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Algorithms in Superconducting Circuits
Axel Dahlberg, Marco Roth
April 24, 2015
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 1 / 30
Motivation
1 Implementing Grover’s algorithm.
2 Superconducting circuits.3 Control over parameters.
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 2 / 30
Motivation
1 Implementing Grover’s algorithm.2 Superconducting circuits.
3 Control over parameters.
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 2 / 30
Motivation
1 Implementing Grover’s algorithm.2 Superconducting circuits.3 Control over parameters.
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 2 / 30
Outline
1 Theoretical aspect of Grover’s algorithm
2 Circuit Implementation
3 Results
4 Conclusion
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 3 / 30
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)
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 4 / 30
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)
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 4 / 30
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)
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 4 / 30
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)
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 4 / 30
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.
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 5 / 30
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.
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 5 / 30
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.
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 5 / 30
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〉
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 6 / 30
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〉
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 7 / 30
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〉
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 8 / 30
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〉
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 9 / 30
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〉
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 10 / 30
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〉
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 11 / 30
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]).
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 12 / 30
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:
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 13 / 30
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:
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 14 / 30
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:
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 14 / 30
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)
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 15 / 30
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)
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 15 / 30
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)
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 15 / 30
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)
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 15 / 30
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)
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 15 / 30
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〉) =
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 16 / 30
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〉) =
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 17 / 30
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〉) =
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 18 / 30
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〉) =
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 19 / 30
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+⟩
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 20 / 30
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+⟩
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 21 / 30
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
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
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
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
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
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
Fidelity of Results
Fidelity F (ρ, ψ) = 〈ψ| ρ |ψ〉 of final states of Grover’s algorithm (Figure taken from [1]).
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 28 / 30
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.
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 29 / 30
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.
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 29 / 30
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.
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 29 / 30
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)
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 30 / 30
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
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 1 / 2
Creating Phase Gates
Creating U01
Axel Dahlberg, Marco Roth Algorithms in Superconducting Circuits April 24, 2015 2 / 2