Realization of a single-ion controlled-NOT gateOutline• Background of trapped ion based quantum computation.• How does a single-ion controlled-NOT quantum logic gate work• Verification of the CNOT gate• Comments and conclusionsAdvantages1. Decoherence can be small (using hyperfine state of an ion as thequbit)2. Extension to large registers is relatively straightforward3. The qubit readout can have near unit efficiencyUniversal quantum logic gates in Cirac and Zoller’s proposal• Single qubit rotation: using the individual addressing laser beam• CNOT between mthand nth ions:– Map the information of mthion to the “bus-mode”– Use the “bus-mode” to control the flip of nthion (analog to a single-ion CNOT)– Map the “bus-mode” back to mthion and also reset the “bus-mode”CNOT gate truth table|0> |↓>|0> |↑>|1> |↓>|1> |↑>|0> |↓> |0> |↑> |1> |↓> |1> |↑>⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛0100100000100001BeforeAfterTarget qubitControl qubit~ 1.25 GHz|F=2, mF=0> |F=2, mF=2>The two qubits of the single-ion CNOT gateControl qubit: first two quantum harmonic oscillator states of a single 9Be+confined in a rfPaul trap. |0>=|n=0> |1>=|n=1>ωx ~ 11 MHz < ωy, ωzhνTarget qubit: two internal hyperfine states of the 9Be+ion |↓>= 2S1/2|F=2,mF=2> |↑>= 2S1/2|F=1,mF=1>9Be+: S=1/2 I=3/2 F=1,2Modified from C. Monroe, et al. PRL 75, 4714|F=1, mF=1>|F=3, mF=3>Initialization of the 9Be+ion statesFrom C. Monroe, et al. PRL 75, 4011The ion is cooled to the 3D zero-point energy state using Doppler and resolved-sideband Raman cooling. |0>|↓> is occupied with 95% probability.Resolved-sideband absorption spectrumXYZBloch sphere representation of the manipulation of a two-level system-z: |↑>+z: |↓>+y:1/√2(|↓>+i|↑>)-y:1/√2(|↓>-i|↑>)+x:1/√2(|↓>+|↑>)Rx(θ)= e-iθX/2Ry(θ)= e-iθY/2Rz(θ)= e-iθZ/2A π/2 pulse creates an equal superposition of the two states == Ry (π/2)A π pulse corresponds to 100% transfer between the two states== Ry (π)-x: 1/√2(|↓>-|↑>)X, Y, Z are Pauli matrices.Carrier ωoRed sidebandωx|1>|↑>|0>|↑>|1>|↓>|0>|↓>Single qubit operations to prepare the four basis states|0>|↓> Æ |0>|↑>: using a π-pulse on carrier transition to flip the electronic state|0>|↓> Æ |1>|↑>: using a π-pulse on the blue sideband transition.|0>|↓> Æ |1>|↓>: using a π-pulse on the blue sideband transition followed by a π-pulse on carrier transition Blue sidebandRealization of the CNOT gate1) A π/2 pulse on the carrier transition: Ry(π/2) rotation on the internal state qubit.2) Controlled phase-flip gate: 2π pulse on the blue sideband transition between |↑> and another electronic level |aux>, Rx(2π)3) A π/2 pulse on the carrier transition with a π phase shift relative to 1):Ry(-π/2) rotation on the internal state qubit.Step 1: Ry(π/2) operation|0>|↓> |0>|↑>|1>|↓>|1>|↑>|0>(|↓>+|↑>) |0>(|↓>-|↑>)|1>(|↓>+|↑>)|1>(|↓>-|↑>)Before step 1After step 1XYZ+x:1/√2(|↓>+|↑>)-x:1/√2(|↓>-|↑>)|↑>|↓>Driving frequency = ω0No change of the motional states.Step2: Controlled phase-flip operation: Rx(2π)ωoωx|1>|↑>|0>|↑>|1>|↓>|0>|↓>|0>|aux>|1>|aux>Only result of step 2: |1>|↑> -|1>|↑>|aux>= 2S1/2|F=2,mF=0>2 π pulse on blue sidebandStep 3: Ry(-π/2) operation: No change of the motional states.|0>|↓> |0>|↑>|1>|↓>|1>|↑>|0>(|↓>+|↑>) |0>(|↓>-|↑>)|1>(|↓>+|↑>)|1>(|↓>-|↑>)Before step 3After step 3Ry(-π/2) operation is nothing but the reverse of Ry(π/2) operationXYZ-x: 1/√2(|↓>-|↑>)|↑>|↓>A π phase shift relative to step 1+x:1/√2(|↓>+|↑>)Whole process of a CNOT gate|0>|↓> |0>|↑>|1>|↓>|1>|↑>|0>(|↓>+|↑>) |0>(|↓>-|↑>)|1>(|↓>+|↑>)|1>(|↓>-|↑>)|0>(|↓>+|↑>)|0>(|↓>-|↑>)|1>(|↓>-|↑>)|1>(|↓>+|↑>)|0>|↓> |0>|↑>|1>|↑>|1>|↓>Initial statesRy(π/2) Rx(2π)Ry(-π/2)Detection of the target qubit output stateThe probability of the target qubit in |↓> state P(|↓>) the ion fluorescence.∝+ circularly polarized light (linewidth ~ 19.4 MHz) doesn’t couple |↑> state appreciably.Detection efficiency is close to 100%.Detection of the control qubit output state|1>|↑>|0>|↑>|1>|↓>|0>|↓>If P(|↓>) = 1π pulse on red sidebandRepeat the fluorescence measurement:Presence Control qubit=|0>Absence Control qubit=|1>|1>|↑>|0>|↑>|1>|↓>|0>|↓>If P(|↓>) = 0π pulse on blue sidebandRepeat the fluorescence measurement:Presence Control qubit=|1>Absence Control qubit=|0>Experiment resultsFrom C. Monroe, et al. PRL 75, 4714 and NIST ion storage group webpage= P(|↓>) Target= P(|1>) ControlComments & Conclusions• Measured probability is not exactly 100%. • Imperfect laser-cooling (95% at |0>|↓>)• Imperfect state preparation and detection preparation• In this experiment, CNOT gate operating time ~ 50 μs. The measured coherence time ~ hundreds to thousands of μs. Far blow the required error correction threshold.• A single-ion CNOT is not enough for quantum computation. Need CNOT between different ions (demonstrated by Ferdinand Schmidt-Kaler, et al, Nature 422, 408,