Lie algebraic aspects of quantum control in interacting spin

.
.
Lie algebraic aspects of quantum control
in interacting spin-1/2 (qubit) chains
Vladimir M. Stojanovi´
c
Condensed Matter Theory Group
HARVARD UNIVERSITY
September 16, 2014
V. M. Stojanovi´
c (Harvard)
TI conference @ CMSA
September 16, 2014
1 / 25
. Outline of the talk
Introduction to quantum (coherent) control
Local quantum control of Heisenberg spin-1/2 chains
R. Heule, C. Bruder, D. Burgarth, and VMS, PRA 82, 052333 (2010);
Eur. Phys. J. D 63, 41 (2011).
VMS, A. Fedorov, A. Wallraff, and C. Bruder, PRB 85, 054504 (2012).
Pulse-sequence generated dynamics of stabilizer operators:
application in measurement-based QC, i.e., one-way QC
T. Tanamoto, D. Becker, VMS, and C. Bruder, PRA 86, 032327 (2012);
T. Tanamoto, VMS, C. Bruder, and D. Becker, PRA 87, 052305 (2013).
Summary
. Quantum control: generalities
State-to-state ( state-selective ) control: How to steer a
quantum system from a given initial- to a desired final state?
Operator ( state-independent ) control: How to realize a
pre-determined unitary transformation (target quantum gate)?
H(t) = H0 +
p
X
fj (t)Hj
fj (t) – control fields
j=1
The system is completely controllable if H(t) can give rise
to an arbitrary unitary transformation on its Hilbert space H
i.e., the reachable set R is equal to U (n) or SU (n) ( n = dim H )
. General controllability theorems
U˙ (t) = −i[H0 +
p
X
fj (t)Hj ]U (t)
,
U (0) =
1
(#)
j=1
Lie algebra rank condition
.
Theorem
.
The reachable set R of a quantum control system described by Eq. (#)
is the connected Lie group associated with the Lie algebra L0 generated
L0
by
. −iH0 , −iH1 , . . . , −iHp , i.e., R = e .
⇒ complete (operator) controllability
.
Theorem
.
A system described by Eq. (#) is completely (operator) controllable iff
L0 = u(n) [ or L0 = su(n) ], where L0 is the Lie algebra generated
.by −iH0 , −iH1 , . . . , −iHp .
. Interacting qubit arrays
Coupling between qubits:
Hint =
XX
αβ α β
Jij
σi σj
( α, β = x, y, z )
i<j α,β
qubit-qubit interaction
Ising
XY
Heisenberg
qubit system
charge
flux, charge-flux, phase, cavity
spin, donor atom
couplings beyond nearest neighbors can be induced using a
“quantum bus” (e.g., cavity) [J. Majer et al., Nature (2007)]
. Local control in interacting systems: general aspects
¯ with controls acting only on C
composite system S = C ∪ C
total Hamiltonian: H = HS +
p
X
fjC (t)HjC
j=1
S is completely controllable iff −iHS and −iHjC (j = 1, . . . , p)
generate the Lie algebra L(S) of all skew-Hermitian operators on S
hiHS , L(C)i = L(S)
L(C) = {−iH1C , . . . , −iHpC }L
hA, Bi – algebraic closure of the operator sets A and B
. Local control in qubit arrays with “always-on” interactions
. Complete controllability of XXZ Heisenberg chains
H0 = J
N
−1
X
Si,x Si+1,x + Si,y Si+1,y + ∆Si,z Si+1,z
i=1
Hc (t) = hx (t) S1x + hy (t) S1y
| {z } |{z} | {z } |{z}
f1 (t) H1
f2 (t) H2
Htotal (t) = H0 + Hc (t)
Acting on the x- and y-components of an end spin in an XXZ
Heisenberg spin chain renders the chain completely controllable!
sufficient to show that the dimension of the dynamical Lie algebra
Lxy generated by {−iH0 , −iS1x , −iS1y } is d2 − 1 (d ≡ 2N )
⇒ Lxy ∼
= su(d) ⇒ eLxy ∼
= SU (d) (complete controllability)
. Generalization to graphs
Local controllability on a graph G = (S, E) by acting on C ⊆ S
H = HS +
p
X
fjC (t)HjC
j=1
HS =
X
Hnm
n,m∈E
graph criterion of controllability (sufficient condition):
algebraic property of Hnm + topological property of G
HS is algebraically propagating if for all n ∈ S and (n, m) ∈ E
h[iHnm , L(n)], L(n)i = L(n, m)
Heisenberg and Affleck-Kennedy-Lieb-Tasaki (AKLT) couplings are A.P. !
S is controllable by acting on C if HS is A.P. and C is infecting
D. Burgarth et. al., PRA 79, 060305(R) (2009)
. Infecting and non-infecting subgraphs
rules of infection:
. Control objectives (target gates)
controlled-NOT on the last two qubits of the chain:
CNOTN ≡ I ⊗ . . . ⊗ I ⊗ |0ih0| ⊗ I + |1ih1| ⊗ X
|
{z
}
|
{z
}
N −2
CNOT
( X ≡ σx )
flip (NOT) of the last qubit XN ≡
requires only an x control!
1 ⊗ ... ⊗ 1 ⊗ X
√
√
SWAP on the last two qubits:
SWAPN −1,N
√
π
π
reminder:
SWAP ≡ ei 8 e−i 8 (X⊗X+Y ⊗Y +Z⊗Z)
N = 3, ∆ = 1 case: dim Lx = 18 , basis {−iH0 , . . . , −iH17 }
√
Is there A ∈ Lx such that SWAP2,3 = eA ?
1XX + 1Y Y
+ 1ZZ = 12 (H0 − H3 + H6 − H16 + H17 )
. Control pulses and fidelity maximization
alternate x and y (or x only !) piecewise-constant controls:
full time evolution (total time tf ≡ Nt T ):
U (tf ) = Uy,Nt /2 Ux,Nt /2 . . . Uy,1 Ux,1
h
gate fidelity: F (tf ) =
Uj,n ≡ e−iHj,n T (j = x, y)
1 †
tr U (tf )Utarget d
h
0 ≤ F (tf ) ≤ 1
i
i
maximize F = F ({hx,n ; hy,n }) numerically
frequency-filtered control fields:
e j (t) = F −1 f (ω)F [hj (t)]
h
ideal low-pass filter:
f (ω) = θ(ω + ω0 ) − θ(ω − ω0 )
. Toffoli-gate realization with superconducting qubits
state of the art: two-qubit gates with F > 90% [ DiCarlo et al., (2009) ]
TOFFOLI ≡ controlled-controlled-NOT
a
a
trapped-ion [F ≈ 71%],
photonic [F ≈ 81%] realizations in 2009!
b
b
conventional 6 CNOTs + 10 single-qubit operations
approach not feasible due to long gate times!
1
1+ab
Way out: use third level
A. Fedorov et al., Nature 2012 : F = 64.5 ± 0.5 %
M. D. Reed et al., Science 2012 : F = 78 ± 1 %
Can quantum control be of help?
F
decoherence
−→
F ∗ exp(−tg /T2 )
. Three-qubit (transmon) circuit QED setup
effective XY -type model:
H0 =
X
Jij (σix σjx + σiy σjy )
i<j
J12 = J23 = J ≈ 30 MHz, J13 ≈ 5 MHz
Hc (t) =
3 h
X
Ω(i)
(t) σix + Ω(i)
(t) σiy
x
y
i
q
Ω2x + Ω2y . 130 MHz
i=1
VMS, A. Fedorov, A. Wallraff, and C. Bruder, PRB 85, 054504 (2012)
. Toffoli gate in circuit QED: results
cutoff frequency:
ω0 = 500 MHz
ω0 ≈ 17J !
tg ≈ 140 ns
F ≈ 99%
tg = 75 ns
F ≈ 92%
. Measurement-based quantum computation (MBQC)
R. Raussendorf and H. J. Briegel, PRL 86, 5188 (2001)
with local (single-qubit)
measurements:
MBQC → one-way QC
2D cluster state is a
universal resource for MBQC!
Other candidates, e.g., the
AKLT state, are difficult to
produce in solid-state systems!
H. J. Briegel et al., Nature Phys. 5, 19 (2009)
. Graph- and cluster states
one-way quantum computing
H. J. Briegel and R. Raussendorf, PRL 86, 910 (2001)
initial preparation:
Y
(ij)
|Gi =
UPG
|+i⊗N
{i,j}
√
|+i = (|0i + |1i)/ 2
UPG = diag(1, 1, 1, −1)
correlation operators Ki ≡ σix
O
j∈nghd(i)
σjz satisfy Ki |ψc i = ±|ψc i
. One-way QC: an illustration
two-qubit cluster state: K1 = σ1x σ2z , K2 = σ1z σ2x
1
|ψc i = √ (|+i|0i + |−i|0i)
2
measure qubit 1 in a basis rotated by a SU(2) matrix:
cos θ
eiφ sin θ
U (θ, φ) ≡
e−iφ sin θ − cos θ
measurement results: m = 0 −→ qubit 2 in |ψ0 i = P (φ)R(θ)|0i
m = 1 −→ |ψ1 i = σ1z P (−φ)σ1x R(θ)|1i
1
P (φ) ≡ √
2
1 eiφ
1 −eiφ
R(θ) ≡
cos θ
sin θ
sin θ − cos θ
arbitrary single-qubit rotation on the second qubit can be
performed by measuring the first one in an appropriate basis!
. Stabilizer Hamiltonian
cluster states are eigenstates of the stabilizer Hamiltonian
P
Hstab = − i Ki
How to “generate” a stabilizer Hamiltonian starting from
a “natural” two-body spin-1/2 (qubit) Hamiltonian?
H = H0 + Hint
Ising: Hint = J
H0 =
X
P
x
i (Ωx σi
+ Ωy σiy + εi σiz )
z
σiz σi+1
i
XY : Hint = J
X
y
x
(σix σi+1
+ σiy σi+1
)
i
Heisenberg: Hint = J
X
i
y
x
z
(σix σi+1
+ σiy σi+1
+ σiz σi+1
)
. Stabilizer Hamiltonian from Ising-type interactions
What e−iθ
P
z
σiz σi+1
i
Hs eiθ
P
i
z
σiz σi+1
amounts to?
basic relations:
−iθσ1z σ2z
e
σ1x,y e
iθσ1z σ2z
= cos(2θ)σ1x,y ± sin(2θ)σ1y,x σ2z
special case θ = π/4 – increasing the order of the Pauli-matrix terms:
e−i 4 σ1 σ2 σ1x ei 4 σ1 σ2 = σ1y σ2z
π
z
z
π
z
z
;
e−i 4 σ1 σ2 σ1y ei 4 σ1 σ2 = −σ1x σ2z
π
z
z
π
z
z
⇒ 1D stabilizer Hamiltonian; 1D → 2D straightforward!
Q: But how can we physically “generate” (≡ induce effective dynamics of)
Hstab = e−i(π/4)
P
i
z
σiz σi+1
Hs ei(π/4)
P
i
z
σiz σi+1
?
. Stabilizer Hamiltonian as the effective Hamiltonian
(a)
Ising-interaction pulses with τ1 ≡ π/(4J ):
Hs
Hint
Hstab = e−i 4
π
τ
τ1
τ1
i
z
σiz σi+1
π
Hs ei 4
P
i
z
σiz σi+1
state evolution:
τ1 Hint τ Hs −τ1 Hint
Hstab
t
e−iτ Hstab
P
ρ(0) −→ −→ −→
ρ(t = τ + 2τ1 )
pulse-induced effective evolution:
!
!
X
πX z z
π
= exp −i
σ σ
σz σz
e−iτ Hs exp i
4 i i i+1
4 i i i+1
unitary tsf. with a generator S, analytic operator function f (A):
eS f (A) e−S = f (eS A e−S )
. Stabilizer Hamiltonian from XY -type interactions
main difference from the Ising case:
x
[σix σi+1
+
e−iθ
y
x σx
σiy σi+1
, σi+1
i+2
+
P
y y
x x
i (σi σi+1 +σi σi+1 )
y
y
σi+1
σi+2
]
does not factorize as
2D
6= 0 ⇒ Hstab
step by step:
1. generate Hstab for adjacent qubit pairs
1D
2. connect the pairs to obtain Hstab
1D
3. generate multiple Hstab
and connect
them pairwise into ladders
2D
4. connect the ladders to obtain Hstab
(b)
(c)
σz
σx
σx
σz
σz
σz
σx
σz
σz
the obtained Hstab is twisted!
the corresponding cluster state
is twisted too!
. Case with always-on interactions
Q: What if H0 and Hint cannot be switched on/off at will?
A: The method can be applied in a stroboscopic fashion!
based on the Baker-Campbell-Hausdorff (BCH) formula: eA eB =
= exp A + B + 12 [A, B] +
1
[A, [A, B]]
12
+
1
[B, [B, A]]
12
+ ...
realizing interaction pulse ( “extracting” Hint ):
A = i(H0 + Hint )τ , B = i(−H0 + Hint )τ
2
exp 2iHint τ + (iτ ) [H0 , Hint ] +
(iτ )3
3
⇒
eA eB =
[H0 , [H0 , Hint ]] + . . .
repeating n-times with nΩτ = π/4: in (eA eB )n
π k
k-th term ∼
⇒ the effects of H0 are eliminated!
4n
. Cluster-state fidelity: numerical results in the XY case
Fst (τ ) ≡ |hΨc |Uτ (δ)|Ψc i|2
Uτ (δ) ≡ e−iτ Hstab (π/4+δ)
1.00
analytical (perturbative) result:
Fst
0.95
0.90
N= 6
N= 8
N = 10
.
1 − Fst ∝ δ 2
( δ π/4 )
δ → δi (i = 1, . . . , N − 1)
0 1 2 3 4 5 6 7 8 9 10
10.2 σ
averaged over 10000 random realizations of the δi
taken from a Gaussian distribution of width σ
T. Tanamoto, D. Becker, VMS, and C. Bruder, PRA 86, 032327 (2012)
. Summary
Local-control approach allows for an efficient realization of quantum
gates in qubit arrays with XXZ Heisenberg interaction!
2D cluster states, a universal resource for MBQC, can be preserved
with high fidelity in XY - and Ising-coupled qubit arrays!
The two approaches to quantum control in interacting qubit arrays
are complementary!
V. M. Stojanovi´
c (Harvard)
TI conference @ CMSA
September 16, 2014
25 / 25

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