Tensors and complexity in quantum information science

Transcription

1 Tensors in CS and geometry workshop, Nov. 11 (2014) Tensors and complexity in quantum information science Akimasa Miyake 1 CQuIC, University of New Mexico

2 Outline Subject: math of tensors <-> physics of entanglement Aim: talk by a quantum information theorist about elementary concepts/results and many open questions 1. Classification of multipartite entanglement - geometry of complex projective spaces - polynomial invariants as entanglement measures 2. Entanglement and its complexity - complexity of quantum computation in terms of tensors - coarser classification of multipartite entanglement

3 Classification of multipartite entanglement Miyake, Phys. Rev. A 67, (2003):dual variety, hyperdeterminants Miyake, Verstraete, Phys. Rev. A 69, (2004): 2x2x4 case

4 Stochastic LOCC Entanglement: quantum correlation, never increasing on average under Local Operations and Classical Communication(LOCC) single copy of a pure state (ray) in k 1 1,,k n 1 i 1,,i n =0 Ψ = ψ i1!i n i 1! i n H =! k 1! "k n Ψ Ψ! = M 1! M n Ψ Ψ! SLOCC: probabilistic M = SL ( ) j k j cf. LOCC: deterministic M j = U( k j ) etc. (invertible) CP map, followed by the postselection of a successful outcome SLOCC classification: classification of orbits of SL(k 1 )! SL(k n ) (in bipartite cases: classification by the Schmidt rank )

5 Bipartite entanglement under SLOCC two k-level qudits system (pure states) on H =! k! k Ψ = k 1 ψ ii i 12 1 i ( ) G= M ( ) 1 M2 T ψ 2 ii #### M 12 1 ψii M 12 2 i, i = basic matrix transformation the rank (Schmidt local rank) r s is invariant under invertible G: k equivalence classes decreasing under noninvertible G: totally ordered monotonicity!p k 2 1 Secant variety Segre variety

6 Encounter with tensors What is the SLOCC classification of 2x2x2 (3-qubit) case? [Gelfand, Kapranov, Zelevinsky Discriminants, Resultants, and Multidimensional Determinants (1994), Chapter 14, Example 4.5] [rediscovery in QIS: Dür, Vidal, Cirac, PRA 62, (2000); over 1300 citations] PHYSICAL REVIEW A, VOLUME 62, Three qubits can be entangled in two inequivalent ways W. Dür, G. Vidal, and J. I. Cirac Institut für Theoretische Physik, Universität Innsbruck, A-6020 Innsbruck, Austria Received 26 May 2000; published 14 November 2000 GHZ 1/ ) W 1/ ),

7 Dual variety point: a state hyperplane: a set of states Φ, orthogonal to Ψ F(Ψ,Φ) = ψ i1 ϕ i n i1 = 0 i n i 1,,i n Ψ dual set X x Xs.t. hyperplane F( ΨΦ, ) = 0 is tangent at x & ( ' ( X: set of completely separable states F(Ψ, x) = k 1 1,,k n 1 ψ i1 x (1) (n) i n i1!x in = 0 i 1,,i n =0 ( x ) i j have at least a nonzero solution x = (x (1),, x (n) ) X ( j ) F(Ψ, x) = 0, j = 1,,n, i j = 0,,k j 1 X : Det Ψ = 0 set of degenerate entangled states (if codim X =1, i.e., k j 1 (k j! 1) j) j" j

8 Hyperdeterminat Det 22Ψ = det Ψ = ψ00ψ11 ψ01ψ10 deg. 2 Det Ψ = ψ ψ + ψ ψ + ψ ψ + ψ ψ 2( ψ ψ ψ ψ ψ ψ ψ ψ + ψ ψ ψ ψ + ψ ψ ψ ψ + ψ ψ ψ ψ ψ ψ ψ ψ ) + 4( ψ ψ ψ ψ + ψ ψ ψ ψ ) ! SLOCC invariant entanglement monotones for genuine multipartite component C = 2 Det 22 Ψ τ = 4 Det 222 Ψ concurrence for 2-qubit pure states 3-tangle for 3-qubit pure states! the degree of n-qubit hyperdeterminants grows very rapidly deg. 4 n deg ,816 60, ,312 6,384,384...! generating function for the entanglement structure singularities of the hypersurface! degenerate entangled classes

9 3-qubit (2x2x2) SLOCC classification GHZ W B2 7 M 6 X 4 B1 3 X S= X node(1) B3 Xcusp Xnode(2) Xnode(3) X X X node X cusp dim 7: , GHZ M(= CP 7 ) X. dim 6: , W X X sing = X X cusp. dim 4: , , , biseparable B j Xnode (j) X for j =1, 2, 3. Xnode (j) =CP j-th 1 CP 3 are three closed irreducible components of Xsing = X cusp. dim 3: 000, completely separable S X = j=1,2,3 X node (j) =CP 1 CP 1 CP 1.

10 2x2x4 SLOCC classification! SLOCC orbits are added outside.! Duality suggests one subsystem is too large, compared to the rest. ( ) 2 2 n n 4 ; 2 2 4! LOCC conversions of distributed 4 qubits. parties classes 4 9 coarser finer

11 Entanglement measures in 2x2x4 case 1,1, n 1 Ψ = ψ i j k i, j, k= 0 ijk A B C By calculating local ranks ( r1, r2, r3) ofρi = tr j i Ψ Ψ! (2,2,4) generic class (dim 15)! (2,2,3) Clare s local operation: ψ ijk = 0( k 3) ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ Det Ψ = ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ " # = $ 0 dim14 0 dim13! (2,2,2) Clare s local operation: ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ijk = 0( k 2) Det Ψ = ( ψ ψ ψ ψ + ψ ψ ψ ψ + ψ ψ ψ ψ + ψ ψ ψ ψ ψ010ψ100ψ011ψ101) + 4( ψ000ψ011ψ101ψ110 + ψ001ψ010ψ100ψ111) " 0 GHZ (dim11) # $ = 0 W (dim10)! (1,2,2), (2,1,2), (2,2,1) Biseparable 1,2,3 (dim 8,8,6)! (1,1,1) Separable (dim 5)

12 SLOCC conversions among different classes noninvertible local operations unique maximally entangled class, lying on the top. Its representative is 2 Bell pairs: ( ) ( ) C AC BC 1 2 A B 5 graded partial order of 9 entangled classes

13 4-qubit hyperdeterminant Det 2222 Ψ = c 1 2 c 2 2 c 3 2 4c 0 c 2 3 c 3 2 4c 1 2 c 2 3 c 4 4c 1 3 c 3 3 6c 0 c 1 2 c 3 2 c 4 +16c 0 c 2 4 c 4 α γ +18c 0 c 1 c 2 c c 1 3 c 2 c 3 c 4 27c 0 2 c c 1 4 c c 0 c 1 c 2 2 c 3 c 4 128c 0 2 c 2 2 c c 0 c 1 2 c 2 c c 0 2 c 2 c 3 2 c 4 192c 0 2 c 1 c 3 c c 0 3 c qubit generic entangled states: ( ) + β( ) ( ) + δ( ) ( α β )( α γ )( α δ )( β γ )( β δ )( γ δ ) ( ) 2 Det Ψ = 0 (Vandermonde determinant) 2! infinitely many SLOCC orbits in generic entanglement! segmentation on the top of the partial order deg. 24 ( x + x ) c j = j j 4 j j x 222Ψ ψ ψ 0 x1 (4 )!! Det iii iii ! GHZ and W are not generic (i.e., Det n 2 Ψ = 0) for n >= 4 qubits

14 Asymptotic bipartite case Classification of entanglement when there are infinitely many identical copies A B A B LOCC Ψ + M = ( λ λ 11 ) M Φ + N = ( ) N LOCC is reversible when N = ME(Ψ) = M( λ logλ (1 λ)log(1 λ)) Entanglement entropy (uniqueness of entanglement measures) embedding into a symmetric subspace (cf. Veronese mapping)

15 Asymptotic case and MREGS Classification of multipartite entanglement with infinitely many identical copies is still open. Minimally Reversibly Entanglement Generating Set (MREGS)? Ψ M $ & & % & & '& ( ) n 1 ( ) n 2 ( ) n 3 Does it exist? If so, is it made by finite generators??

16 Outline Subject: math of tensors <-> physics of entanglement Aim: talk by a quantum information theorist about elementary concepts/results and many open questions 1. Classification of multipartite entanglement - geometry of complex projective spaces - polynomial invariants as entanglement measures 2. Entanglement and its complexity - complexity of quantum computation in terms of tensors - coarser classification of multipartite entanglement

17 Entanglement and its complexity Van den Nest, Miyake, Dür, Briegel, Phys. Rev. Lett. 97, (2006) Van den Nest, Dür, Miyake, Briegel, New. J. Phys. 9, 204 (2007)

18 quantum circuit as tensor composition entanglement can be generated by a quantum circuit a 1-qubit gate: SU(2) unitary map a 2-qubit gate: SU(4) unitary map changing the quality of entanglement! 2 "! 2! 2! 2 "! 2! 2

19 2-qubit Controlled-Z gate CZ = = σ z C = C = C = { 1, 2 }: { 1, 2,3 }: { 1, 2,3, 4 }: ( 0 1 ) + + = ( 0 1 ) = z x z x ( 0 1 ) = x z x x z x ( 0 1 σ z ) ( ) + z z x z x x z x Bell state GHZ state 1D cluster state (not GHZ) = z x z x z x z x z x z x z x z x

20 2D cluster state GHZ state 7-qubit codeword Multipartite entanglement as graphs [ review: Hein et al., quant-ph/ ] 1. For a graph G, vertices = qubits, edges = Ising-type interaction pattern. degree is the number of edges from a vertex. ( ab, ) edges ( ab, ) m ( a) ( b) z z 1 + = ( ) 2 G = CZ +, where CZ = diag(1,1,1, 1) σ σ 2. joint eigenstate of N commuting correlation operators for N qubits. K G = G, K = σ σ ( a) ( b) a a x b N z 3. any stabilizer state can be written as a graph state, up to local unitaries. a

21 Universality of quantum computation Using single-qubit projective measurements and classical feedforward of measurement outcomes (LOCC), it s capable to simulate any unitary gate operation on a (known) n-qubit input state deterministically it s capable to produce any corresponding output state ψ! N LOCC!!! φ! measured! #" ## $ 2D cluster states n { C, k = 1,2,... } k N -n were shown to be universal Definition of Universal resource (without encoding): A set of states Ψ = ψ 1, ψ 2,..., is universal for QC, if { } ψ k! N LOCC """ φ! measured! #" ## $ n N -n

22 ! entanglement width: distinguish 1D and 2D cluster states! geometric measure! Schmidt measure (logarithm of tensor rank)! localizable entanglement Criterion for universality Idea: if any significant entanglement feature exhibited by a set of universal resource states (ex. the 2D cluster states) is not available from another set Ψ, then it cannot be universal E( φ ) E( φ" ) whenever φ φ' LOCC A set of states Ψ cannot be universal for MQC if sup E( φ ) > sup E( ψ ) φ ψ Ψ sup E( φ ) sup E( C ) φ = k k

23 Entanglement width Any universal resource must have a diverging entanglement width. maximal entropy of entanglement associated with certain bipartitions Ewd( ψ )=min max ( ψ ) T E et e A T edge e subcubic tree T! non-increasing under LOCC (i.e. fulfills (P1))! equivalent to the rank-width in graph theory Ewd( C ) log ( 2) 1 k k 2 k+

24 Criterion by localizable entanglement (a,b) EL ( ψ ) [ Verstraete et al., PRL 92, (2004); Gour et al., PRA 72, (2005) ] entanglement (as measured by the concurrence) that can be at most created between two qubits (a,b) by LOCC! non-increasing under LOCC (fulfills (P1))! maximum number N LE of qubits such that E ( ψ ) = 1, ( a, b) N ( ab, ) L LE E ( C ) = 1, ( a, b) ( ab, ) L k k 2 NLE ( Ck k ) = k = m : unbounded Non-universal resource states:! all states such that E L < 1 (e.g. W states)! all states such that E L decays with distance (N LE is bounded )

25 Non-universal graph states Any set of the graph states whose entanglement-width is bounded for the number of qubits N is not a universal resource. interaction geometry determines the value as a resource! 1D cluster states (Ewd = 1, SM=N/2)! GHZ (tree) states, fully connected graphs! graphs with bounded tree-width or clique-width complementary to classical simulatibility (note universal QC is simply believed not to be classically simulatable efficiently) Measurement-based computation on some resource states (e.g. 1D cluster states and GHZ states) [Nielsen 05; Markov & Shi 05; Jozsa 06; Van den Nest et al. 06; ]

26 Tensor network as quantum computation SIAM J. COMPUT. c 2008 Society for Industrial and Applied Mathematics Vol. 38, No. 3, pp SIMULATING QUANTUM COMPUTATION BY CONTRACTING TENSOR NETWORKS IGOR L. MARKOV AND YAOYUN SHI Abstract. The treewidth of a graph is a useful combinatorial measure of how close the graph is to a tree. We prove that a quantum circuit with T gates whose underlying graph has a treewidth d can be simulated deterministically in T O(1) exp[o(d)] time, which, in particular, is polynomial in T if d = O(log T ). Among many implications, we show efficient simulations for log-depth circuits whose gates SIAM J. COMPUT. Vol. 39, No. 7, pp c 2010 Society for Industrial and Applied Mathematics QUANTUM COMPUTATION AND THE EVALUATION OF TENSOR NETWORKS ITAI ARAD AND ZEPH LANDAU Abstract. We present a quantum algorithm that additively approximates the value of a tensor network to a certain scale. When combined with existing results, this provides a complete problem for quantum computation. The result is a simple new way of looking at quantum computation in which unitary gates are replaced by tensors and time is replaced by the order in which the tensor network is swallowed. We use this result to derive new quantum algorithms that approximate the partition function of a variety of classical statistical mechanical models, including the Potts model.

27 Examples of universal resources graph states by 2D regular tilings are universal resources. hexagonal lattice square lattice (2D cluster state) triangular lattice degree = 3 degree = 4 degree = 6 minimum possible degree for uniform lattices to be universal More by other tensor network states [Gross, Eisert PRL 98, (2007); Gross et.al., PRA 76, (2008)]

28 Tensor network states Workshops Spring 2014 Tensor Networks and Simulations Apr. 21 Apr. 24, 2014 Program: Quantum Hamiltonian Complexity Add to Calendar View schedule & video» View abstracts» Organizers: Ignacio Cirac (Max Planck Institute, Garching), Frank Verstraete (University of Vienna). This workshop will focus on tensor network (MPS, PEPS, MERA) based simulations of strongly correlated quantum many body systems, relation to area laws, quantum spin glasses, as well as related work in mathematics, quantum chemistry and quantum information theory.

29 entropic entanglement width Schmidt-rank width Complexity of entanglement Universal resources are necessarily macroscopic entanglement! classically O(log( N)) efficiently simulatable universal resource families ON ( ) Schmidt measure (logarithm of tensor rank) maximum size of unit localizable entanglement geometric measure

30 Summary: open questions Subject: math of tensors <-> physics of entanglement Aim: talk by a quantum information theorist about elementary concepts/results and many open questions 1. Classification of multipartite entanglement - complete characterization and monogamy constraints (cf. two previous talks) - asymptotic case: minimally generating set 2. Entanglement and its complexity - necessary and sufficient condition for graph states to be universal - classes of tensor networks which are efficiently computable