Algorithms in Superconducting Circuits · 2015. 7. 27. · Motivation 1 Implementing Grover’s...

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.


Motivation

1 Implementing Grover’s algorithm.2 Superconducting circuits.

3 Control over parameters.


Motivation

1 Implementing Grover’s algorithm.2 Superconducting circuits.3 Control over parameters.


Outline

1 Theoretical aspect of Grover’s algorithm

2 Circuit Implementation

3 Results

4 Conclusion


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)




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






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)





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




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.



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.



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〉



a b


|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〉



a b c


|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〉



a b c

d


|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〉



a b c

d e


|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〉



a b c

d e f


|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〉


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]).


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:



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:



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)


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〉) =




|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〉) =




|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〉) =




|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+⟩


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

Fidelity of Results

Fidelity F (ρ, ψ) = 〈ψ| ρ |ψ〉 of final states of Grover’s algorithm (Figure taken from [1]).


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.


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.


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.


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)


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


Creating Phase Gates

Creating U01


Algorithms in Superconducting Circuits · 2015. 7. 27. · Motivation 1 Implementing Grover’s...

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