Quantum gate control · Chapter 01

From specifying a gate to learning how to calibrate it

An ideal gate is easy to write down as a matrix. The harder question is how to realize many such matrices when the physical evolution differs from the ideal model.

The target and the device are different objects

A target gate is a unitary matrix, but a noisy implementation is a channel acting on density matrices. The distinction matters: dissipation can turn pure states into mixed states.

Definition

States, targets, and channels

Here $\rho$ is a two-qubit state and $\mathcal{U}_\star$ the ideal channel. A physically admissible noisy map $\mathcal{E}$ is completely positive and trace-preserving (CPTP).

\begin{gathered}\rho\in\mathbb{C}^{4\times4},\quad \rho\succeq0,\quad \operatorname{Tr}\rho=1,\\ U_\star^\dagger U_\star=I_4,\quad \det U_\star=1,\\ \mathcal{U}_\star(\rho)=U_\star\rho U_\star^\dagger.\end{gathered}

Definition

Average gate fidelity

The integral uses normalized Haar measure over pure two-qubit states. $F_{\mathrm e}$ is the process fidelity with the target unitary channel. This score evaluates a gate on all input directions, not just one trajectory on a sphere.

\begin{aligned}F_{\mathrm{avg}}(\mathcal{E},U_\star)&=\int \langle\psi|U_\star^\dagger\mathcal{E}(|\psi\rangle\langle\psi|)U_\star|\psi\rangle\,d\psi\\&=\frac{4F_{\mathrm e}+1}{5}.\end{aligned}

Current manuscript: Gate Fidelity; standard average/process-fidelity identity for dimension four.

A controlled model of uncertainty

Relaxation and dephasing are modeled by a Lindblad equation. Independent Gaussian detunings are resampled between tasks, but fixed during every pulse evaluation within a task. The controller is not given their values.

Model

Open-system dynamics and quasi-static detuning

We set $\hbar=1$ . $Z_1=\sigma_z\otimes I_2$ and $Z_2=I_2\otimes\sigma_z$ . The nonnegative rates $\gamma_\ell$ are fixed; jump operators are local lowering and Pauli-Z operators. Braces denote the anticommutator.

\begin{aligned}\dot\rho&=-i[H(t),\rho]+\sum_\ell\gamma_\ell\mathcal{D}[L_\ell](\rho),\\ \mathcal{D}[L](\rho)&=L\rho L^\dagger-\tfrac12\{L^\dagger L,\rho\},\\ H_0&=\delta_1Z_1+\delta_2Z_2,\quad\delta_q\overset{\mathrm{iid}}{\sim}\mathcal{N}(0,\sigma_\delta^2).\end{aligned}

Current manuscript: open-system formulation and noise configuration.

Learn a procedure, not a library of pulses

Optimizing each requested gate independently is one possible solution. This project instead trains across randomly sampled targets and asks whether the resulting policies can guide calibration of new target–noise pairs. The reusable object is a decision rule for proposing corrections, not one universal waveform.

Here, zero-shot generalization has a precise scope: the policy weights stay fixed at evaluation time, but the controller still evaluates pulses and receives feedback. It does not mean open-loop control, zero measurements, or proven transfer to noise distributions absent from training.

Model

The ideal single-qubit limit behind the explorer

The ideal reference rotates around the constant control vector $\mathbf u=(u_x,u_y,u_z)$ at angular speed $\Omega_{\mathrm{ref}}\|\mathbf u\|$ . Each control is dimensionless and ranges from −2 to 2. The fixed scale $\Omega_{\mathrm{ref}}$ converts it to physical angular frequency; changing the pulse duration does not change that scale. The explorer defaults to 210 ns and allows 10–1,000 ns, independently of animation playback speed. A Bloch sphere represents one qubit and cannot represent a general two-qubit state.

\begin{gathered}H_{\mathrm{ideal}}=\tfrac{\Omega_{\mathrm{ref}}}{2}\mathbf u\cdot\boldsymbol\sigma,\quad \Omega_{\mathrm{ref}}=\frac{2\pi}{210\,\mathrm{ns}},\quad U(t)=e^{-itH_{\mathrm{ideal}}},\\ \rho=\tfrac12(I+\mathbf r\cdot\boldsymbol\sigma),\qquad\dot{\mathbf r}=\Omega_{\mathrm{ref}}\mathbf u\times\mathbf r,\\ P(0)=\tfrac12(1+r_z),\qquad P(1)=\tfrac12(1-r_z).\end{gathered}

Proof / derivation

Insert the Bloch representation of $\rho$ into $\dot\rho=-i[H,\rho]$ , and use the Pauli commutator identity. Exact Rodrigues rotation gives the ideal trajectory.

[\mathbf a\cdot\boldsymbol\sigma,\mathbf b\cdot\boldsymbol\sigma]=2i(\mathbf a\times\mathbf b)\cdot\boldsymbol\sigma.

See the same pulse under a fixed noise realization

The explorer compares the ideal trajectory with a single noisy trajectory, or with 24 trajectories having different Gaussian detunings. It uses the manuscript's relaxation, dephasing, and quasi-static detuning mechanisms in a separate constant-control, single-qubit model. It does not run the learned policy or reproduce the two-qubit calibration experiment.

Each detuning remains constant throughout a pulse. Changing controls or resetting playback preserves the realization; New realization draws another set. Switching presets reuses the same underlying Gaussian samples, and changing the detuning standard deviation rescales them. The chosen dissipative rates are fixed throughout each pulse and shared by every ensemble member.

Model

The noisy single-qubit Bloch equation

This follows the research code's explicit jump-operator convention: relaxation drives the undriven qubit toward $|1\rangle$ , the south pole, rather than $|0\rangle$ . The detuning term is $\delta\sigma_z$ , so its contribution to Bloch angular velocity is $2\delta$ . With the dissipator defined above, Pauli-Z dephasing contributes $2\gamma_z$ to transverse decay. Rates are in inverse seconds, detuning in radians per second, and physical time is converted from nanoseconds to seconds.

\begin{gathered}H_\delta=\tfrac{\Omega_{\mathrm{ref}}}{2}\mathbf u\cdot\boldsymbol\sigma+\delta\sigma_z,\qquad \delta\sim\mathcal N(0,\sigma_\delta^2),\\ \dot\rho=-i[H_\delta,\rho]+\gamma_1\mathcal D[\sigma_-](\rho)+\gamma_z\mathcal D[\sigma_z](\rho),\\ \sigma_-=|1\rangle\langle0|,\qquad\sigma_z=\operatorname{diag}(1,-1),\\ \mathbf b=\Omega_{\mathrm{ref}}\mathbf u+2\delta\mathbf e_z,\qquad\Gamma_2=\tfrac{\gamma_1}{2}+2\gamma_z,\\ \dot{\mathbf r}=\mathbf b\times\mathbf r-\begin{pmatrix}\Gamma_2r_x\\\Gamma_2r_y\\\gamma_1(r_z+1)\end{pmatrix}.\end{gathered}

Current manuscript: open-system formulation and noise configuration; research implementation: Pauli and jump-operator definitions. The constant-control explorer is a teaching adaptation.

Identity

An ensemble mean is a density matrix

The ensemble view shows sample paths, their mean, and the purity of the mean state. Different coherent detunings can shorten the mean Bloch vector even when every member stays pure; dissipation can also mix each member. The displayed ensemble purity is not the average of the individual purities. Twenty-four samples illustrate variation and do not establish a converged disorder average or gate fidelity.

\begin{gathered}\bar\rho(t)=\frac1N\sum_{j=1}^N\rho_j(t)=\tfrac12(I+\bar{\mathbf r}(t)\cdot\boldsymbol\sigma),\quad \bar{\mathbf r}(t)=\frac1N\sum_{j=1}^N\mathbf r_j(t),\quad N=24,\\ \operatorname{Tr}(\bar\rho^2)=\tfrac12(1+\|\bar{\mathbf r}\|^2),\qquad P_{\bar\rho}(0)=\tfrac12(1+\bar r_z).\end{gathered}

Standard Bloch representation and purity identity; finite ensemble used only for the interactive illustration.

Explorer noise presets and their provenance
Preset	Relaxation γ₁ (s⁻¹)	Dephasing γz (s⁻¹)	Detuning SD (rad/s)
None	0	0	0
Paper Q1	2,653	2,411	5 × 10⁶
Paper Q2	5,305	31,831	5 × 10⁶
Amplified (100× Q2 dissipation)	530,500	3,183,100	5 × 10⁶
Custom	Editable	Editable	Editable

Paper Q1 and Q2 use the respective qubit's fixed manuscript rates and the lower reported detuning standard deviation. They are alternative parameters for one displayed qubit. Amplified multiplies only the Q2 dissipative rates by 100 to make their effect visible; it is not a paper setting. Custom edits remain fixed during a pulse.

What would count as a useful improvement?

The natural starting point is an analytical pulse evaluated under the same noisy dynamics as the learned controller. Improvement should then be judged alongside the number of pulse evaluations, the information supplied to each method, and performance on held-out tasks. These distinctions keep better feedback, more search, and genuinely better learned decisions from being mistaken for the same achievement.

Last updated September 27, 2026