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