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 Motivation Generalizing the Pauli stabilizer formalism 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 all subfields of QIT: Quantum error-correction, one-way QC, classical simulations, entanglement purification, information-theoretic protocols, … Aim of this work Why is PSF so successful? What are disadvantages of PSF? Stabilizer picture offers efficient description Interesting quantities can be efficiently computed from this description (e.g. local observables, entanglement entropy, …) More generally: understand properties of states by manipulating their stabilizers 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…. Outline I. Monomial stabilizers: definitions + examples II. Main characterizations III. Computational complexity & efficiency IV. Outlook and conclusions I. Monomial stabilizers Definitions + examples M-states/spaces Observation: Pauli operators are monomial unitary matrices 0 1 X= 1 0 0 i Y= -i 0 Precisely one nonzero entry per row/column Nonzero entries are complex phases M-state/space: arbitrary monomial unitary stabilizer operators Ui Ui | = | 1 0 Z= 0 -1 i = 1, …, m Restrict to Ui with efficiently computable matrix elements E.g. k-local, poly-size quantum circuit of monomial operators, … 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 … 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 ψ1 01 10 ψ3 12 21 ψ2 02 20 ψ4 00 11 22 Ground level = zero energy: all |ψ with Hi,i+1 |ψ = |ψ Consider monomial unitary U: 01 10 12 21 02 20 00 11 22 00 Ground level = all |ψ with Ui,i+1 |ψ = |ψ and thus M-space II. Main characterizations How are properties of state/space reflected in properties of stabilizer group? Notation: computational basis |x, |y, … 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 D diagonal 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 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: ψ ψ ψx = 1 |G| Ux UG = c y Ox y y 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) 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 ψ ξ y Ox x (y) y 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 Which orbit is the right one? ψ ξ y Ox x (y) y For every |x let x be the subgroup of all U which have |x as eigenvector. Then: Ox is the correct orbit iff x|U|x = 1 for all U x Example: GHZ state with stabilizers Zi Zi+1 and X1 …Xn. Ox = {|x , |x + d } where d = (1, …, 1) x generated by Zi Zi+1 for every x Therefore O0 = {|0 , |d } is correct orbit 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 |ψ1 |ψ2 … |ψd Computational basis Example: AKLT model (n even) Recall: monomial stabilizer for particles i and i+1 01 10 12 21 02 20 00 11 22 00 Generators of permutation group: replace +1 by -1 There are 4 Orbits: All All All All basis basis basis basis states states states states with with with with even number of |0s, |1s and |2s odd number of |0s and even number of |1s, |2s odd number of |1s and even number of |0s, |2s odd number of |2s and even number of |0s, |1s Corollary: ground level at most 4-fold degenerate Example: AKLT model (n even) 01 10 12 21 02 20 00 11 22 00 Orbit basis for open boundary conditions: ψ = Tr σσ ...σ σ a1 an a ...a 1 n σ = I, X, Y,Z σ0 = X, σ1 = Y, σ 2 = Z Unique ground state for periodic boundary conditions: ψ = Tr σ a1 ...σ an a1...an III. Computational complexity and efficiency 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? Efficient classical simulations Consider M-state |ψ Then |x|ψ|2 can be sampled efficiently classically if the following problems have efficient classical solutions: Additional conditions to ensure that local expectation values can be estimated efficiently classically Find an arbitrary |x such that ψ|x 0 Generate uniformly random element from the orbit of |x 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) 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 IV. Conclusions and outlook 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 … Thank you!