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Operational perspective of quantum simulations usingquantum walks
C. M. Chandrashekar(Arindam Mallick and Shivani Singh)
The Institute of Mathematical Sciences, Chennai, India
QIPA 2018Harish-Chandra Research Institute, Allahabad
Operational perspective of quantum simulations using quantum walks
Outline
Quantum simulations : Operational approach and why is it important ?
Discrete-time quantum walks (DTQW)
DTQW with Dirac equation
Disorder QW and localization of quantum state
Dynamic localizationAnderson localization
Accelerate DTQW
one particle casetwo particle localization and entanglement generation
Operational perspective of quantum simulations using quantum walks
Outline
Quantum simulations : Operational approach and why is it important ?
Discrete-time quantum walks (DTQW)
DTQW with Dirac equation
Disorder QW and localization of quantum state
Dynamic localizationAnderson localization
Accelerate DTQW
one particle casetwo particle localization and entanglement generation
Operational perspective of quantum simulations using quantum walks
Outline
Quantum simulations : Operational approach and why is it important ?
Discrete-time quantum walks (DTQW)
DTQW with Dirac equation
Disorder QW and localization of quantum state
Dynamic localizationAnderson localization
Accelerate DTQW
one particle casetwo particle localization and entanglement generation
Operational perspective of quantum simulations using quantum walks
Outline
Quantum simulations : Operational approach and why is it important ?
Discrete-time quantum walks (DTQW)
DTQW with Dirac equation
Disorder QW and localization of quantum state
Dynamic localizationAnderson localization
Accelerate DTQW
one particle casetwo particle localization and entanglement generation
Operational perspective of quantum simulations using quantum walks
Outline
Quantum simulations : Operational approach and why is it important ?
Discrete-time quantum walks (DTQW)
DTQW with Dirac equation
Disorder QW and localization of quantum state
Dynamic localizationAnderson localization
Accelerate DTQW
one particle casetwo particle localization and entanglement generation
Operational perspective of quantum simulations using quantum walks
Quantum simulation
• Simulation of quantum physics is a difficult computational problem :when dealing with large system or an high energy quantum phenomenon
How to overcome the difficulty ?
Using some controllable quantum system to study less controllable /accessiblequantum system (Quantum Simulation)
• Application : study of condensed matter physics, high-energy physics, atomicphysics, quantum chemistry and cosmology
• Can be implemented in quantum computer and also with simpler analog devicesthat require less control and easier to constructFor example :neutral atoms, ion, polar molecules, electron in semiconductor,superconducting circuits, nuclear spins and photons
Operational perspective of quantum simulations using quantum walks
Quantum simulation
• Simulation of quantum physics is a difficult computational problem :when dealing with large system or an high energy quantum phenomenon
How to overcome the difficulty ?
Using some controllable quantum system to study less controllable /accessiblequantum system (Quantum Simulation)
• Application : study of condensed matter physics, high-energy physics, atomicphysics, quantum chemistry and cosmology
• Can be implemented in quantum computer and also with simpler analog devicesthat require less control and easier to constructFor example :neutral atoms, ion, polar molecules, electron in semiconductor,superconducting circuits, nuclear spins and photons
Operational perspective of quantum simulations using quantum walks
Quantum simulation
• Simulation of quantum physics is a difficult computational problem :when dealing with large system or an high energy quantum phenomenon
How to overcome the difficulty ?
Using some controllable quantum system to study less controllable /accessiblequantum system (Quantum Simulation)
• Application : study of condensed matter physics, high-energy physics, atomicphysics, quantum chemistry and cosmology
• Can be implemented in quantum computer and also with simpler analog devicesthat require less control and easier to constructFor example :neutral atoms, ion, polar molecules, electron in semiconductor,superconducting circuits, nuclear spins and photons
Operational perspective of quantum simulations using quantum walks
Quantum simulation
• Simulation of quantum physics is a difficult computational problem :when dealing with large system or an high energy quantum phenomenon
How to overcome the difficulty ?
Using some controllable quantum system to study less controllable /accessiblequantum system (Quantum Simulation)
• Application : study of condensed matter physics, high-energy physics, atomicphysics, quantum chemistry and cosmology
• Can be implemented in quantum computer and also with simpler analog devicesthat require less control and easier to constructFor example :neutral atoms, ion, polar molecules, electron in semiconductor,superconducting circuits, nuclear spins and photons
Operational perspective of quantum simulations using quantum walks
Quantum simulation
• Simulation of quantum physics is a difficult computational problem :when dealing with large system or an high energy quantum phenomenon
How to overcome the difficulty ?
Using some controllable quantum system to study less controllable /accessiblequantum system (Quantum Simulation)
• Application : study of condensed matter physics, high-energy physics, atomicphysics, quantum chemistry and cosmology
• Can be implemented in quantum computer and also with simpler analog devicesthat require less control and easier to constructFor example :neutral atoms, ion, polar molecules, electron in semiconductor,superconducting circuits, nuclear spins and photons
Operational perspective of quantum simulations using quantum walks
Quantum Simulation :Operational approach (algorithmic) approach
using a qubit
Operational perspective of quantum simulations using quantum walks
Qubit : Basic Element
• Qubit - Ideal two-state quantum system : | ↑〉 =
[10
]and | ↓〉 =
[01
]logical /computational states
photons (V and H polarization, transmission mode - path encoding)
electrons or other spin- 12 systems (spin up and down)
systems defined by two energy levels of atoms or ions
Allows superposition : |Ψ〉 = α| ↑〉+ β| ↓〉α, β ∈ C ; |α|2 + |β|2 = 1 ; |Ψ〉 = e iη|Ψ〉
Operational perspective of quantum simulations using quantum walks
Discrete-time quantum walk
• Walk is defined on the Hilbert space H = Hc ⊗Hp
Hc (particle) is spanned by | ↑〉 and | ↓〉Hp (position) is spanned by |j〉, j ∈ Z
• Initial state :|Ψin〉 = [cos(δ)| ↑〉+ e iη sin(δ)| ↓〉]⊗ |j = 0〉
• Evolution :
Coin operation : C (θ) =
[cos(θ) − i sin(θ)
−i sin(θ) cos(θ)
]Conditional unitary shift operation S :
S =∑
j∈Z
[| ↑〉〈↑ | ⊗ |j − 1〉〈j |+ | ↓〉〈↓ | ⊗ |j + 1〉〈j |
]state | ↑〉 moves to the left and state | ↓〉 moves to the right
Operational perspective of quantum simulations using quantum walks
Discrete-time quantum walk
• Walk is defined on the Hilbert space H = Hc ⊗Hp
Hc (particle) is spanned by | ↑〉 and | ↓〉Hp (position) is spanned by |j〉, j ∈ Z
• Initial state :|Ψin〉 = [cos(δ)| ↑〉+ e iη sin(δ)| ↓〉]⊗ |j = 0〉
• Evolution :
Coin operation : C (θ) =
[cos(θ) − i sin(θ)
−i sin(θ) cos(θ)
]Conditional unitary shift operation S :
S =∑
j∈Z
[| ↑〉〈↑ | ⊗ |j − 1〉〈j |+ | ↓〉〈↓ | ⊗ |j + 1〉〈j |
]state | ↑〉 moves to the left and state | ↓〉 moves to the right
Operational perspective of quantum simulations using quantum walks
Discrete-time quantum walk
• Walk is defined on the Hilbert space H = Hc ⊗Hp
Hc (particle) is spanned by | ↑〉 and | ↓〉Hp (position) is spanned by |j〉, j ∈ Z
• Initial state :|Ψin〉 = [cos(δ)| ↑〉+ e iη sin(δ)| ↓〉]⊗ |j = 0〉
• Evolution :
Coin operation : C (θ) =
[cos(θ) − i sin(θ)
−i sin(θ) cos(θ)
]Conditional unitary shift operation S :
S =∑
j∈Z
[| ↑〉〈↑ | ⊗ |j − 1〉〈j |+ | ↓〉〈↓ | ⊗ |j + 1〉〈j |
]state | ↑〉 moves to the left and state | ↓〉 moves to the right
Operational perspective of quantum simulations using quantum walks
Discrete-time quantum walk
• Walk is defined on the Hilbert space H = Hc ⊗Hp
Hc (particle) is spanned by | ↑〉 and | ↓〉Hp (position) is spanned by |j〉, j ∈ Z
• Initial state :|Ψin〉 = [cos(δ)| ↑〉+ e iη sin(δ)| ↓〉]⊗ |j = 0〉
• Evolution :
Coin operation : C (θ) =
[cos(θ) − i sin(θ)
−i sin(θ) cos(θ)
]
Conditional unitary shift operation S :
S =∑
j∈Z
[| ↑〉〈↑ | ⊗ |j − 1〉〈j |+ | ↓〉〈↓ | ⊗ |j + 1〉〈j |
]state | ↑〉 moves to the left and state | ↓〉 moves to the right
Operational perspective of quantum simulations using quantum walks
Discrete-time quantum walk
• Walk is defined on the Hilbert space H = Hc ⊗Hp
Hc (particle) is spanned by | ↑〉 and | ↓〉Hp (position) is spanned by |j〉, j ∈ Z
• Initial state :|Ψin〉 = [cos(δ)| ↑〉+ e iη sin(δ)| ↓〉]⊗ |j = 0〉
• Evolution :
Coin operation : C (θ) =
[cos(θ) − i sin(θ)
−i sin(θ) cos(θ)
]Conditional unitary shift operation S :
S =∑
j∈Z
[| ↑〉〈↑ | ⊗ |j − 1〉〈j |+ | ↓〉〈↓ | ⊗ |j + 1〉〈j |
]state | ↑〉 moves to the left and state | ↓〉 moves to the right
Operational perspective of quantum simulations using quantum walks
Quantum walk
• Each step of QW : W = S(C (θ)⊗ 1)
−100 −80 −60 −40 −20 0 20 40 60 80 1000
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Particle position
Pro
babili
ty
Quantum walkClassical random walk
100 step of CRW and QW [S(C(π/4)⊗ 1)]100on a particle with initial state1√2(| ↑〉+ i | ↓〉)
• G. V. Riazanov (1958), R. Feynman (1986)• K.R. Parthasarathy, Journal of applied probability 25, 151-166 (1988)•Y. Aharonov, L. Davidovich and N. Zugury, Phys. Rev. A, 48, 1687 (1993)• Use of word Quantum random walk
Operational perspective of quantum simulations using quantum walks
Quantum walks
Superposition and interference : Evolution (dynamics) exploits superpositionand interference aspects of quantum mechanics entangling the Hilbert spaceinvolved in the dynamics ( particle and position space)Explores multiple possible paths simultaneously with amplitude correspondingto different paths interfering
The variance grows quadratically with number of steps (t) compared to thelinear growth for CRW :σ2 ∝ t2 (QW), σ2 ∝ t (CRW)
Experimentally implemented and control over the dynamics demonstratedNMR system, ion traps, photons in optical waveguide, neutral atoms onoptical lattice.......
Operational perspective of quantum simulations using quantum walks
Quantum walks
Superposition and interference : Evolution (dynamics) exploits superpositionand interference aspects of quantum mechanics entangling the Hilbert spaceinvolved in the dynamics ( particle and position space)Explores multiple possible paths simultaneously with amplitude correspondingto different paths interfering
The variance grows quadratically with number of steps (t) compared to thelinear growth for CRW :σ2 ∝ t2 (QW), σ2 ∝ t (CRW)
Experimentally implemented and control over the dynamics demonstratedNMR system, ion traps, photons in optical waveguide, neutral atoms onoptical lattice.......
Operational perspective of quantum simulations using quantum walks
Quantum walks
Superposition and interference : Evolution (dynamics) exploits superpositionand interference aspects of quantum mechanics entangling the Hilbert spaceinvolved in the dynamics ( particle and position space)Explores multiple possible paths simultaneously with amplitude correspondingto different paths interfering
The variance grows quadratically with number of steps (t) compared to thelinear growth for CRW :σ2 ∝ t2 (QW), σ2 ∝ t (CRW)
Experimentally implemented and control over the dynamics demonstratedNMR system, ion traps, photons in optical waveguide, neutral atoms onoptical lattice.......
Operational perspective of quantum simulations using quantum walks
Continuum limit and simulation of Dirac equation
Dirac equation
(i~∂
∂t− HD
)Ψ =
(i~∂
∂t+ i~cα · ∂
∂x− βmc2
)Ψ = 0
From DTQW
ψ↑x,t+1 = cos(θ)ψ↑x+1,t − i sin(θ)ψ↓x−1,t
ψ↓x,t+1 = cos(θ)ψ↓x−1,t − i sin(θ)ψ↑x+1,t
when θ = 0, the expression in continuum limit takes the form[i~∂
∂t− i~σ3
∂
∂x
]Ψ(x , t) = 0
Massless Dirac equation
David Mayer (1996) ; Fredrick Strauch (2006) ; CMC (2010)
Operational perspective of quantum simulations using quantum walks
Continuum limit and simulation of Dirac equation
Dirac equation
(i~∂
∂t− HD
)Ψ =
(i~∂
∂t+ i~cα · ∂
∂x− βmc2
)Ψ = 0
From DTQW
ψ↑x,t+1 = cos(θ)ψ↑x+1,t − i sin(θ)ψ↓x−1,t
ψ↓x,t+1 = cos(θ)ψ↓x−1,t − i sin(θ)ψ↑x+1,t
when θ = 0, the expression in continuum limit takes the form[i~∂
∂t− i~σ3
∂
∂x
]Ψ(x , t) = 0
Massless Dirac equation
David Mayer (1996) ; Fredrick Strauch (2006) ; CMC (2010)
Operational perspective of quantum simulations using quantum walks
DE with mass term : Split-step / 2-period QW
C (θ1) =
(cos(θ1) −i sin(θ1)
−i sin(θ1) cos(θ1)
),
C (θ2) =
(cos(θ2) −i sin(θ2)
−i sin(θ2) cos(θ2)
)and a two half-shift operators,
S− =
(T− 00 I
), S+ =
(I 00 T+
)S =
(T− 00 T+
)
T− = |j − 1〉〈j | ; T+ = |j + 1〉〈j |
USQW = S+
(C (θ2)⊗ I
)S−(C (θ1)⊗ I
)≡ S
(C (θ2)⊗ I
)S(C (θ1)⊗ I
)
Operational perspective of quantum simulations using quantum walks
DE with mass term : Split-step / 2-period QW
C (θ1) =
(cos(θ1) −i sin(θ1)
−i sin(θ1) cos(θ1)
),
C (θ2) =
(cos(θ2) −i sin(θ2)
−i sin(θ2) cos(θ2)
)and a two half-shift operators,
S− =
(T− 00 I
), S+ =
(I 00 T+
)S =
(T− 00 T+
)
T− = |j − 1〉〈j | ; T+ = |j + 1〉〈j |
USQW = S+
(C (θ2)⊗ I
)S−(C (θ1)⊗ I
)≡ S
(C (θ2)⊗ I
)S(C (θ1)⊗ I
)Operational perspective of quantum simulations using quantum walks
DCA and SS-QW
−100 −50 0 50 1000
0.02
0.04
0.06
0.08
Position
Pro
babili
ty
DTQW
SS−DTQW
−100 0 1000
0.05
−100 −50 0 50 1000
0.02
0.04
0.06
0.08
0.1
Position
Pro
babili
ty
θ1 = π/12
θ1=π/3
θ1 = 5π/12
θ2 = π/4
SSQW (θ1 = 0, θ2 = π/4) = DCA α = β = 1√2
Substituting
θ1 = φ1 = δ1 = δ2 = 0 we get,
USQW =
(cos(θ2)T− −i sin(θ2)I−i sin(θ2)I cos(θ2)T+
)which is in the same form as UDA where β = sin(θ2) ≡ mca
~ and α = cos(θ2).
Operational perspective of quantum simulations using quantum walks
SS-QW and DE
[∂
∂t− cos(θ2)
[cos(θ1) −i sin(θ1)i sin(θ1) − cos(θ1)
]∂
∂x−[cos(θ1 + θ2)− 1 −i sin(θ1 + θ2)−i sin(θ1 + θ2) cos(θ1 + θ2)− 1
] ] [ψ↓
x,t
ψ↑x,t
]= 0
1 Setting θ1 = 0 and θ2 to a small value (mass of sub-atomic particles) :
i~
[∂
∂t−(
1− θ22
2
)[1 00 −1
]∂
∂x+ iθ2
[0 11 0
]] [ψ↓x,tψ↑x,t
]≈ 0
2 Setting cos(θ1 + θ2) = 1 :
i~
[∂
∂t− cos(θ2)
[cos(θ1) −i sin(θ1)i sin(θ1) − cos(θ1)
]∂
∂x
] [ψ↓x,tψ↑x,t
]= 0
3 Setting θ1 to be extremely small and cos(θ1 + θ2) = 1 :
i~
[∂
∂t− cos(θ2)
[1 00 −1
]∂
∂x
] [ψ↓x,tψ↑x,t
]≈ 0
Scientific Reports 6, 25779 (2016); Phys. Rev. A 97, 012116 (2018)
Operational perspective of quantum simulations using quantum walks
SS-QW and DE
[∂
∂t− cos(θ2)
[cos(θ1) −i sin(θ1)i sin(θ1) − cos(θ1)
]∂
∂x−[cos(θ1 + θ2)− 1 −i sin(θ1 + θ2)−i sin(θ1 + θ2) cos(θ1 + θ2)− 1
] ] [ψ↓
x,t
ψ↑x,t
]= 0
1 Setting θ1 = 0 and θ2 to a small value (mass of sub-atomic particles) :
i~
[∂
∂t−(
1− θ22
2
)[1 00 −1
]∂
∂x+ iθ2
[0 11 0
]] [ψ↓x,tψ↑x,t
]≈ 0
2 Setting cos(θ1 + θ2) = 1 :
i~
[∂
∂t− cos(θ2)
[cos(θ1) −i sin(θ1)i sin(θ1) − cos(θ1)
]∂
∂x
] [ψ↓x,tψ↑x,t
]= 0
3 Setting θ1 to be extremely small and cos(θ1 + θ2) = 1 :
i~
[∂
∂t− cos(θ2)
[1 00 −1
]∂
∂x
] [ψ↓x,tψ↑x,t
]≈ 0
Scientific Reports 6, 25779 (2016); Phys. Rev. A 97, 012116 (2018)
Operational perspective of quantum simulations using quantum walks
SS-QW and DE
[∂
∂t− cos(θ2)
[cos(θ1) −i sin(θ1)i sin(θ1) − cos(θ1)
]∂
∂x−[cos(θ1 + θ2)− 1 −i sin(θ1 + θ2)−i sin(θ1 + θ2) cos(θ1 + θ2)− 1
] ] [ψ↓
x,t
ψ↑x,t
]= 0
1 Setting θ1 = 0 and θ2 to a small value (mass of sub-atomic particles) :
i~
[∂
∂t−(
1− θ22
2
)[1 00 −1
]∂
∂x+ iθ2
[0 11 0
]] [ψ↓x,tψ↑x,t
]≈ 0
2 Setting cos(θ1 + θ2) = 1 :
i~
[∂
∂t− cos(θ2)
[cos(θ1) −i sin(θ1)i sin(θ1) − cos(θ1)
]∂
∂x
] [ψ↓x,tψ↑x,t
]= 0
3 Setting θ1 to be extremely small and cos(θ1 + θ2) = 1 :
i~
[∂
∂t− cos(θ2)
[1 00 −1
]∂
∂x
] [ψ↓x,tψ↑x,t
]≈ 0
Scientific Reports 6, 25779 (2016); Phys. Rev. A 97, 012116 (2018)
Operational perspective of quantum simulations using quantum walks
SS-QW and DE
[∂
∂t− cos(θ2)
[cos(θ1) −i sin(θ1)i sin(θ1) − cos(θ1)
]∂
∂x−[cos(θ1 + θ2)− 1 −i sin(θ1 + θ2)−i sin(θ1 + θ2) cos(θ1 + θ2)− 1
] ] [ψ↓
x,t
ψ↑x,t
]= 0
1 Setting θ1 = 0 and θ2 to a small value (mass of sub-atomic particles) :
i~
[∂
∂t−(
1− θ22
2
)[1 00 −1
]∂
∂x+ iθ2
[0 11 0
]] [ψ↓x,tψ↑x,t
]≈ 0
2 Setting cos(θ1 + θ2) = 1 :
i~
[∂
∂t− cos(θ2)
[cos(θ1) −i sin(θ1)i sin(θ1) − cos(θ1)
]∂
∂x
] [ψ↓x,tψ↑x,t
]= 0
3 Setting θ1 to be extremely small and cos(θ1 + θ2) = 1 :
i~
[∂
∂t− cos(θ2)
[1 00 −1
]∂
∂x
] [ψ↓x,tψ↑x,t
]≈ 0
Scientific Reports 6, 25779 (2016); Phys. Rev. A 97, 012116 (2018)
Operational perspective of quantum simulations using quantum walks
Examples of simulations
500 1500 2500 3500 45000
0.2
0.4
0.6
0.8
1
Number of steps
Oscill
atio
n P
rob
ab
ility
νe(0) → ν
e(t)
νe(0) → ν
τ(t)
νe(0) → ν
µ(t)
(1) Neutrino flavour oscillation Eur. Phys. J. C (Particles and Fields), 77: 85 (2017)
(2) Dirac Hamiltonian in curved space with U(1) gauge potential arXiv:1712.03911
Operational perspective of quantum simulations using quantum walks
Disorder and localization
Operational perspective of quantum simulations using quantum walks
Disorder induced by randomized unitary evolution
• Temporal disorder
S(B(θt)⊗ 1
)· · · S
(B(θj)⊗ 1
)· · · S
(B(θ0)⊗ 1
)|Ψin〉
randomly chosen parameter θ ∈ {−π/2, π/2} for each step.
• Spatial disorder [S
(∑x
B(θx)⊗ |x〉〈x |
)]t|Ψin〉
randomly chosen parameters θ ∈ {−π/2, π/2} for each position.
• Spatio-temporal disorder (fluctuating)
S
(∑x
B(θx,t)⊗ |x〉〈x |
)· · · S
(∑x
B(θx,j)⊗ |x〉〈x |
)· · · S
(∑x
B(θx,1)⊗ |x〉〈x |
)|Ψin〉
randomly chosen parameters θ ∈ {−π/2, π/2} for each position and step.
Operational perspective of quantum simulations using quantum walks
Disorder induced by randomized unitary evolution
• Temporal disorder
S(B(θt)⊗ 1
)· · · S
(B(θj)⊗ 1
)· · · S
(B(θ0)⊗ 1
)|Ψin〉
randomly chosen parameter θ ∈ {−π/2, π/2} for each step.
• Spatial disorder [S
(∑x
B(θx)⊗ |x〉〈x |
)]t|Ψin〉
randomly chosen parameters θ ∈ {−π/2, π/2} for each position.
• Spatio-temporal disorder (fluctuating)
S
(∑x
B(θx,t)⊗ |x〉〈x |
)· · · S
(∑x
B(θx,j)⊗ |x〉〈x |
)· · · S
(∑x
B(θx,1)⊗ |x〉〈x |
)|Ψin〉
randomly chosen parameters θ ∈ {−π/2, π/2} for each position and step.
Operational perspective of quantum simulations using quantum walks
Disorder induced by randomized unitary evolution
• Temporal disorder
S(B(θt)⊗ 1
)· · · S
(B(θj)⊗ 1
)· · · S
(B(θ0)⊗ 1
)|Ψin〉
randomly chosen parameter θ ∈ {−π/2, π/2} for each step.
• Spatial disorder [S
(∑x
B(θx)⊗ |x〉〈x |
)]t|Ψin〉
randomly chosen parameters θ ∈ {−π/2, π/2} for each position.
• Spatio-temporal disorder (fluctuating)
S
(∑x
B(θx,t)⊗ |x〉〈x |
)· · · S
(∑x
B(θx,j)⊗ |x〉〈x |
)· · · S
(∑x
B(θx,1)⊗ |x〉〈x |
)|Ψin〉
randomly chosen parameters θ ∈ {−π/2, π/2} for each position and step.
Operational perspective of quantum simulations using quantum walks
Disorder induced by randomized unitary evolution
• Temporal disorder
S(B(θt)⊗ 1
)· · · S
(B(θj)⊗ 1
)· · · S
(B(θ0)⊗ 1
)|Ψin〉
randomly chosen parameter θ ∈ {−π/2, π/2} for each step.
• Spatial disorder [S
(∑x
B(θx)⊗ |x〉〈x |
)]t|Ψin〉
randomly chosen parameters θ ∈ {−π/2, π/2} for each position.
• Spatio-temporal disorder (fluctuating)
S
(∑x
B(θx,t)⊗ |x〉〈x |
)· · · S
(∑x
B(θx,j)⊗ |x〉〈x |
)· · · S
(∑x
B(θx,1)⊗ |x〉〈x |
)|Ψin〉
randomly chosen parameters θ ∈ {−π/2, π/2} for each position and step.
Operational perspective of quantum simulations using quantum walks
Disorder and localization
20% disorder 100% disorder
−100 −50 0 50 1000
0.02
0.04
0.06
0.08
0.1
0.12
Position
Pro
babili
ty
θ = π/4
SD
TD
S−TD
−100 −50 0 50 1000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Position
Pro
babili
ty
θ = π/4
SD
TD
S−TD
0 20 40 60 80 1000
10
20
30
40
50
60
Percentage of Disorder
Sta
ndard
Devia
tion
SD
TD
S−TD
0 50 100 150 2000
20
40
60
80
100
120
Steps
Sta
ndard
Devia
tion (
σ)
θ = π/4
SD
TD
S−TD
Operational perspective of quantum simulations using quantum walks
Dynamic localization from temporal disorder
vg (k, θt) = ± k cos(θt)√k2 cos(θt) + A(θ, t)
A(θ, t) = 1 + sin(θt)[
csc(θt±1)− cot(θt±1)]− cos(θt)
For each instance of time t the group velocity will be a random value in the range:−1 ≤ vg (k, θt) ≤ 1. The average group velocity :
vTDg (k , θt , t) =
1
t
t∑t=0
vg (k , θt) ≈ 0
−600 −400 −200 0 200 400 6000
0.02
0.04
0.06
0.08
0.1
Position
Pro
bab
ilit
y
100 steps QW
200 steps QW
300 steps QW
600 steps QW
100 steps TDQW
200 steps TDQW
300 steps TDQW
600 steps TDQW
−50 0 500
0.02
0.04
0.06
0.08
Position
Pro
bab
ilit
y
Distribution remains localized near the origin irrespective of the time (t)
Operational perspective of quantum simulations using quantum walks
Entanglement in localized states
Probability distribution Entanglement
−100 −50 0 50 1000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Position
Pro
babili
ty
θ = π/4
SD
TD
S−TD
0 20 40 60 80 1000
0.2
0.4
0.6
0.8
1
Steps
Enta
ngle
ment
θ = π/4
SD
TD
S−TD
0 20 40 60 80 1000.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Percentage of Disorder
Enta
ngle
ment
SD
TD
S−TD
N (ρ) =∑i
|λi | − λi2
where λi are the eigenvalues.
Entanglement is robust against disorder
Operational perspective of quantum simulations using quantum walks
Entanglement in localized states
Probability distribution Entanglement
−100 −50 0 50 1000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Position
Pro
babili
ty
θ = π/4
SD
TD
S−TD
0 20 40 60 80 1000
0.2
0.4
0.6
0.8
1
Steps
Enta
ngle
ment
θ = π/4
SD
TD
S−TD
0 20 40 60 80 1000.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Percentage of Disorder
Enta
ngle
ment
SD
TD
S−TD
N (ρ) =∑i
|λi | − λi2
where λi are the eigenvalues.
Entanglement is robust against disorder
Operational perspective of quantum simulations using quantum walks
Entanglement in localized states
Probability distribution Entanglement
−100 −50 0 50 1000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Position
Pro
babili
ty
θ = π/4
SD
TD
S−TD
0 20 40 60 80 1000
0.2
0.4
0.6
0.8
1
Steps
Enta
ngle
ment
θ = π/4
SD
TD
S−TD
0 20 40 60 80 1000.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Percentage of Disorder
Enta
ngle
ment
SD
TD
S−TD
N (ρ) =∑i
|λi | − λi2
where λi are the eigenvalues.
Entanglement is robust against disorder
Operational perspective of quantum simulations using quantum walks
Controlling localization - two period QW
Probability distribution Entanglement
0 50 100 150 200t
04080
120160
σ
b)θ1 = π/12
θ2 = π/3
θ1, θ2
−200 −100 0 100 200Position
0.00
0.04
0.08
0.12Proba
blity
a)
0 50 100 150 200t
04080
120160
σ
d)θ1 = π/6
θ2 =5π/12
θ1, θ2
−200 −100 0 100 200Position
0.00
0.02
0.04
0.06
Proba
blity
c)
0 50 100 150 200t
04080
120160
σ
f)θ1 = π/3
θ2 = π/6
θ1, θ2
−200 −100 0 100 200Position
0.00
0.02
0.04
0.06
Proba
blity
e)
0 π/4 π/2 3π/4 π0
50
100
θ1
σ
θ2 = θ
1
θ2 = π/3
θ2 = π/6
π/3
π/6
0 π/4 π0
0.5
1
θ1
Llo
c (
50 tim
e a
ve.)
Spatial disorder
temporal disorder
θ2 = random value
0 20 40 60 80 1000
1
2
3
Time
Llo
c
θ1 = π/16
θ1 = π/4
θ1 = π/16
θ1 = π/4
Red − Lloc
T
Blue − Lloc
S
Operational perspective of quantum simulations using quantum walks
Controlling localization - two period QW
Probability distribution Entanglement
0 50 100 150 200t
04080
120160
σ
b)θ1 = π/12
θ2 = π/3
θ1, θ2
−200 −100 0 100 200Position
0.00
0.04
0.08
0.12Proba
blity
a)
0 50 100 150 200t
04080
120160
σ
d)θ1 = π/6
θ2 =5π/12
θ1, θ2
−200 −100 0 100 200Position
0.00
0.02
0.04
0.06
Proba
blity
c)
0 50 100 150 200t
04080
120160
σ
f)θ1 = π/3
θ2 = π/6
θ1, θ2
−200 −100 0 100 200Position
0.00
0.02
0.04
0.06
Proba
blity
e)
0 π/4 π/2 3π/4 π0
50
100
θ1
σ
θ2 = θ
1
θ2 = π/3
θ2 = π/6
π/3
π/6
0 π/4 π0
0.5
1
θ1
Llo
c (
50 tim
e a
ve.)
Spatial disorder
temporal disorder
θ2 = random value
0 20 40 60 80 1000
1
2
3
Time
Llo
c
θ1 = π/16
θ1 = π/4
θ1 = π/16
θ1 = π/4
Red − Lloc
T
Blue − Lloc
S
Operational perspective of quantum simulations using quantum walks
Accelerating quantum walks
Replace θ in coin operation by θ(t) = θ(a, t) = θ0e−at
Operational perspective of quantum simulations using quantum walks
Accelerating quantum walks
Replace θ in coin operation by θ(t) = θ(a, t) = θ0e−at
Operational perspective of quantum simulations using quantum walks
Accelerated quantum walks : disorder and localization
Φφ =
(1 00 e iφ
)⊗∑x
|x〉 〈x |
W = SxΦ(φ)C (θ0) ≡ SxB(θ0, φ).
Operational perspective of quantum simulations using quantum walks
Accelerated 2 particle quantum walks
Entanglement
|Ψ〉AB = |a〉A ⊗ |b〉B = |a〉A|b〉B|a〉A|b〉B →
∑cab|a〉A|b〉B
|Ψ〉AB = α|0〉A|0〉B ± β|1〉A|1〉B
|Ψin〉 = |00〉 → localised QW on two particle → entangled statecoin operation
C [θ(t)] =
cos[θ(t)] 0 0 −i sin[θ(t)]
0 cos[θ(t)] −i sin[θ(t)] 00 −i sin[θ(t)] cos[θ(t)] 0
−i sin[θ(t)] 0 0 cos[θ(t)]
shift-operator
S1 ⊗ S2
Operational perspective of quantum simulations using quantum walks
Accelerated 2 particle quantum walks
Entanglement
|Ψ〉AB = |a〉A ⊗ |b〉B = |a〉A|b〉B|a〉A|b〉B →
∑cab|a〉A|b〉B
|Ψ〉AB = α|0〉A|0〉B ± β|1〉A|1〉B
|Ψin〉 = |00〉 → localised QW on two particle → entangled state
coin operation
C [θ(t)] =
cos[θ(t)] 0 0 −i sin[θ(t)]
0 cos[θ(t)] −i sin[θ(t)] 00 −i sin[θ(t)] cos[θ(t)] 0
−i sin[θ(t)] 0 0 cos[θ(t)]
shift-operator
S1 ⊗ S2
Operational perspective of quantum simulations using quantum walks
Accelerated 2 particle quantum walks
Entanglement
|Ψ〉AB = |a〉A ⊗ |b〉B = |a〉A|b〉B|a〉A|b〉B →
∑cab|a〉A|b〉B
|Ψ〉AB = α|0〉A|0〉B ± β|1〉A|1〉B
|Ψin〉 = |00〉 → localised QW on two particle → entangled statecoin operation
C [θ(t)] =
cos[θ(t)] 0 0 −i sin[θ(t)]
0 cos[θ(t)] −i sin[θ(t)] 00 −i sin[θ(t)] cos[θ(t)] 0
−i sin[θ(t)] 0 0 cos[θ(t)]
shift-operator
S1 ⊗ S2
Operational perspective of quantum simulations using quantum walks
Two-particle Entanglement Generation
a = 0.002 and a = 0.02
00.001
0.002 0
50
100
00.5
1
Stepsa
Operational perspective of quantum simulations using quantum walks
Summary
With a careful choice of evolution parameters we can engineer thelocalization of quantum walk for effective use in QIP protocols and simulationof various accessible and inaccessible quantum phenomenon with disorder inboth, high energy and low energy physical systems.
Operational perspective of quantum simulations using quantum walks