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:


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





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
UG
=
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


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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:

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All
All
All
All
basis
basis
basis
basis
states
states
states
states
with
with
with
with
even number of |0s, |1s and |2s
odd number of |0s and even number of |1s, |2s
odd number of |1s and even number of |0s, |2s
odd number of |2s and even number of |0s, |1s
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:

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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

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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:
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Construct new state families that can be treated with MSF
2D version of AKLT
Connection to MPS/PEPS
Physical meaning of monomiality
…
Thank you!