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Experimental implementation of Grover's
algorithm with transmon qubit architecture
Andrea AgazziZuzana Gavorova
Quantum systems for information technology, ETHZ
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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
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The oracle
• The oracle MARKS the correct solution
Dilution operator(interpreter)
• Solution is more recognizable
x
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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
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Geometric visualization
• Preparation of the state
O
Michael A. Nielsen and Isaac L. Chuang, Quantum computation and quantum information, 2011
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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
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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!
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Grover's algorithm Circuit
Preparation
X
X∣0 ⟩
∣0 ⟩
for t=2
Dewes, A; Lauro, R; Ong, FR; et al. arXiv:1109.6735 (2011)
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Grover's algorithm Oracle
for t=2
Will get 2 cases:
Dewes, A; Lauro, R; Ong, FR; et al. arXiv:1109.6735 (2011)
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Grover's algorithm Decoding
for t=2X
∣01⟩X
Dewes, A; Lauro, R; Ong, FR; et al. arXiv:1109.6735 (2011)
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Experimental setup
ωq= qubit frequencyωr= resonator frequencyΩ = drive pulse amplitude
Dewes, A; Lauro, R; Ong, FR; et al. arXiv:1109.6735 (2011)
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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
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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)
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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
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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
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Pulse sequence
Dewes, A; Lauro, R; Ong, FR; et al. arXiv:1109.6735 (2011)
X
X
X
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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
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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
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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
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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)
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Errors
Nonzero probability of incorrect mappingCrosstalk
Dewes, A; Lauro, R; Ong, FR; et al. arXiv:1109.6735 (2011)
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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)
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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)
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Thank you for your attention
Special acknowledgements:
S. FilippA. Fedorov