Continuous Monitoring and Feedback Control
of Superconducting Qubits

Justin Dressel
Institute for Quantum Studies
Schmid College of Science and Technology
Chapman University

UT Arlington, 2025/03/26

Outline

Superconducting Qubits
- Circuit QED Overview
- Transmon Qubits
Continuous Measurement
- Dispersive Readout
- Quantum Trajectories
A Tour of Recent Results
- State Tracking
- Zeno-dragging Gates
- Feedback State Stabilization
- Continuous Error Correction
- LSTM RNN State Tracking

Superconducting Qubits

Mesoscopic coherence of collective charge motion at \(\mu\)m scale, mK temperature

EM Fields of charge motion described by Circuit QED

Anharmonic oscillator potentials treated as artificial atoms, with lowest 2 levels as qubit

Qubit levels controlled by resonant microwave field drives

Qubit levels measured via dispersive frequency-shift of coupled microwave resonator mode

Circuit QED Overview

Canonically conjugate dynamical variables: \([\hat{\Phi}, \hat{Q}] = i\hbar\)

inductive (magnetic) flux: \(\hat{\Phi} = \Phi_0\,\hat{\phi}\), \(\Phi_0 = \hbar/2e\)
capacitive (electric) charge: \(\hat{Q} = (2e)\,\hat{n}\)

Dimensionless conjugate variables: \( [\hat{\phi},\,\hat{n}] = i\)

Example: Harmonic Oscillator

Useful circuit, but bad qubit:
Can't isolate specific level pairs since
all energy gaps are identical

Josephson Junction: \(\displaystyle \hat{H}_J = E_J\,\left(1 - \cos\hat{\phi}\right) \approx \frac{E_J}{2}\,\hat{\phi}^2 - \frac{E_J}{4!}\hat{\phi}^4 + \cdots \)

\(\displaystyle \hat{\phi} = \frac{\hat{\Phi}}{\Phi_0}, \quad E_J = \frac{\Phi_0^2}{L_J} \) Acts as nonlinear inductance => anharmonic oscillator
Shunting with large capacitor shields from charge noise

\(E_J\)

\(E_C\)

Quantum Pendulum (Transmon)

Nonlinearity makes energy gaps different
Energy level pairs now addressable as qubits

In reality, ~7 levels are bound in the cosine well
The bottom two levels are the most stable qubit

Microwave drive resonant with qubit energy gap
induces single-qubit gates (controlled Rabi oscillations)

E_1 - E_0 = \hbar \omega_q \quad\qquad\qquad \omega_q/2\pi \sim \text{4--7 GHz}

\(\displaystyle \hat{H} = E_0\,\hat{1} + \hbar\omega_q\,|1\rangle\!\langle 1| + \hbar(\omega_q - \delta_q)\,|2\rangle\!\langle 2|+ \cdots \)

E_2 - E_1 = \hbar (\omega_q - \delta_q) \qquad \delta_q/2\pi \sim \text{100--300 MHz}

Transmon Qubits as Artificial Atoms

Cosine potential acts like a binding potential well

Below an ionization threshold, energies are discretized

(Above the threshold, unbound energies are continuous)

Distinct behavior from optical regime with real atoms:

Engineered chips permit ultra-strong and deep-strong coupling regimes that are difficult to achieve with atoms
Lower frequencies than optics make transients more relevant
Emission can be directionally controlled down waveguides to minimize collection loss and increase detection efficiency
These atoms are super-cool (~4 mK)

\hbar\omega_q

\hbar(\omega_q - \delta_q)

\omega_q/2\pi \sim \text{4--7GHz}

\delta_q/2\pi \sim \text{100--300MHz}

\(\displaystyle \hat{H} \approx E_0\,\hat{1} + \hbar\omega_q\,|1\rangle\!\langle 1| + \hbar(\omega_q - \delta_q)\,|2\rangle\!\langle 2| \)

Measuring a Transmon Qubit

Problem:

Qubit is on a chip inside a fridge near absolute zero.
How does one "measure the energy eigenstate" of the qubit without causing unwanted changes to those energy states?

C_J

L_J

C_c

Solution:

Indirectly peek at the qubit energy by dispersively coupling
the qubit to a strongly detuned resonator, then
probing that resonator with a microwave tone.

The frequency shift of the leaked and amplified tone stores information about the energy state of the qubit without allowing energy transitions between the qubit and resonator.

This type of measurement that does not disturb energy eigenstates in the process of measuring the energy is called a
"quantum non-demolition (QND) measurement".

Dispersive Readout: Quantum Filtering

Reconstructing qubit information from an amplified and noisy homodyne record is a challenging filtering problem!

Need to process stochastic voltage records to isolate tiny phase-differences
(classical signal filtering)

The corresponding qubit/resonator state evolution must then be inferred
(quantum state filtering)

Amazingly, all you need is Bayes' Rule from probability theory
and the Born rule from quantum mechanics to derive how
the state partially collapses with information gain!

Problem:

The microwave probe tone must be compared to the original source, using homodyne or heterodyne measurements.

These signals are noisy due to the intrinsic vacuum noise of the source, which dominates the tiny amount of qubit information per unit time.

Dispersive Readout: Amplification

Without quantum-limited on-chip pre-amplification, this technique wouldn't work due to the weak signal.

With pre-amplification, the weak signal is boosted above noise thresholds for later HEMTs.

Quantum-limited amplifiers (built from Josephson junctions using their nonlinear inductance):

Josephson Parametric Amplifier (JPA):
- narrow-band reflected 3-wave mixer
- can operate in phase-sensitive (squeezed, above) or phase-preserving (unsqueezed) modes
Traveling Wave Parametric Amplifier (TWPA):
- broad-band amplification over transmission-line propagation
- only operates in phase-preserving (unsqueezed) mode

Outgoing signal is further amplified to enhance phase difference in steady-state resonator modes

One (informational) quadrature encodes the qubit state information as a displacement of the signal distribution

The other (phase) quadrature encodes photon number fluctuations inside the resonator

Single shot "projective" readout
using phase-sensitive amplifier to isolate informational quadrature

Integrated noisy signals clearly distinguish definite qubit states
and can detect "quantum jumps" between them (useful for syndrome detection)

Prevents normal dynamics from occurring (quantum Zeno effect)

Strong Continuous Monitoring

Stronger coupling yields more distinguishable resonator states

More information per unit time yields more rapid projection to the stationary eigenstates of the coupling

=> Strong continuous measurement

For each trajectory, the qubit state is affected less per unit time
but the same average information can be collected over an ensemble

Weak coupling can "gently" monitor average information during dynamics,
yielding a gradual collapse to an eigenstate as information accumulates.

Weak Continuous Monitoring

Weaker coupling yields
less distinguishable resonator states

Less information per unit time yields
slower projection to the coupling eigenstates

=> Weak continuous measurement

Following Bayes' rule, the qubit state randomly walks as probabilities are updated with each small amount of information in the signal

P(k|r) = \frac{P(r|k)P(k)}{P(r|1)P(1) + P(r|0)P(0)} \\ \; \\ |\psi\rangle = \sqrt{P(0)}|0\rangle + \sqrt{P(1)}e^{i \phi}|1\rangle \; \\ |\psi\rangle \xrightarrow{r} \sqrt{P(0|r)}|0\rangle + \sqrt{P(1|r)}e^{i \phi}|1\rangle

Modern UC Berkeley Planar Chip: 8 Transmons

Bus doubles as Purcell filter,
coupled to traveling wave parametric amplifier (TWPA)

(Stripline) Resonator couples capacitively to both bus and transmon for qubit readout

Double-junction Transmon couples to both readout resonator and a separate control line

Control uses a resonant (Rabi) drive where envelope area is the rotation angle

TWPA

Bus is pumped to excite resonator. Reflected part of the field and leaked resonator field are amplified by the TWPA, which is phase-preserving

Distinguishable resonator phase is made visible in a heterodyne measurement

A Brief Tour of Recent Work using
Continuous Measurements

Monitored Rabi Drive

JD and Siddiqi Group, Nature 511, 570 (2014)

Partial collapses compete with unitary dynamics

Ensemble-averaging the stochastic evolution recovers smooth dissipative (Lindblad) dynamics

Post-selected State Dynamics

JD and Siddiqi Group, Nature 511, 570 (2014)

Experimental most probable path matches ODE solution derived from stochastic path integral

JD and Jordan Group, PRA 88, 042110 (2013)

Siddiqi group, JD, Nature Communications 13, 2307 (2022)

2 parity syndromes of 3 qubits monitored continuously and strongly
Zeno effect protects code space, readout alerts when flip needs correction

Continuous Quantum (Bit-flip) Error Correction

Siddiqi group, JD, Nature Communications 13, 2307 (2022)

Continuous Quantum (Bit-flip) Error Correction

2 parity syndromes of 3 qubits monitored continuously and strongly
Zeno effect protects code space, readout alerts when flip needs correction

Feedback State Stabilization

Linear feedback (with very small temporal delays) can stabilize the qubit state to targeted regions of the Bloch sphere.

JD and Jordan Group, PRA 96, 022311 (2017)

Levenson-Falk, JD, PRApplied 21, 024022 (2024)

Two drives, parity measurement, and feedback stabilize Bell state

Non-Markovian noise sources suppressed by measurement and dynamical decoupling from drives (theory + simulations)

Entangled State Stabilization with Feedback

Symmetrically detuned pumps
Beats stroboscopically measure rotating qubit
Yields displacement coupling
Allows tunable measurement axis
Multiple cavity modes = multiple observables

\hat{H}^{\text{rf}}_{\text{q-r}} = \chi\,(\hat{a}+\hat{a}^\dagger)\,\hat{\sigma}_{\delta}

Displacement coupling:

\hat{\sigma}_{\delta} = \cos\delta\,\hat{\sigma}^{\text{rf}}_x + \sin\delta\,\hat{\sigma}^{\text{rf}}_y

Siddiqi group, Nature 538, 491 (2016)

Stroboscopic Displacement Coupling

\hat{H}_{\text{q}} = \frac{\hbar\Omega_R}{2}\hat{\sigma}_x

Rotating frame:

Incoherent Zeno-Dragging Qubit Gate

Idea : use time-varying measurement axes to drag the quantum state around the Bloch sphere using the quantum Zeno effect

The record tracks the state well in this regime, so can be used as a herald for high-fidelity gates

Non-unitary gate

(measurement-based)

\hat{H}_{\text{q-r}}(t) = \chi\,(\hat{a}+\hat{a}^\dagger)\,\hat{\sigma}_{\delta(t)}

\hat{\sigma}_{\delta(t)} = \cos\delta(t)\,\hat{\sigma}_x + \sin\delta(t)\,\hat{\sigma}_y

Stroboscopic displacement coupling can be time-varying

JD, Siddiqi group, PRL 120, 020505 (2018)

Jumps : Faster drag speeds allow trajectories to jump to the opposite pole, decreasing ensemble-averaged dragging fidelity

Jump-axis : Dragging dynamics causes lag of actual Zeno-pinned behind the measurement axis by a fixed angle

\Omega = 2\pi\,v : \text{dragging frequency}

\Gamma_D = 1/2\tau_m\eta : \text{measurement dephasing rate}

JD, Siddiqi group, PRL 120, 020505 (2018)

Pinned to poles : Other than the jumps, state remain pinned to lagged measurement poles

Incoherent Zeno-Dragging Qubit Gate

Siddiqi group, Nature 538, 491 (2016)

4 pumps, symmetrically detuned from 2 resonator modes

2 simultaneous noncommuting observables

Partial collapses compete with each other, preventing full collapse to a stationary state

If observables are maximally non-commuting, creates persistent phase-diffusion in Bloch sphere

Noncommuting Observable Measurement

\hat{H}^{\text{rf}}_{\text{q-r}} = \chi_1\,(\hat{a}+\hat{a}^\dagger)\,\hat{\sigma}_{\delta_1} + \chi_2\,(\hat{b}+\hat{b}^\dagger)\,\hat{\sigma}_{\delta_2}

\hat{\sigma}_{\delta_k} = \cos\delta_k\,\hat{\sigma}^{\text{rf}}_x + \sin\delta_k\,\hat{\sigma}^{\text{rf}}_y

Siddiqi group, Nature 538, 491 (2016)

State purifies, but diffuses randomly

Basins of attraction if measurement axes are nearly aligned