jacob biamonte- penrose graphical calculus for tensor network states

8/3/2019 Jacob Biamonte- Penrose Graphical Calculus for Tensor Network States

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Penrose Graphical Calculusfor Tensor Network States

Jacob BiamonteCQT SingaporeTensor network states course homepage

http://www.qubit.org/iqc2011
http://www.qubit.org/iqc2011http://www.qubit.org/iqc2011


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Current collaborators

John Baez

Ville Bergholm

Stuart BroadfootStephen Clark

Oscar Dahlsten

Sam Denny

Dieter Jaksch

Tomi Johnson

Ann Kallin (UW)

Marco LanzagortaSebastian Meznaric

Alex Parent (UW)

Chris Wood (IQC, PI)

et al.


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Overview take I

Quantum Circuits

Classical CircuitsTensor Network StatesIsing Spin Models

Penrose Tensor Networks

Categorical Algebra


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Quantum Circuits

Temporally ordered or time sequencedAll maps are unitary so # inputs ='s # outputs

Describe quantum algorithmsUniversality result: every quantum state approximately prepared by a quantumcircuitA model of quantum computationConceptual understanding (in some cases compared to evolution under H)Complexity bounds (gate counts to simulate H)

Quantum Circuits are normally written backwards


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Examples of quantumcircuits

(With James Whitfield and A. Aspuru-Guzik) Molecular Physics,Volume 109, Issue 5 March 2011 , pages 735 - 750


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Quantum programminglanguge?

A programming language written across the page

using lines of text (1D), needs to describe theinherently two-dimensional nature of quantuminteractions in the plane.

Quantum circuits are inherently 2D.


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Quantum Circuit Logic

Gate familiesMatch gatesStabiliser gates

Rewrite rulesGate identities (these are symmetries)


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Classical Circuits

In mathematics, a (finitary) Boolean function (or switching function) is afunction of the form : B^k B, where B = {0, 1} is a Boolean domainand k is the arity of the function.

Asynchronous circuits for every such Boolean functionUniversal gate families (need boolean non-linearity)A model of computationComplexity bounds on circuit families

Decomposition methods, synthesis, Shannon & Davio expansions


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Classical Circuit Example(adder)

I t ti ( l i l


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Intersection (classicalquantum circuits ~ quantum

classical circuits)

The intersection between quantum and classicalcircuits is currently taken to be reversiblecircuits.

...However, we will go past this!


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Tensor Network States

Algorithms 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)
http://www.qubit.org/iqc2011http://pirsa.org/11060004/http://pirsa.org/11060004/http://www.qubit.org/iqc2011


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Tensor Network StatesExamples


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Ising Spin Models

Energy penalties

Spin configurationsEach spin can take either of two values


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Penrose Tensor Networks

Graphical depiction of tensors

CompositionalityDiagrams 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).
http://www.qubit.org/content/biamonte/penrose-applications-of-negative-dimensional-tensors-1971.pdfhttp://www.qubit.org/content/biamonte/penrose-applications-of-negative-dimensional-tensors-1971.pdfhttp://www.qubit.org/content/biamonte/penrose-applications-of-negative-dimensional-tensors-1971.pdf


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Seeing tensors[Penrose, 1971]




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Cups, caps, snake equation




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Emphasis of input/outputequivalence




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Tensors for algorithms




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Graphical rewrite system




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Graphical Calculus forQuantum Theory [Penrose]

Page 659

Page 802


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Categorical Algebra

Duality, Pairing, abstraction as a uniting tool.Precise, clear definitions

Pay entrance fee to join the conversation

Baez-Dolan Dagger Compact Categories describeQuantum Theory [1995]

Refining Penrose Tensor


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Refining Penrose TensorCalculus

[Lafont]

Y. Lafont, Penrose diagrams and 2-dimensional rewriting, in Applications of Categories in Computer Science, LondonMathematical Society Lecture Note Series 177, p. 191-201, Cambridge University Press (1992).

Ab i
http://www.google.co.uk/url?sa=t&source=web&cd=4&sqi=2&ved=0CC8QFjAD&url=http%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fdownload%3Fdoi%3D10.1.1.53.3029%26rep%3Drep1%26type%3Dps&rct=j&q=Y.%20Lafont%2C%20Penrose%20diagrams%20and%202-dimensional%20rewriting&ei=DacOTpiINsr9sQKWybmdCg&usg=AFQjCNFLoOhjNp-pnjkSzsYPN9G9F8YKhg&cad=rjahttp://www.google.co.uk/url?sa=t&source=web&cd=4&sqi=2&ved=0CC8QFjAD&url=http%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fdownload%3Fdoi%3D10.1.1.53.3029%26rep%3Drep1%26type%3Dps&rct=j&q=Y.%20Lafont%2C%20Penrose%20diagrams%20and%202-dimensional%20rewriting&ei=DacOTpiINsr9sQKWybmdCg&usg=AFQjCNFLoOhjNp-pnjkSzsYPN9G9F8YKhg&cad=rjahttp://www.google.co.uk/url?sa=t&source=web&cd=4&sqi=2&ved=0CC8QFjAD&url=http%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fdownload%3Fdoi%3D10.1.1.53.3029%26rep%3Drep1%26type%3Dps&rct=j&q=Y.%20Lafont%2C%20Penrose%20diagrams%20and%202-dimensional%20rewriting&ei=DacOTpiINsr9sQKWybmdCg&usg=AFQjCNFLoOhjNp-pnjkSzsYPN9G9F8YKhg&cad=rja


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Abstract tensor rewritesystem [Lafont]

Y. Lafont,Penrose diagrams and 2-dimensional rewriting, inApplications of Categories in Computer Science,London Mathematical Society Lecture Note Series177, p. 191-201, Cambridge University Press(1992).
http://www.google.co.uk/url?sa=t&source=web&cd=4&sqi=2&ved=0CC8QFjAD&url=http%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fdownload%3Fdoi%3D10.1.1.53.3029%26rep%3Drep1%26type%3Dps&rct=j&q=Y.%20Lafont%2C%20Penrose%20diagrams%20and%202-dimensional%20rewriting&ei=DacOTpiINsr9sQKWybmdCg&usg=AFQjCNFLoOhjNp-pnjkSzsYPN9G9F8YKhg&cad=rjahttp://www.google.co.uk/url?sa=t&source=web&cd=4&sqi=2&ved=0CC8QFjAD&url=http%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fdownload%3Fdoi%3D10.1.1.53.3029%26rep%3Drep1%26type%3Dps&rct=j&q=Y.%20Lafont%2C%20Penrose%20diagrams%20and%202-dimensional%20rewriting&ei=DacOTpiINsr9sQKWybmdCg&usg=AFQjCNFLoOhjNp-pnjkSzsYPN9G9F8YKhg&cad=rjahttp://www.google.co.uk/url?sa=t&source=web&cd=4&sqi=2&ved=0CC8QFjAD&url=http%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fdownload%3Fdoi%3D10.1.1.53.3029%26rep%3Drep1%26type%3Dps&rct=j&q=Y.%20Lafont%2C%20Penrose%20diagrams%20and%202-dimensional%20rewriting&ei=DacOTpiINsr9sQKWybmdCg&usg=AFQjCNFLoOhjNp-pnjkSzsYPN9G9F8YKhg&cad=rja


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A Prehistory of n-Categorical PhysicsAuthors: John C. Baez,Aaron Lauda http://arxiv.org/abs/0908.2469

Quantum groups, Christian Kassel, Springer, 1995

Frobenius algebras and 2D topological quantum field theories, Joachim Kock,

Cambridge University Press, 2004
http://arxiv.org/find/hep-th/1/au:+Baez_J/0/1/0/all/0/1http://arxiv.org/find/hep-th/1/au:+Lauda_A/0/1/0/all/0/1http://arxiv.org/find/hep-th/1/au:+Lauda_A/0/1/0/all/0/1http://arxiv.org/abs/0908.2469http://arxiv.org/abs/0908.2469http://arxiv.org/find/hep-th/1/au:+Lauda_A/0/1/0/all/0/1http://arxiv.org/find/hep-th/1/au:+Baez_J/0/1/0/all/0/1


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Unification

The network models we have considered are alldifferent (it would seem)...

...How can we relate them?


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Overview take II

Classical Circuits + Spin Models

Quantum CircuitsTensor Network States


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Ground State Spin Logic

JB, Physical Review A 77 052331. 2008.


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Composing Gates


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We are dealing with spans


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Quantum Networks

Penrose (Wire Bending)


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Penrose (Wire Bending)Duality


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Bell states vs Pauli basis


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Boolean States


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Boolean Tensor Networks


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AND-tensors


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COPY-, XOR-tensors


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Quantum AND-tensors

W


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W-state

Boolean States vs Spin


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Boolean States vs SpinModels

Spins

States

A li ti 3SAT


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Application: 3SAT

(with Tomi Johnson, Stephen Clark, Dieter Jaksch)

Connection to quantum


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Connection to quantumcircuits

Connection to Vidal's MERA


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Connection to Vidal s MERA

Connection to Vidal's MERA

The category of quantum


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g y qcircuits

Connection to quantum


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qcircuits

Return to Penrose's


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graphical denity state

Page 802



Diagrammatic SVD


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Diagrammatic SVD

Algebraic Invariant Theory for Matrix Product States(with Ville Berghlom and Marco Lanzagortat)

Map state duality


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Map state duality


Purification


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Purification


Entanglement topology


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Entanglement topology


MPS


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MPS

Polynomial Invariants


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Polynomial Invariants


Invariants of mixed states


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Invariants of mixed states

Pure vs mixed invarinats


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Pure vs mixed invarinats

Applications


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Applications

Entanglement Spectrum

Reyni Entropy

Estimating Rank

(with Ann Kallin and others)

General methods to factor


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states

Stabilizer Tensors


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Stabilizer Tensors

(with Oscar Dahlsten and others)

Preparing states AlexP t


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Parent

We are currently considered applications ofthese methods to state preparation usingquantum circuits

Strong optimality: Have one degree of freedom inthe circuit, for every degree of freedom in thestate.Gate rewrites are equivalent to symmetries in a

state

(with Alex Parent and others)

Open quantum systems Ch i W d


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Chris Wood

Tensor networks for open systems

2D tensor networks SamD


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Denny

Algebraically contractible topological tensor network states,

S. J. Denny, JB, Jaksch and Clark. (2011). 1108.0888

Invariants and covariants ofsymmetric tensors


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symmetric tensors

Thanks to currentcollaborators


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collaborators

John Baez Ville Bergholm

Stuart Broadfoot

Stephen Clark

Oscar Dahlsten

Sam Denny Dieter Jaksch

Tomi Johnson Ann Kallin (UW)

Marco Lanzagorta

Sebastian Meznaric

Alex Parent (UW)

Chris Wood (IQC, PI) et al.

A Benchmark for Species


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jacob biamonte- penrose graphical calculus for tensor network states

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