García
Journal: Annals of Physics 325, 2153 (2010) [arXiv:1001.3807].
We introduce a framework for characterizing Matrix Product States (MPS) and Projected Entangled Pair States (PEPS) in terms of symmetries. This allows us to understand how PEPS appear as ground states of local Hamiltonians with finitely degenerate ground states and to characterize the ground state subspace. Subsequently, we apply our framework to show how the topological properties of these ground states can be explained solely from the symmetry: We prove that ground states are locally indistinguishable and can be transformed into each other by acting on a restricted region, we explain the origin of the topological entropy, and we discuss how to renormalize these states based on their symmetries. Finally, we show how the anyonic character of excitations can be understood as a consequence of the underlying symmetries.
3. The inverse eigenvalue problem for quantum channels
Authors: Michael M. Wolf, David Perez-Garcia
Journal: J. Math. Phys. (in press, 2011) [arXiv:1005.4545].
Given a list of n complex numbers, when can it be the spectrum of a quantum channel, i.e., a completely positive trace preserving map? We provide an explicit solution for the n=4 case and show that in general the characterization of the non-zero part of the spectrum can essentially be given in terms of its classical counterpart - the non-zero spectrum of a stochastic matrix. A detailed comparison between the classical and quantum case is given. We discuss applications of our findings in the analysis of time-series and correlation functions and provide a general characterization of the peripheral spectrum, i.e., the set of eigenvalues of modulus one. We show that while the peripheral eigen-system has the same structure for all Schwarz maps, the constraints imposed on the rest of the spectrum change immediately if one departs from complete positivity.
4. Entanglement can completely defeat quantum noise
Authors: Jianxin Chen, Toby S. Cubitt, Aram W. Harrow, Graeme Smith
Phys. Rev. Lett. 107, 250504 (2011)
Abstract: We describe two quantum channels that individually cannot send any information, even classical, without some chance of decoding error. But together a single use of each channel can send quantum information perfectly reliably. This proves that the zero-error classical capacity exhibits superactivation, the extreme form of the superadditivity phenomenon in which entangled inputs allow communication over zero capacity channels. But our result is stronger still, as it even allows zero-error quantum communication when the two channels are combined. Thus our result shows a new remarkable way in which entanglement across two systems can be used to resist noise, in this case perfectly. We also show a new form of superactivation by entanglement shared between sender and receiver.
b. Criteria which separate quantum from classical evolutions
1. M. M. Wolf, D. Pérez-García, Assessing Quantum Dimensionality from Observable Dynamics, Phys. Rev. Lett. 102, 190504 (2009).
Using tools from classical signal processing, we show how to determine the dimensionality of a quantum system as well as the effective size of the environment's memory from observable dynamics in a model-independent way. We discuss the dependence on the number of conserved quantities, the relation to ergodicity and prove a converse showing that a Hilbert space of dimension D+2 is sufficient to describe every bounded sequence of measurements originating from any D-dimensional linear equations of motion. This is in sharp contrast to classical stochastic processes which are subject to more severe restrictions: a simple spectral analysis shows that the gap between the required dimensionality of a quantum and a classical description of an observed evolution can be arbitrary large.
2. Inverting the central limit theorem
Miguel Navascues, David Perez-Garcia, Ignacio Villanueva
arXiv:1110.2394v2
The central limit theorem states that the sum of N independently distributed n-tuples of real variables (subject to appropriate normalization) tends to a multivariate gaussian distribution for large N. Here we propose to invert this argument: given a set of n correlated gaussian variables, we try to infer information about the spectrum of the discrete microscopic probability distributions whose convolution generated such a macroscopic behaviour. The techniques developed along the article are applied to prove that the classical description of certain macroscopic optical experiments is infinitely more complex than the quantum one.
3. Operator Space Theory: A Natural Framework for Bell Inequalities
Journal: Phys. Rev. Lett. 104, 170405 (2010)
Authors: M. Junge, C. Palazuelos, D. Perez-Garcia, I. Villanueva, M.M. Wolf
Abstract: In this Letter we show that the field of operator space theory provides a general and powerful mathematical framework for arbitrary Bell inequalities, in particular, regarding the scaling of their violation within quantum mechanics. We illustrate the power of this connection by showing that bipartite quantum states with local, Hilbert space dimension n can violate a Bell inequality by a factor of order √n/(log2n) when observables with n possible outcomes are used. Applications to resistance to noise, Hilbert space dimension estimates, and communication complexity are given.
c. Quantum counterparts of results for convergence of Markov chains
1. The semigroup structure of Gaussian channels
Journal: Quantum Inf. Comp. 10: 0619-0635 (2010)
Author: T. Heinosaari, A.S. Holevo, M.M. Wolf
Abstract: We investigate the semigroup structure of bosonic Gaussian quantum channels. Particular focus lies on the sets of channels which are divisible, idempotent or Markovian (in the sense of either belonging to one-parameter semigroups or being infinitesimal divisible). We show that the non-compactness of the set of Gaussian channels allows for remarkable differences when comparing the semigroup structure with that of finite dimensional quantum channels. For instance, every irreversible Gaussian channel is shown to be divisible in spite of the existence of Gaussian channels which are not infinitesimal divisible. A simpler and known consequence of non-compactness is the lack of generators for certain reversible channels. Along the way we provide new representations for classes of Gaussian channels: as matrix semigroup, complex valued positive matrices or in terms of a simple form describing almost all one-parameter semigroups.