Jacob Biamonte- Penrose Graphical Calculus for Tensor Network States
Transcript of 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]
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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Cups, caps, snake equation
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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Emphasis of input/outputequivalence
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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Tensors for algorithms
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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Graphical rewrite system
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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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
Applications of negative dimensional tensors, Rodger Penrose
in Combinatorial Mathematics and its Applications, Academic Press (1971).
Diagrammatic SVD
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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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
Algebraic Invariant Theory for Matrix Product States(with Ville Berghlom and Marco Lanzagortat)
Purification
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Purification
Algebraic Invariant Theory for Matrix Product States(with Ville Berghlom and Marco Lanzagortat)
Entanglement topology
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Entanglement topology
Algebraic Invariant Theory for Matrix Product States(with Ville Berghlom and Marco Lanzagortat)
MPS
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MPS
Polynomial Invariants
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Polynomial Invariants
Algebraic Invariant Theory for Matrix Product States(with Ville Berghlom and Marco Lanzagortat)
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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p