2014 04 22 wits presentation oqw
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The document discusses open quantum walks, focusing on their properties, dynamics, and applications in quantum computing. It provides an analytic derivation of the probability distribution and explores the role of noncommuting jump operators in these processes. The conclusion highlights that the asymptotic form of the probability distribution is either normal or solitonic, with its behavior determined by the spectrum of the jump operators.
![Open Quantum Walks
M j
i
=Bj
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: Bi
j
● Normalization:
● Time evolution:
● Probability
distribution:
∑
i
B j
i†
B j
i
=I
ρi
[n+1]
=∑
j
B j
i
ρj
[n]
B j
i †
Pi
[n+1]
=Tr (ρi
[n+1]
)](https://image.slidesharecdn.com/20140422witspresentationoqw-140610180401-phpapp01/85/2014-04-22-wits-presentation-oqw-7-320.jpg)
![Homogeneous open quantum walks on
the line
ρ0
[0]
=∑
j=1
N
P j∣j〉〈 j∣+ ∑
j, l=1
N
qjl∣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](https://image.slidesharecdn.com/20140422witspresentationoqw-140610180401-phpapp01/85/2014-04-22-wits-presentation-oqw-10-320.jpg)
![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 ,∣bj∣
2
+∣cj∣
2
=1
̂C=∑
k=1
N
√1−∣bk
2
∣∣π(k)〉〈k∣̂B=∑
j=1
N
b j∣j 〉〈 j∣
Pk
[n]
=∑
j=1
N
( n
(n−k)/2)∣bj
2
∣
(n−k)/2
∣cj
2
∣
(n+k)/2
P j
̄xj=n(∣bj∣
2
−∣cj∣
2
),σ(xj)=2√n∣bj∣∣cj∣](https://image.slidesharecdn.com/20140422witspresentationoqw-140610180401-phpapp01/85/2014-04-22-wits-presentation-oqw-24-320.jpg)
![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)(∣bj∣
2
)
(n−k)/2
(1−∣bj∣
2
)
(n+k)/2
P j
̄xj=n(2∣bj∣2
−1),σ(xj)=2√n∣bj∣√1−∣bj∣2](https://image.slidesharecdn.com/20140422witspresentationoqw-140610180401-phpapp01/85/2014-04-22-wits-presentation-oqw-25-320.jpg)
2014 04 22 wits presentation oqw
- 1. Open Quantum Walks with Noncommuting Jump Operators R. C. F. Caballar, I. Sinayskiy and F. Petruccione Caballar@ukzn.ac.za, ilsinay@gmail.com, petruccione@ukzn.ac.za
- 2. Outline ● Open Quantum Walks: A Very Short Review ● Homogeneous Open Quantum Walks ● Dynamical Properties ● Analytic Derivation of the Probability Distribution ● Conclusion
- 3. Open Quantum Walks: A Very Short Review
- 4. 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
- 5. Open Quantum Walks ● Total Hilbert Space: HP x HI ● 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
- 6. 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
- 7. Open Quantum Walks M j i =Bj 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: Bi j ● Normalization: ● Time evolution: ● Probability distribution: ∑ i B j i† B j i =I ρi [n+1] =∑ j B j i ρj [n] B j i † Pi [n+1] =Tr (ρi [n+1] )
- 8. 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 excitations Primary references: Attal et al, Phys. Lett. A 376, 1545 (2012)) and Sinayskiy and Petruccione, Quant. Inf. Quan. Proc. 11, 1301-1309 (2012)
- 10. Homogeneous open quantum walks on the line ρ0 [0] =∑ j=1 N P j∣j〉〈 j∣+ ∑ j, l=1 N qjl∣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
- 11. Dynamics of Homogeneous OQWs Consider 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 2d ̂Ai ρ ̂ Ai † m=∑i=1 N Tr( ̂Ai ρ∞ ̂ Ai † )ei ( ̂L−L * ( ̂L))=∑i=1 2d ̂ Ai † ̂Ai(ei l)−(ml)I
- 12. Dynamics of Homogeneous OQWs Theorem (Attal et al, arXiv: 1206.1472) Consider the stationary open quantum walk on Zd associated to the operators {A1 ,...,A2d }. Assume that the completely positive map admits a unique invariant state Let (rn , Xn )n>0 be the quantum trajectory process associated to this open quantum walk. L(ρ)=∑i=1 2d ̂Ai ρ ̂ Ai † ρ∞ .
- 13. Dynamics of Homogeneous OQWs Then converges in law to the Gaussian distribution N(0,C) in Rd with covariance matrix: Cij=δij (Tr( ̂Aiρ∞ ̂Ai † )+Tr( ̂Ai+d ρ∞ ̂Ai+d † ))−mi m j +(Tr( ̂Ai ρ∞ ̂ Ai † ̂L j)+Tr( ̂Ajρ∞ ̂ Aj † ̂Li)−Tr( ̂Ai+d ρ∞ ̂ Ai+d † ̂L j)−Tr( ̂Aj+d ρ∞ ̂ Aj+d † ̂Li)) −(mi Tr(ρ∞ ̂Lj)+m jTr (ρ∞ ̂Li)) X n −nm √n
- 14. Dynamics of Homogeneous OQWs Particular case: d = 1, jump operators A1 and A2 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)
- 16. 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 bj∣j〉〈 j∣ ̂B† ̂B+ ̂C† ̂C=I ̂C= ∑ j , k=1 N √1−∣bk 2 ∣U jk∣j〉〈k∣,∑ j=1 N U jn ✳ U jk=δn ,k
- 17. 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 ρjk∣j〉〈k∣ ρjk=b j bk * ρjk+∑l , m=1 N √(1−∣bl∣ 2 )(1−∣bm∣ 2 )U jl U km * ρlm ρ∞= 1 R ∑ j N 1 1−∣bj∣ 2 ∣j〉〈 j∣ R=∑ j N 1 1−∣b j∣ 2 m=∑j=1 N 2∣bj∣ 2 −1 1−∣b j∣ 2
- 18. 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, Uj is a scalar equal to 1 or 0 ̂U = ∑ j , k=1 m √1−∣bk 2 ∣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 )
- 19. N=7, n=500 ● U1 , U2 and U3 are 2 x 2 irreducible unitary blocks. ● U4 = 1 ● Eigenvalues: b1 =0.35, b2 =0.4, b3 =0.55, b4 =0.65, b5 =0.8, b1 =0.85, b7 =1 ● Initial state is diagonal
- 20. N=5, n=500 ● U1 and U2 are 2 x 2 irreducible unitary blocks. ● U3 = 1 ● Eigenvalues: b1 =0.23, b2 =0.28, b3 =0.84, b4 =0.87, b5 =1 ● Initial state is diagonal
- 21. N=7, n=500 ● U1 , U2 and U3 are 2 x 2 irreducible unitary blocks. ● U4 = 1 ● Eigenvalues: b1 =0.21, b2 =0.26, b3 =0.7, b4 =0.71, b5 =0.8, b1 =0.85, b7 =1 ● Initial state is diagonal
- 22. Analytic Derivation of the Probability Distributions
- 23. 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.
- 24. 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 ,∣bj∣ 2 +∣cj∣ 2 =1 ̂C=∑ k=1 N √1−∣bk 2 ∣∣π(k)〉〈k∣̂B=∑ j=1 N b j∣j 〉〈 j∣ Pk [n] =∑ j=1 N ( n (n−k)/2)∣bj 2 ∣ (n−k)/2 ∣cj 2 ∣ (n+k)/2 P j ̄xj=n(∣bj∣ 2 −∣cj∣ 2 ),σ(xj)=2√n∣bj∣∣cj∣
- 25. 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)(∣bj∣ 2 ) (n−k)/2 (1−∣bj∣ 2 ) (n+k)/2 P j ̄xj=n(2∣bj∣2 −1),σ(xj)=2√n∣bj∣√1−∣bj∣2
- 26. Time evolution and probability distribution ● (Konno and Yoo, J. Stat. Phys. 150, 299 (2012)) Given an initial state r0 , the probability distribution can be computed as follows: Px (n) = 1 2π ∫K dk e−ikx Tr(ρ0Y n(k)) Y n(k)=(eik ̂B† ̂B+e−ik ̂C† ̂C)n I
- 27. Probability Distribution, General Case
- 28. Probability Distribution, General Case
- 29. 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
- 30. 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.
- 31. 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.