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 Alexandra Carpentier Thomas Monz

Quantum state tomography � Prepare i.i.d. quantum systems � Take measurements � 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,

Quantum state tomography � Prepare i.i.d. quantum systems � Take measurements � Reconstruct unknown density operator , ρ ∈ S ( H ) � ’Curse of dimensionality’ in quantum state estimation and tomography � Yes! Significantly so � More pronounced if confidence sets and statistical aspects considered � In a way… � For -dimensional quantum systems, expectation values are required d � Can we do better? � Structure is the key � E.g., 1.099.511.627.775 expectation values for 20 spins,

Mind map of the talk Compressed sensing and matrix completion Estimators and experimental Compressed sensing compressed sensing Direct certification Structured problems: Quantum field tomography Into the wild: tensor completion and bilinear problems

Compressed sensing and matrix completion

Gene expression data matrices Observations � Columns observations (samples of DNA arrays) � Typical numbers 6-10K genes, 
 50-150 samples, many entries missing Genes � Predict missing entries?

Netflix prize Users � Netflix offered $1M for practical solution to rating problem Movies � User rate videos... but obviously only some � Predict missing entries? � People's tastes not random, linear dependencies � Matrix is low rank � Can one reconstruct a low rank matrix from knowing a few entries? r Candes, Plan, arXiv:0903.3131 Candes, Plan, arXiv:0903.3131 Toescher, Jahrer, Bell (2009) Toescher, Jahrer, Bell (2009)

Matrix completion [ [ r � Knowing the structure allows for recovery: entries 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

Matrix completion [ [ ρ = � General measurements ( A j , ρ ) = tr( A j ρ ) = c j , j = 1 , . . . , m � Find Hermitian under low-rank assumption rank( ρ ) = r ρ ∈ C D × D � Say, form unitary operator basis (incoherence property) { A j } � Can one reconstruct a low rank matrix from knowing a few entries? r

Rank minimization [ [ ρ = � First idea: Minimise rank Min rank( ρ ) Subject to ( A j , ρ ) = c j , j = 1 , . . . , m � Unfortunately, this is an NP-hard problem � Can one reconstruct a low rank matrix from knowing a few entries? r

Convex relaxation [ [ ρ = � Second idea: Minimise trace-(nuclear) norm: Convex optimisation problem Min Subject to ( A j , ρ ) = c j , j = 1 , . . . , m � Allows for reconstruction! � Can one reconstruct a low rank matrix from knowing a few entries? r

A theorem � Theorem � Any rank matrix can be recovered using randomly 
 r ρ chosen measurements , unitary operator basis ( A j , ρ ) { A i } � Reconstruction will fail only with exponentially small probability � Recovery is exact � Recovery is efficient : SDP (hugely improved using thresholding) 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

T his is the tomography problem � Theorem � Any rank matrix can be recovered using randomly 
 r ρ chosen measurements , unitary operator basis ( A j , ρ ) { A i } � Reconstruction will fail only with exponentially small probability � Recovery is exact � Recovery is efficient : SDP (hugely improved using thresholding) Gross, Liu, Flammia, Becker, Eisert, Phys Rev Lett 105, 150401 (2010) Gross, IEEE Trans Inf Th 57, 1548 (2011) � E.g., take for words of Pauli matrices { A j }  0  0  1 � � � 1 − i 0 σ 1 = , σ 2 = , σ 3 = 1 0 i 0 0 1 � So simple Pauli measurements do � Exponential improvement for

Geometry of the problem � Theorem � Any rank matrix can be recovered using randomly 
 r ρ chosen measurements , unitary operator basis ( A j , ρ ) { A i } � Reconstruction will fail only with exponentially small probability � Recovery is exact � Recovery is efficient : SDP (hugely improved using thresholding) Y "Data" ρ Trace-norm ball B "Orthogonal deviations" ∆ ∈ range R ⊥

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 Low rank states All quantum states

Estimators and experimental compressed sensing (This is not quite how it works)

Pauli bases � One gets “clicks”, not perfect expectation values � Say, choose Pauli words labeled by x ∈ { 1 , 2 , 3 } N m � For each Pauli σ x = π x + − π x − � Obtain valued random variable C x P ( C x = j ) = tr(( π x 1 j 1 ⊗ · · · ⊗ π x N j N ) ρ ) tr(( σ x 1 ⊗ · · · ⊗ σ x N ) ρ ) = X � Then χ ( j )tr( ρ ( π x 1 j 1 ⊗ · · · ⊗ π x N j N )) with parity j ∈ { − 1 , 1 } N � Sampling operator collects projector exps � known Bernoulli error model Flammia, Gross, Liu, Eisert, New J Phys 14, 095022 (2012)

Estimators � Estimators efficient in dimension � Matrix Lasso: � Trace minimisation with positivity constraint � 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 � Lesson Compare Nigg, Mueller, Martinez, Schindler, Hennrich, Monz, Martin- � Compressed sensing nicely workable in experimental settings Delgado, Blatt, arXiv:1403.5426 � 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 Compare Butucea, Guta, Kypraios, arXiv:1504.08295

More structured problems: 
 Quantum field tomography

Quantum many-body systems � Tomography in quantum many-body systems? � Necessarily identify right "data structure” � Bring down effort from to "Physical" corner" All quantum states Low rank states

Matrix-product state tomography � Matrix-product states (MPS) X , A ∈ C b × b | ψ i = tr( A [ s 1 ] . . . A [ s n ]) | s 1 , . . . , s n i s 1 ,...,s n X � "bond dimension" b H = h j j � Low entanglement states, approximate ground states of local Hamiltonians 
 provably well, dense in state space "Physical" corner" All quantum states Low rank states 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)

Continuous matrix-product state tomography � Continuous matrix-product states (cMPS) R L 0 dx ( Q ⊗ 1+ R ⊗ Ψ † ( x ) ) | ø i | ψ Q,R i = tr aux ( P e 0 L � Natural continuum limit of MPS, low entanglement states "Physical" corner" All quantum states Low rank states 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 � Continuous matrix-product states (cMPS) R L 0 dx ( Q ⊗ 1+ R ⊗ Ψ † ( x ) ) | ø i | ψ Q,R i = tr aux ( P e 0 L � Vacuum in Fock space | ø i � matrices, finite-dimensional auxiliary system Q, R : b × b � Are variational parameters, "bond dimension" b R ⊗ R ∈ C d 2 × d 2 � Transfer matrix T = ¯ Q ⊗ 1 + 1 ⊗ Q + ¯ 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 � Continuous matrix-product states (cMPS) R L 0 dx ( Q ⊗ 1+ R ⊗ Ψ † ( x ) ) | ø i | ψ Q,R i = tr aux ( P e 0 L � Theorem � [Under reasonable assumptions] any MPS and cMPS can be 
 reconstructed based on 2- and 3-point correlation functions only Huebener, Mari, Eisert, Phys Rev Lett 110, 040401 (2013)