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Algorithms in Superconducting Circuits · 2015. 7. 27. · Motivation 1 Implementing Grover’s...
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Algorithms in Superconducting Circuits
Axel Dahlberg, Marco Roth
April 24, 2015
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Motivation
1 Implementing Grover’s algorithm.
2 Superconducting circuits.3 Control over parameters.
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Motivation
1 Implementing Grover’s algorithm.2 Superconducting circuits.
3 Control over parameters.
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Motivation
1 Implementing Grover’s algorithm.2 Superconducting circuits.3 Control over parameters.
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Outline
1 Theoretical aspect of Grover’s algorithm
2 Circuit Implementation
3 Results
4 Conclusion
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Grover’s Algorithm
1 A quantum algorithm for finding an element in an unstructured set of N = 2n
elements.
2 Assumption: Given a function f (x) : {0, 1}n → {0, 1} such that
f (x) ={
1 x = xk0 else , (1)
for some k.3 Task: Find xk with the least amount of calls to the function f .4 To evaluate the function f a gate called the oracle Of is implemented. The
gate does the following|x〉 → (−1)f (x) |x〉 . (2)
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Grover’s Algorithm
1 A quantum algorithm for finding an element in an unstructured set of N = 2n
elements.2 Assumption: Given a function f (x) : {0, 1}n → {0, 1} such that
f (x) ={
1 x = xk0 else , (1)
for some k.
3 Task: Find xk with the least amount of calls to the function f .4 To evaluate the function f a gate called the oracle Of is implemented. The
gate does the following|x〉 → (−1)f (x) |x〉 . (2)
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Grover’s Algorithm
1 A quantum algorithm for finding an element in an unstructured set of N = 2n
elements.2 Assumption: Given a function f (x) : {0, 1}n → {0, 1} such that
f (x) ={
1 x = xk0 else , (1)
for some k.3 Task: Find xk with the least amount of calls to the function f .
4 To evaluate the function f a gate called the oracle Of is implemented. Thegate does the following
|x〉 → (−1)f (x) |x〉 . (2)
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Grover’s Algorithm
1 A quantum algorithm for finding an element in an unstructured set of N = 2n
elements.2 Assumption: Given a function f (x) : {0, 1}n → {0, 1} such that
f (x) ={
1 x = xk0 else , (1)
for some k.3 Task: Find xk with the least amount of calls to the function f .4 To evaluate the function f a gate called the oracle Of is implemented. The
gate does the following|x〉 → (−1)f (x) |x〉 . (2)
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Grover’s Algorithm
1 In classical computation the task can only be completed with O(N) calls tothe oracle.
2 Grover’s algorithm does this with O(√
N) calls.
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Grover’s Algorithm
1 In classical computation the task can only be completed with O(N) calls tothe oracle.
2 Grover’s algorithm does this with O(√
N) calls.
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Grover’s Algorithm
1 In classical computation the task can only be completed with O(N) calls tothe oracle.
2 Grover’s algorithm does this with O(√
N) calls.
|0〉⊗n H⊗n Uf H⊗n Uf0 H⊗n Readout
Repeat
The function f0 outputs a 1 if the input is 0⊗n and otherwise 0.
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Grover’s Algorithm
a
One iteration of Grover’s algorithm ontwo qubits.
|ϕa〉 = |0, 0〉
↓ H⊗2
|ϕb〉 = (|0, 0〉+ |0, 1〉+ |1, 0〉+ |1, 1〉)/2
↓ Uf
|ϕc〉 = (|0, 0〉+ |0, 1〉 − |1, 0〉+ |1, 1〉)/2
↓ H⊗2
|ϕd〉 = (|0, 0〉 − |0, 1〉+ |1, 0〉+ |1, 1〉)/2
↓ Uf0
|ϕe〉 = (− |0, 0〉 − |0, 1〉+ |1, 0〉+ |1, 1〉)/2
↓ H⊗2
|ϕf〉 = − |1, 0〉
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Grover’s Algorithm
a b
One iteration of Grover’s algorithm ontwo qubits.
|ϕa〉 = |0, 0〉
↓ H⊗2
|ϕb〉 = (|0, 0〉+ |0, 1〉+ |1, 0〉+ |1, 1〉)/2
↓ Uf
|ϕc〉 = (|0, 0〉+ |0, 1〉 − |1, 0〉+ |1, 1〉)/2
↓ H⊗2
|ϕd〉 = (|0, 0〉 − |0, 1〉+ |1, 0〉+ |1, 1〉)/2
↓ Uf0
|ϕe〉 = (− |0, 0〉 − |0, 1〉+ |1, 0〉+ |1, 1〉)/2
↓ H⊗2
|ϕf〉 = − |1, 0〉
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Grover’s Algorithm
a b c
One iteration of Grover’s algorithm ontwo qubits.
|ϕa〉 = |0, 0〉
↓ H⊗2
|ϕb〉 = (|0, 0〉+ |0, 1〉+ |1, 0〉+ |1, 1〉)/2
↓ Uf
|ϕc〉 = (|0, 0〉+ |0, 1〉 − |1, 0〉+ |1, 1〉)/2
↓ H⊗2
|ϕd〉 = (|0, 0〉 − |0, 1〉+ |1, 0〉+ |1, 1〉)/2
↓ Uf0
|ϕe〉 = (− |0, 0〉 − |0, 1〉+ |1, 0〉+ |1, 1〉)/2
↓ H⊗2
|ϕf〉 = − |1, 0〉
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Grover’s Algorithm
a b c
d
One iteration of Grover’s algorithm ontwo qubits.
|ϕa〉 = |0, 0〉
↓ H⊗2
|ϕb〉 = (|0, 0〉+ |0, 1〉+ |1, 0〉+ |1, 1〉)/2
↓ Uf
|ϕc〉 = (|0, 0〉+ |0, 1〉 − |1, 0〉+ |1, 1〉)/2
↓ H⊗2
|ϕd〉 = (|0, 0〉 − |0, 1〉+ |1, 0〉+ |1, 1〉)/2
↓ Uf0
|ϕe〉 = (− |0, 0〉 − |0, 1〉+ |1, 0〉+ |1, 1〉)/2
↓ H⊗2
|ϕf〉 = − |1, 0〉
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Grover’s Algorithm
a b c
d e
One iteration of Grover’s algorithm ontwo qubits.
|ϕa〉 = |0, 0〉
↓ H⊗2
|ϕb〉 = (|0, 0〉+ |0, 1〉+ |1, 0〉+ |1, 1〉)/2
↓ Uf
|ϕc〉 = (|0, 0〉+ |0, 1〉 − |1, 0〉+ |1, 1〉)/2
↓ H⊗2
|ϕd〉 = (|0, 0〉 − |0, 1〉+ |1, 0〉+ |1, 1〉)/2
↓ Uf0
|ϕe〉 = (− |0, 0〉 − |0, 1〉+ |1, 0〉+ |1, 1〉)/2
↓ H⊗2
|ϕf〉 = − |1, 0〉
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Grover’s Algorithm
a b c
d e f
One iteration of Grover’s algorithm ontwo qubits.
|ϕa〉 = |0, 0〉
↓ H⊗2
|ϕb〉 = (|0, 0〉+ |0, 1〉+ |1, 0〉+ |1, 1〉)/2
↓ Uf
|ϕc〉 = (|0, 0〉+ |0, 1〉 − |1, 0〉+ |1, 1〉)/2
↓ H⊗2
|ϕd〉 = (|0, 0〉 − |0, 1〉+ |1, 0〉+ |1, 1〉)/2
↓ Uf0
|ϕe〉 = (− |0, 0〉 − |0, 1〉+ |1, 0〉+ |1, 1〉)/2
↓ H⊗2
|ϕf〉 = − |1, 0〉
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Circuit ImplementationSuperconducting circuit with two qubits used for realising Grover’s algorithm.
A four-port device with a coplanar waveguide cavity bus coupling two transmon qubits(Fig. taken from [1]).
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Implementing the C-Phase gate
Flux dependence of transition frequencies (Fig. taken from [1]).
• Cavity-qubit interactions produce a frequency shift between the |1, 1〉 and the|0, 2〉 state
• This shift can be exploited to create a C-Phase gate:
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Implementing the C-Phase gate
Flux dependence of transition frequencies (Fig. taken from [1]).
1 Cavity-qubit interactions produce a frequency shift between the |1, 1〉 and the|0, 2〉 state
2 This shift can be exploited to create a C-Phase gate:
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Implementing the C-Phase gate
Flux dependence of transition frequencies (Fig. taken from [1]).
1 Cavity-qubit interactions produce a frequency shift between the |1, 1〉 and the|0, 2〉 state
2 This shift can be exploited to create a C-Phase gate:
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Implementing the C-Phase gate
1 The interaction between the |1, 1〉 state and the non-computational |0, 2〉state introduces a phase difference:
12 (|0, 0〉+ |0, 1〉+ |1, 0〉+ |1, 1〉+ 0 |0, 2〉)
flux pulse−−−−−→ 12 (|0, 0〉+ |0, 1〉+ |1, 0〉+ e−iπ |1, 1〉+ e+iπ0 |0, 2〉)
2 Which effectively produces the gate:
U =
1 0 0 00 1 0 00 0 1 00 0 0 −1
(3)
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Implementing the C-Phase gate
1 The interaction between the |1, 1〉 state and the non-computational |0, 2〉state introduces a phase difference:
12 (|0, 0〉+ |0, 1〉+ |1, 0〉+ |1, 1〉+ 0 |0, 2〉)
flux pulse−−−−−→ 12 (|0, 0〉+ |0, 1〉+ |1, 0〉+ e−iπ |1, 1〉+ e+iπ0 |0, 2〉)
2 Which effectively produces the gate:
U =
1 0 0 00 1 0 00 0 1 00 0 0 −1
(3)
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Implementing the C-Phase gate
1 The interaction between the |1, 1〉 state and the non-computational |0, 2〉state introduces a phase difference:
12 (|0, 0〉+ |0, 1〉+ |1, 0〉+ |1, 1〉+ 0 |0, 2〉)
flux pulse−−−−−→ 12 (|0, 0〉+ |0, 1〉+ |1, 0〉+ e−iπ |1, 1〉+ e+iπ0 |0, 2〉)
2 Which effectively produces the gate:
U =
1 0 0 00 1 0 00 0 1 00 0 0 −1
(3)
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Implementing the C-Phase gate
1 The interaction between the |1, 1〉 state and the non-computational |0, 2〉state introduces a phase difference:
12 (|0, 0〉+ |0, 1〉+ |1, 0〉+ |1, 1〉+ 0 |0, 2〉)
flux pulse−−−−−→ 12 (|0, 0〉+ |0, 1〉+ |1, 0〉+ e−iπ |1, 1〉+ e+iπ0 |0, 2〉)
2 Which effectively produces the gate:
U =
1 0 0 00 1 0 00 0 1 00 0 0 −1
(3)
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Implementing the C-Phase gate
1 The interaction between the |1, 1〉 state and the non-computational |0, 2〉state introduces a phase difference:
12 (|0, 0〉+ |0, 1〉+ |1, 0〉+ |1, 1〉+ 0 |0, 2〉)
flux pulse−−−−−→ 12 (|0, 0〉+ |0, 1〉+ |1, 0〉+ e−iπ |1, 1〉+ e+iπ0 |0, 2〉)
2 Which effectively produces the gate:
U =
1 0 0 00 1 0 00 0 1 00 0 0 −1
(3)
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Creating entanglement
The C-Phase gate can be used to createentanglement (Fig. taken from [1]).
|ϕa〉 = |0, 0〉
↓ Rπ/2y
|ϕb〉 = 12 (|0, 0〉+ |0, 1〉+ |1, 0〉+ |1, 1〉)
↓ U10
|ϕc〉 = 12 (|0, 0〉+ |0, 1〉 − |1, 0〉+ |1, 1〉)
↓ Rπ/2y
|ϕd〉 = 1√2
(|0, 0〉+ |1, 1〉) =
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Creating entanglement
The C-Phase gate can be used to createentanglement (Fig. taken from [1]).
|ϕa〉 = |0, 0〉
↓ Rπ/2y
|ϕb〉 = 12 (|0, 0〉+ |0, 1〉+ |1, 0〉+ |1, 1〉)
↓ U10
|ϕc〉 = 12 (|0, 0〉+ |0, 1〉 − |1, 0〉+ |1, 1〉)
↓ Rπ/2y
|ϕd〉 = 1√2
(|0, 0〉+ |1, 1〉) =
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Creating entanglement
The C-Phase gate can be used to createentanglement (Fig. taken from [1]).
|ϕa〉 = |0, 0〉
↓ Rπ/2y ⊗ Rπ/2
y
|ϕb〉 = 12 (|0, 0〉+ |0, 1〉+ |1, 0〉+ |1, 1〉)
↓ U10
|ϕc〉 = 12 (|0, 0〉+ |0, 1〉 − |1, 0〉+ |1, 1〉)
↓ Rπ/2y
|ϕd〉 = 1√2
(|0, 0〉+ |1, 1〉) =
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Creating entanglement
The C-Phase gate can be used to createentanglement (Fig. taken from [1]).
|ϕa〉 = |0, 0〉
↓ Rπ/2y ⊗ Rπ/2
y
|ϕb〉 = 12 (|0, 0〉+ |0, 1〉+ |1, 0〉+ |1, 1〉)
↓ U10
|ϕc〉 = 12 (|0, 0〉+ |0, 1〉 − |1, 0〉+ |1, 1〉)
↓ Rπ/2y
|ϕd〉 = 1√2
(|0, 0〉+ |1, 1〉) =
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Creating entanglement
The C-Phase gate can be used to createentanglement (Fig. taken from [1]).
|ϕa〉 = |0, 0〉
↓ Rπ/2y ⊗ Rπ/2
y
|ϕb〉 = 12 (|0, 0〉+ |0, 1〉+ |1, 0〉+ |1, 1〉)
↓ U10
|ϕc〉 = 12 (|0, 0〉+ |0, 1〉 − |1, 0〉+ |1, 1〉)
↓ Rπ/2y
|ϕd〉 = 1√2
(|0, 0〉+ |1, 1〉) =∣∣Ψ+⟩
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Creating entanglement
The C-Phase gate can be used to createentanglement (Fig. taken from [1]).
|ϕa〉 = |0, 0〉
↓ Rπ/2y ⊗ Rπ/2
y
|ϕb〉 = 12 (|0, 0〉+ |0, 1〉+ |1, 0〉+ |1, 1〉)
↓ U10
|ϕc〉 = 12 (|0, 0〉+ |0, 1〉 − |1, 0〉+ |1, 1〉)
↓ Rπ/2y
|ϕd〉 = 1√2
(|0, 0〉+ |1, 1〉) =∣∣Ψ+⟩
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Grover’s algorithm
Grover’s algorithm with two transmon qubits (Fig. taken from [1]).Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 22 / 30
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Grover’s algorithm
Grover’s algorithm with two transmon qubits (Fig. taken from [1]).Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 23 / 30
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Grover’s algorithm
Grover’s algorithm with two transmon qubits (Fig. taken from [1]).Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 24 / 30
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Grover’s algorithm
Grover’s algorithm with two transmon qubits (Fig. taken from [1]).Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 25 / 30
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Grover’s algorithm
Grover’s algorithm with two transmon qubits (Fig. taken from [1]).Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 26 / 30
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Grover’s algorithm
Grover’s algorithm with two transmon qubits (Fig. taken from [1]).Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 27 / 30
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Fidelity of Results
Fidelity F (ρ, ψ) = 〈ψ| ρ |ψ〉 of final states of Grover’s algorithm (Figure taken from [1]).
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Conclusion
1 The on-demand creation and detection of entangled states is possible.
2 Basic algorithms have been implemented in superconducting circuit systems.3 The present architecture can easily be expanded to several qubits.
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Conclusion
1 The on-demand creation and detection of entangled states is possible.2 Basic algorithms have been implemented in superconducting circuit systems.
3 The present architecture can easily be expanded to several qubits.
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Conclusion
1 The on-demand creation and detection of entangled states is possible.2 Basic algorithms have been implemented in superconducting circuit systems.3 The present architecture can easily be expanded to several qubits.
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Sources
1 L. DiCarlo, J. M. Chow, J. M. Gambetta, Lev S. Bishop, B. R. Johnson, D. I.Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin & R. J. SchoelkopfDemonstration of two-qubit algorithms with a superconducting quantumprocessorNature 460, 7252 (2009)
2 Clarke, J. & Wilhelm, F.K.Superconducting quantum bitsNature 453, 1031 (2008)
3 Schoelkopf, R.J. & Girvin, S.M.Wiring up quantum systemsNature 451, 664 (2008)
4 Devoret, M.H. & Martinis, J.M. Implementing Qubits with SuperconductingIntegrated CircuitsQuant. Inf. Proc, 3 163 (2004)
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Creating Phase Gates
U =
1 0 0 00 e iΦ01 0 00 0 e iΦ10 00 0 0 e iΦ11
(4)
Φ01 is adjusted by tuning the rising or falling edge of the pulseΦ10 is adjusted by varying the amplitude of a simultaneous VL pulse
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Creating Phase Gates
Creating U01
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