## General Introduction

The generic problem treated by Q-CTRL involves the controllability (or performance) of a quantum system (single-qubit, multi-qubit, etc.), given available controls in the presence of relevant noise sources. The quantum system is described by

consisting of a control Hamiltonian, $H_{c}(t)$, describing perfect control of the system, e.g. via an ideal external driving field, and the noise Hamiltonian, $H_{n}(t)$, capturing undesirable time-varying interactions with a noise process. The evolution of the quantum system is described by the time evolution operator, $U_{tot}(t,0)$, obtained by solving the Schrödinger equation.

The generic time-dependent control Hamiltonian may be written as

To unpack this notation we introduce some nomenclature useful for defining various
concepts in quantum control. Let $\mathcal{H}$ be a $d$-dimensional Hilbert space for the controlled quantum system.
We define the *control operators*

formally separated into three classes:

The $A_{l}$ and $C_{j}$ are referred to as *shift control operators* and *drive control operators*, respectively.
The operator $D$ is a time-independent operator, which we refer to as the *drift operator*. The corresponding time-dependent control functions $\alpha_{l}(t)$ and $\gamma_{j}(t)$ are the *real* and the *complex control pulses*, respectively. Thus

In this framework, the evolution of the quantum system may be viewed as a combination of dynamic rotations, each driven by
real or complex pulses about the effective *control axis* defined by the associated operator.
At times, it will be convenient to express the description of the quantum system more compactly. To do so, we vectorize the controls by defining

where real and complex pulses are listed as row vectors, and the corresponding operators are listed as column vectors. Using matrix notation we therefore write

and express the control Hamiltonian as

Given a set of control operators, $\boldsymbol{A}$ and $\boldsymbol{C}$, the most general description of control is therefore specified by the set of functions $\boldsymbol{\alpha}(t)$ and $\boldsymbol{\gamma}(t)$, defined on the time interval $t\in[0,\tau]$ specifying the duration over which the control is applied. We refer to this structure as a control solution. In particular, we provide tools to evaluate control solutions, including a variety of familiar predefined control solutions. Below we apply this generic structure fo some familiar special cases.

## Conventions

Here we introduce notational conventions followed by Q-CTRL to define complex control pulses, their decomposition, and their relationship to shift and drive operators. Since the following structure applies to every pulse-operator pair $(\gamma_{j}(t), C_{j})$, we drop the subscript $j$ for simplicity. The complex pulses $\gamma(t)\in\mathbb{C}$ may be written in polar or Cartesian form. Namely,

allowing us to decompose the complex pulses into constituent control variables via the coordinate mappings

where we establish the following definitions

and note the drive phase $\phi(t) = +\text{Arg}(\gamma(t))$ is defined as the *positive* complex argument.
The term in the control Hamiltonian corresponding to $\gamma(t)$ is therefore expressed as

where

define *shift* operators. Each complex pulse-control pair $(\gamma(t), C)$ therefore
decomposes into real pulse-control pairs $(I(t), A_{I})$ and $(Q(t), A_{Q})$. That is, the quadrature (Cartesian)
controls $(I(t),Q(t))$ are equivalent to additional real pulses driving the shift operators $(A_{I},A_{Q})$ in the quantum system. These are related to the drive operator as

resembling a *lowering operator* with respect to the $(A_{I}, A_{Q})$ pair.
The *drive modulus* $\Omega(t)$ sets the pulse-driven rotation rate, while the *drive phase* $\phi(t)$ sets the direction of
the control axis, oriented between the control axes defined by $(A_{I}, A_{Q})$.

## Single-Qubit Control Hamiltonian

For a single qubit with transition frequency $\omega_0$ and bare Hamiltonian

a near-resonant drive interaction is described in lab frame by the Hamiltonian

where $\Omega(t)$ and $\phi(t)$ map to the amplitude (modified by the drive modulus) and phase of the near-resonant qubit driving field. Moving to the interaction picture rotating with the qubit frequency, and performing the rotating-wave approximation, the combined system $H_\text{qubit}+H_\text{drive}$ is described by the Hamiltonian

where $\Delta(t)$ is the time-dependent detuning from resonance

and the Pauli operators are defined by

Absorbing the coupled $xy$ controls into a single pulse, $H_{int}(t)$ may be cast in the standard form used at Q-CTRL yielding the control Hamiltonian

where

In particular, we find

In the case of RF or microwave systems such as this, the $I-Q$ controls map to the $\sigma_{x}$ and $\sigma_{y}$ operators (again, accounting for the field-qubit coupling strength). This can be seen by recalling the controls $(I(t),Q(t))$ are equivalent to additional real pulses driving the shift operators $(A_{I},A_{Q})$ defined by

That is, the control Hamiltonian may be expressed as

where

and the real pulses and corresponding shift operators are defined by

With these definitions, we may equivalently write

where $\Omega’(t)$ is the modified Rabi rate describing the instantaneous rate of rotation (in angular frequency) about the rotation axis $\hat{\boldsymbol{n}}(t)\in\mathbb{R}^3$ associated with the rotation generator $\hat{\boldsymbol{n}}(t)\boldsymbol{\sigma} \equiv n_x(t)\sigma_x+n_y(t)\sigma_y+n_z(t)\sigma_z$. Details on the production of control solutions for this generic single-qubit Hamiltonian are presented in Single-Qubit Driven-Control Formats.

## Two-Qubit Parametric Control Hamiltonian

Parametrically-driven two-qubit gates may be implemented between two capacitively-coupled transmon qubits consisting of one fixed- and one tunable-frequency transmon. A control flux drive $\Phi(t)$ is applied to the tunable-frequency transmon. This modulates the transition frequency $\omega_{T}(t)$ of the tuneable qubit and, via the capacitive coupling, generates a modulated effective two-qubit coupling. Target two-qubit gates are then driven by tuning the flux modulation resonantly with the desired transition. A detailed description of the underlying physical system and the derivation of the associated Hamiltonians can be found in Didier, 2017 , Caldwell, 2018 , and Reagor, 2018 . Key relationships are summarized as follows.

### Overview of Physical System

A flux drive $\Phi(t)$ with frequency $\omega_{p}$ and phase offset $\theta_{p}$ parametrically modulates the transition frequency of the tunable-frequency qubit, ideally resulting in the modulation

where is the average shift in qubit frequency and is the amplitude of the modulation caused by the applied flux drive. The Hamiltonian for the system under this modulation, transforming to an interaction picture, takes the form

Here $g(t)$ describes the capacitive coupling between the transmon qubits; $\eta_{T}(\eta_{F}$) are the positively-defined anharmonicities for the tunable-frequency (fixed-frequency) transmons; is the detuning between the average transition frequency of the tunable-frequency qubit and the fixed transition frequency of the fixed-frequency qubit; and $J_{n}(x)$ is the $n$th-order Bessel function of the first kind. For typical experimental parameters, the time-dependent phase factors on the Hamiltonian operators above lead to rapidly-oscillating terms in the system evolution that effectively suppress the coupling rate to the associated transitions. Activation of a target transition is achieved by resonantly tuning the drive frequency to cancel the associated phase factor. In particular:

### iSWAP Subspace

The iSWAP interaction is turned on by resonantly driving the term, for example by setting the 1st-order ($n=1$) resonance condition $\omega_{p} = \Delta/2$. In this case the remaining rapidly-oscillating terms may be ignored, and $H_{int}(t)$ reduces to the iSWAP control Hamiltonian

where the parametric coupling rate and the parametric drive phase .

Single-qubit controls in the the two-qubit space may be incorporated via additional Hamiltonian terms. For example controls on qubit 1 are simply expressed via the control Hamiltonian where is any single-qubit control Hamiltonian on the 2-dimensional Hilbert space of qubit 1, and $\mathbb{I}$ is the identity associated with the 2-dimensional Hilbert space of qubit 2. In particular, a resonant drive on qubit 1 takes the form

The full control Hamiltonian, , including both iSWAP and single-qubit controls is therefore compactly expressed

where

with shift operator counterparts

and where we have restricted attention to the relevant iSWAP subspace, spanned by the eigenstates

These iSWAP and single-qubit pulses yield the control parameters

Our formulation assumes these relationships are calibrated for a particular system and the effective control operators may be implemented.

## Mølmer-Sørensen Control Hamiltonian

The Mølmer-Sørensen drive implements entangling gates between two trapped-ion qubits. An entangling phase is generated by engineering effective spin-spin couplings, mediated by shared motional modes, after irradiating the ions with a bichromatic laser. Phase- and amplitude-modulated qubit controls may be used to implement robust entangling gates, bringing the advantage of improved performance against relevant noise sources. A detailed description of the underlying physical system and the derivation of the associated Hamiltonians can be found in Milne, 2017 . Key relationships are summarized as follows. For a system with $N$ ions and $M$ motional modes the Mølmer-Sørensen Hamiltonian takes the form ($\hbar=1$)

where is the spin operator for the $\mu$th ion (embedded in the $N$-ion Hilbert space); () are the annihilation (creation) operators for the $k$th mode (embedded in the $M$-mode Hilbert space); and is a time-dependent scalar function defined by

Here $\Omega(t)$ and $\phi(t)$ are the Rabi rate and drive phase for resonantly-driven qubit transitions; $\delta_{k}$ is a laser detuning parameter referenced from the $k$th mode; and $\eta_{k}^{(\mu)}$ is Lamb-Dicke parameter capturing the mode-laser coupling. The mapping between these control variables and the associated physical parameters is described below. Thus $\kappa_{k}^{(\mu)}(t)$ captures both the global qubit controls and mode-laser interactions. With some slight rearrangement this may be cast in the standard form used at Q-CTRL, resulting in the control Hamiltonian

where the complex pulse $\gamma_\text{MS}(t)$ and drive operator $C_\text{MS}$ are defined as

The control parameters for Mølmer-Sørensen drives are therefore defined

Here the parameters $I(t)$ and $Q(t)$ are the real control pulses for shift operators terms associated with the position and momentum quadratures in phase space.

### Physical Parameters

Here we define the relationship between the control parameters appearing in the Mølmer-Sørensen control Hamiltonian, and the physical parameters associated with underlying ion-laser interactions. These details are presented with a view to clarifying the notation and (importantly) sign conventions used above.

Qubit transitions
are mediated by a virtual level
in a two-photon Raman transition, driven by two separate Raman lasers (R1 and R2) with distinct frequencies, wavevectors and phases. We assume the convention R1 drives the higher-energy transition $|0\rangle\leftrightarrow |v\rangle$ and R2 nominally drives the lower-energy transition $|1\rangle\leftrightarrow |v\rangle$. That is, the laser frequencies satisfy $\omega^\text{(R1)}>\omega^\text{(R2)}$. Spin-motional coupling is engineered by simultaneously driving red-sideband (RSB) and blue-sideband (BSB) transitions in the dressed qubit-oscillator picture. This may be done by replacing the nominal carrier frequency of R2 with a bichromat beam with frequency components $\omega^\text{(R2)}\pm\delta^\text{(SB)}$ where $\delta^{(\text{SB})}$ is the absolute value of the bichromat frequency detunings from $\omega^\text{(R2)}$. This may be done by adding sidebands (SB) to R2 and suppressing the component at the carrier frequency. We refer to $\delta^{(\text{SB})}$ simply as the *laser detuning*. Below we identify the relevant laser frequencies, transition frequencies, and detuning parameters. These are always measured in angular frequency units.

To engineer Mølmer-Sørensen interactions the nominal carrier detuning $\delta^{(c)}$ is tuned resonantly with the qubit transition frequency, though the spectral component of R2 at the carrier frequency is suppressed, such that only the $(\pm)$ bichromat components drive transitions. The RSB transition is driven by the $(-)$ component of the bichromat laser, red-detuned from $\omega_{r,k}$ by an additional amount $\delta_{k}$; and the BSB transition is driven by the $(+)$ component of the bichromat laser, blue-detuned from $\omega_{b,k}$ by the same amount $\delta_{k}$. Thus

Substituting in the expressions for $\delta^{(\pm)}$, $\omega_{r,k}$ and $\omega_{b,k}$ defined above, and using the carrier-resonance condition $\delta^{(c)} = \omega_\text{qubit}$, we obtain

The bichromat phases set the basis for engineering spin-spin coupling, and also govern the drive phase in the complex control pulse. We define:

The spin-motional coupling also requires momentum exchange in the ion-laser interactions. This is imparted by the difference wavevector, simply referred to as the *laser wavevector*, due to intersecting the two Raman beams. We define:

where the $\hat{n}$ are the unit vectors describing the direction of the wave vectors. For an optically driven Raman transition to $|v\rangle$, and a microwave qubit, the relative detunings are very small: $\delta^{(c)}/\omega^\text{(R1)}\ll1$ and $\delta^{(c)}/\omega^\text{(R2)}\ll1$. In this case the effective wavelength is well approximated as $\lambda\approx\lambda^\text{(R1)}\approx\lambda^\text{(R2)}$.