Quantum annealing and the Sign Problem

23 Pages·2012·1.73 MB·English

Quantum annealing and the Sign Problem
Gustavo Du¨ring and Jorge Kurchan
PMMH-ESPCI,Paris
[email protected]
http://www.pmmh.espci.fr/∼jorge
IHP 2012
J.Kurchan (PMMH-ESPCI) Signproblem 1/1
1. The sign problem
J.Kurchan (PMMH-ESPCI) Signproblem 2/1
We are asked to compute averages
over configurations s
(cid:80)
e−W(s)O(s)
(cid:104)O(cid:105) = s
(cid:80)
e−W(s)
s
and W = W + iW is not real.
R I
J.Kurchan (PMMH-ESPCI) Signproblem 3/1
Quantum mechanics / Field Theory with real
time or θ-terms
Hubbard-Stratonovich decoupling W = W −b(cid:80)C2
o α
Z = (cid:88)e−Wo(s)−b(cid:80)Cα2 = (cid:88)(cid:90) dλ e−Wo(s)+√bλαCα−λ22α
s s
TheHubbard-Stratonovichtransformationaboveintroducesan
(cid:112) (cid:80)
imaginarytermW = |b| λ C intherepulsivecaseb<0.
I α α
J.Kurchan (PMMH-ESPCI) Signproblem 4/1
In solid state theory the sign problem is the main
obstacle for giving a numerical answer to very
urgent questions.
e.g. HubbardmodelH =−t(cid:80) (c† c +h.c.)+U(cid:80) n n
<ij>,σ iσ jσ i i↑ i↓
J.Kurchan (PMMH-ESPCI) Signproblem 5/1
For definiteness: W = ih M(s)
I I
whereM(s)isaninteger-valued:
—————-
(cid:88)
Z = Z e−ihIM
M
M
(cid:88)
Z = e−βF(M) = δ(M(s)−M) e−WR
M
s
——————
• Fieldtheorywithθ terms. M =atopologicalnumber
’tHooft,Haldane,...
• Fermionsystems: M =Fermionworld-linecrossings
Muramatsuetal
• Hubbardmodel: M =thenumberofupHirschspins
[withavariantofHubbard-Stratonovichtransformation,see:DeForcrand,Batrouni]
J.Kurchan (PMMH-ESPCI) Signproblem 6/1
Also: If we knew how to deal with the sign problem, we would
know how to compute numerically averaged disordered models
using identities of the form
1 =−lim 1−e−λx =−lim (cid:80)∞(−1)nλnxn−1
x λ→∞ x λ→∞ 1 n!
———————-
(cid:104)E(cid:105) = −Z−1 ∂Z = −lim ∂ (cid:80)∞ (−1)nλnZn
∂β λ→∞∂β 1 n n!
J.Kurchan (PMMH-ESPCI) Signproblem 7/1
In practice, what we can do is to compute
(cid:104)O(cid:105) = (cid:104)e−iWI O(cid:105)R ; where (cid:104)•(cid:105) = P •e−WR(s)
(cid:104)e−iWI(cid:105)R R Pe−WR(s)
Note that Monte Carlo is only really good to calculate (cid:104)O(cid:105) for O
non-exponential
and not for
(cid:104)e−B O(cid:105)
(cid:104)O(cid:105) = A
(cid:104)e−B(cid:105)
A
if, e.g. B is exponential in N
J.Kurchan (PMMH-ESPCI) Signproblem 8/1
An example:
Non-interacting spins in a magnetic field h +iπ
R 2
Z = (cid:16)e−h−iπ2 +eh+iπ2(cid:17)N = e−iN2π (cid:0)e−h−eh(cid:1)N
J.Kurchan (PMMH-ESPCI) Signproblem 9/1
(q+p)N (q − p)N
(q+p)N = (cid:80) N! qN−rpr
r r!(N−r)!
(q−p)N = (cid:80) N! (−1)rqN−rpr
r r!(N−r)!
J.Kurchan (PMMH-ESPCI) Signproblem 10/1