D - EPIQ

epiq.physique.usherbrooke.ca

D - EPIQ

The continuum limit of aquantum circuit: variationalclasses for quantum fieldsTobias J. Osborne, withIgnacio Cirac, Jens Eisert, Jutho Haegeman, HenriVerschelde, and Frank VerstraetearXiv:1102.5524arXiv:1005.1268http://tjoresearchnotes.wordpress.com


Outline• Quantum fields• Reminder of MPS and MERA• The passage to the continuum• Properties• Area laws• Variational method


Fundamentale.g. Klein-GordonDiracEffectivee.g. Lieb-LinigerBCS, Impurities(x)[⇥(x), ⇥ † (y)] = (x y)


How to solve?


Perturbation theoryMonte CarloVariational methodinf||H| ⇥


Finite fermion densityStrong interactions(no sign problem)Real time evolutionThe variationalprinciple (QFT)Need good trial statePoor reproduction ofcorrelationsUV sensitivity


“... it is no damn good at all!”R. P. Feynman, Proceedings of the InternationalWorkshop on Variational Calculations in Quantum FieldTheory. Wangerooge, West Germany. 1-4 Sept. 1987


AKLT model(1987)DMRG(1992)Matrix productstates (1992)Strongly correlatedEntanglement(2001)systemsDynamics(2003)2D & beyond(2004+)


Variational classes &quantum circuits


Matrix product statesM. Fannes, B. Nachtergaele, and R. F. Werner,Comm. Math. Phys. 144, 443-490 (1992).


Hilbert spaceB 1 2 3 4 5 ···C D ⌦ C 2 ⌦ C 2 ⌦ ···⌦ C 2


• Stage 0: spins initialised in “all 0”s state• Stage 1: subject B & 1st spin to interaction• Stage 2: subject B & 2nd spin to interation• Stage 3: repeat until Nth spin• Stage 4: measure B


|···|0 |0 |0 |0 ···|0


DX1XA j ↵ |j 1| B=1j=0U|↵ B |0 1


1X| ⇥ =! L |A j 1A j2 ···A j n|0⇥|j 1 j 2 ···j n ⇥j 1 ,...,j n =0


#variationalparameters of :2ND 2 |Expectation values:|A| ⇥Complexity: O(ND 4 )MPS obeyentropy/area law:tr(⇢ L log(⇢ L )) ⇠ O(log(D))


MPS capture relevantphysics


Example:H =n 1j=1h jwhereh j = I 1···j 1 h I j+2···n


Theorem (Hastings).If the spectral gap isthen the (assumed) unique ground state|⌦iof H can be well-approximated by an MPSwith|⌦ 0 iD ⇠ poly(n, 1/✏, e 1/ )Matthew B. Hastings, JSTAT, P08024 (2007)


Theorem (TJO).If|t| appleO(log(n))and| (t)i = e itH | (0)ithen| (t)ican be well-approximated by an MPSwithD ⇠ poly(n, 1/✏)| 0 (t)iTJO, Phys. Rev. Lett. 97, 157202 (2006)


The multiscaleentanglementrenormalisation ansatzG. Vidal, Phys. Rev. Lett. 99, 220405 (2007)


• Stage 0: spins initialised in “all 0”s state|0• Stage 1: subject spins to local interactionU 1• Stage 2: transform scale by factor of 2• Stage 3: introduce new uncorrelated spins viaR• Stage 4: repeat


||0 |0 |0 |0 |0 |0 |0 |0|0 |0 |0 |0 |0 |0 |0 |0|0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0


| MERA = U m RU m 1 R···RU 1 |0


MERA = dilation +local interaction


Algebraiccorrelation decayApplies to 2D, 3DMERA |Excellent numericsCorrections toentropy area lawarXiv:0912.1651


The passage to thecontinuum


(x)✏✏j j ⇥ j = a j / p[⇥(x), [a [a j ,a j ,a ⇥ † k (y)]k ]= = jk (x y)jk


Idea: make everythinginfinitesimal


U = e iMU = e i✏MO(1)O(✏)


cMPS


✏B⇥ j = a j / p


| ⇥ =1Xj 1 ,...,j n =0! L |A j 1A j2 ···A j n|! R ⇥|j 1 j 2 ···j n ⇥|j 1 j 2 ···j n =( † 1 )j 1( † 2 )j2 ···( † n) j n|✏A 0 = I + ✏Q A 1 = ✏R A j = j R j /j!| ⇥ = ! L |T e R l0 Q⌦I+R⌦ † (s) dx |! R ⇥| ⇥


| ⇥ = ! L |T e R l0 Q⌦I+R⌦ † (s) dx |! R ⇥| ⇥Analyticallycalculate allcorrelationfunctionsInbuilt UVcutoffGenericclustering ofcorrelations1. F. Verstraete and J. I. Cirac, Phys. Rev. Lett. 104, 190405 (2010)2. TJO, J. Eisert, and F. Verstraete, Phys. Rev. Lett. 105, 260401 (2010)


cMERA


|0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0Stage 0: initial state✏|⌦(x)| =0|0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0


Stage 1: local interactionZ✏K =dx k(x)U 1 = e i K


Stage 2: scale transform,generator:L = i 2Z† (x)x d (x)dxx d † (x)dx(x) dxRe i L


Stage 3: new degrees offreedom?


Impose UV cutoff


Why?Gives lengthscale✏ ⇠ 1


(Bad) example:ZK =† 2 (x)+ 2 (x)dx


uncorrelated degrees offreedomcome from high mtm viaRe i L


(1) (2)(3) (4)cMERA steps in momentum space


cMERA| ⇤ (e i L e i K ) 1/ ⇥!0T e i R s✏s ⇠K(s)+Lds|⇥⇤(K can depend on the “time” parameter s)


1st infinitesimal layere i L e i K(s ⇠)correlates at lengthscale⇠ = e s ⇠


Last infinitesimal layere i L e i K(s ✏)correlates at lengthscale✏ = e s ✏


cMERA have fluctuationsfrom⇠ ✏to⇠ ⇠


Main difference betweencMERA & MERA:flexibility in UV cutoff


MERA = lattice cutoffcMERA = smooth cutoff


Entropy/area laws


1D critical systems:S Ac log |A|1D noncritical systems:S A ⇠ c


Standard MPS:S A apple log(D)


cMPS?


DX| =⇥p↵ |u ↵ |v ↵↵=1


S A apple log(D)


Standard MERA:S Ac log |A|


cMERA?


egion A(s) shrinks below cutoffs ✏sA(s)xEntropy of a region A/area of red region


Entanglement generated by K:dS A (t)dtc|@A|


Z s✏S A apple c(L e s s ✏) d 1 dss ✏ log(L )=(c log(L ), d =1⇣ ⌘cd 1 (L )d 1 11(L ) d 1, d > 1.


Variational methodinf| ⇥ V|H| ⇥


Non-rel. bosonic groundstate:H =Z apple d⇥†dxd⇥dx + µ⇥† ⇥ (⇥ †2 + ⇥ 2 ) dx


Ground state admits cMERA description withZhb † (k) b† ( k) b ( k) b (k)iK(s) = i 2g( k , s)dkwhereg(k/⇥, s) =(s) (|k|/⇥)and(apple)is a cutoff function and⇥(s) = 2( / 2 )e 2s [(e 2s + µ/ 2 ) 2 4 2 / 4 ]1


Conclusions• Quantum circuits as variational classes• Take continuum limit• Entropy/area laws• Exact representations for QFTs


Free Dirac fieldZappleH =dxi † ↵ d dx+ m †


Ground state admits cMERA description withK(s) =iZdk g( k , s)h ib †1 (k) b† 2 ( k)+ b 1(k) b 2( k)where⇥(s) =2 dd1 arcsin"2 p 2+(m/ ) 2 #apple=e s


Klein-Gordon fieldZh⇡ 2 +(r ) 2 + m 2 2iH =d d x 1 2


Ground state admits cMERA description withK(s) = 1 2Zd d kg( ~ k , s)hb(k)b⇡( k)+b⇡( k) b (k)iwhere(s) =e 2s /2/[e 2s +(m/ ) 2 ]


Gross-Neveu modelĥ GN = i 2 ˆ†a y d ˆadx + h.c. g 22 :(ˆ†a z ˆa) 2 :,


Gross-Neveu modelSpontaneous symmetrybreakingAsymptotic freedom


!(Λ)"/ Λ10.1exact a D = 6fit b D = 8D = 10D = 160.010.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7! -1 (Λ) = [(N - 1)g(Λ) 2 ] -1Expectation value of = h | ˆ† z ˆ| i

More magazines by this user
Similar magazines