Experimental implementation of Grover's

algorithm with transmon qubit architecture

Andrea AgazziZuzana Gavorova

Quantum systems for information technology, ETHZ

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

• 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) stepsO(N) bits needed

O( ) stepsO(log(N)) qubits needed

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

.a

The oracle

• The oracle MARKS the correct solution

Dilution operator(interpreter)

• Solution is more recognizable

x

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

Geometric visualization

• Preparation of the state

O

Michael A. Nielsen and Isaac L. Chuang, Quantum computation and quantum information, 2011

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

Grover's algorithm 2 qubits

N=4Oracle marks one state M=1

After a single run and a projection measurement will get target state with probability 1!

Grover's algorithm Circuit

Preparation

X

X∣0 ⟩

∣0 ⟩

for t=2

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

Grover's algorithm Oracle

for t=2

Will get 2 cases:

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

Grover's algorithm Decoding

for t=2X

∣01⟩X

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

Experimental setup

ωq= qubit frequencyωr= resonator frequencyΩ = drive pulse amplitude

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

Single qubit manipulation• Qubit frequency

controlvia flux bias

• Rotations around z axis:detuning Δ

• Rotations around x and y axes:resonant pulses with amplitude Ω Δ

Filipp S., Wallraff A., Lecture notes “quantum system for information technology”, ETHZ, 2012a

Experimental setup

ωq,I = 1st qubit frequencyωq,II= 2nd qubit frequencyg = coupling strength

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

Qubit capacitive coupling

In the rotating frame whereω = ωq,II

the coupling Hamiltonian is:

Bialczak, RC; Ansmann, M; Hofheinz, M; et al., Nature Physics 6, 409 (2007) a

iSWAP gate• Controlled

interaction between ⟨10∣ and ⟨01∣

• By letting the two states interact for

t = π/gwe obtain an iSWAP gate!

|ge⟩

|eg⟩

Filipp S., Wallraff A., Lecture notes “quantum system for information technology”, ETHZ, 2012

Pulse sequence

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

X

X

X

Linear transmission line and single-shot measurement

Quality factor of empty linear transmission lineQ=

νrΔ νr

With resonant frequency νr

Presence of a transmon in state shifts the resonant frequency of the transmission lineIf microwave at full transmission for state, partial for state

Current corresponding to transmitted EM

ω0=2 π ν r→ ω0−χ∣g ⟩

ω0−χ ∣g ⟩ ∣e ⟩

Filipp S., Wallraff A., Lecture notes “quantum system for information technology”, ETHZ, 2012

Linear transmission line and single-shot measurement

Single shotIf there was no noise, would get either blue or red curveReal curve so noisy that cannot tell whether or

∣g ⟩ ∣e ⟩

Cannot do single-shot readout

We need an amplifier which increases the area between and curves, but does not amplify the noise

∣g ⟩ ∣e ⟩

Filipp S., Wallraff A., Lecture notes “quantum system for information technology”, ETHZ, 2012

Josephson Bifurcation Amplifier (JBA)

Nonlinear transmission line due to

•Josephson junctionResonant frequency

ω0

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:1109.6735 (2011)E. Boaknin, V. Manucharyan, S. Fissette et al: arXiv:cond-mat/0702445v1

Josephson Bifurcation Amplifier (JBA)

Switching probability p: probability that the JBA changes to the second solution

ExciteChoose power corresponding to the biggest difference of switching probabilities

ErrorsNonzero probability of incorrect mappingCrosstalk

∣1 ⟩ →∣2 ⟩

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

Errors

Nonzero probability of incorrect mappingCrosstalk

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

Conclusions•Gate operations of Grover algorithm successfully implemented with capacitively coupled transmon qubits

•Arrive at the target state with probability 0.62 – 0.77 (tomography)

•Single-shot readout with JBA (no quantum speed-up without it)

•Measure the target state in single shot with prob 0.52 – 0.67 (higher than 0.25 classically)

Sources• Dewes, A; Lauro, R; Ong, FR; et al.,

“Demonstrating quantum speed-up in a superconducting two-qubit processor”,arXiv:1109.6735 (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)

Thank you for your attention

Special acknowledgements:

S. FilippA. Fedorov