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!