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A Monomial matrix formalism to describe quantum many-body states

A Monomial matrix formalism to describe quantum many-body states. Maarten Van den Nest Max Planck Institute for Quantum Optics. arXiv:1108.0531. Montreal, October 19 th 2011. Motivation Generalizing the Pauli stabilizer formalism. The Pauli stabilizer formalism (PSF).

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A Monomial matrix formalism to describe quantum many-body states

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  1. A Monomial matrix formalism to describe quantum many-body states Maarten Van den Nest Max Planck Institute for Quantum Optics arXiv:1108.0531 Montreal, October 19th 2011

  2. MotivationGeneralizing the Pauli stabilizer formalism

  3. The Pauli stabilizer formalism (PSF) • The PSF describes joint eigenspaces of sets of commuting Pauli operators i: i| = | i = 1, …, k • Encompasses important many-body states/spaces: cluster states, GHZ states, toric code, … • E.g. 1D cluster state:i = Zi-1 Xi Zi+1 • The PSF is used in virtually allsubfields of QIT: • Quantum error-correction, one-way QC, classical simulations, entanglement purification, information-theoretic protocols, …

  4. Aim of this work • Why is PSF so successful? • Stabilizer picture offers efficient description • Interestingquantities can be efficiently computedfrom this description (e.g. local observables, entanglement entropy, …) • More generally: understand properties of states by manipulating their stabilizers • What are disadvantages of PSF? • Small class of states • Special properties: entanglement maximal or zero, cannot occur as unique ground states of two-local hamiltonians, commuting stabilizers, (often) zero correlation length… • Aim of this work: Generalize PSF by using larger class of stabilizer operators + keep pros and get rid of cons….

  5. Outline I. Monomial stabilizers: definitions + examples II. Main characterizations III. Computational complexity & efficiency IV. Outlook and conclusions

  6. I. Monomial stabilizersDefinitions + examples

  7. M-states/spaces • Observation:Pauli operators are monomial unitarymatrices • Precisely one nonzero entry per row/column • Nonzero entries are complex phases • M-state/space: arbitrary monomial unitary stabilizer operators Ui Ui| = | i = 1, …, m • Restrict toUiwith efficiently computable matrix elements • E.g. k-local, poly-size quantum circuit of monomial operators, …

  8. Examples • M-states/spaces encompass many importantstate families: • All stabilizer states and codes (also for qudits) • AKLT model • Kitaev’s abelian + nonabelian quantum doubles • W-states • Dicke states • Coherent probabilistic computations • LME states (locally maximally entanglable) • Coset states of abelian groups • …

  9. Example: AKLT model • 1D chain of spin-1 particles (open or periodic boundary conditions) • H = I-Hi,i+1 where Hi,i+1 is projector on subspace spanned by • Ground level = zero energy: all |ψ withHi,i+1 |ψ = |ψ • Considermonomial unitary U: • Ground level = all |ψ withUi,i+1 |ψ = |ψ and thus M-space

  10. II. Main characterizationsHow are properties of state/space reflected in properties of stabilizer group? Notation: computational basis |x, |y, …

  11. Two important groups M-space Ui| = | i = 1, …, m • Stabilizer group = (finite) group generated by Ui • Permutation group  • Every monomial unitary matrix can be written as U = PD with P permutation matrix and Ddiagonal matrix. Call U := P • Define  := {U : U }= group generated by Ui • Orbits:Ox= orbit of comp. basis state |x under action of • |y Ox iff there exists U  and phase  s.t. U|x = |y

  12. Characterizing M-states • Consider M-state |ψ and fix arbitrary |x such that ψ|x  0 • Claim 1: All amplitudes are zero outside orbit Ox: • Claim 2:All nonzero amplitudes y|ψ have equal modulus • For all |y Ox there exists U  and phase  s.t. U|x = |y • Then y|ψ =  x|U*|ψ =  x|ψ • Phase  is independent of U:  = x(y)

  13. M-states are uniform superpositions • Fix arbitrary |x such that ψ|x  0 • All amplitudes are zero outside orbit Ox • All nonzero amplitudes have equal modulus with phase x(y) |ψ is uniform superposition over orbit • Recipe to compute x(y): • Find any U  such that s.t. U|x = |y for some ; then  = x(y) • (Almost) complete characterization in terms of stabilizer group

  14. Which orbit is the right one? • For every |x let x be the subgroup of all U which have |xas eigenvector. Then: Oxis the correct orbitiffx|U|x = 1 for all U x • Example: GHZ state with stabilizers Zi Zi+1and X1 …Xn. • Ox = {|x, |x + d} where d = (1, …, 1) • xgenerated by Zi Zi+1for every x • Therefore O0 = {|0, |d} is correct orbit

  15. M-spaces and the orbit basis • Use similar ideas to construct basis of any M-space (orbit basis) B = {|ψ1, … |ψd } • Each basis state is uniform superposition over some orbit • These orbits are disjoint ( dimension bounded by total # of orbits!) • Phases x(y) + “good” orbits can be computed analogous to before Computational basis |ψ1 |ψ2 … |ψd

  16. Example: AKLT model (n even) • Recall: monomial stabilizer for particles i and i+1 • Generators of permutation group: replace +1 by -1 • There are 4 Orbits: • All basis states with even number of |0s, |1s and |2s • All basis states with odd number of |0s and even number of |1s, |2s • All basis states with odd number of |1s and even number of |0s, |2s • All basis states with odd number of |2s and even number of |0s, |1s • Corollary: ground level at most 4-fold degenerate

  17. Example: AKLT model (n even) • Orbit basis for open boundary conditions: • Unique ground state for periodic boundary conditions:

  18. III. Computational complexityand efficiency

  19. NP hardness • Consider an M-state |ψ described in terms of diagonal unitary stabilizers acting on at most 3 qubits. • Problem 1: Compute (estimate) single-qubit reduced density operators (with some constant error) • Problem 2: Classically sample the distribution |x|ψ|2 • Both problems are NP-hard (Proof: reduction to 3SAT) • Under which conditions are efficient classical simulations possible?

  20. Efficient classical simulations • Consider M-state |ψ Then |x|ψ|2can be sampled efficiently classicallyif the following problems have efficient classical solutions: • Find an arbitrary |x such that ψ|x  0 • Generate uniformly random element from the orbit of |x • Additional conditions to ensure that local expectation values can be estimated efficiently classically • Given y, does |x belong to orbit of x? • Given y in the orbit of x, compute x(y) • Note: Simulations via sampling (weak simulations)

  21. Efficient classical simulations • Turns out: this general classical simulation method works for all examples given earlier • Pauli stabilizer states (also for qudits) • AKLT model • Kitaev’s abelian + nonabelian quantum doubles • W-states • Dicke states • LME states (locally maximally entanglable) • Coherent probabilistic computations • Coset states of abelian groups • Yields unified method to simulate a number of state families

  22. IV. Conclusions and outlook

  23. Conclusions & Outlook • Goal of this work was to demonstrate that: (1) M-states/spaces contain relevant state families, well beyond PSF (2) Properties of M-states/-spaces can transparently be understood by manipulating their monomial stabilizer groups (3) NP-hard in general but efficient classical simulations for interesting subclass • Many questions: • Construct new state families that can be treated with MSF • 2D version of AKLT • Connection to MPS/PEPS • Physical meaning of monomiality • …

  24. Thank you!

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