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![Page 1: The relationship between continuous and discrete …amchilds/talks/squint09.pdfThe relationship between continuous- and discrete-time quantum walk Andrew Childs Department of Combinatorics](https://reader035.vdocuments.net/reader035/viewer/2022081405/5f0a9f177e708231d42c8876/html5/thumbnails/1.jpg)
The relationship betweencontinuous- and discrete-time
quantum walkAndrew Childs
Department of Combinatorics & Optimizationand Institute for Quantum Computing
University of Waterloo
arXiv:0810.0312 PacNQuInT 2009
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Quantum walk algorithms
• Black box graph traversal [CCDFGS 03]
• Hidden sphere problem [CSV 07]
• Search on graphs [Shenvi, Kempe, Whaley 02], [CG 03, 04], [Ambainis, Kempe, Rivosh 04]
• Element distinctness [Ambainis 03]
• Triangle finding [Magniez, Santha, Szegedy 03]
• Checking matrix multiplication [Buhrman, Špalek 04]
• Testing group commutativity [Magniez, Nayak 05]
• Formula evaluation [Farhi, Goldstone, Gutmann 07], [ACRŠZ 07], [Cleve, Gavinsky, Yeung 08], [Reichardt, Špalek 08]
• Unstructured search [Grover 96] (+ many applications)
Exponential speedups
Polynomial speedups
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Two models, both alike in dignity
Continuous Discrete
state space
simplicity
ease of implementation
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Two models, both alike in dignity
Continuous Discrete
state space
simplicity
ease of implementation
vertices directed edges(“quantum coin”)
![Page 5: The relationship between continuous and discrete …amchilds/talks/squint09.pdfThe relationship between continuous- and discrete-time quantum walk Andrew Childs Department of Combinatorics](https://reader035.vdocuments.net/reader035/viewer/2022081405/5f0a9f177e708231d42c8876/html5/thumbnails/5.jpg)
Two models, both alike in dignity
Continuous Discrete
state space
simplicity
ease of implementation
vertices directed edges(“quantum coin”)
![Page 6: The relationship between continuous and discrete …amchilds/talks/squint09.pdfThe relationship between continuous- and discrete-time quantum walk Andrew Childs Department of Combinatorics](https://reader035.vdocuments.net/reader035/viewer/2022081405/5f0a9f177e708231d42c8876/html5/thumbnails/6.jpg)
Two models, both alike in dignity
Continuous Discrete
state space
simplicity
ease of implementation
vertices directed edges(“quantum coin”)
![Page 7: The relationship between continuous and discrete …amchilds/talks/squint09.pdfThe relationship between continuous- and discrete-time quantum walk Andrew Childs Department of Combinatorics](https://reader035.vdocuments.net/reader035/viewer/2022081405/5f0a9f177e708231d42c8876/html5/thumbnails/7.jpg)
Walks on lines
!100 !50 0 50 100
!100 !50 0 50 100
Continuous
Discrete
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Dueling algorithmsContinuous Discrete
searching a grid (d dimensions)
glued trees
element distinctness
balanced binary AND-OR trees
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Dueling algorithmsContinuous Discrete
searching a grid (d dimensions)
glued trees
element distinctness
balanced binary AND-OR trees
for d > 4 [CG 03]O(!
N)
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Dueling algorithmsContinuous Discrete
searching a grid (d dimensions)
glued trees
element distinctness
balanced binary AND-OR trees
for d > 4 [CG 03]O(!
N)
for d > 2 [AKR 04]O(!
N)
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Dueling algorithmsContinuous Discrete
searching a grid (d dimensions)
glued trees
element distinctness
balanced binary AND-OR trees
for d > 4 [CG 03]O(!
N)
for d > 2 [AKR 04]O(!
N)for d > 2 [CG 04]O(
!N)
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Dueling algorithmsContinuous Discrete
searching a grid (d dimensions)
glued trees
element distinctness
balanced binary AND-OR trees
for d > 4 [CG 03]O(!
N)
for d > 2 [AKR 04]O(!
N)for d > 2 [CG 04]O(
!N)
exponential speedup over classical
[CCDFGS 03]
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Dueling algorithmsContinuous Discrete
searching a grid (d dimensions)
glued trees
element distinctness
balanced binary AND-OR trees
for d > 4 [CG 03]O(!
N)
for d > 2 [AKR 04]O(!
N)for d > 2 [CG 04]O(
!N)
exponential speedup over classical
[CCDFGS 03]?
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Dueling algorithmsContinuous Discrete
searching a grid (d dimensions)
glued trees
element distinctness
balanced binary AND-OR trees
for d > 4 [CG 03]O(!
N)
for d > 2 [AKR 04]O(!
N)for d > 2 [CG 04]O(
!N)
exponential speedup over classical
[CCDFGS 03]?
O(N2/3) [Ambainis 03]
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Dueling algorithmsContinuous Discrete
searching a grid (d dimensions)
glued trees
element distinctness
balanced binary AND-OR trees
for d > 4 [CG 03]O(!
N)
for d > 2 [AKR 04]O(!
N)for d > 2 [CG 04]O(
!N)
exponential speedup over classical
[CCDFGS 03]?
? O(N2/3) [Ambainis 03]
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Dueling algorithmsContinuous Discrete
searching a grid (d dimensions)
glued trees
element distinctness
balanced binary AND-OR trees
for d > 4 [CG 03]O(!
N)
for d > 2 [AKR 04]O(!
N)for d > 2 [CG 04]O(
!N)
exponential speedup over classical
[CCDFGS 03]?
? O(N2/3) [Ambainis 03]
[FGG 07]O(!
N)
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Dueling algorithmsContinuous Discrete
searching a grid (d dimensions)
glued trees
element distinctness
balanced binary AND-OR trees
for d > 4 [CG 03]O(!
N)
for d > 2 [AKR 04]O(!
N)for d > 2 [CG 04]O(
!N)
exponential speedup over classical
[CCDFGS 03]?
? O(N2/3) [Ambainis 03]
[FGG 07]O(!
N)
in circuit model [CCJY 07]
N12 +o(1)
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Dueling algorithmsContinuous Discrete
searching a grid (d dimensions)
glued trees
element distinctness
balanced binary AND-OR trees
for d > 4 [CG 03]O(!
N)
for d > 2 [AKR 04]O(!
N)for d > 2 [CG 04]O(
!N)
exponential speedup over classical
[CCDFGS 03]?
? O(N2/3) [Ambainis 03]
[FGG 07]O(!
N)[ACRŠZ 07]O(
!N) in circuit
model [CCJY 07]N
12 +o(1)
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A formal equivalence?
Is there a single framework describing both kinds of walks?
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A formal equivalence?
E.g., do the walks behave the same in some limit?
Is there a single framework describing both kinds of walks?
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A formal equivalence?
E.g., do the walks behave the same in some limit?
Of course not! The state spaces aren’t even the same!
Is there a single framework describing both kinds of walks?
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Reconciliation
In fact, there is a close correspondence between the continuous- and discrete-time models (suitably defined).
In particular:
• There is a sequence of discrete-time quantum walks whose behavior (in an appropriate subspace) converges to the dynamics of the continuous-time quantum walk.
• By applying phase estimation instead of taking that limit, we can obtain the continuous-time quantum walk more efficiently. (⇒ improved simulations of Hamiltonian dynamics)
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Outline• Models- Classical and quantum, continuous- and discrete-time- Szegedy’s theorem- Szegedizing Hamiltonians
• Continuous-time walk as a limit of discrete-time walks• Hamiltonian simulation• Applications- Algorithms- Hamiltonian oracles
• Open question: A sign problem for Hamiltonian simulation
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Models
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Random walkA Markov process on a graph G = (V, E).
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Random walkA Markov process on a graph G = (V, E).
In discrete time:
Wkj ! 0,!
k Wkj = 1
probability of taking a step from j to k
Stochastic matrix ( )W ! R|V |!|V |
with iff (j, k) ! EWkj != 0
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Random walkA Markov process on a graph G = (V, E).
In discrete time:
Wkj ! 0,!
k Wkj = 1
probability of taking a step from j to k
Stochastic matrix ( )W ! R|V |!|V |
with iff (j, k) ! EWkj != 0
Dynamics: pt = Wtp0 pt ! R|V | t = 0, 1, 2, . . .
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Random walkA Markov process on a graph G = (V, E).
In discrete time:
Wkj ! 0,!
k Wkj = 1
probability of taking a step from j to k
Stochastic matrix ( )W ! R|V |!|V |
with iff (j, k) ! EWkj != 0
Ex: Simple random walk. Wkj =
!1
deg j (j, k) ! E
0 (j, k) "! E
Dynamics: pt = Wtp0 pt ! R|V | t = 0, 1, 2, . . .
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Random walkA Markov process on a graph G = (V, E).
In continuous time:
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Random walkA Markov process on a graph G = (V, E).
In continuous time:!
k Mkj = 0
probability per unit time of taking a step from j to k
Generator matrix ( )M ! R|V |!|V |
with iffMkj != 0 (j, k) ! E
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Random walkA Markov process on a graph G = (V, E).
In continuous time:!
k Mkj = 0
probability per unit time of taking a step from j to k
Generator matrix ( )M ! R|V |!|V |
with iffMkj != 0 (j, k) ! E
Dynamics: d
dtp(t) = Mp(t) p(t) ! R|V | t ! R
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Random walkA Markov process on a graph G = (V, E).
In continuous time:!
k Mkj = 0
probability per unit time of taking a step from j to k
Generator matrix ( )M ! R|V |!|V |
with iffMkj != 0 (j, k) ! E
Dynamics: d
dtp(t) = Mp(t) p(t) ! R|V | t ! R
Ex: Laplacian walk. Mkj =
!"#
"$
! deg j j = k
1 (j, k) ! E
0 (j, k) "! E
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Continuous-time quantum walkQuantum analog of a random walk on a graph G = (V, E).
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Continuous-time quantum walkQuantum analog of a random walk on a graph G = (V, E).
Idea: Replace probabilities by quantum amplitudes.
amplitude for vertex v at time t
|!(t)! =!
v!V
qv(t)|v!
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Continuous-time quantum walkQuantum analog of a random walk on a graph G = (V, E).
Idea: Replace probabilities by quantum amplitudes.
amplitude for vertex v at time t
|!(t)! =!
v!V
qv(t)|v!
Define time-homogeneous, local dynamics on G.
id
dt|!(t)! = H|!(t)!
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Continuous-time quantum walkQuantum analog of a random walk on a graph G = (V, E).
Idea: Replace probabilities by quantum amplitudes.
amplitude for vertex v at time t
|!(t)! =!
v!V
qv(t)|v!
Define time-homogeneous, local dynamics on G.
id
dt|!(t)! = H|!(t)!
with iffH = H† Hkj != 0 (j, k) ! E
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Continuous-time quantum walkQuantum analog of a random walk on a graph G = (V, E).
Idea: Replace probabilities by quantum amplitudes.
amplitude for vertex v at time t
|!(t)! =!
v!V
qv(t)|v!
Define time-homogeneous, local dynamics on G.
id
dt|!(t)! = H|!(t)!
with iffH = H† Hkj != 0 (j, k) ! E
Ex: Adjacency matrix. Hkj =
!1 (j, k) ! E
0 (j, k) "! E
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Discrete-time quantum walkWe can also define a quantum walk with discrete steps. [Watrous 99]
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Discrete-time quantum walkWe can also define a quantum walk with discrete steps. [Watrous 99]
Unitary matrix with iffU ! C|V |!|V | Ukj != 0 (j, k) ! E
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Discrete-time quantum walkWe can also define a quantum walk with discrete steps. [Watrous 99]
Unitary matrix with iffU ! C|V |!|V | Ukj != 0 (j, k) ! E
Ex: On an infinite line, |x! "! 1#2(|x ! 1!+ |x + 1!)
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Discrete-time quantum walkWe can also define a quantum walk with discrete steps. [Watrous 99]
Unitary matrix with iffU ! C|V |!|V | Ukj != 0 (j, k) ! E
Ex: On an infinite line, |x! "! 1#2(|x ! 1!+ |x + 1!)
but then |x + 2! "! 1#2(|x + 1!+ |x + 3!)
which is not orthogonal!
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Discrete-time quantum walk
[Meyer 96], [Severini 03]
In general, we must enlarge the state space.
We can also define a quantum walk with discrete steps. [Watrous 99]
Unitary matrix with iffU ! C|V |!|V | Ukj != 0 (j, k) ! E
Ex: On an infinite line, |x! "! 1#2(|x ! 1!+ |x + 1!)
but then |x + 2! "! 1#2(|x + 1!+ |x + 3!)
which is not orthogonal!
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Szegedy’s discrete-time quantum walk
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Szegedy’s discrete-time quantum walkState space: span|j, k!, |k, j! : (j, k) " E
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Szegedy’s discrete-time quantum walkState space: span|j, k!, |k, j! : (j, k) " E
Let W be a stochastic matrix (a discrete-time random walk).
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Szegedy’s discrete-time quantum walkState space: span|j, k!, |k, j! : (j, k) " E
Let W be a stochastic matrix (a discrete-time random walk).
|!j! :=!
k!V
!Wkj|j, k!Define !!j|!k" = "j,k(note )
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Szegedy’s discrete-time quantum walkState space: span|j, k!, |k, j! : (j, k) " E
Let W be a stochastic matrix (a discrete-time random walk).
|!j! :=!
k!V
!Wkj|j, k!Define !!j|!k" = "j,k(note )
R := 2!
j!V
|!j!"!j| ! I
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Szegedy’s discrete-time quantum walkState space: span|j, k!, |k, j! : (j, k) " E
Let W be a stochastic matrix (a discrete-time random walk).
|!j! :=!
k!V
!Wkj|j, k!Define !!j|!k" = "j,k(note )
S|j, k! := |k, j!
R := 2!
j!V
|!j!"!j| ! I
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Szegedy’s discrete-time quantum walkState space: span|j, k!, |k, j! : (j, k) " E
Let W be a stochastic matrix (a discrete-time random walk).
|!j! :=!
k!V
!Wkj|j, k!Define !!j|!k" = "j,k(note )
Then a step of the walk is the unitary operator U := iSR.
S|j, k! := |k, j!
R := 2!
j!V
|!j!"!j| ! I
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Szegedy’s spectral theorem
Let .T :=!
j |!j!"j|
are eigenvectors of with eigenvalues .U := iS(2TT † ! I) ±e±i arcsin !
ThenI ! e±i arccos !S!
2(1 ! !2)T |!!
Theorem. Let where .|!j! :=!
k
!Wkj|j, k!
!k |Wkj| = 1
Suppose the matrix has an eigenvector ∣λ⟩ with eigenvalue λ.
!j,k
!W!
jkWkj|k!"j|
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Szegedizing a HamiltonianIdea: Let H be a Hermitian matrix. If we find a matrix W with and!
k |Wkj| = 1
for some real number h, then W defines a discrete-time quantum walk closely related to H.
Hjk = h!
WjkW!kj
[ACRŠZ 07]
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Szegedizing a HamiltonianIdea: Let H be a Hermitian matrix. If we find a matrix W with and!
k |Wkj| = 1
for some real number h, then W defines a discrete-time quantum walk closely related to H.
Hjk = h!
WjkW!kj
Two strategies:
[ACRŠZ 07]
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Szegedizing a HamiltonianIdea: Let H be a Hermitian matrix. If we find a matrix W with and!
k |Wkj| = 1
for some real number h, then W defines a discrete-time quantum walk closely related to H.
Hjk = h!
WjkW!kj
Two strategies:
Let be the principal eigenvector of abs(H).|d! =!
j dj|j!
Let abs(H) denote the matrix with elements abs(H)jk = ∣Hjk∣.
Wjk =Hjk
!abs(H)!dk
djThen gives h = ∥abs(H)∥.
1.
[ACRŠZ 07]
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Szegedizing a HamiltonianIdea: Let H be a Hermitian matrix. If we find a matrix W with and!
k |Wkj| = 1
for some real number h, then W defines a discrete-time quantum walk closely related to H.
Hjk = h!
WjkW!kj
Two strategies:Let .!H!1 := max
j
!
k
|Hjk|
Introduce another state, denoted ∣∅⟩.
2.
Then gives h = ∥H∥1.W =H
!H!1+
!
k
!1 !
!
j
|Hjk|
!H!1
"|!"#k|
[ACRŠZ 07]
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Continuous-time walk as a limit of discrete-time walks
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Classical caseDiscrete-time random walk: pt+1 = W pt
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Classical caseDiscrete-time random walk:
“Lazy walk”: Only move with probability ɛ, so W → ɛW + (1 - ɛ)I
pt+1 = W pt
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Classical caseDiscrete-time random walk:
“Lazy walk”: Only move with probability ɛ, so W → ɛW + (1 - ɛ)I
pt+1 = [!W + (1 ! !)I]pt
pt+1 = W pt
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Classical caseDiscrete-time random walk:
“Lazy walk”: Only move with probability ɛ, so W → ɛW + (1 - ɛ)I
pt+1 = [!W + (1 ! !)I]pt
pt+1 ! pt
!= (W ! I)pt
pt+1 = W pt
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Classical caseDiscrete-time random walk:
“Lazy walk”: Only move with probability ɛ, so W → ɛW + (1 - ɛ)I
pt+1 = [!W + (1 ! !)I]pt
pt+1 ! pt
!= (W ! I)pt
As ɛ → 0 with τ = ɛt, d
d!p(!) = (W ! I)p(!)
pt+1 = W pt
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Quantum caseWith a suitable “lazy discrete-time quantum walk” Uɛ, defined by an isometry ,T! :=
!j,k
!Wkj(!)|j, k!"j|
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Quantum caseWith a suitable “lazy discrete-time quantum walk” Uɛ, defined by an isometry ,T! :=
!j,k
!Wkj(!)|j, k!"j|
1. Apply to the input state ∣ψ⟩.I + iS!2
T!
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Quantum caseWith a suitable “lazy discrete-time quantum walk” Uɛ, defined by an isometry ,T! :=
!j,k
!Wkj(!)|j, k!"j|
2. Perform ht/ɛ steps of the walk,(where h = ∥abs(H)∥ or ∥H∥1).
U! := iS(2T!T †! ! I)
1. Apply to the input state ∣ψ⟩.I + iS!2
T!
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Quantum caseWith a suitable “lazy discrete-time quantum walk” Uɛ, defined by an isometry ,T! :=
!j,k
!Wkj(!)|j, k!"j|
2. Perform ht/ɛ steps of the walk,(where h = ∥abs(H)∥ or ∥H∥1).
U! := iS(2T!T †! ! I)
1. Apply to the input state ∣ψ⟩.I + iS!2
T!
3. Measure in the basis .!
I + iS!2
T!|j""
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Quantum caseWith a suitable “lazy discrete-time quantum walk” Uɛ, defined by an isometry ,T! :=
!j,k
!Wkj(!)|j, k!"j|
2. Perform ht/ɛ steps of the walk,(where h = ∥abs(H)∥ or ∥H∥1).
U! := iS(2T!T †! ! I)
1. Apply to the input state ∣ψ⟩.I + iS!2
T!
3. Measure in the basis .!
I + iS!2
T!|j""
Then as ɛ → 0, .Pr(j) ! |!j|e!iHt|!"|2
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Quantum caseWith a suitable “lazy discrete-time quantum walk” Uɛ, defined by an isometry ,T! :=
!j,k
!Wkj(!)|j, k!"j|
2. Perform ht/ɛ steps of the walk,(where h = ∥abs(H)∥ or ∥H∥1).
U! := iS(2T!T †! ! I)
1. Apply to the input state ∣ψ⟩.I + iS!2
T!
3. Measure in the basis .!
I + iS!2
T!|j""
Then as ɛ → 0, .Pr(j) ! |!j|e!iHt|!"|2
(We can get error at most δ in steps.)O(ht, (!H!t)3/2/"
!)
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Hamiltonian simulation
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Simulating sparse HamiltoniansProblem: For a given Hamiltonian H, simulate the unitary time evolution e-iHt for any desired t.
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Simulating sparse HamiltoniansProblem: For a given Hamiltonian H, simulate the unitary time evolution e-iHt for any desired t.
Suppose H is sparse: for any x, we can efficiently compute all the nonzero matrix elements ⟨y∣H∣x⟩ (so in particular, there are only polynomially many such y).
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Simulating sparse HamiltoniansProblem: For a given Hamiltonian H, simulate the unitary time evolution e-iHt for any desired t.
Approach: Color the graph of H. Then the simulation breaks into small pieces that are easy to handle. [Aharonov, Ta-Shma 03], [CCDFGS 03]
Suppose H is sparse: for any x, we can efficiently compute all the nonzero matrix elements ⟨y∣H∣x⟩ (so in particular, there are only polynomially many such y).
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Simulating sparse HamiltoniansProblem: For a given Hamiltonian H, simulate the unitary time evolution e-iHt for any desired t.
Approach: Color the graph of H. Then the simulation breaks into small pieces that are easy to handle. [Aharonov, Ta-Shma 03], [CCDFGS 03]
Suppose H is sparse: for any x, we can efficiently compute all the nonzero matrix elements ⟨y∣H∣x⟩ (so in particular, there are only polynomially many such y).
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Simulating sparse HamiltoniansProblem: For a given Hamiltonian H, simulate the unitary time evolution e-iHt for any desired t.
Approach: Color the graph of H. Then the simulation breaks into small pieces that are easy to handle. [Aharonov, Ta-Shma 03], [CCDFGS 03]
Suppose H is sparse: for any x, we can efficiently compute all the nonzero matrix elements ⟨y∣H∣x⟩ (so in particular, there are only polynomially many such y).
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Simulating sparse HamiltoniansProblem: For a given Hamiltonian H, simulate the unitary time evolution e-iHt for any desired t.
Approach: Color the graph of H. Then the simulation breaks into small pieces that are easy to handle. [Aharonov, Ta-Shma 03], [CCDFGS 03]
= + +
Suppose H is sparse: for any x, we can efficiently compute all the nonzero matrix elements ⟨y∣H∣x⟩ (so in particular, there are only polynomially many such y).
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Simulating sparse HamiltoniansProblem: For a given Hamiltonian H, simulate the unitary time evolution e-iHt for any desired t.
Approach: Color the graph of H. Then the simulation breaks into small pieces that are easy to handle. [Aharonov, Ta-Shma 03], [CCDFGS 03]
Any sufficiently sparse graph can be efficiently colored using only local information. [Linial 87]
= + +
Suppose H is sparse: for any x, we can efficiently compute all the nonzero matrix elements ⟨y∣H∣x⟩ (so in particular, there are only polynomially many such y).
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Simulating sparse HamiltoniansProblem: For a given Hamiltonian H, simulate the unitary time evolution e-iHt for any desired t.
Approach: Color the graph of H. Then the simulation breaks into small pieces that are easy to handle. [Aharonov, Ta-Shma 03], [CCDFGS 03]
Any sufficiently sparse graph can be efficiently colored using only local information. [Linial 87]
= + +
Suppose H is sparse: for any x, we can efficiently compute all the nonzero matrix elements ⟨y∣H∣x⟩ (so in particular, there are only polynomially many such y).
So we can simulate H in poly(deg(H), log dim(H), t, 1/ɛ) steps.
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Simulating a sum of HamiltoniansLie product formula: lim
n!"
!e!iA/Ne!iB/n
"n= e!i(A+B)
[C 04], [Berry, Ahokas, Cleve, Sanders 05]
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Simulating a sum of HamiltoniansLie product formula: lim
n!"
!e!iA/Ne!iB/n
"n= e!i(A+B)
[C 04], [Berry, Ahokas, Cleve, Sanders 05]
We can approximately simulate A + B for time t using
To get error ɛ, it suffices to use O(t2/ɛ) steps.
!e!iAt/ne!iBt/n
"n ! e!i(A+B)t.
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Simulating a sum of HamiltoniansLie product formula: lim
n!"
!e!iA/Ne!iB/n
"n= e!i(A+B)
[C 04], [Berry, Ahokas, Cleve, Sanders 05]
We can approximately simulate A + B for time t using
To get error ɛ, it suffices to use O(t2/ɛ) steps.
!e!iAt/ne!iBt/n
"n ! e!i(A+B)t.
We can do better using a higher-order formula:
Then O(t3/2/ɛ1/2) steps suffice.
!e!iAt/2ne!iBt/ne!iAt/2n
"n ! e!i(A+B)t.
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Simulating a sum of HamiltoniansLie product formula: lim
n!"
!e!iA/Ne!iB/n
"n= e!i(A+B)
[C 04], [Berry, Ahokas, Cleve, Sanders 05]
Using even better approximations (systematically constructed by Suzuki), we can simulate A + B for time t in t1 + o(1) steps.
We can approximately simulate A + B for time t using
To get error ɛ, it suffices to use O(t2/ɛ) steps.
!e!iAt/ne!iBt/n
"n ! e!i(A+B)t.
We can do better using a higher-order formula:
Then O(t3/2/ɛ1/2) steps suffice.
!e!iAt/2ne!iBt/ne!iAt/2n
"n ! e!i(A+B)t.
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The no fast-forwarding theorem
[Berry, Ahokas, Cleve, Sanders 05]
Can we simulate H for time t using a number of operations that is sublinear in t?
In special cases, yes! (e.g., whenever for a small τ)e!iH! = I
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The no fast-forwarding theorem
[Berry, Ahokas, Cleve, Sanders 05]
Can we simulate H for time t using a number of operations that is sublinear in t?
In special cases, yes! (e.g., whenever for a small τ)
But this is not possible in general: for some Hamiltonians, Ω(t) operations are required.
Proof is by reduction of parity to simulating a Hamiltonian.
e!iH! = I
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Phase estimation
1!n
n!1!
x=0
|x" • QFT!1n |!"
|"" Ux |""
U|!! = ei!|!!
Precision δ with error probability at most ɛ using O(1/δɛ) applications of U.
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Hamiltonian simulation by discrete-time quantum walkTo simulate H for time t:
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Hamiltonian simulation by discrete-time quantum walk
1. Apply T to the input state ∣ψ⟩.To simulate H for time t:
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Hamiltonian simulation by discrete-time quantum walk
1. Apply T to the input state ∣ψ⟩.To simulate H for time t:
2. Perform phase estimation with , estimating a phase for the component of T∣ψ⟩ corresponding to an eigenvector of H with eigenvalue λ.
U = iS(2TT † ! I)±e±i arcsin !
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Hamiltonian simulation by discrete-time quantum walk
1. Apply T to the input state ∣ψ⟩.To simulate H for time t:
2. Perform phase estimation with , estimating a phase for the component of T∣ψ⟩ corresponding to an eigenvector of H with eigenvalue λ.
U = iS(2TT † ! I)±e±i arcsin !
3. Use the estimate of arcsin λ to estimate λ, and apply the phase .e!i!t
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Hamiltonian simulation by discrete-time quantum walk
1. Apply T to the input state ∣ψ⟩.To simulate H for time t:
2. Perform phase estimation with , estimating a phase for the component of T∣ψ⟩ corresponding to an eigenvector of H with eigenvalue λ.
U = iS(2TT † ! I)±e±i arcsin !
3. Use the estimate of arcsin λ to estimate λ, and apply the phase .e!i!t
4. Uncompute the phase estimation and T, giving an approximation of .e!iHt|!!
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Hamiltonian simulation by discrete-time quantum walk
1. Apply T to the input state ∣ψ⟩.To simulate H for time t:
2. Perform phase estimation with , estimating a phase for the component of T∣ψ⟩ corresponding to an eigenvector of H with eigenvalue λ.
U = iS(2TT † ! I)±e±i arcsin !
3. Use the estimate of arcsin λ to estimate λ, and apply the phase .e!i!t
4. Uncompute the phase estimation and T, giving an approximation of .e!iHt|!!
This is linear in t, and works even in cases where H is not sparse!
Theorem. To achieve fidelity 1 - ɛ, it suffices to use steps of the discrete-time quantum walk.O(!abs(H)!t/!3/2)
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Applications
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in out
AlgorithmsGlued trees
Element distinctness
There is a discrete-time quantum walk that travels from “in” to “out” in polynomial time.
Given a black box for f : 0, 1, ..., n → S, are there are distinct indices x,y such that f(x) = f(y)?There is a continuous-time quantum walk algorithm that can be implemented with O(N2/3) queries.Walk takes place on a Johnson graph (not sparse).
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Hamiltonian query model
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Hamiltonian query modelConventional quantum query model:
Qx|i, b! = |i, b" xi!• Query operator Qx, where . • Unitary operators U0, U1, ..., Un.• Algorithm is UnQx...QxU1QxU0.
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Hamiltonian query model
Hamiltonian query model:• Hamiltonian Hx that generates Qx.• Driving Hamiltonian HD(t).• Algorithm is HD(t) + Hx, from t = 0 to T.
Conventional quantum query model:Qx|i, b! = |i, b" xi!• Query operator Qx, where .
• Unitary operators U0, U1, ..., Un.• Algorithm is UnQx...QxU1QxU0.
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Hamiltonian query model
Hamiltonian query model:• Hamiltonian Hx that generates Qx.• Driving Hamiltonian HD(t).• Algorithm is HD(t) + Hx, from t = 0 to T.
Conventional quantum query model:Qx|i, b! = |i, b" xi!• Query operator Qx, where .
• Unitary operators U0, U1, ..., Un.• Algorithm is UnQx...QxU1QxU0.
The Hamiltonian model is potentially more powerful!
![Page 95: The relationship between continuous and discrete …amchilds/talks/squint09.pdfThe relationship between continuous- and discrete-time quantum walk Andrew Childs Department of Combinatorics](https://reader035.vdocuments.net/reader035/viewer/2022081405/5f0a9f177e708231d42c8876/html5/thumbnails/95.jpg)
Hamiltonian query model
Hamiltonian query model:• Hamiltonian Hx that generates Qx.• Driving Hamiltonian HD(t).• Algorithm is HD(t) + Hx, from t = 0 to T.
Conventional quantum query model:Qx|i, b! = |i, b" xi!• Query operator Qx, where .
• Unitary operators U0, U1, ..., Un.• Algorithm is UnQx...QxU1QxU0.
The Hamiltonian model is potentially more powerful!Theorem. [Cleve, Gottesman, Mosca, Somma, Yonge-Mallo 08]If a function can be evaluated with Hamiltonian queries for time T, then it can be evaluated with O(T log T) discrete queries.
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Hamiltonian query model
Hamiltonian query model:• Hamiltonian Hx that generates Qx.• Driving Hamiltonian HD(t).• Algorithm is HD(t) + Hx, from t = 0 to T.
Conventional quantum query model:Qx|i, b! = |i, b" xi!• Query operator Qx, where .
• Unitary operators U0, U1, ..., Un.• Algorithm is UnQx...QxU1QxU0.
The Hamiltonian model is potentially more powerful!Theorem. [Cleve, Gottesman, Mosca, Somma, Yonge-Mallo 08]If a function can be evaluated with Hamiltonian queries for time T, then it can be evaluated with O(T log T) discrete queries.
Theorem. If HD is time-independent, O(∥abs(HD)∥T) discrete queries suffice.
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Open question:A sign problem for
Hamiltonain simulation
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How hard is it to simulate H?
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How hard is it to simulate H?By the no fast forwarding theorem, Ω(∥H∥t) operations are necessary.
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How hard is it to simulate H?By the no fast forwarding theorem, Ω(∥H∥t) operations are necessary.
We have seen that O(∥abs(H)∥t) steps (of the corresponding discrete-time quantum walk) are sufficient. Is this a fundamental barrier?
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How hard is it to simulate H?By the no fast forwarding theorem, Ω(∥H∥t) operations are necessary.
We have seen that O(∥abs(H)∥t) steps (of the corresponding discrete-time quantum walk) are sufficient. Is this a fundamental barrier?
For , we have
(and these bounds are the best possible).!H! " !abs(H)! "
#N!H!
H ! CN!N
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How hard is it to simulate H?By the no fast forwarding theorem, Ω(∥H∥t) operations are necessary.
We have seen that O(∥abs(H)∥t) steps (of the corresponding discrete-time quantum walk) are sufficient. Is this a fundamental barrier?
For , we have
(and these bounds are the best possible).!H! " !abs(H)! "
#N!H!
H ! CN!N
Simulations using O(∥H∥t) steps would have applications such as• approximately computing exponential sums• breaking pseudorandom generators derived from strongly
regular graphs.