Preparing Topological States on a Quantum Computer. Martin Schwarz (1) , Kristan Temme (1) , Frank Verstraete (1) Toby Cubitt (2) , David Perez-Garcia (2). (1) University of Vienna (2) Complutense University, Madrid. STV, Phys. Rev. Lett. 108, 110502 (2012)
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Preparing Topological States on a Quantum Computer
Martin Schwarz(1), Kristan Temme(1),Frank Verstraete(1)
Toby Cubitt(2), David Perez-Garcia(2)
(1)University of Vienna
(2)Complutense University, Madrid
STV, Phys. Rev. Lett. 108, 110502 (2012)
STVCP-G, (QIP 2012; paper in preparation)
Obtain PEPS by applying maps to maximally entangled pairs
But...
[V, Wolf, P-G, Cirac 2006]
Algorithm
Algorithm
Algorithm
Algorithm
Algorithm
Algorithm
Algorithm
Algorithm
Algorithm
Algorithm
Algorithm
??
QPE
local Hamiltonian ) Hamiltonian simulation )
! Use quantum phase estimation:
measure if energy is < or not
QPE
! Use quantum phase estimation:
measure if energy is < or not
0
0
0
0
c s
1
P0(t) =
P0(t+1) =
0
-s c
0
0
0
0
“Jordan’s lemma” (or “CS decomposition”)
! Use Marriot-Watrous measurement rewinding trick:
! Use Marriot-Watrous measurement rewinding trick:
! Use Marriot-Watrous measurement rewinding trick:
c
! Use Marriot-Watrous measurement rewinding trick:
s
! Use Marriot-Watrous measurement rewinding trick:
c
! Use Marriot-Watrous measurement rewinding trick:
c
s
! Use Marriot-Watrous measurement rewinding trick:
c
s
! Use Marriot-Watrous measurement rewinding trick:
c
s
c
c
! Use Marriot-Watrous measurement rewinding trick:
c
s
c
s
! Use Marriot-Watrous measurement rewinding trick:
c
c
s
s
! Use Marriot-Watrous measurement rewinding trick:
c
c
s
s
c
s
c
s
c
s
! Use Marriot-Watrous measurement rewinding trick:
c
c
s
s
c
s
c
s
c
s
! Use Marriot-Watrous measurement rewinding trick:
c
c
s
s
c
s
Algorithm:
Algorithm:
Run-time:
Condition 1: Spectral gap (Ht) > 1/poly
Condition 2: Unique ground state (= injective PEPS)
Condition 3: Condition number (At ) > 1/poly
Rules out all topological quantum states!
0
0
1
1
P0(t) =
0
0
c1 s1
c2 s2
-s2 c2
-s1 c1
0
0
P0(t+1) =
“Jordan’s lemma” (or “CS decomposition”)
s
s
Idea:
Algorithm
Idea:
Algorithm
For (suitable representation of) trivial group G = 1,G-isometric PEPS = maximally entangled pairs!recover original algorithm
Algorithm
G-isometric PEPS = quantum double models! algorithms known for preparing these exactly[e.g. Aguado, Vidal, PRL 100, 070404 (2008)]
Key Lemma:If initial state is already in G-invariant subspace, prob. successful measurement is condition number restricted to G-invariant subspace
Algorithm
! Marriot-Watrous measurement rewinding trick works!