Download Quantum information processing on tensor-network states

Quantum information processing on
tensor-network states
Tomoyuki Morimae (Univ. Paris-Est)
V. Dunjko and E. Kashefi (Univ. Edinburgh)
1
Outline of this talk
2
Part I
Review of matrix product states
and tensor-network states
3
Quantum many-body system
• A system of N particles.
• Each particle · · · Hd. The total system · · · (Hd)⊗N .
• d is small (d ∼ 2), and N is large (N ∼ 1023).
4
Quantum many-body state
• Spin-1/2 particles (d = 2).
• General pure state |ψ⟩ ∈ (H2)⊗N , where H2 = {|0⟩, |1⟩}:
|ψ⟩ =
1
∑
z1=0
...
1
∑
c(z1, ..., zN )|zN ⟩ ⊗ ... ⊗ |z1⟩.
zN =0
{0, 1}N ∋ (z1, ..., zN ) 7→ c(z1, ..., zN ) ∈ C.
• We must specify O(2N ) complex numbers c(z1, ..., zN ) to specify |ψ⟩.
• N ∼ 1023: Large memory and long CPU time are required.
5
Matrix product state (MPS)
• General pure state |ψ⟩ =
∑1
∑1
...
z1=0
zN =0 c(z1 , ..., zN )|zN ⟩ ⊗ ... ⊗ |z1 ⟩
• {0, 1}N ∋ (z1, ..., zN ) 7→ c(z1, ..., zN ) ∈ C is assumed to be
c(z1, ..., zN ) = ⟨L|A[zN ]...A[z1]|R⟩
where |L⟩, |R⟩ are D-dimensional vecs, A[0], A[1] are D × D matrices.
• We have only to specify |L⟩, |R⟩, A[0], and A[1], (i.e., 2D2 + 2D
complex numbers)!
• For many states in physics, D is very small (D ∼ 2)!
(For review, see Verstraete, et. al. 2008 and Cirac, et. al. 2009)
6
Example
• Three-qubit state
|ψ⟩ = ⟨L|A[0]A[0]A[0]|R⟩ |000⟩
+⟨L|A[0]A[0]A[1]|R⟩ |001⟩
+⟨L|A[0]A[1]A[0]|R⟩ |010⟩
+⟨L|A[0]A[1]A[1]|R⟩ |011⟩
+⟨L|A[1]A[0]A[0]|R⟩ |100⟩
+⟨L|A[1]A[0]A[1]|R⟩ |101⟩
+⟨L|A[1]A[1]A[0]|R⟩ |110⟩
+⟨L|A[1]A[1]A[1]|R⟩ |111⟩
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Graphical representation
∑
⟨L|A[zN ]...A[z1]|R⟩ = j,k,l,m,...s,t LtA[zN ]t,s...A[z3]m,l A[z2]l,k A[z1]k,j Rj
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Tensor network state (TNS)
C(T [z1]T [z2]T [z3]T [z4]T [z5]) =
∑
T [z1]j T [z2]j,k,l T [z3]k,mT [z4]l,m,nT [z5]n
j,k,l,m,n
∑1
∑1
|ψ⟩ = z =0 ... z =0 C(T [zN ]...T [z1])|zN ⟩ ⊗ ... ⊗ |z1⟩
1
N
(For review, see Verstraete, et. al. 2008 and Cirac, et. al. 2009)
9
Outline of this talk
10
Part II
Review of measurement-based
quantum computation on
tensor-network states
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Measurement-based quantum computation
Universal resource state → universal quantum computation
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Resource states for MBQC
• Cluster state (R. Raussendorf and H. J. Briegel, PRL86,5188(2001); R.
Raussendorf, D. E. Browne, and H. J. Briegel, PRA68, 022312 (2003))
• 1d AKLT state (G. K. Brennen and A. Miyake, PRL101,010502(2008))
• 2d AKLT state (A. Miyake, arXiv: 1009.3491; T. C. Wei, I. Affleck, and R.
Raussendorf, arXiv: 1009.2840)
• Weighted-graph state (D. Gross and J. Eisert, PRL98,220503(2007))
• Tri-cluster state (X. Chen, B. Zeng, Z. Gu, B. Yoshida, and I. L. Chuang,
PRL102, 220501 (2009))
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What is the general strategy?
• With these different kinds of resources, the natural question is
How to understand the general MBQC on general resource states?
• One strategy to answer this question: MPS/TNS
• Resouce states are many-body states → Why don’t we represent
them in MPS/TNS?
MBQC on MPS
• The resource state, one-dimensional qubit chain:
|ψ⟩ =
1
∑
⟨L|A[zN ]...A[z1]|R⟩|zN ⟩ ⊗ ... ⊗ |z1⟩,
z1,...,zN =0
where D = 2.
• If we project the first qubit onto |θ1, ϕ1⟩ = cos θ21 |0⟩ + sin θ21 eiϕ1 |1⟩,
the state becomes
1
∑
⟨L|A[zN ]...A[z2]A[θ1, ϕ1]|R⟩|zN ⟩ ⊗ ... ⊗ |z2⟩ ⊗ |θ1, ϕ1⟩
z2 ,...,zN =0
where A[θ1, ϕ1] = cos θ21 A[0] + e−iϕ1 sin θ21 A[1].
(D. Gross and J. Eisert, PRL98,220503(2007))
14
1
∑
⟨L|A[zN ]...A[z2]A[θ1, ϕ1]|R⟩|zN ⟩ ⊗ ... ⊗ |z2⟩ ⊗ |θ1, ϕ1⟩
z2 ,...,zN =0
If we further project the second qubit onto |θ2, ϕ2⟩,
1
∑
⟨L|A[zN ]...A[z3]A[θ2, ϕ2]A[θ1, ϕ1]|R⟩|zN ⟩ ⊗ ... ⊗ |z3⟩ ⊗ |θ2, ϕ2⟩ ⊗ |θ1, ϕ1⟩
z3,...,zN =0
And finally...
⟨L|A[θN , ϕN ]...A[θ1, ϕ1]|R⟩|θN , ϕN ⟩ ⊗ ... ⊗ |θ1, ϕ1⟩
(D. Gross and J. Eisert, PRL98,220503(2007))
The virtual space
|ψ⟩ =
∑1
∑1
...
z1 =0
zN =0 ⟨L|A[z1 ]...A[zN ]|R⟩|z1⟩ ⊗ ... ⊗ |zN ⟩
Quantum computation is performed in the virtual space.
(D. Gross and J. Eisert, PRL98,220503(2007))
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Example 1: cluster state
• Its MPS is
A[0] = (
1 0
),
1 0
A[1] = (
0 1
)
0 −1
• The projection on |0⟩ + eiϕ|1⟩ gives
A[0] + e−iϕA[1] = HeiZϕ/2 ≡ J(ϕ)
• J(ϕ) is universal (SU (2)) since
eiZϕ/2 = J(0)J(ϕ)
eiXϕ/2 = J(0)J(ϕ)J(0)
(R. Raussendorf and H. J. Briegel, PRL86,5188(2001))
16
Example 2: AKLT chain
• AKLT model: spin-1 Hamiltonian
N [
∑
1
Sj · Sj+1 + (Sj · Sj+1)2
3
j=1
]
• It has gapped ground state whose MPS is
|AKLT ⟩ =
3
∑
z1=1
...
3
∑
⟨L|A[zN ]...A[z1]|R⟩|zN ⟩ ⊗ ... ⊗ |z1⟩,
zN =1
where D = 2, A[1] = X, A[2] = XZ, and A[3] = Z.
• Universal MBQC is possible on |AKLT ⟩.
(G. K. Brennen and A. Miyake, PRL101,010502(2008))
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• If we do measurement {|αθ ⟩, |βθ ⟩, |γ⟩}, where
1
1
iθ
|αθ ⟩ =
(1 + e )|1⟩ + (1 − eiθ )|2⟩
2
2
1
1
|βθ ⟩ =
(1 − eiθ )|1⟩ + (1 + eiθ )|2⟩
2
2
|γ⟩ = |3⟩,
then we can implement
A[αθ ] = XeiZθ/2
A[βθ ] = XZeiZθ/2
A[γ] = Z
(G. K. Brennen and A. Miyake, PRL101,010502(2008))
• Apply V = |3⟩⟨1| + |1⟩⟨2| + |2⟩⟨3|
If we do the same measurement {|αθ ⟩, |βθ ⟩, |γ⟩}, where
1
1
(1 + eiθ )|1⟩ + (1 − eiθ )|2⟩
|αθ ⟩ =
2
2
1
1
|βθ ⟩ =
(1 − eiθ )|1⟩ + (1 + eiθ )|2⟩
2
2
|γ⟩ = |3⟩,
then we can implement
A[αθ ] = XZe−iXθ/2
A[βθ ] = Ze−iXθ/2
A[γ] = X
(G. K. Brennen and A. Miyake, PRL101,010502(2008))
Outline of this talk
18
Part III
Uploading a physical state into
the virtual space
(TM, arXiv:1007.5018)
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The concept of the virtual space
|ψ⟩ =
∑1
∑1
...
z1 =0
zN =0 ⟨L|A[z1 ]...A[zN ]|R⟩|z1⟩ ⊗ ... ⊗ |zN ⟩
(D. Gross and J. Eisert, PRL98,220503(2007))
20
Can we do upload and download?
Can we do the upload and download between RS and VS?
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Motivations
Fault-tolerant QC
Distributed QC
Bob
Alice
U
H
U
H
VS
VS
DL
H
H
DL
UL
RS
H
H
H
H
UL
QC
RS
VS
RS
(TM, arXiv:1007.5018)
22
Download from VS to RS
• Downloading a state from VS into RS.
∑
⟨L|A[zN ]...A[z1]|ψ⟩|zN , ..., z1⟩ → |Φ⟩ ⊗ |ψ⟩
z1 ,...,zN
(J. Cai, W. Dür, M. Van den Nest, A. Miyake, and H. J. Briegel, PRL103,050503(2009))
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• For the simple case
A[m0] = |ϕ0⟩⟨0|
A[m1] = |ϕ1⟩⟨1|,
where A[mi] = ⟨mi|0⟩A[0]+⟨mi|1⟩A[1], and {|m0⟩, |m1⟩} and {|ϕ0⟩, |ϕ1⟩}
are certain orthonormal vectors, the deterministic download is possible.
• For the most general case,
A[0] ∝ W
A[1] ∝ W diag(e−iα, eiα),
where W ∈ SU (2), the probabilistic download is possible with the
filtering {F , F̄ }.
(J. Cai, W. Dür, M. Van den Nest, A. Miyake, and H. J. Briegel, PRL103,050503(2009))
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Upload from RS to VS
One possibility · · · do the inverse of the downloading method
(b)
(a)
5
4
3
2
1
5
4
(c)
5
4
3
3
2
1
2
1
(d)
2
1
5
4
3
• Entanglement must be kept; If entanglement is destroied, the upload
fails, and the original state is destroied.
• New filtering {G, Ḡ} is required.
(TM, arXiv: 1007.5018)
25
No-cloning between RS and VS
Cloning between RS and VS contradicts to the no-cloning theorem.
(TM, arXiv: 1007.5018)
26
Upload: teleportation-like method
Cloning is impossible → teleportation!
VS
VS
RS
RS
(TM, arXiv: 1007.5018)
27
|ϕ⟩ = α|0⟩ + β|1⟩

|ψ⟩ ⊗ |ϕ⟩ = 
=
→

∑
⟨L|A[zN ]...A[z1]|R⟩|zN , ..., z1⟩ ⊗ |ϕ⟩
z1 ,...,zN
[
∑
z2,...,zN
∑
⟨L|A[zN ]...A[z2]A[0]|R⟩|zN , ..., z2⟩ ⊗ |0⟩
+⟨L|A[zN ]...A[z2]A[1]|R⟩|zN , ..., z2⟩ ⊗ |1⟩
]
⊗ |ϕ⟩
⟨L|A[zN ]...A[z2](αA[0]|R⟩ + βA[1]|R⟩)|zN , ..., z2⟩ ⊗ |Bell⟩
z2,...,zN
If not orthogonal, the same filterling as Cai’s can be used!
(TM, arXiv:1007.5018)
Advantage of our uploading method
• Simple and intuitive
• stable
• no new filtering
(TM, arXiv:1007.5018)
28
Applications of our uploading method
• Entanglement swapping in VS
• Gottesman-Chuang gate teleportation trick between RS and VS
VS
VS
VS
VS
RS
RS
RS
RS
(TM, arXiv:1007.5018)
29
Part IV
Blind MBQC with AKLT
(TM, arXiv:1009.3486)
(TM, V. Dunjko, E. Kashefi, in preparation)
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Blind quantum computation
Alice
Bob
• Classical: Feigenbaum 1986.
• Quantum circuit: A. Childs, Quant. Inf. Compt. 5, 456 (2005).
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Blind QC with cluster state
(A. Broadbent, J. Fitzsimons, and E. Kashefi, FOCS 2009, pp.517)
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Blind QC with AKLT chain
• Similar blind QC can be possible with the AKLT chain
• AKLT is physically motivated resource
• Provides many new features for the proof of the blindness
(TM, arXiv:1009.3486; TM, V. Dunjko, and E. Kashefi, in preparation)
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(TM, arXiv:1009.3486; TM, V. Dunjko, and E. Kashefi, in preparation)
Idea: blind z-rotation
∑3
∑3
• l =1 ... l=1⟨L|A[lN ]...A[l1]|R⟩|lN ⟩⊗...⊗|l1⟩, where A[1] = X, A[2] =
1
XZ, and A[3] = Z.
• Measurement {|αθ ⟩, |βθ ⟩, |γ⟩}, where |αθ ⟩ = (1 + eiθ )|1⟩ + (1 − eiθ )|2⟩,
|βθ ⟩ = (1 − eiθ )|1⟩ + (1 + eiθ )|2⟩, and |γ⟩ = |3⟩.
• A[αθ ] = XeiZθ/2, A[βθ ] = XZeiZθ/2, and A[γ] = Z.
iξ
iξ
iξ
iξ
1−e |2⟩)⟨1|+( 1−e |1⟩+ 1+e |2⟩)⟨2|+|3⟩⟨3|.
|1⟩+
• Apply U (ξ) = ( 1+e
2
2
2
2
• Tell Bob ϕ = θ + ξ + rπ. Then A[αϕ] = XeiZ(ϕ−ξ)/2, A[βϕ] =
XZeiZ(ϕ−ξ)/2, and A[αϕ] = Z.
(TM, arXiv:1009.3486)
34
Idea: blind x-rotation
• Apply V = |3⟩⟨1| + |1⟩⟨2| + |2⟩⟨3|.
• Bob does measurement {|αθ ⟩, |βθ ⟩, |γ⟩}.
• A[αθ ] = XZe−iXθ/2, A[βθ ] = Ze−iXθ/2, and A[γ] = X.
• Send U (ξ)V |Ψ⟩ and transmit ϕ = θ + ξ + rπ. Then A[αϕ] =
XZe−iX(ϕ−ξ)/2, A[βϕ] = Ze−iX(ϕ−ξ)/2, and A[γ] = X.
• V is applied periodically → Bob cannot know when Alice does xrotation.
(TM, arXiv:1009.3486)
35
Idea: blind CZ
• Brickwork state (A. Broadbent, et. al. 2008).
(c)
(a)
j1
(b)
W
(d)
j2
36
PEPS
Another way of representing a quantum many-body state
P P P P P
P ≡
1 ∑
1 ∑
1
∑
A[z]a,b|z⟩⟨a, b|
z=0 a=0 b=0
(F. Verstraete and I. Cirac, arXiv: 0407066)
37
Overall protocol
(TM, arXiv:1009.3486)
38
Detecting evil Bob
Alice
Bob
(A. Broadbent, J. Fitzsimons, and E. Kashefi, 2008)
(TM, arXiv:1009.3486)
39
MBQC with Haldane gapped states
∑N
• j=1 [Sj · Sj+1 − β(Sj · Sj+1)2], where (−1 < β < 1).
• z-rotation: Adiabatically turning off the interaction between sites j
and j + 1, and do the measurement {αθ ⟩, |βθ ⟩, |γ⟩} on site j.
• x-rotation: Before the measurement, do V rotation.
(A. Miyake, PRL105,040501(2010))
40
Blind MBQC with Haldane gapped states
• After the adiabatic turning off, Alice applies U or U V , and sends it
to Bob.
• Bob does the MBQC.
(TM, arXiv:1009.3486)
41
Summary
• We have briefly reviewed the MPS, TNS, and QCTN.
• We have proposed a method of uploading a quantum state into the
correlation space.
(TM, arXiv:1007.5018)
• We have proposed a method of the blind QC with AKLT chains.
(TM, arXiv:1009.3486; TM, V. Dunjko, and E. Kashefi, in preparation)
42
Part V
QC on string-net condensate
(TM, arXiv:1012.1000)
43
What is the string-net condensate
• Phase transition · · · central of condmat and statphys
• Traditional Landau’s symmetry-breaking theory
• Quantum topological order (e.g., fractional quantum Hall system)
× Landau’s theory
• String-net condensate (Wen)
It can describe the quantum topological order
44
String-net condensate on the 2d hexagonal lattice
(a)
e
(b)
p
v
∑ ⊗
∑ ⊗
∑ ⊗
H = − p i∈Sp Xi − v i∈Sv Zi − e i∈Se Zi
|Hex⟩ =
∑
s∈CL |s⟩
45
Tensor-network representation of |Hex⟩
(b)
T[ 0 0 ]=
0
0
(a)
0
0
T[ 0 11 ]=
0
T[ 1 0 1 ]=
1
T[ 1 1 0 ]=
1
1
1
1
0
0
1
(X. Chen, B. Zeng, Z. C. Gu, I. L. Chuang, and X. G. Wen, PRB 82, 165119(2010))
46
How to do QC on |Hex⟩
(a)
Space
Time
(TM, arXiv:1012.1000)
47
z-rotation
Information flow
(TM, arXiv:1012.1000)
48
x-rotation
Information flow
(TM, arXiv:1012.1000)
49
CZ-gate
Information flow
(TM, arXiv:1012.1000)
50
Summary
• We have briefly reviewed the MPS, TNS, and QCTN.
• We have proposed a method of uploading a quantum state into the
correlation space.
(TM, arXiv:1007.5018)
• We have proposed a method of the blind QC with AKLT chains.
(TM, arXiv:1009.3486; TM, V. Dunjko, and E. Kashefi, in preparation)
• We have studied QC on a string net condensate.
(TM, arXiv:1012.1000)
51
QCTN renormalization
(S. D. Bartlett, G. K. Brennen, A. Miyake, and J. M. Renes, PRL105,110502(2010))
52
Fault-torelant QCTN
(M. Silva, V. Danos, E. Kashefi, and H. Ollivier, NJP9,192,(2007))
(J. Joo and D. L. Feder, PRA80,032312(2009))
53
Noise in CS and RS
54
2d system
A. Miyake, arXiv:1009.3491
T. C. Wei, I. Affleck, and R. Raussendorf, arXiv:1009.2840
J. Cai, A. Miyake, W. Dür, and H. J. Briegel, PRA82,052309(2010).
55
Reduction
X. Chen, R. Duan, Z. Ji, and B. Zeng, PRL105,020502(2010).
56