This video contains the talk that I would have given at the Optimal Quantum Control session of the APS March Meeting 2020, in Denver, linked here http://meetings.aps.org/Meeting/MAR20/Session/S38.1
Youtube video: https://www.youtube.com/watch?v=FViq1-WOeVk
Understanding Partial Differential Equations: Types and Solution Methods
Variational quantum control for single- and two-qubit transmon gates
1. Variational quantum control for
single- and two-qubit
transmon gates
APS March Meeting 2020
Session S38: Optimal Quantum Control
Andrés Ruiz Chamorro
Érik Torrontegui
Juanjo García Ripoll
arXiv:2002.10320
2. 𝐶 𝐸𝐽 Φ
Φ
J. Koch et al, Phys. Rev. A 76, 042319;
J. A. Schreier et al Phys. Rev. B 77, 180502(R) (2008);
L. DiCarlo et al., Nature 460, 240–244 (2009);
R. Barends et al., Nature 508, 500–503 (2014);
M. Rol et al., PRL 123, 120502 (2019);
C. K. Andersen et al, arXiv:1912.09410v1
3. 𝐶 𝐸𝐽 Φ
Φ
J. Koch et al, Phys. Rev. A 76, 042319
𝜔12
𝜔01
|0⟩
|1⟩
|2⟩
|3⟩
𝜔01 = 𝜔02 + 𝛼
9. 𝜔0𝑛
CZ gate
𝜔 𝑎(𝑡)
𝑡𝑖𝑚𝑒
L. DiCarlo et al., Nature 460, 240–244 (2009);
R. Barends et al., Nature 508, 500–503 (2014);
M. Rol et al., PRL 123, 120502 (2019);
C. K. Andersen et al, arXiv:1912.09410v1
|0,0⟩
|0,1⟩
|1,0⟩
|1,1⟩
11 → 11 𝑒 𝑖𝜙12
19. Conclusions
• Arbitrary many controls to implement a CZ gate with
tuneable frequency qubits
• Only two parameters to tune
‒ Waiting time
‒ Destination frequency
• Errors below 0.01% for upcoming qubits with T1 > 100
μs
• Finite bandwidth control, suited for filtered experiments.
Hello, my name is Juanjo García Ripoll and I work at the Spanish Research Council in Madrid.
I would like to welcome you to this online presentation of our work on quantum control for transmon qubits.
This work has been done in collaboration with Andrés Ruiz Chamorro and Erik Torrontegui, a master student and postdoctoral researcher in our group.
An e-print is available at the arXiv, as shown below.
A transmon qubit is a superconducting quantum circuit made of a Josephson junction and a capacitor.
It sports a good balance between reproducibility, speed and coherence, and it appears in most superconducting quantum computers nowadays.
A transmon qubit is modeled as a weakly nonlinear harmonic oscillator.
The Josephson junction provides a nonlinear inductance, creating this cosine potential, with an anharmonic spacing between energy levels.
The anharmonicity of the transmon can be quite large, of hundreds of MHz.
This allows us to address the ground and excited state separate from all other configurations, using them as a qubit.
One of the things we can do, is using microwaves to implement single-qubit rotations in this reduced Hilbert space
In many experiments we can also tune the frequency of the transmons.
Injecting a little bit of magnetic field through the SQUID, we change the Josephson energy and the qubit energy gap.
However, to make a real quantum computer, we also need to implement two-qubit gates.
For transmons, these gates appear naturally from the capacitive coupling between nearby qubits.
We can shape and connect and disconnect these interactions in various ways.
One of them, by shifting the relative frequency between two coupled transmons.
As shown in this picture, there are two resonance conditions for implementing a gate.
When transmons are degenerate, they have the same gap, the capacitive coupling supports the exchange of interactions, or SWAP gates.
But if we increase the gap of one transmon, we find another resonance condition between the 11 state and a state with two excitations in one transmon.
When we diagonalize the Hamiltonian of both coupled transmons, this coupling translates into an avoided crossing between both levels.
We can use this splitting to implement phase gates.
The method, introduced by DiCarlo and coworkers, starts with two highly detuned transmons.
We then perform an adiabatic ramp of the frequency down to the resonance condition.
If the ramp is adiabatic, the system stays in the instantaneous eigenstates of the coupled transmons.
The result is just phase shifts in all levels, with a particular phase shift in the |11> state which is sufficient to impelemtn a two-qubit gate.
It is usually ignored the fact that the transmon wavefunctions change when we squeeze the qubit frequency, enlarging or decreasing the Josephson energy.
This squeezing lays behind errors, such as leakage to higher excitations.
We can model this change, approximating how the ground state evolves by means of a variational wavefunction.
When we do so, we find that the width of the wavefunction depends on our external control, the qubit frequency ω.
Most important, we can apply a reverse logic and deduce the change in transmon frequency that takes us from an initial width to any other final state, with 100% fidelity, in an non-adiabatic fashion.
This idea can be tested with an isolated transmon, as shown here.
The variational method perform much better than a simple ramp of the flux in the squid, or than a fast quasiadiabatic method (FAQUAD).
This is because our method ensures that the initial and final states are eigenstates of the problem and does not worry about intermediate stages.
Moreover, the controls that we obtain lead to smooth flux pulses with greater differntiability and smaller bandwidth.
If we try to apply this idea to the two-qubit gate, results are mixed. We get bad fidelities for the variational method, while the quasiadiabatic method preserves some fidelity.
Why?
The first obvious source of error is phase over or under shooting.
The second source of error is more subtle. FAQUAD prepares eigenstates, but variational method prepares good isolated transmon states. These are not eigenstates and are rotated by the interaction, causing leakage.
The solution is simple
1. We will introduce a waiting time. This time allows the states we prepare to return to the |11> configuration.
We will adjust the target frequency, to fine tune the total phase, accounting
We can apply this optimization to all possible controls.
In all controls we find orders of magnitude improvement in the error of the gate.
However, the variational method still performs best, as it controls the effects related to squeezing, and is faster.
This excellent performance survives even in the presence of decoherence.
Note that ensuring instantaneous quasi-eigenstates does not make the gate better.
This is because those eigenstates include admixtures of |02> and |20> which already have larger decay rates.
- We have developed multiple analytic controls for a CZ gate between tuneable frequency transmons.
- We find that with simple optimizations, the errors are substantially reduced in all gates.
- The speed of the gate is competitive and sports very good fidelities for existing transmons, approaching the fault tolerance limit when we consider upcoming coherence times of 100's of milliseconds.
I would like to close by thanking our funding agencies,
the support of CSIC
and most of all the work of my collaborators in this long journey.
