1. Quantum systems for information technology, ETHZ
Experimental implementation of Grover's algorithm with transmon qubit architecture
Andrea Agazzi
Zuzana Gavorova
Quantum systems for information technology, ETHZ

2. What is Grover's algorithm?
Quantum search algorithm
Task: In a search space of dimension N, find those 0<M<N elements displaying some given characteristics (being in some given states).
Classical search (random guess)
Grover’s algorithm
Classical search (random guess)
Grover’s algorithm
Guess randomly the solution
Control whether the guess is actually a solution
Apply an ORACLE, which marks the solution
Decode the marked solution, in order to recognize it
Classical search (random guess)
Grover’s algorithm
Guess randomly the solution
Control whether the guess is actually a solution
Apply an ORACLE, which marks the solution
Decode the marked solution, in order to recognize it
O(N) steps
O(N) bits needed
O( ) steps
O(log(N)) qubits needed a .

3. The oracle x The oracle MARKS the correct solution
Dilution operator (interpreter)
Solution is more recognizable x

4. Grover's algorithm Procedure
Preparation of the state
Oracle application
Dilution of the solution
Readout O
Filipp S., Wallraff A., Lecture notes “quantum system for information technology”, ETHZ, 2012

5. Geometric visualization
Preparation of the state
Change uw with oracle O
Michael A. Nielsen and Isaac L. Chuang, Quantum computation and quantum information, 2011

6. Grover's algorithm Performance
Every application of the algorithm is a rotation of θ
The Ideal number of rotations is:
Michael A. Nielsen and Isaac L. Chuang, Quantum computation and quantum information, 2011

7. Grover's algorithm 2 qubits
N=4
Oracle marks one state M=1
After a single run and a projection measurement
will get target state with probability 1!

8. Grover's algorithm Circuit
Preparation X X
for t=2
Dewes, A; Lauro, R; Ong, FR; et al. arXiv: (2011)

9. Grover's algorithm Oracle
Will get 2 cases:
for t=2
Dewes, A; Lauro, R; Ong, FR; et al. arXiv: (2011)

10. Grover's algorithm DecodingX X
for t=2
Dewes, A; Lauro, R; Ong, FR; et al. arXiv: (2011)

11. Dewes, A; Lauro, R; Ong, FR; et al. arXiv:1109.6735 (2011)
Experimental setup
ωq= qubit frequency
ωr= resonator frequency
Ω = drive pulse amplitude
Dewes, A; Lauro, R; Ong, FR; et al. arXiv: (2011)

12. Single qubit manipulation
Qubit frequency control via flux bias
Rotations around z axis: detuning Δ
Rotations around x and y axes: resonant pulses with amplitude Ω Δ a
Filipp S., Wallraff A., Lecture notes “quantum system for information technology”, ETHZ, 2012

13. Dewes, A; Lauro, R; Ong, FR; et al. arXiv:1109.6735 (2011)
Experimental setup
ωq,I = 1st qubit frequency
ωq,II= 2nd qubit frequency
g = coupling strength
Dewes, A; Lauro, R; Ong, FR; et al. arXiv: (2011)

14. Qubit capacitive coupling
In the rotating frame where
ω = ωq,II
the coupling Hamiltonian is:
Values for g and delta a
Bialczak, RC; Ansmann, M; Hofheinz, M; et al., Nature Physics 6, 409 (2007)

15. iSWAP gate Controlled interaction between ⟨10∣ and ⟨01∣
By letting the two states interact for t = π/g we obtain an iSWAP gate!
|eg⟩
|ge⟩
Filipp S., Wallraff A., Lecture notes “quantum system for information technology”, ETHZ, 2012

16. Dewes, A; Lauro, R; Ong, FR; et al. arXiv:1109.6735 (2011)
Pulse sequence X X X
Dewes, A; Lauro, R; Ong, FR; et al. arXiv: (2011)

17. Linear transmission line and single-shot measurement
Quality factor of empty linear
transmission line
With resonant frequency
Presence of a transmon in state
shifts the resonant frequency of the transmission line
If microwave at full transmission for state, partial for state
Current corresponding to transmitted EM
Filipp S., Wallraff A., Lecture notes “quantum system for information technology”, ETHZ, 2012

18. Linear transmission line and single-shot measurement
If there was no noise, would get either blue or red curve
Real curve so noisy that cannot tell whether or
Cannot do single-shot readout
We need an amplifier which increases the area between and curves, but does not amplify the noise
Filipp S., Wallraff A., Lecture notes “quantum system for information technology”, ETHZ, 2012

19. Josephson Bifurcation Amplifier (JBA)
Nonlinear transmission line due to
Josephson junction
Resonant frequency
At Pin = PC max. slope diverges
Bifurcation: at the correct (Pin ,ωd) two stable solutions, can map the collapsed state of the qubit to them
Dewes, A; Lauro, R; Ong, FR; et al. arXiv: (2011)
E. Boaknin, V. Manucharyan, S. Fissette et al: arXiv:cond-mat/ v1

20. Josephson Bifurcation Amplifier (JBA)
Switching probability p: probability that the JBA changes to the second solution
Excite
Choose power corresponding to the biggest difference of switching probabilities
Errors
Nonzero probability of incorrect mapping
Crosstalk
Dewes, A; Lauro, R; Ong, FR; et al. arXiv: (2011)

21. Dewes, A; Lauro, R; Ong, FR; et al. arXiv:1109.6735 (2011)
Errors
Nonzero probability of incorrect mapping
Crosstalk
Dewes, A; Lauro, R; Ong, FR; et al. arXiv: (2011)

22. Conclusions
Gate operations of Grover algorithm successfully implemented with capacitively coupled transmon qubits
Arrive at the target state with probability 0.62 – (tomography)
Single-shot readout with JBA (no quantum speed-up without it)
Measure the target state in single shot with prob 0.52 – (higher than 0.25 classically)

23. Sources
Dewes, A; Lauro, R; Ong, FR; et al., “Demonstrating quantum speed-up in a superconducting two-qubit processor”, arXiv: (2011)
Bialczak, RC; Ansmann, M; Hofheinz, M; et al., “Quantum process tomography of a universal entangling gate implemented with Josephson phase qubits”, Nature Physics 6, 409 (2007)

24. Thank you for your attention
Special acknowledgements:
S. Filipp
A. Fedorov