2014 04 22 wits presentation oqw
TRANSCRIPT
Open Quantum Walks with Noncommuting Jump Operators
R. C. F. Caballar, I. Sinayskiy and F. Petruccione
Outline
● Open Quantum Walks: A Very Short Review● Homogeneous Open Quantum Walks● Dynamical Properties● Analytic Derivation of the Probability
Distribution● Conclusion
Open Quantum Walks: A Very Short Review
Open Quantum Walks
● Classical Random Walk – system is in a superposition of position probabilities
● Quantum Random Walk – system is in a superposition of position and internal states
– Wide applications in quantum computing
– Have been experimentally realized
● Open Quantum Random Walk – evolution is due to interaction between system and environment
Open Quantum Walks
● Total Hilbert Space: HP x H
I
● Evolution: discrete timesteps (via unitary U) or continuous (Schrodinger equation)
● Quantum computing algorithms based on open quantum walks
– Unitary operator U or Hamiltonian– Measurement of position at the end of the
evolution– Decoherence effects for control of walker
Open Quantum Walks
● If a model of computation is universal, it can be used to simulate any other model of computation.
● Quantum computation via continuous time quantum walk (Childs, Phys. Rev. Lett. 102, 180501 (2009)) or discrete time quantum walk Lovett et al, Phys. Rev. A 81(4), 042330 (2010)) is universal
● Basis: universal gate sets can be simulated by a quantum walk on different graphs
Open Quantum Walks
M ji=B j
i׿¿
Walker's internal state changes as it jumps from node to node(fig. from Sinayskiy and Petruccione, Phys. Scr. T151, 014077 (2012)
● Jump operator: Bij
● Normalization:
● Time evolution:
● Probability distribution:
∑i
B ji †B j
i=I
ρi[n+1]
=∑j
B jiρ j
[n ]B ji †
P i[n+1]=Tr (ρi
[n+1 ])
Applications of Open Quantum Walks
● Dissipative Quantum computing algorithms– Two nodes: Single-qubit operations
– More nodes: Multi-qubit operations
● Dissipative quantum state preparation– Two nodes: Single-qubit states
– More nodes: Multiple Qubit States
● Efficient quantum transport of excitationsPrimary references: Attal et al, Phys. Lett. A 376, 1545 (2012)) and Sinayskiy and Petruccione, Quant. Inf. Quan. Proc. 11, 1301-1309 (2012)
Homogeneous Open Quantum Walks
Homogeneous open quantum walks on the line
ρ0[0]
=∑j=1
N
P j∣ j ⟩ ⟨ j∣+ ∑j , l=1
N
q jl∣ j ⟩ ⟨ l∣
● Initial state of the system:
● Time evolution:● Normalization:
(fig. from Sinayskiy and Petruccione, Phys. Scr. T151, 014077 (2012)
ρi[n+1]
=Bρi+1[n] B†
+Cρi−1[n ] C †
B† B+C †C=I
Dynamics of Homogeneous OQWsConsider an open quantum walk on Zd given by
Let where ei is part of a canonical
basis in Zd, ei+d
= -ei . m is the mean value.
Lemma (Attal et al, arXiv: 1206.1472)
For every l in Rd, there is a solution to
The difference between any two solutions of this equation is a multiple of the identity.
L(ρ)=∑i=1
2dAiρ
Ai†
m=∑i=1
NTr ( Aiρ∞
Ai†)e i
( L−L*( L))=∑i=1
2d Ai† Ai(ei l)−(ml ) I
Dynamics of Homogeneous OQWs
Theorem (Attal et al, arXiv: 1206.1472)
Consider the stationary open quantum walk on Zd associated to the operators {A
1,...,A
2d}. Assume
that the completely positive map
admits a unique invariant state Let (rn, X
n)
n>0
be the quantum trajectory process associated to this open quantum walk.
L(ρ)=∑i=1
2dAiρ
Ai†
ρ∞ .
Dynamics of Homogeneous OQWs
Then
converges in law to the Gaussian distribution N(0,C) in Rd with covariance matrix:
C ij=δij (Tr ( Aiρ∞ Ai†)+Tr ( Ai+dρ∞
Ai+d† ))−mim j
+(Tr ( Aiρ∞Ai† L j)+Tr ( A jρ∞
A j† L i)−Tr ( Ai+dρ∞
Ai+d† L j)−Tr ( A j+dρ∞
A j+d† Li))
−(miTr (ρ∞ L j)+m jTr (ρ∞ Li))
X n−nm
√n
Dynamics of Homogeneous OQWs
Particular case: d = 1, jump operators A1 and A
2
m=Tr ( A1ρ∞ A1†)−Tr ( A2ρ∞ A2
†)
L−L*( L)=A1† A1− A2
† A2−mI=2 A1† A1−(1+m)I
σ2=Tr ( A1ρ∞ A1†)+Tr ( A2 ρ∞ A2
†)−m2
+2(Tr ( A1ρ∞A1† L)−Tr ( A2ρ∞
A2† L))−2mTr(ρ∞ L)
Dynamical Properties
General Form of the Jump Operators
● B is diagonalizable:
● Invariance of normalization condition:
● General form of C:
– Ujk are elements of an irreducible unitary matrix
U.
B† B+C †C=I
B=V †BV=∑j=1
N
b j∣ j ⟩ ⟨ j∣
B† B+C † C=I
C= ∑j , k=1
N
√1−∣bk2∣U jk∣ j ⟩ ⟨ k∣,∑
j=1
N
U jn✳ U jk=δn ,k
Steady State of the Jump Operators● General form:
● Steady state condition:
● One solution to this equation:
● Attal et al's central limit theorem can be applied
ρ∞=Bρ∞B†+Cρ∞C
†
ρ∞= ∑j , k=1
N
ρ jk∣ j ⟩ ⟨ k∣
ρ jk=b jbk*ρ jk+∑l , m=1
N
√(1−∣bl∣2)(1−∣bm∣
2)U jlU km
*ρlm
ρ∞=1R∑j
N1
1−∣b j∣2∣ j ⟩ ⟨ j∣ R=∑
j
N1
1−∣b j∣2
m=∑ j=1
N 2∣b j∣2−1
1−∣b j∣2
General Case●
● Large n: Distribution becomes normal
● Uj are irreducible
● Number of irreducible block diagonals Uj in C =
number of peaks in distribution
● Solitonic component → bj=0 or 1, U
j is a scalar
equal to 1 or 0
U= ∑j , k=1
m
√1−∣bk2∣U jk∣ j ⟩ ⟨k∣,
∑j=1
m
U jn✳ U jk=δn , k
C=(U 1 0 0…0
0 U 2 0…0⋮ ⋮ ⋮⋱⋮
0 0 0…U N)
N=7, n=500
● U1, U
2 and U
3 are 2 x
2 irreducible unitary blocks.
● U4 = 1
● Eigenvalues: b1=0.35,
b2=0.4, b
3=0.55,
b4=0.65, b
5=0.8,
b1=0.85, b
7=1
● Initial state is diagonal
N=5, n=500
● U1 and U
2 are 2 x 2
irreducible unitary blocks.
● U3 = 1
● Eigenvalues: b1=0.23,
b2=0.28, b
3=0.84,
b4=0.87, b
5=1
● Initial state is diagonal
N=7, n=500
● U1, U
2 and U
3 are 2 x
2 irreducible unitary blocks.
● U4 = 1
● Eigenvalues: b1=0.21,
b2=0.26, b
3=0.7,
b4=0.71, b
5=0.8,
b1=0.85, b
7=1
● Initial state is diagonal
Analytic Derivation of the Probability Distributions
What do these distributions tell us?
● The spectrum of B and the structure of C determines the asymptotic probability distribution of the system.
● Knowing the analytic form of the distribution will tell us immediately what the system's dynamical behavior will be at any instant of time.
● Given the spectrum of B and the structure of C, we can immediately construct the system's probability distribution to tell us its dynamical behavior at any instant of time.
Special cases: Exact results● Case 1: Commuting jump operators (I. Sinayskiy and F.
Petruccione, Phys. Scr. T151, 014077 (2012))
● Case 2: Noncommuting jump operators (R. C. F. Caballar, I. Sinayskiy and F. Petruccione, in preparation)
● Probability distribution, case 1:
B=∑j=1
N
b j∣ j ⟩ ⟨ j∣ C=∑j=1
N
c j∣ j ⟩ ⟨ j∣ ∀ j ,∣b j∣2+∣c j∣
2=1
C=∑k=1
N
√1−∣bk2∣∣π(k ) ⟩ ⟨k∣B=∑
j=1
N
b j∣ j ⟩ ⟨ j∣
P k[n ]=∑
j=1
N
( n(n−k )/2)∣b j
2∣(n− k)/2
∣c j2∣
(n+ k)/2P j
x j=n(∣b j∣2−∣c j∣
2) ,σ( x j)=2√n∣b j∣∣c j∣
Time evolution and probability distribution
● Probability distribution, case 2:
● For both cases, the distribution is normal even for small timesteps n
– Limiting case: bj = 0 or 1 (solitonic)
Pk[n ]=∑
j=1
N
( n(n−k )/2)(∣b j∣
2)(n−k)/2
(1−∣b j∣2)
(n+k)/2P j
x j=n(2∣b j∣2−1) ,σ (x j)=2√n∣b j∣√1−∣b j∣
2
Time evolution and probability distribution
● (Konno and Yoo, J. Stat. Phys. 150, 299 (2012)) Given an initial state r
0, the
probability distribution can be computed as follows:
P x(n )
=1
2π∫Kdk e−ikxTr (ρ0Y n(k ))
Y n(k )=(e ik B† B+e−ik C † C)n I
Probability Distribution, General Case
Probability Distribution, General Case
What we have done and what remains to be done
● Without having to numerically evolve the system, we can determine, at any instant of time, the dynamical behavior of the system by looking at the analytic form of the distribution.
– Distribution is expressed in terms of the spectrum of B and the matrix elements of C
● What remains to be done:– Compute for the covariance of the distribution
What we have done and what remains to be done
● If C has N irreducible block diagonals Uj then
the probability distribution will have N components, each of the form given by P(n)
x
● There will then be N peaks in the distribution.● Form of C will determine the number of peaks
present in the distribution.
Conclusion
● For a homogeneous open quantum walk on the line, the distribution's asymptotic form is either a normal or solitonic distribution.
● The distribution's analytic form is a function of the spectrum of B and the matrix elements of C.
● Distribution's analytic form tells us at any instant of time what the system's dynamical behavior will be.