jacob biamonte- quantum versus classical network structure and function
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Quantum versus Classical Network Structure and FunctionJacob Biamonte CQT Singapore ISI Foundation, Torino Italy Tensor network states course homepage http://www.qubit.org/iqc2011
Current collaboratorsJohnBaez VilleBergholm StuartBroadfoot StephenClark SamDenny DieterJaksch TomiJohnson MarcoLanzagorta SebastianMeznaric AlexParent(UW) ChrisWood(IQC,PI) etal. NOTE:thisisoutdated!
We'll cover results from the following references[1] Categorical Tensor Network States with Stephen R. Clark and Dieter Jaksch, accepted, AIP Advances (2011). arXiv:1012.0531 [2] Categorical Quantum Circuits with Ville Bergholm, In Journal of Physics A: Mathematical and Theoretical, Vol. 44, No. 17, pages 25304-25324, (2011). arXiv:1010.4840 [3] Algebraically Contractible Topological Tensor Network States, with S. J. Denny, D. Jaksch and S. R. Clark, accepted Journal of Physics A:Mathematical and Theoretical (2011). arXiv:1108.0888 [4] Youtube series, Lectures on Tensor Network States QIC 890/891 Selected Advanced Topics in Quantum Information, The University of Waterloo, Waterloo Ontario, Canada, (2011). http://www.qubit.org/iqc2011
Overview take I
Quantum Circuits Classical Circuits Tensor Network States Ising Spin Models Penrose Tensor Networks Categorical Algebra
Quantum CircuitsTemporally ordered or time sequenced All maps are unitary so # inputs ='s # outputs Describe quantum algorithms Universality result: every quantum state approximately prepared by a quantum circuit A model of quantum computation Conceptual understanding (in some cases compared to evolution under H) Complexity bounds (gate counts to simulate H)
Quantum Circuits are normally written backwards
Examples of quantum circuits
(With James Whitfield and A. Aspuru-Guzik) Molecular Physics, Volume 109, Issue 5 March 2011 , pages 735 - 750
Quantum programming languge?A programming language written across the page using lines of text (1D), needs to describe the inherently two-dimensional nature of quantum interactions in the plane. Quantum circuits are inherently 2D.
Quantum Circuit Logic
Gate families Match gates Stabiliser gates Rewrite rules Gate identities (these are symmetries)
Classical CircuitsIn mathematics, a (finitary) Boolean function (or switching function) is a function of the form : B^k B, where B = {0, 1} is a Boolean domain and k is the arity of the function. Asynchronous circuits for every such Boolean function Universal gate families (need boolean non-linearity) A model of computation Complexity bounds on circuit families Decomposition methods, synthesis, Shannon & Davio expansions
Classical Circuit Example (adder)
Intersection (classical quantum circuits ~ quantum classical circuits)The intersection between quantum and classical circuits is currently taken to be reversible circuits. ...However, we will go past this!
Tensor Network StatesAlgorithms to describe many-body physics using classical computers Data compression methods (different than those already present in AI) Uses diagrammatic language to describe networks of contracted tensors
At PI: Lukasz Cincio, Robert Pfeifer, Guifre Vidal Tensor Network States IQC/UW Course http://www.qubit.org/iqc2011 http://pirsa.org/11060004/ (RP)
Tensor Network States Examples
Ising Spin Models
Energy penalties Spin configurations Each spin can take either of two values
Penrose Tensor Networks
Graphical depiction of tensors Compositionality Diagrams to reason about equations and physics Algorithms to solve problems [1971]Applications of negative dimensional tensors, Rodger Penrose in Combinatorial Mathematics and its Applications, Academic Press (1971).
Seeing tensors [Penrose, 1971]
Applications of negative dimensional tensors, Rodger Penrose in Combinatorial Mathematics and its Applications, Academic Press (1971).
Cups, caps, snake equation
Applications of negative dimensional tensors, Rodger Penrose in Combinatorial Mathematics and its Applications, Academic Press (1971).
Emphasis of input/output equivalence
Applications of negative dimensional tensors, Rodger Penrose in Combinatorial Mathematics and its Applications, Academic Press (1971).
Tensors for algorithms
Applications of negative dimensional tensors, Rodger Penrose in Combinatorial Mathematics and its Applications, Academic Press (1971).
Graphical rewrite system
Applications of negative dimensional tensors, Rodger Penrose in Combinatorial Mathematics and its Applications, Academic Press (1971).
Graphical Calculus for Quantum Theory [Penrose]
Page 659
Page 802
Categorical AlgebraDuality, Pairing, abstraction as a uniting tool. Precise, clear definitions Pay entrance fee to join the conversation Baez-Dolan Dagger Compact Categories describe Quantum Theory [1995]
Refining Penrose Tensor Calculus [Lafont]
Y. Lafont, Penrose diagrams and 2-dimensional rewriting, in Applications of Categories in Computer Science, London Mathematical Society Lecture Note Series 177, p. 191-201, Cambridge University Press (1992).
Abstract tensor rewrite system [Lafont]
Y. Lafont, Penrose diagrams and 2-dimensional rewriting, in Applications of Categories in Computer Science, London Mathematical Society Lecture Note Series 177, p. 191-201, Cambridge University Press (1992).
Frobenius algebras and 2D topological quantum field theories, Joachim Kock, Cambridge University Press, 2004
Quantum groups, Christian Kassel, Springer, 1995
A Prehistory of n-Categorical PhysicsAuthors: John C. Baez, Aaron Lauda
http://arxiv.org/abs/0908.2469
Unification
The network models we have considered are all different (it would seem)... ...How can we relate them?
Overview take II
Classical Circuits + Spin Models Quantum Circuits Tensor Network States
Ground State Spin Logic
JB, Physical Review A 77 052331. 2008.
Composing Gates
We are dealing with spans
Quantum Networks
Penrose (Wire Bending) Duality
Bell states vs Pauli basis
Boolean States
Boolean Tensor Networks
AND-tensors
COPY-, XOR-tensors
Quantum AND-tensors
W-state
Boolean States vs Spin Models
Spins
States
Application: 3SAT
(with Tomi Johnson, Stephen Clark, Dieter Jaksch)
Connection to quantum circuits
Connection to Vidal's MERA
Connection to Vidal's MERA
The category of quantum circuits
Connection to quantum circuits
Return to Penrose's graphical denity state
Page 802
Applications of negative dimensional tensors, Rodger Penrose in Combinatorial Mathematics and its Applications, Academic Press (1971).
Diagrammatic SVD
Algebraic Invariant Theory for Matrix Product States (with Ville Berghlom and Marco Lanzagortat)
Map state duality
Algebraic Invariant Theory for Matrix Product States (with Ville Berghlom and Marco Lanzagortat)
Purification
Algebraic Invariant Theory for Matrix Product States (with Ville Berghlom and Marco Lanzagortat)
Entanglement topology
Algebraic Invariant Theory for Matrix Product States (with Ville Berghlom and Marco Lanzagortat)
MPS
Polynomial Invariants
Algebraic Invariant Theory for Matrix Product States (with Ville Berghlom and Marco Lanzagortat)
Invariants of mixed states
Pure vs mixed invarinats
Applications
Entanglement Spectrum Reyni Entropy Estimating Rank (with Ann Kallin and others)
General methods to factor states
Stabilizer Tensors
(with Oscar Dahlsten and others)
Preparing states Alex ParentWe are currently considered applications of these methods to state preparation using quantum circuits Strong optimality: Have one degree of freedom in the circuit, for every degree of freedom in the state. Gate rewrites are equivalent to symmetries in a state
(with Alex Parent and others)
Open quantum systems Chris Wood
Tensor networks for open systems
2D tensor networks Sam Denny
Algebraically contractible topological tensor network states, S. J. Denny, JB, Jaksch and Clark. (2011). 1108.0888
Invariants and covariants of symmetric tensors
Thanks to current collaborators
JohnBaez VilleBergholm StuartBroadfoot StephenClark OscarDahlsten SamDenny DieterJaksch
TomiJohnson AnnKallin(UW) MarcoLanzagorta SebastianMeznaric AlexParent(UW) ChrisWood(IQC,PI) etal.
A Benchmark for SpeciesQuantum versus Classical Network Structure and Function[1] Categorical Tensor Network States, with Stephen R. Clark and Dieter Jaksch, accepted, AIP Advances (2011). arXiv:1012.0531 [2] Categorical Quantum Circuits, with Ville Bergholm, In Journal of Physics A: Mathematical and Theoretical, Vol. 44, No. 17, pages 25304-25324, (2011). arXiv:1010.4840 [3] Algebraically Contractible Topological Tensor Network States, with S. J. Denny, D. Jaksch and S. R. Clark, accepted Journal of Physics A: Mathematical and Theoretical (2011). arXiv:1108.0888 [4] Youtube series, Lectures on Tensor Network States, QIC 890/891 Selected Advanced Topics in Quantum Information, The University of Waterloo, Waterloo Ontario, Canada, (2011). http://www.qubit.org/iqc2011
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