Less is more: Exploiting structure in high-dimensional quantum - - PowerPoint PPT Presentation
Less is more: Exploiting structure in high-dimensional quantum tomography and other problems Jens Eisert FU Berlin Mentions joint work with several people, of which are here Adrian Steffens Carlos Riofrio David Gross Richard Nickl
Jens Eisert FU Berlin
Adrian Steffens Carlos Riofrio David Gross Richard Nickl Alexandra Carpentier Thomas Monz
Mentions joint work with several people, of which are here
Quantum state tomography
Prepare i.i.d. quantum systems Reconstruct unknown density operator ,
ρ ∈ S(H)
For -dimensional quantum systems, expectation values are required
d
E.g., 1.099.511.627.775 expectation values for 20 spins, Take measurements
Quantum state tomography
Prepare i.i.d. quantum systems Reconstruct unknown density operator ,
ρ ∈ S(H)
For -dimensional quantum systems, expectation values are required
d
E.g., 1.099.511.627.775 expectation values for 20 spins, Take measurements ’Curse of dimensionality’ in quantum state estimation and tomography More pronounced if confidence sets and statistical aspects considered Can we do better? Yes! Significantly so In a way… Structure is the key
Mind map of the talk
Compressed sensing Estimators and experimental compressed sensing Structured problems: Quantum field tomography Direct certification Into the wild: tensor completion and bilinear problems
Compressed sensing and matrix completion
Gene expression data matrices
Columns observations (samples of DNA arrays) Typical numbers 6-10K genes,
50-150 samples, many entries missing
Predict missing entries?
Genes Observations
Netflix offered $1M for practical solution to
rating problem
User rate videos... but obviously only some
Candes, Plan, arXiv:0903.3131 Toescher, Jahrer, Bell (2009)
Netflix prize
Movies Users
Predict missing entries? People's tastes not random, linear dependencies Matrix is low rank
Candes, Plan, arXiv:0903.3131 Toescher, Jahrer, Bell (2009)
Can one reconstruct a low rank matrix from knowing a few entries?
r
Matrix completion
Knowing the structure allows for recovery: entries
r
O(rn)
Can one reconstruct a low rank matrix from knowing a few entries?
r
Matrix completion
Not all pairs of matrices and measurements allow for reconstruction
Needs some incoherence Can one reconstruct a low rank matrix from knowing a few entries?
r
General measurements
ρ =
Find Hermitian under low-rank assumption rank(ρ) = r
ρ ∈ CD×D
Matrix completion
Say, form unitary operator basis (incoherence property)
{Aj}
(Aj, ρ) = tr(Ajρ) = cj, j = 1, . . . , m
Can one reconstruct a low rank matrix from knowing a few entries?
r
Rank minimization
First idea: Minimise rank
(Aj, ρ) = cj, j = 1, . . . , m
rank(ρ)
Min Subject to
Unfortunately, this is an NP-hard problem
ρ =
Can one reconstruct a low rank matrix from knowing a few entries?
r
Convex relaxation
Second idea: Minimise trace-(nuclear) norm: Convex optimisation problem
ρ =
Min
Allows for reconstruction!
(Aj, ρ) = cj, j = 1, . . . , m
Subject to
Can one reconstruct a low rank matrix from knowing a few entries?
r
A theorem
Gross, Liu, Flammia, Becker, Eisert, Phys Rev Lett 105, 150401 (2010) Gross, IEEE Trans Inf Th 57, 1548 (2011) Recht, Fazel, Parrilo, SIAM 52, 471 (2010), arXiv:0706.4138 Candes, Recht, arXiv:0805.4471 Candes, Tao, arXiv:0903.1476
Any rank matrix can be recovered using randomly
chosen measurements , unitary operator basis
Reconstruction will fail only with exponentially small probability Recovery is exact Recovery is efficient: SDP (hugely improved using thresholding)
r ρ
{Ai}
Theorem
(Aj, ρ)
Gross, Liu, Flammia, Becker, Eisert, Phys Rev Lett 105, 150401 (2010) Gross, IEEE Trans Inf Th 57, 1548 (2011)
This is the tomography problem
E.g., take for words of Pauli matrices
{Aj}
So simple Pauli measurements do Any rank matrix can be recovered using randomly
chosen measurements , unitary operator basis
Reconstruction will fail only with exponentially small probability Recovery is exact Recovery is efficient: SDP (hugely improved using thresholding)
r ρ
{Ai}
Theorem
(Aj, ρ)
σ1 = 0 1 1
0 −i i
1 1
Geometry of the problem
Y
Trace-norm ball
"Data" "Orthogonal deviations"
∆ ∈ rangeR⊥
ρ B
Any rank matrix can be recovered using randomly
chosen measurements , unitary operator basis
Reconstruction will fail only with exponentially small probability Recovery is exact Recovery is efficient: SDP (hugely improved using thresholding)
r ρ
{Ai}
Theorem
(Aj, ρ)
Quantum state tomography of low-rank states
Lesson Approximately low-rank states can be tomographied much more
efficiently than naively - but with same type of measurements All quantum states Low rank states
Estimators and experimental compressed sensing
(This is not quite how it works)
Pauli bases
Flammia, Gross, Liu, Eisert, New J Phys 14, 095022 (2012)
One gets “clicks”, not perfect expectation values Obtain valued random variable Say, choose Pauli words labeled by
m
σx = πx
+ − πx −
Cx
tr((σx1 ⊗ · · · ⊗ σxN )ρ) = X
j∈{−1,1}N
χ(j)tr(ρ(πx1
j1 ⊗ · · · ⊗ πxN jN ))
P(Cx = j) = tr((πx1
j1 ⊗ · · · ⊗ πxN jN )ρ)
x ∈ {1, 2, 3}N
For each Pauli Then
with parity
Sampling operator collects projector exps known Bernoulli error model
Estimators
Matrix Lasso: Trace minimisation with positivity constraint Estimators efficient in dimension Can equip low-rank matrix recovery with confidence sets (see Richard’s talk)
Carpentier, Eisert, Gross, Nickl, arXiv:1504.0323 Flammia, Gross, Liu, Eisert, New J Phys 14, 095022 (2012)
Experimental implementation and model selection
Low-rank state recovery in 7-ion experiment in Innsbruck
Riofrio, Gross, Flammia, Monz, Roos, Blatt, Eisert, arXiv:Soon
Topological color code encoding - works well Significant overlap in leading subspaces, cross-validation
Compare Nigg, Mueller, Martinez, Schindler, Hennrich, Monz, Martin- Delgado, Blatt, arXiv:1403.5426 Compare Butucea, Guta, Kypraios, arXiv:1504.08295
Lesson Compressed sensing nicely workable in experimental settings Continuous and discrete model selection, in spirit of AIC, but
efficient in physical dimension
Does not make sense to “give back full state”: High dimensional tomography
should give advice on what observables to measure
More structured problems: Quantum field tomography
Tomography in quantum many-body systems?
Quantum many-body systems
All quantum states Low rank states
Necessarily identify right "data structure” Bring down effort from to
"Physical" corner"
"bond dimension"
All quantum states Low rank states "Physical" corner"
Matrix-product state tomography
Matrix-product states (MPS)
|ψi = X
s1,...,sn
tr(A[s1] . . . A[sn])|s1, . . . , sni
Low entanglement states, approximate ground states of local Hamiltonians
provably well, dense in state space
Cramer, Plenio, Flammia, Somma, Gross, Bartlett, Landon-Cardinal, Poulin, Liu, Nature Comm 1, 149 (2010) Baumgratz, Gross, Cramer, Plenio, Phys Rev Lett 111, 020401 (2013) Huebener, Mari, Eisert, Phys Rev Lett 110, 040401 (2013)
, A ∈ Cb×b b H = X
j
hj
All quantum states Low rank states "Physical" corner"
Continuous matrix-product state tomography
Continuous matrix-product states (cMPS) Natural continuum limit of MPS, low entanglement states
L
|ψQ,Ri = traux(Pe
R L
0 dx(Q⊗1+R⊗Ψ†(x))|øi
Verstraete, Cirac, Phys Rev Lett 104, 190405 (2010) Osborne, Eisert, Verstraete, Phys Rev Lett 105, 260401 (2010) Haegeman, Cirac, Osborne, Verstraete, Phys Rev B 88, 085118 (2013)
Continuous matrix-product state tomography
Vacuum in Fock space |øi matrices, finite-dimensional auxiliary system Are variational parameters, "bond dimension"
L
|ψQ,Ri = traux(Pe
R L
0 dx(Q⊗1+R⊗Ψ†(x))|øi
b
Q, R : b × b
Continuous matrix-product states (cMPS) Transfer matrix T = ¯
Q ⊗ 1 + 1 ⊗ Q + ¯ R ⊗ R ∈ Cd2×d2
Verstraete, Cirac, Phys Rev Lett 104, 190405 (2010) Osborne, Eisert, Verstraete, Phys Rev Lett 105, 260401 (2010) Haegeman, Cirac, Osborne, Verstraete, Phys Rev B 88, 085118 (2013)
Continuous matrix-product state tomography
L
|ψQ,Ri = traux(Pe
R L
0 dx(Q⊗1+R⊗Ψ†(x))|øi
[Under reasonable assumptions] any MPS and cMPS can be
reconstructed based on 2- and 3-point correlation functions only
Theorem
Huebener, Mari, Eisert, Phys Rev Lett 110, 040401 (2013)
Continuous matrix-product states (cMPS)
Given:
Matrix pencils and prony methods
C(n)(τ1, . . . , τn−1) =
d2
X
k1,...,kn−1=1
ρk1,...,kn−1eλk1τ1 . . . eλkn−1τn−1
Build Hankel matrices
decompose as and and estimate from points where is rank-deficient
Matrix pencil methods for discretized 2-point function Cj =
d2
X
k=1
ρkeλk∆τj
C(1) = C0 C1 . . . CN/2−1 C1 C2 . . . CN/2 . . . . . . . . . CN/2−1 CN/2 . . . CN−2 C(2) =
C1 C2 . . . CN/2 C2 C3 . . . CN/2+1 . . . . . . . . . CN/2 CN/2+1 . . . CN−1
{λk} C(1) = V1AV2 C(1) = V1AV0V2
C(2) − γC(1)
Hua, Hershman, Cheng, High resolution and robust signal processing, Marcel Dekker (2004)
L L
τ1 τ2
Given:
Matrix pencils and prony methods
C(n)(τ1, . . . , τn−1) =
d2
X
k1,...,kn−1=1
ρk1,...,kn−1eλk1τ1 . . . eλkn−1τn−1
Matrix pencil methods for {λk} Prony's method for residues
2-3-point functions
Reconstruct
is in basis of transfer
ρk1,k2 = M1,k1Mk1,k2Mk2,1 M
Reconstruct Q, R Predict all quantum field correlation functions
Steffens, Riofrio, Huebener, Eisert, New J Phys 16, 123020 (2014) Huebener, Mari, Eisert, Phys Rev Lett 110, 040401 (2013)
Quantum field tomography
Quantum field states - the collection of all correlation functions - can
be efficiently reconstructed within a cMPS approach
Lesson Applied to experiments with cold atoms on atom chips out of equilibrium
Steffens, Friesdorf, Langen, Rauer, Schweigler, Huebener, Schmiedmayer, Riofrio, Eisert, arXiv:1406.3632, Nature Communications, in press (2015)
Direct certification
(Wanting even less)
Take measurements
State certification
Again, prepare quantum systems
A-priori knowledge about preparation: "We know what we want"
Is certification of state preparations any easier?
Naive approach Robust certification Not robust, but can be practical
Mindset of state certification
Show that target fidelity with anticipated state
∆
%p
1 − F
%t
1 − FT
%p
1 − F
%t
1 − FT
%p
1 − F
%t
1 − FT
Efficient certification for most practical optical preparations
For (a) linear optical, (b) Gaussian, (c) post-selected preparations, one
can perform robust state certification with preparations, efficient in mode number and max failure prob
Theorem
O ✓ poly(m, 1/∆) log(1/(1 − α)) ◆
Aolita, Gogolin, Kliesch, Eisert, arXiv:1407.4817, Nature Communications, in press (2015) Compare also Flammia, Gross, Liu, Eisert, New J Phys 14, 095022 (2012) Flammia, Liu, Phys Rev Lett 106, 23050 (2011)
Efficient certification possible in practical settings where efficient
tomography is way out of scope
Lesson
Methods: Interactive proof system, extremality of Gaussian operations, non-Gaussian nullifiers
Into the wild
(Bi-linear forms and tensor completion)
Into the wild
Acquisition of data Bi-linear problems: Uncalibrated detectors, deconvolution probs Tensor completion: Many problems, in fact, but marred by
hardness of even computing even the tensor rank
Beyond vector and matrix compressed sensing
?
Into the wild
Ask the author about the slides
All quantum states Low rank states "Physical" corner"
Quantum state tomography and estimation for large quantum systems
Certification