a monomial matrix formalism to describe quantum many-body states maarten van den nest max planck...
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A Monomial matrix formalism to
describe quantum many-body states
Maarten Van den Nest
Max Planck Institute for Quantum Optics
Montreal, October 19th 2011
arXiv:1108.0531
Motivation
Generalizing the Pauli stabilizer formalism
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 Encompasses important many-body states/spaces: cluster states, states,
GHZ states, toric code, …GHZ states, toric code, … E.g. 1D cluster state:E.g. 1D cluster state: i = Zi-1 Xi Zi+1
The PSF is used in virtually allThe PSF is used in virtually all subfields of QIT:subfields of QIT: Quantum error-correction, one-way QC, classical simulations, Quantum error-correction, one-way QC, classical simulations,
entanglement purification, information-theoretic protocols, …entanglement purification, information-theoretic protocols, …
Aim of this work Why is PSF so successful?Why is PSF so successful?
Stabilizer picture offers Stabilizer picture offers efficient description InterestingInteresting quantities can be can be efficiently computed from this from this
description (e.g. local observables, entanglement entropy, …)description (e.g. local observables, entanglement entropy, …) More generally: More generally: understand properties of states by properties of states by
manipulating their stabilizersmanipulating their stabilizers
What are disadvantages of PSF?What are disadvantages of PSF? Small class of states class of states Special properties: entanglement maximal or zero, cannot occur as : entanglement maximal or zero, cannot occur as
unique ground states of two-local hamiltonians, commuting stabilizers, unique ground states of two-local hamiltonians, commuting stabilizers,
(often) zero correlation length…(often) zero correlation length…
Aim of this workAim of this work: Generalize PSF by using larger class of : Generalize PSF by using larger class of
stabilizer operators + keep pros and get rid of cons….stabilizer operators + keep pros and get rid of cons….
Outline
I.I. Monomial stabilizers: definitions + Monomial stabilizers: definitions + examplesexamples
II.II. Main characterizationsMain characterizations
III.III. Computational complexity & efficiencyComputational complexity & efficiency
IV.IV. Outlook and conclusionsOutlook and conclusions
I. Monomial stabilizers
Definitions + examples
M-states/spaces
Observation: Pauli operators are Pauli operators are monomial unitary matricesmatrices
Precisely one nonzero entry per row/columnPrecisely one nonzero entry per row/column Nonzero entries are complex phasesNonzero entries are complex phases
M-state/space: M-state/space: arbitraryarbitrary monomial unitary stabilizer operators monomial unitary stabilizer operators UUii
UUii | | = | = | i = 1, …, m i = 1, …, m
Restrict toRestrict to UUii with efficiently computable matrix elements 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, …
0 1X=
1 0
0 iY=
-i 0
1 0Z=
0 -1
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)1D chain of spin-1 particles (open or periodic boundary conditions) H = H = I-H I-Hi,i+1 where HHi,i+1 is projector on subspace spanned by
Ground level = zero energy: all ||ψψ withwith HHi,i+1 ||ψψ = | = |ψψ ConsiderConsider monomial unitary monomial unitary UU::
Ground level = all ||ψψ withwith UUi,i+1 ||ψψ = | = |ψψ and thus and thus M-spaceM-space
1ψ 01 10
2ψ 02 20
3ψ 12 21
4ψ 00 11 22
01 10
02 20
12 21
00 11 22 00
II. Main characterizations
How are properties of state/space reflected in properties of stabilizer group?
Notation:Notation: computational basis computational basis |x, , |y, , ……
Two important groups
M-space M-space UUii | | = | = | i = 1, …, m 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 under action of |y Ox iff there exists U iff there exists U and phase and phase s.t. U s.t. U|x = = |y
Characterizing M-states
Consider M-stateConsider M-state |ψ and fix arbitrary fix arbitrary |x such that such that ψψ||xx 0 0
Claim 1:Claim 1: All amplitudes are zero outside orbit All amplitudes are zero outside orbit Ox:
Claim 2:Claim 2: All nonzero amplitudes y|ψ have equal modulus
For all |y Ox there exists U there exists U and phase and phase s.t. U s.t. U|x = = |y Then Then yy||ψψ = = xx||UU**||ψψ = = xx||ψψ
Phase Phase is independent of U: is independent of U: = = xx(y)(y)
1
| |
xUy
y O
ψ ψ x = U xψ = c y GG
M-states are uniform superpositions
Fix arbitrary Fix arbitrary |x such that such that ψψ||xx 0 0 All amplitudes are zero outside orbit All amplitudes are zero outside orbit Ox
All nonzero amplitudes have equal modulus with phase All nonzero amplitudes have equal modulus with phase xx(y)(y)
||ψψ is is uniform superpositionuniform superposition over orbit over orbit
Recipe to compute Recipe to compute xx(y): (y): Find any U Find any U such that such that s.t. Us.t. U|x = = |y for some for some ; then ; then = = xx(y) (y)
(Almost) complete characterization in terms of stabilizer group(Almost) complete characterization in terms of stabilizer group
x
xy O
ψ ξ (y) y
Which orbit is the right one?
For every For every |x let let x be the subgroup of all U be the subgroup of all U which have |x as eigenvector. Then:. Then:
Ox is the correct orbit iff x|UU|x = 1 for all U = 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
x
xy O
ψ ξ (y) y
M-spaces and the orbit basis
Use similar ideas to construct basis of any M-space (orbit Use similar ideas to construct basis of any M-space (orbit basis)basis)
= {= {||ψψ11, … |, … |ψψdd }}
Each basis state is Each basis state is uniform superpositionuniform superposition over some orbit over some orbit These orbits are These orbits are disjointdisjoint ( ( dimension bounded by total # of dimension bounded by total # of
orbits!)orbits!) Phases Phases xx(y)(y) + “good” orbits can be computed analogous to + “good” orbits can be computed analogous to
before before Computational basis
||ψψ11
||ψψ22
||ψψdd
……
Example: AKLT model (n even)
Recall: monomial stabilizer for particles i and i+1Recall: monomial stabilizer for particles i and i+1
Generators of permutation group: replace +1 by -1Generators of permutation group: replace +1 by -1
There areThere are 4 Orbits: 4 Orbits: All basis states with even number of |0All basis states with even number of |0s, |1s, |1s and |2s and |2ss All basis states with odd number of |0All basis states with odd number of |0s and even number of |1s and even number of |1s, |2s, |2ss All basis states with odd number of |1All basis states with odd number of |1s and even number of |0s and even number of |0s, |2s, |2ss All basis states with odd number of |2All basis states with odd number of |2s and even number of |0s and even number of |0s, |1s, |1ss
Corollary: ground level Corollary: ground level at most 4-foldat most 4-fold degenerate degenerate
01 10
02 20
12 21
00 11 22 00
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:
1 na aσ1 nψ = Tr σσ ...σ a ...a
σ=I,X,Y,Z
0 1 2σ =X, σ =Y, σ =Z
01 10
02 20
12 21
00 11 22 00
1 na a1 nψ = Tr σ ...σ a ...a
III. Computational complexityand efficiency
NP hardness
Consider an M-state Consider an M-state ||ψψ described in terms of described in terms of diagonaldiagonal unitary unitary
stabilizers acting on at most 3 qubits. stabilizers acting on at most 3 qubits.
Problem 1:Problem 1: Compute (estimate) single-qubit reduced Compute (estimate) single-qubit reduced densitydensity
operators (with some constant error)operators (with some constant error)
Problem 2:Problem 2: Classically sample the distribution | Classically sample the distribution |x|x|ψψ||22
Both problems are Both problems are NP-hardNP-hard (Proof: reduction to 3SAT) (Proof: reduction to 3SAT)
Under which conditions are efficient classical simulations Under which conditions are efficient classical simulations possible?possible?
Efficient classical simulations
Consider M-state |Consider M-state |ψψ
Then |Then |x|x|ψψ||22 can be sampled efficiently classically can be sampled efficiently classically if the if the
following problems have efficient classical solutions:following problems have efficient classical solutions:
Find an arbitrary Find an arbitrary |x such that such that ψ|xψ|x 0 0 Generate uniformly random element from the orbit of Generate uniformly random element from the orbit of |x
Additional conditions to ensure that Additional conditions to ensure that local expectation local expectation
valuesvalues can be estimated efficiently classically can be estimated efficiently classically
Given Given y, does , does |x belong to orbit of x? belong to orbit of x? Given y in the orbit of x, compute Given y in the orbit of x, compute xx(y)(y)
Note: Simulations via Note: Simulations via samplingsampling (weak simulations) (weak simulations)
Efficient classical simulations
Turns out: this general classical simulation method works for Turns out: this general classical simulation method works for
allall examples given earlier 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 Yields unified methodunified method to simulate a number of state families to simulate a number of state families
IV. Conclusions and outlook
Conclusions & Outlook
Goal of this work was to demonstrate that:Goal of this work was to demonstrate that:
(1)(1) M-states/spaces contain relevant state families, well beyond M-states/spaces contain relevant state families, well beyond PSFPSF
(2)(2) Properties of M-states/-spaces can transparently be Properties of M-states/-spaces can transparently be understood by manipulating their monomial stabilizer groupsunderstood by manipulating their monomial stabilizer groups
(3)(3) NP-hard in general but efficient classical simulations for NP-hard in general but efficient classical simulations for interesting subclassinteresting subclass
Many questions:Many questions: Construct new state families that can be treated with MSFConstruct new state families that can be treated with MSF 2D version of AKLT2D version of AKLT Connection to MPS/PEPSConnection to MPS/PEPS Physical meaning of monomialityPhysical meaning of monomiality ……
Thank you!