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

2. Motivation Generalizing the Pauli stabilizer formalism

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

4. Aim of this work Why is PSF so successful? Why is PSF so successful? Stabilizer picture offers Stabilizer picture offers efficient description Interesting can be from this Interesting quantities can be efficiently computed from this description (e.g. local observables, entanglement entropy, …) More generally: properties of states by More generally: understand properties of states by manipulating their stabilizers What are disadvantages of PSF? What are disadvantages of PSF? class of states Small class of states : entanglement maximal or zero, cannot occur as 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 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 stabilizers Definitions + examples

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

8. Examples M-states/spaces encompass many important state 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) 1D chain of spin-1 particles (open or periodic boundary conditions) H =   I-HH H =   I-H i,i+1 where H i,i+1 is projector on subspace spanned by | ψ  with H| ψ  = | ψ  Ground level = zero energy: all | ψ  with H i,i+1 | ψ  = | ψ  Consider monomial unitary U: Consider monomial unitary U: | ψ  with U| ψ  = | ψ  and thus M-space Ground level = all | ψ  with U i,i+1 | ψ  = | ψ  and thus M-space

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

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

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

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

14. Which orbit is the right one? For every  let be the subgroup of all U  . Then: For every |x  let  x be the subgroup of all U   which have |x  as eigenvector. Then:  U  = 1 for all U  O x is the correct orbit iff  x|U|x  = 1 for all U   x Example: GHZ state with stabilizers Z i Z i+1 and X 1 …X n.   O x = {|x , |x + d  } where d = (1, …, 1)  x generated by Z i Z i+1 for every x   Therefore O 0 = {|0 , |d  } is correct orbit

15. M-spaces and the orbit basis Use similar ideas to construct basis of any M-space (orbit basis) Use similar ideas to construct basis of any M-space (orbit basis) B = {| ψ 1 , … | ψ d  } Each basis state is uniform superposition over some orbit Each basis state is uniform superposition over some orbit These orbits are disjoint (  dimension bounded by total # of orbits!) These orbits are disjoint (  dimension bounded by total # of orbits!) Phases  x (y) + “good” orbits can be computed analogous to before Phases  x (y) + “good” orbits can be computed analogous to before Computational basis |ψ1|ψ1|ψ1|ψ1 |ψ2|ψ2|ψ2|ψ2 |ψd|ψd|ψd|ψd …

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

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

18. III. Computational complexity and efficiency

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

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

21. Efficient classical simulations Turns out: this general classical simulation method works for 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 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: 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 understood by manipulating their monomial stabilizer groups (3) NP-hard in general but efficient classical simulations for interesting subclass interesting subclass Many questions: Many questions: Construct new state families that can be treated with MSF Construct new state families that can be treated with MSF 2D version of AKLT 2D version of AKLT Connection to MPS/PEPS Connection to MPS/PEPS Physical meaning of monomiality Physical meaning of monomiality …

24. Thank you!