# Error channels in quantum nondemolition measurements on spin systems

Benjamin Joecker  Centre for Quantum Computation and Communication Technology,
School of Electrical Engineering & Telecommunications, UNSW Sydney, NSW, 2052,
Australia  Holly G. Stemp  Centre for Quantum Computation and Communication
Technology, School of Electrical Engineering & Telecommunications, UNSW
Sydney, NSW, 2052, Australia  Irene Fernández de Fuentes  Centre for Quantum
Computation and Communication Technology, School of Electrical Engineering &
Telecommunications, UNSW Sydney, NSW, 2052, Australia  Mark A. I. Johnson
Centre for Quantum Computation and Communication Technology, School of
Electrical Engineering & Telecommunications, UNSW Sydney, NSW, 2052, Australia
Andrea Morello  Centre for Quantum Computation and Communication Technology,
School of Electrical Engineering & Telecommunications, UNSW Sydney, NSW, 2052,
Australia

###### Abstract

Quantum nondemolition (QND) measurements are a precious resource for quantum
information processing. Repetitive QND measurements can boost the fidelity of
qubit preparation and measurement, even when the underlying single-shot
measurements are of low fidelity. However, this fidelity boost is limited by
the degree in which the physical system allows for a truly QND process –
slight deviations from ideal QND measurement result in bit flip errors
(‘quantum jumps’) if the measurement is repeated too often. Here, we develop a
theoretical framework to understand and quantify the resulting error arising
from deviation from perfect QND measurement in model spin qubit systems. We
first develop our model on the ubiquitous example of exchange-coupled electron
spins qubits tunnel-coupled to a charge reservoir. We then extend it to
electron-nuclear spin systems, to illustrate the crucial similarities and
differences between the two limits. Applied to the well-understood platform of
a donor nuclear spin in silicon, the model shows excellent agreement with
experiments. For added generality, we conclude the work by considering the
effect of anisotropic spin couplings.

##  I Introduction

The measurement postulate of quantum mechanics is usually described as
follows: Upon measuring some physical quantity $\mathcal{A}$, described by a
Hermitian operator $\hat{A}$, the outcome of the measurement can only be one
of the eigenvalues $a_{n}$ of $\hat{A}$; immediately after a measurement has
occurred, the system will be found in the eigenstate $\ket{\phi_{n}}$
associated with the eigenvalue $a_{n}$ [1]. Despite being found in all
textbooks, the second part of the postulate does not describe all (or even the
majority of) practical situations. For example, measuring the presence or
absence of a photon can destroy the photon completely – if a photon is
registered by the detector, we know that one photon existed before the
measurement, but after the measurement it no longer does. A similar scenario
is found in the measurement of electron spins in semiconductors, via the
mechanism of energy-dependent tunneling [2, 3, 4]. A high-energy electron
(spin-up, in materials with positive Landé $g$-factor) can tunnel into a cold
charge reservoir, leaving an imprint in a nearby charge sensor. After the
measurement, however, the spin-up electron is entirely lost – all we are left
with is the knowledge that the now-lost electron was in the spin-up state.

The textbook example, where the system remains in the post-measurement state
corresponding to the observed eigenvalue, describes what is otherwise known as
a quantum nondemolition (QND) measurement[5]. Its key property is that, since
the first measurement with outcome $a_{n}$ projects the system in the
eigenstate $\ket{\phi_{n}}$ of the observable under study, every subsequent
measurement will return the same outcome $a_{n}$ with certainty. In realistic
experiments, where there may be noise affecting the apparatus and reducing the
single-shot fidelity, the ability to perform repetitive QND measurements and
averaging over the results can greatly enhance the overall fidelity of the
outcome[6, 7, 8, 9, 10]. A high-fidelity QND readout enables equally reliable
state preparation[11]. Other applications of robust QND measurements include
quantum error correction[12], entanglement by measurement [13], or observation
of the quantum Zeno effect[14, 15].

The experimental realization of a QND measurement usually consists of three
components: (i) the quantum system of interest (here assumed to be a qubit),
described by the Hamiltonian $\hat{H}_{\rm Q}$, (ii) an ancillary quantum
system, $\hat{H}_{\rm A}$, which can be read out (destructively or otherwise),
and (iii) a coupling between the two systems, $\hat{H}_{\rm C}$. The condition
a measurement must fulfill to behave in QND measurement manner is that
$\hat{H}_{\rm Q}$ commutes with the interaction $\hat{H}_{\rm C}$[5, 9]

$\displaystyle\left[\hat{H}_{\rm C},\hat{H}_{\rm Q}\right]=0.$ (1)

A faithful implementation of this condition can be achieved in the context of
cavity quantum electrodynamics (cQED), by coupling the qubit to photons in a
high-quality resonator [16, 17]. In the dispersive limit, the qubit state only
shifts the cavity frequency, which is measurable without affecting the qubit
state itself[18]. A wide range of QND experiments in solid-state cQED setups
have been demonstrated on superconducting [6, 7] and spin-based[19, 20]
qubits. The cQED architecture has been extremely successful in enabling
intermediate-scale quantum computation [21, 22], but requires a large
footprint due to the size of the resonator.

Here, we study an approach to realize a QND readout, where both the qubit and
the ancillary system are spins hosted in a solid-state device. In the context
of semiconductor spin qubits, this approach has the benefit of retaining the
small footprint and high qubit density that is typical of such platforms[23].
This kind of QND measurement has been experimentally demonstrated with
electrons in quantum dots[24, 25, 11], and in various electron-nuclear spin
systems including Nitrogen-Vacancy (NV) centers in diamond[26, 8, 27, 28, 29,
30], donors in silicon[9, 10, 31], and a quantum dot electron spin coupled to
a nuclear spin ensemble[32].

An ideal QND measurement would require the interaction between the qubit and
the ancilla to be of Ising type, or a similar interaction that fulfills Eq.
(1). Unfortunately, the native coupling mechanism in semiconductor spin
systems is either the Heisenberg exchange (between pairs of electron spins) or
the hyperfine interaction (between an electron and a nuclear spin). These do
not commute with the Hamiltonian of the qubit, and thus violate the QND
condition in Eq. (1). During the QND protocol, these interactions repeatedly
weakly entangle the qubit to the ancilla, which in turn is measured
projectively. This process can lead to unwanted flips of the data qubit, and
constitutes an error channel for the QND readout.

The goal of this paper is to develop a general theoretical framework to
understand the error channels in this type of QND measurements, and quantify
the corresponding error rates. The paper is organized as follows. In section 2
we review how to perform QND measurements on the idealized example of two
Ising exchange-coupled spins, and establish theoretical models describing the
relevant processes. Section 3 discusses error channels in a realistic system
of two Heisenberg exchange-coupled spins. Section 4 applies our models to
donor spin systems, covering both exchange-coupled donor electrons and a
nuclear spin hyperfine-coupled to a bound electron, in the isotropic and the
anisotropic case. Section 5 concludes and summarizes the results.

##  II Ising exchange: Ideal QND readout scheme

Figure 1: Illustration of the quantum nondemolition (QND) readout scheme of an
electron spin qubit. (a) The model under consideration. A data qubit (with
Hamiltonian $\hat{H}_{\rm D}$) is coupled to an ancilla ($\hat{H}_{\rm A}$)
via a coupling $\hat{H}_{\rm C}$. The ancilla is tunnel-coupled ($\hat{H}_{\rm
T}$) to a lead. The state of the ancilla can be flipped by driving a spin
resonance (SR) antenna ($\hat{H}_{\rm SR}$). (b) The energy levels of the two-
particle (top) and one-particle (bottom) states are split by the ancilla on-
site energy and the mutual charging energy with the data qubit. For Ising-
coupled spins (Eq. (5)) $\hat{H}_{\rm T}$ only allows the indicated
transitions (Eq. (10)). (c) Exemplary Fermi distribution of the lead with
respect to the chemical potentials from (b). Placing the Fermi energy of the
lead between the $\uparrow_{\rm A}$ and $\downarrow_{\rm A}$ transitions
allows for initialization (Load) and readout of the ancilla (Read) via spin-
dependent tunneling from and to the lead. (d) Going from left to right
illustrates the different stages of the QND measurement, where the blue
(green) dot indicates the state evolution in case of a $\uparrow_{\rm D}$
($\downarrow_{\rm D}$). The flip of the ancilla can be conditional on the
$\downarrow_{\rm D}$ (CR$(\downarrow_{\rm D})$) or $\uparrow_{\rm D}$
(CR$(\uparrow_{\rm D})$).

The setup needed to perform the QND measurement scheme is shown in Fig. 1a. A
data qubit, described by the Hamiltonian $\hat{H}_{\rm D}$, is connected to an
ancilla ($\hat{H}_{\rm A}$) via a coupling $\hat{H}_{\rm C}$. The ancilla in
turn is tunnel-coupled ($\hat{H}_{\rm T}$) to a lead, i.e. a cold reservoir of
electrons, for the purpose of performing spin readout based on energy-
dependent tunneling[2, 3, 4]. The same mechanism ensures that, after readout,
the ancilla is reset to the low-energy spin state. The ancilla spin can be
controlled via spin resonance ($\hat{H}_{\rm SR}$). In the following, we
establish the model for the data-ancilla system and treat $\hat{H}_{\rm T}$ as
a perturbation to determine tunnel rates from and to the lead.

The data qubit and the ancilla are described by

$\displaystyle\hat{H}_{\rm D}=\sum_{\sigma}(E_{\rm D}+\epsilon_{\rm
D}\sigma)\hat{d}_{\sigma}^{\dagger}\hat{d}_{\sigma}$ (2)
$\displaystyle\hat{H}_{\rm A}=\sum_{\sigma}(E_{\rm A}+\epsilon_{\rm
A}\sigma)\hat{a}_{\sigma}^{\dagger}\hat{a}_{\sigma}$ (3)

where $\hat{d}_{\sigma}^{\dagger}$ ($\hat{a}_{\sigma}^{\dagger}$) creates a
particle with spin $\sigma\in\\{\uparrow,\downarrow\\}$ on the data (ancilla).
$E_{\rm D/A}$ is the on-site energy and $\epsilon_{\rm D/A}$ the corresponding
spin splitting. We assume that $\epsilon_{\rm D}$ and $\epsilon_{\rm A}$ are
close in value but sufficiently different to ensure that both spins can be
individually addressed via spin resonance. This is easily achieved in quantum
dot systems by using gradient magnetic fields from micromagnets[33] or
exploiting different $g$-factors between dots[34]; in donor systems, it can be
achieved by orienting the donors’ nuclear spins in opposite directions[35,
36]. Since the data qubit electron is never removed, in the following we may
drop the constant energy offset $E_{\rm D}$. Allowing no more than single
particle occupation, the Ising coupling between two spins is described by

$\displaystyle\hat{H}_{\rm C}=\sum_{\sigma,\sigma^{\prime}}(E_{\rm C}+J_{\rm
I}\sigma\sigma^{\prime})\hat{n}_{\rm a,\sigma}\hat{n}_{\rm
d,\sigma^{\prime}},$ (4)

where $\hat{n}_{\rm a,\sigma}=\hat{a}_{\sigma}^{\dagger}\hat{a}_{\sigma}$ is
the number operator for an ancilla with spin $\sigma$ and $\hat{n}_{\rm
d,\sigma}$ the equivalent for the data. $E_{\rm C}$ is the mutual charging
energy and $J_{\rm I}$ the magnitude of the Ising spin-coupling. Throughout
this paper we express energies in units of frequency by dividing by the Planck
constant $h$. The resulting energy levels are sketched in Fig. 1b for $E_{\rm
C}+E_{\rm A}\gg\epsilon_{\rm A}\gtrsim\epsilon_{\rm D}\gg J_{\rm I}$. The
energy splitting between the center of the two-particle (top) and one-particle
(bottom) states is given by $E_{\rm C}+E_{\rm A}$.

From the point of view of the data-ancilla system, the tunneling of a particle
to (from) the lead is the annihilation (creation) of an ancilla particle. A
full picture including an explicit treatment of the lead can be found in App.
A. Assuming the tunnel coupling $t_{0}$ is constant over the relevant energy
range, i.e. the tunnel barrier is not changing much with energy (E), the rate
of a transition between the one-particle (1P) and two-particle (2P) states
scales with $|\braket{{\rm 1P}}{\sum_{\sigma}t_{0}\hat{a}_{\sigma}}{{\rm
2P}}|^{2}$ (see App. A). The 1P states are simply
$\ket{\uparrow_{\textrm{D}}}$ and $\ket{\downarrow_{\textrm{D}}}$, while the
2P states can be easily found by solving the effective spin Hamiltonian in the
two-particle subspace

$\displaystyle\hat{H}_{\rm A}+\hat{H}_{\rm D}+\hat{H}_{\rm C}=\epsilon_{\rm
A}\hat{S}_{\textrm{A},z}+\epsilon_{\rm D}\hat{S}_{\textrm{D},z}+J_{\rm
I}\hat{S}_{\textrm{A},z}\hat{S}_{\textrm{D},z}$ (5)

Here, $\hat{S}_{\textrm{A/D},z}$ are the spin operators of the ancilla/data
qubit in the basis
$\\{\ket{\uparrow_{\textrm{A/D}}},\ket{\downarrow_{\textrm{A/D}}}\\}$. Since
all terms in Eq. (5) commute, the eigenstates of the data-ancilla system are
separable, i.e. there is no entanglement between the two spins. The only
allowed transitions are those that preserve the spin of the data qubit
(colored arrows in Fig. 1b) as required in a perfect QND measurement.

The chemical potentials $\mu_{{\rm 2P}\leftrightarrow{\rm 1P}}$, i.e. the
energy change of the system that the ancilla particle carries as it tunnels
into the lead, are indicated by the length of the colored arrows in Fig. 1b.
Energetically, a particle can only tunnel off (on) the ancilla, if a state at
this energy is unoccupied (occupied) in the lead. To quantify this, we treat
the lead as a continuum of states with density $n(E)$. The fraction of states
in the lead occupied at energy $E$ for a given temperature $T$ is given by the
Fermi distribution,

$\displaystyle f(E)=\frac{1}{1+e^{(E-\mu_{\rm L})/k_{\rm B}T}},$ (6)

where $\mu_{\rm L}$ is the chemical potential of the lead and $k_{\rm B}$
Boltzmann’s constant (in units of Hz/K, given our choice of units for the
energy). The full transition rates at which a particle can tunnel between the
lead and the ancilla are then given by the golden rule[37, 38, 39, 40]

$\displaystyle\Gamma^{\rm in}_{\rm 1P\rightarrow 2P}$
$\displaystyle=|\braket{{\rm 1P}}{\sum_{\sigma}t_{0}\hat{a}_{\sigma}}{{\rm
2P}}|^{2}n(\mu_{{\rm 1P}\leftrightarrow{\rm 2P}})f(\mu_{\rm 1P\leftrightarrow
2P})$ (7a) $\displaystyle\Gamma^{\rm out}_{\rm 2P\rightarrow 1P}$
$\displaystyle=|\braket{{\rm
2P}}{\sum_{\sigma}t_{0}\hat{a}^{\dagger}_{\sigma}}{{\rm 1P}}|^{2}n(\mu_{{\rm
1P}\leftrightarrow{\rm 2P}})(1-f(\mu_{\rm 1P\leftrightarrow 2P})).$ (7b)

Here, Eq. (7a) describes the tunneling of an electron onto the ancilla while
Eq. (7b) is the inverse process. Fig. 1c shows an exemplary Fermi distribution
with respect to the transition energies to the right. Electrostatically tuning
the lead with respect to the ancilla, i.e. tuning $\mu_{\rm 2P\leftrightarrow
1P}-\mu_{\rm L}$ in Eq. (6), shifts the ratio of occupied and empty states. A
transition in tune with occupied states (dark grey shade) will result in
$\Gamma^{\rm in}_{\rm 1P\rightarrow 2P}\gg\Gamma^{\rm out}_{\rm 2P\rightarrow
1P}$, while a transition in tune with empty states (light grey shade) leads to
$\Gamma^{\rm out}_{\rm 2P\rightarrow 1P}\gg\Gamma^{\rm in}_{\rm 1P\rightarrow
2P}$.

Finally, the spin resonance antenna[41] depicted on the right in Fig. 1a
provides an oscillating magnetic field $B_{1}$ that drives spin resonance
transitions via the Hamiltonian $\hat{H}_{\rm SR}=\gamma B_{1}\cos{(2\pi\nu
t)}(\hat{S}_{\textrm{A},x}+\hat{S}_{\textrm{D},x})$, where $\nu$ is a
frequency corresponding to one of the conditional rotations (CR) shown in Fig.
1d and $\gamma$ the gyromagentic ratio of the spin (in units of Hz/T).

The QND measurement scheme now follows the steps outlined in Fig. 1d. At the
start, a previous QND measurement has left the data qubit either in the
$\uparrow$ or $\downarrow$ state, marked by the blue and green circles. In the
first step, we load a $\downarrow$ electron onto the ancilla by tuning the
corresponding chemical potential below that of the lead, while keeping the
$\uparrow$ chemical potential well above. Subsequently, we tune (‘plunge’) to
a regime where tunnel-off events are suppressed, and perform a $\pi$-pulse on
the ancilla conditional on the state of the data qubit. Here, we can choose to
either drive a CR dependent on the data qubit being $\downarrow$
(CR$(\downarrow_{\rm D})$) or $\uparrow$ (CR$(\uparrow_{\rm D})$), or
alternating between the two (CR$(\updownarrow_{\rm D})$). Finally, we readout
the ancilla spin by pulsing back to the load/read configuration, where now
only an $\uparrow$ ancilla will cause a tunnel event, detectable by a charge
sensor. In the CR$(\downarrow_{\rm D})$ case, a tunnel event confirms that the
data qubit was previously in the $\downarrow$ state; in the CR$(\uparrow_{\rm
D})$ case, finding a $\uparrow$ ancilla is associated with a $\uparrow$ data
qubit. Note that if no tunnel event occurs, we have already performed the load
stage for the next QND measurement, while if a tunnel event occurs a
$\downarrow$ electron will automatically be reloaded onto the ancilla if we
wait for a sufficiently long time at the load/read tuning.

The readout protocol with Ising-coupled spins described above can suffer from
various imperfections, such as state preparation and measurement errors on the
ancilla due to the nonzero temperature of the lead[39], or imperfect
$\pi$-pulses in the implementation of the conditional rotations. Fortunately,
such errors preserve the QND nature of the measurement and leave the data
qubit unaffected. Therefore, it is possible to repeat the QND measurement
cycle multiple times to increase the confidence in determining the qubit’s
state[8, 9, 24, 25].

##  III Heisenberg exchange: Breakdown of the QND condition

Figure 2: Error channels in the QND measurement for Heisenberg exchange
coupled spins. (a) Energy levels of the two-particle (top) and one-particle
(bottom) states. For exchange-coupled spins (Eq. (8)) the antiparallel spin
states are hybridized (Eq. (9)) and $\hat{H}_{\rm T}$ allows the additional
transitions indicated by the yellow and green double arrow (Eq. (10)). (b)
Chemical potentials with respect to an exemplary Fermi distribution in the
lead for the read/load tuning. The resulting rates in Eq. (11) belonging to an
up ($\mu_{\uparrow}$) or down ($\mu_{\downarrow}$) spin particle tunneling in
or out are indicated. (c) Simulations of the QND measurement with an initial
$\uparrow_{\rm D}$, and driving a conditional rotation on the ancilla
dependent on the data being $\uparrow_{\rm D}$ (CR$(\uparrow_{\rm D})$) for
$T=$0\text{\,}$$ (Eq. (41)) with $\Gamma^{\rm in}_{\downarrow}=\Gamma^{\rm
out}_{\uparrow}=\Gamma_{0}$, and $s^{2}=$2.5\text{\times}{10}^{-3}\text{\,}$$.
The first two panels show the state distribution as a function of time during
the first two read/load periods. The last panel gives the final distribution
after the Nth repetition, i.e. the last point of each QND step (indicated by
the small dots in the first two panels) spaced by $t_{\rm R/L}=5/\Gamma_{0}$.
In total $N=1000$ QND cycles are simulated. (d) Probability of finding the
data in a $\uparrow_{\rm D}$ state (Eq. (17)) after repeated QND measurements
as a function of time on a logarithmic scale. The cases of driving the
CR$(\uparrow_{\rm D})$, CR$(\downarrow_{\rm D})$, and alternating between the
two (CR$(\updownarrow_{\rm D})$) are shown. The case of CR$(\uparrow_{\rm D})$
and an initial $\uparrow_{\rm D}$ discussed in (c) is highlighted by the black
dotted line. The data qubit flipping rates on the right are (in multiples of
the bare tunnel rate $\Gamma_{0}$) extracted by fitting Eq. (46). The
evolution of the data qubit is well described by these rates, independently of
its initial state. In the $T=0$ limit only an initial $\uparrow_{\rm D}$ can
be flipped. (e) and (f) are the same as (c) and (d), but for $T>0$, i.e.
$\Gamma^{\rm in}_{\downarrow}=\Gamma^{\rm out}_{\uparrow}=\Gamma_{0}(1-f)$ and
$\Gamma^{\rm in}_{\uparrow}=\Gamma^{\rm out}_{\downarrow}=\Gamma_{0}f$ with
$f=0.03$. (e) Evolution of an initial $\downarrow_{\rm D}$, using
CR$(\downarrow_{\rm D})$. (f) In the $T>0$ case, an initial $\downarrow_{\rm
D}$ can be flipped as well.

The idealized QND measurement described in the previous section relied upon an
Ising-type interaction between the spins. In the near-totality of practical
applications, however, the true interaction takes the form of Heisenberg
exchange:

$\displaystyle\hat{H}_{\rm A}+\hat{H}_{\rm D}+\hat{H}_{\rm C}=\epsilon_{\rm
A}\hat{S}_{\textrm{A},z}+\epsilon_{\rm D}\hat{S}_{\textrm{D},z}+J_{\rm
H}\vec{S}_{\textrm{A}}\cdot\vec{S}_{\textrm{D}},$ (8)

where $J_{\rm H}$ is the amplitude of the Heisenberg exchange interaction and
$\vec{S}_{\textrm{A/D}}=(\hat{S}_{\textrm{A/D},x},\hat{S}_{\textrm{A/D},y},\hat{S}_{\textrm{A/D},z})$
the vector of spin operators. The Heisenberg interaction Hamiltonian contains
terms that do not commute with the data qubit Hamiltonian, and therefore
violates the QND condition Eq. (1).

Fig. 2a shows the 1P and 2P energy level in this case for $\epsilon_{\rm
A}\gtrsim\epsilon_{\rm D}\gg J_{\rm H}$. Crucially, the exchange coupling now
weakly entangles the antiparallel spin states, resulting in eigenstates of the
form

$\displaystyle\ket{\widetilde{\uparrow_{\rm A}\downarrow_{\rm D}}}$
$\displaystyle=c\ket{\uparrow_{\rm A}\downarrow_{\rm D}}-s\ket{\downarrow_{\rm
A}\uparrow_{\rm D}}$ (9a) $\displaystyle\ket{\widetilde{\downarrow_{\rm
A}\uparrow_{\rm D}}}$ $\displaystyle=c\ket{\downarrow_{\rm A}\uparrow_{\rm
D}}+s\ket{\uparrow_{\rm A}\downarrow_{\rm D}},$ (9b)

where $c=\cos(\theta)$ and $s=\sin(\theta)$ with $\tan(2\theta)=J_{\rm
H}/\Delta\epsilon$ and $\Delta\epsilon=\epsilon_{\rm A}-\epsilon_{\rm D}$.
This hybridization enables additional spin transitions marked by the double-
headed yellow and green arrows in Fig. 2a associated with flipping the data
qubit spin, i.e. a QND violation. To analyze the probability with which these
flips occur while the ancilla is loaded and read out, we need to find the
associated tunnel rates in Eq. (7). As discussed in the previous section,
these have two main contributions.

First, we have the amplitudes of the transition matrix elements coupling the
1P and 2P states, given by

$\displaystyle M_{\rm 1P,2P}=|\braket{{\rm
1P}}{\hat{a}_{\uparrow}+\hat{a}_{\downarrow}}{{\rm 2P}}|^{2}.$ (10)

These can be viewed as a selection rule, as transitions in Fig. 2a are only
allowed if $M_{\rm 1P,2P}$ is nonzero.

Secondly, we have the selection through energy. At the read/load position
shown in Fig. 2b, we now have three high-energy transitions associated with an
$\uparrow$ particle tunneling, and three low-energy transitions where a
$\downarrow$ particle tunnels. Note that, since
$\Delta\epsilon\ll\epsilon_{\rm A},\epsilon_{\rm D}$, transitions where the
data and ancilla are exchanged in the 2P state have roughly the same energy.
For the rest of this section, we will assume that the density of states in the
lead is approximately constant over the relevant energy scale and we may
define a bare tunneling rate $\Gamma_{0}=|t_{0}|^{2}n(\mu)$ with
$n(\mu)=const$. Assuming further that $f(E)$ is approximately the same for all
the three $\uparrow$ as well as all three $\downarrow$ transitions, the
relevant rates are

$\displaystyle\Gamma^{\rm in}_{\uparrow}=\Gamma_{0}f(\mu_{\uparrow})$ (11a)
$\displaystyle\Gamma^{\rm out}_{\uparrow}=\Gamma_{0}(1-f(\mu_{\uparrow}))$
(11b) $\displaystyle\Gamma^{\rm
in}_{\downarrow}=\Gamma_{0}f(\mu_{\downarrow})$ (11c)
$\displaystyle\Gamma^{\rm
out}_{\downarrow}=\Gamma_{0}(1-f(\mu_{\downarrow})),$ (11d)

where $\mu_{\uparrow}$ ($\mu_{\downarrow}$) represents the high (low) energy
chemical potentials. Finally, we also assume that $\mu_{\rm L}$ is perfectly
centered between $\mu_{\uparrow}$ and $\mu_{\downarrow}$, so that
$f=f(\mu_{\uparrow})=1-f(\mu_{\downarrow})$.

###  III.1 Zero-temperature limit

We begin by analyzing the $T=0$ limit, where $f(E)$ is a step function and
only $\Gamma^{\rm in}_{\downarrow}$ and $\Gamma^{\rm out}_{\uparrow}$ are non-
zero at the read/load position. To understand the breakdown of the QND
measurement, we consider the isolated process of a particle being loaded into
the ancilla while the data qubit is $\uparrow_{\rm D}$. The relevant
transitions rates are:

$\displaystyle\Gamma^{\rm in}_{\uparrow_{\rm
D}\rightarrow\widetilde{\uparrow_{\rm A}\downarrow_{\rm D}}}$
$\displaystyle=\Gamma^{\rm in}_{\downarrow}M_{\uparrow_{\rm
D},\widetilde{\uparrow_{\rm A}\downarrow_{\rm D}}}=\Gamma^{\rm
in}_{\downarrow}s^{2},$ (12a) $\displaystyle\Gamma^{\rm in}_{\uparrow_{\rm
D}\rightarrow\widetilde{\downarrow_{\rm A}\uparrow_{\rm D}}}$
$\displaystyle=\Gamma^{\rm in}_{\downarrow}M_{\uparrow_{\rm
D},\widetilde{\downarrow_{\rm A}\uparrow_{\rm D}}}=\Gamma^{\rm
in}_{\downarrow}c^{2}.$ (12b)

The time evolution of the system then obeys the rate equations[39, 40]

$\displaystyle\dot{\vec{\rho}}=\mathcal{L}\vec{\rho},$ (13)

where $\vec{\rho}$ is the vector of the state distribution and $\mathcal{L}$
is the Liouvillian of the system. We first restrict our analysis to the three
states in Eq. (12) and obtain

$\displaystyle\begin{pmatrix}\dot{\rho}_{\widetilde{\uparrow_{\rm
A}\downarrow_{\rm D}}}\\\ \dot{\rho}_{\widetilde{\downarrow_{\rm
A}\uparrow_{\rm D}}}\\\ \dot{\rho}_{\uparrow_{\rm
D}}\end{pmatrix}=\begin{pmatrix}0&0&\Gamma^{\rm in}_{\downarrow}s^{2}\\\
0&0&\Gamma^{\rm in}_{\downarrow}c^{2}\\\ 0&0&-\Gamma^{\rm
in}_{\downarrow}\end{pmatrix}\begin{pmatrix}\rho_{\widetilde{\uparrow_{\rm
A}\downarrow_{\rm D}}}\\\ \rho_{\widetilde{\downarrow_{\rm A}\uparrow_{\rm
D}}}\\\ \rho_{\uparrow_{\rm D}}\end{pmatrix}.$ (14)

For an initial $\rho_{\uparrow_{\rm D}}(0)=1$, we find the probabilities of
the 2P states as

$\displaystyle\rho_{\widetilde{\uparrow_{\rm A}\downarrow_{\rm D}}}(t)=$
$\displaystyle s^{2}\left(1-e^{-\Gamma^{\rm
in}_{\downarrow}t}\right)\xrightarrow{t\rightarrow\infty}s^{2},$ (15a)
$\displaystyle\rho_{\widetilde{\downarrow_{\rm A}\uparrow_{\rm D}}}(t)=$
$\displaystyle c^{2}\left(1-e^{-\Gamma^{\rm
in}_{\downarrow}t}\right)\xrightarrow{t\rightarrow\infty}c^{2}.$ (15b)

After waiting long enough for an electron to tunnel, we have a non-zero
probability to flip the data during the load. For a ratio $J_{\rm
H}/\Delta\epsilon\approx 1/10$, as typically found in experiments[33, 34, 36,
42], the probability is

$\displaystyle s^{2}=\sin^{2}\left(\frac{1}{2}\tan^{-1}\left(\frac{J_{\rm
H}}{\Delta\epsilon}\right)\right)\approx\left(\frac{J_{\rm
H}}{2\Delta\epsilon}\right)^{2}=$2.5\text{\times}{10}^{-3}\text{\,}$.$ (16)

This represents the probability of accidentally flipping the data qubit,
purely as a consequence of it becoming weakly entangled with the ancilla qubit
each time it is loaded. Further errors can arise due to the ancilla electron
tunneling back out, leaving the subspace of states in Eq. (14).

The full QND readout protocol includes repetitive ancilla loading and readout,
and rotation of the ancilla conditional on a specific data qubit state.
Simulating the full protocol requires solving the rate equations including all
states and transitions allowed by Eq. (10) indicated in Fig. 2a. The full
Liouvillian in the zero temperature limit is given in Eq. (41) in App. B. The
QND protocol with an initial $\uparrow_{\rm D}$ data qubit and the
CR$(\uparrow_{D})$ ancilla rotation is described in Fig. 2c. After loading the
$\widetilde{\downarrow_{\rm A}\uparrow_{\rm D}}$ state and performing the
CR$(\uparrow_{D})$ operation the system is in the state $\uparrow_{\rm
A}\uparrow_{\rm D}$. The first panel shows the evolution of the state
distribution as a function of time in the first read/load window. The
$\uparrow_{\rm A}\uparrow_{\rm D}$ probability decreases as the ancilla
tunnels out with rate $\Gamma^{\rm out}_{\uparrow}=\Gamma_{0}$, leading to an
initially increased probability of finding the 1P state $\uparrow_{\rm D}$.
The $\uparrow_{\rm D}$ probability quickly peaks and decreases as the ancilla
is reloaded into the $\widetilde{\downarrow_{\rm A}\uparrow_{\rm D}}$ state.
Here, we chose a read/load window of duration $t_{\rm R/L}=5/\Gamma_{0}$ in
which a $\uparrow_{\rm A}$ ancilla is read out and replaced by a
$\downarrow_{\rm A}$ with a probability of 0.96. Not surprisingly, the shape
of $\rho_{\uparrow_{\rm D}}(t)$ reflects the averaged signal from a charge
sensor detecting an initial $\uparrow_{\rm A}$ electron (see e.g. the
supplementary information of Ref. [4] for an example).

The second panel shows the repetition of this process after the application of
the conditional $\pi$-pulse that swaps the $\widetilde{\downarrow_{\rm
A}\uparrow_{\rm D}}$ and $\uparrow_{\rm A}\uparrow_{\rm D}$ populations. Here,
we notice a small nonzero $\rho_{\downarrow_{\rm A}\downarrow_{\rm D}}$
population from the start, since the data qubit may have already accidentally
flipped. The last panel shows the final state distribution after the Nth
repetition of the QND scheme, where each repetition takes $t=5/\Gamma_{0}$. In
the shown case, the data qubit will eventually flip to $\downarrow_{\rm D}$
and remain in that state.

Fig. 2d shows the probability of finding the data in an $\uparrow_{\rm D}$
state independent of the current state of the ancilla

$\displaystyle P(\uparrow_{\rm D})=\rho_{\uparrow_{\rm A}\uparrow_{\rm
D}}+\rho_{\widetilde{\downarrow_{\rm A}\uparrow_{\rm D}}}+\rho_{\uparrow_{\rm
D}}$ (17)

after the Nth read/load window on a logarithmic scale and for the cases of
driving the CR$(\uparrow_{D})$, CR$(\downarrow_{D})$, and alternating between
the two (CR$(\updownarrow_{D})$). We can see that an initial $P(\uparrow_{\rm
D})=1$ decays exponentially as a function of time. The decay is in fact
fastest when the ancilla rotation is conditioned on the opposite state
(CR$(\downarrow_{D})$) and the system practically idles in the
$\widetilde{\downarrow_{\rm A}\uparrow_{\rm D}}$ state, from which the data
qubit can be flipped via $\widetilde{\downarrow_{\rm A}\uparrow_{\rm
D}}\rightarrow\downarrow_{\rm D}$. In the CR$(\uparrow_{D})$ case, shown in
Fig. 2c, the system spends time in the $\uparrow_{\rm A}\uparrow_{\rm D}$,
which only allows spin-preserving transitions, and the decay is slowed down.
Fitting Eq. (46) to the evolution we extract the flipping rates on the right
of Fig. 2d. An initial $\downarrow_{\rm D}$ data qubit state cannot be flipped
due to the fact that the corresponding transitions require an $\uparrow_{\rm
A}$ ($\downarrow_{\rm A}$) particle to tunnel into (out of) the system. These
transitions are energetically forbidden at zero temperature.

###  III.2 Nonzero temperature

In the $T>0$ case, the Fermi distribution is no longer a step function,
resulting in small but non-zero $\Gamma^{\rm in}_{\uparrow}$ and $\Gamma^{\rm
out}_{\downarrow}$ and the Liouvillian in Eq. (41) needs to be extended by the
elements in Eq. (42). To demonstrate how this introduces flips of a
$\downarrow_{\rm D}$, Fig. 2e shows the same simulations as in Fig. 2c for the
case of driving the CR$(\downarrow_{D})$, an initial $\downarrow_{\rm D}$, and
$T>0$, i.e. $\Gamma^{\rm in}_{\downarrow}=\Gamma^{\rm
out}_{\uparrow}=\Gamma_{0}(1-f)$ and $\Gamma^{\rm in}_{\uparrow}=\Gamma^{\rm
out}_{\downarrow}=\Gamma_{0}f$. We choose $f=0.03$, corresponding to
$\mu_{\uparrow}-\mu_{\downarrow}=2\mu_{B}\times$1\text{\,}\mathrm{T}$\approx$27.97\text{\,}\mathrm{GHz}$$
(for electron spins with Landé $g$-factor $\approx 2$) and
$k_{B}T=k_{B}\times$200\text{\,}\mathrm{mK}$\approx$4\text{\,}\mathrm{GHz}$$.
Here, the read/load process is marginally slower compared to the $T=0$ case,
due to the slight reduction of the fast rates ($\Gamma^{\rm out}_{\uparrow}$
and $\Gamma^{\rm in}_{\downarrow}$) through temperature and the small
probability of accidentally loading a $\uparrow_{\rm A}$ or a loaded
$\downarrow_{\rm A}$ tunneling back out. This reduces the probability to
reinitialize the system to $\downarrow_{\rm A}\downarrow_{\rm D}$ to 0.92,
leaving the ancilla empty with probability 0.07. After repeated measurements,
the probability of finding the data flipped to $\uparrow_{\rm D}$, i.e. the
$\widetilde{\downarrow_{\rm A}\uparrow_{\rm D}}$ state, increases, but then
quickly saturates at $1.2\text{\times}{10}^{-2}\text{\,}$. While a
$\downarrow_{\rm D}$ can now be flipped via processes involving the slow rates
$\Gamma^{\rm in}_{\uparrow}$ and $\Gamma^{\rm out}_{\downarrow}$, the much
faster processes involving $\Gamma^{\rm in}_{\downarrow}$ and $\Gamma^{\rm
out}_{\uparrow}$ will still flip the $\uparrow_{\rm D}$ back down, and the
rate equations converge to an equilibrium state of these two competing
processes.

The $P(\uparrow_{\rm D})$ (see Eq. (17)) as a function of QND repetitions for
$T>0$ in Fig. 2f show that a different equilibrium is reached depending on
whether the CR$(\uparrow_{\rm D})$ or CR$(\downarrow_{\rm D})$ is driven. The
equilibrium ratio $P(\uparrow_{D})/(1-P(\uparrow_{D}))$ equals the ratios of
the $\uparrow_{\rm D}$ and $\downarrow_{\rm D}$ flipping rates (see App. C).
The data qubit flipping rates for $T>0$ on right in Fig. 2f are extracted by
fitting Eq. (46). The $\Gamma_{\downarrow}$ flipping rates are much slower
when driving the CR$(\downarrow_{\rm D})$ transition. This differences can be
understood by looking at the flow chart of the involved processes. In the case
of CR$(\downarrow_{\rm D})$, flipping the data from $\downarrow_{\rm D}$ to
$\uparrow_{\rm D}$ requires a slow and a fast process:

(22)

Conversely, driving CR$(\uparrow_{\rm D})$ requires two slow processes

(25)

which results in a much slower flipping rate. Alternating between the two
(CR$(\updownarrow_{\rm D})$), yields an intermediate situation.

Overall, the above discussion indicates that driving the CR$(\uparrow_{\rm
D})$ transition yields a better QND readout fidelity, and is particularly
effective at protecting the $\downarrow_{\rm D}$ state.

##  IV Application to donor spins and electron-nuclear systems

The previous section assumed the data and ancilla qubits to be exchange-
coupled electron spins in a double quantum dot. In this section we apply the
discussion to the case where the qubits are electron spins bound to donors
atoms in silicon[43, 36], and then extend it to the case where the data qubit
is the donor nuclear spin[9], while the ancilla is the donor-bound
electron[9]. Finally, we extend the latter scenario to the case where the
electron-nuclear hyperfine coupling is anisotropic, which applies to much
wider range of electron-nuclear spin systems, including NV-centers in
diamond[8].

###  IV.1 Exchange-coupled donor electron spins

Figure 3: Example of error channels in the QND measurement of exchange-coupled spins with experimental parameters. (a) Chemical potentials with respect to a Fermi distribution in the lead for an off-centered read/load tuning representing the given set of experimentally-informed spin-dependent tunnel rates from Eq. (11). For an electron in a magnetic field $B_{0}=$1.77\text{\,}\mathrm{T}$$, where $\mu_{\uparrow}-\mu_{\downarrow}=2\mu_{B}\times$1.77\text{\,}\mathrm{T}$\approx$50\text{\,}\mathrm{GHz}$$, the ratios of spin-up and spin-down tunnel rates correspond to a temperature of roughly $600\text{\,}\mathrm{mK}$. (b) Simulations of the QND measurement with an initial $\downarrow_{\rm D}$, and driving a conditional rotation on the ancilla dependent on the data being $\downarrow_{\rm D}$ (CR$(\downarrow_{\rm D})$) for the tunnel rates in (a), and $s^{2}=$2.5\text{\times}{10}^{-3}\text{\,}$$. The first two panels show the state distribution as a function of time during the first two read/load periods. The last panel gives the final distribution after the Nth repetition, i.e. the last point of each QND step spaced by $t_{\rm R/L}=$1\text{\,}\mathrm{ms}$$. (c) Probability of finding the data in an $\uparrow_{\rm D}$ state (Eq. (17)) after repeated QND measurements as a function of time. The cases of driving the CR$(\uparrow_{\rm D})$, CR$(\downarrow_{\rm D})$, and alternating between the two (CR$(\updownarrow_{\rm D})$) are shown. The case of CR$(\downarrow_{\rm D})$ and an initial $\downarrow_{\rm D}$ discussed in (b) is highlighted by the black dotted line. The data qubit flipping rates on the right are extracted by fitting Eq. (46). The evolution of the data qubit is well described by these rates, independently of its initial state. Table 1:  Data qubit flipping rates in the QND measurements using exchange-coupled donor electron spins additionally including a spin relaxation rate $\Gamma^{T_{1}}_{\uparrow}=$1\text{\,}{\mathrm{s}}^{-1}$$. | CR$(\downarrow_{D})$ | CR$(\uparrow_{D})$ | CR$(\updownarrow_{D})$
---|---|---|---
$\Gamma_{\uparrow}$ ($\text{\,}{\mathrm{s}}^{-1}$) | 7.11 | 6.47 | 6.70
$\Gamma_{\downarrow}$ ($\text{\,}{\mathrm{s}}^{-1}$) | 1.10 | 0.18 | 0.78

For the analysis of donor electron spin experiments, we augment the model in
the previous section by relaxing the assumption that the lead has a constant
density of states [44, 45], and its chemical potential is perfectly centered
between the $\mu_{\uparrow}$ and $\mu_{\downarrow}$ transition energies. Fig.
3 describes the repetitive quasi-QND readout protocol, using experimentally-
informed values for the spin-dependent tunneling rates (Eq. (11)) given in
Fig. 3a. These rates are chosen to allow a direct comparison to experiments.
The sketched off-centered tuning of the transition energies with respect to
the Fermi distribution at the readout position can explain the ratio of tunnel
in and out rates, while a non-constant density of states[45] $n(E)$ or tunnel
coupling $t_{0}$ can motivate the differences between spin up and down rates.
For an electron in a field of $B_{0}=$1.77\text{\,}\mathrm{T}$$ as used in
Ref. [9], where
$\mu_{\uparrow}-\mu_{\downarrow}=2\mu_{B}\times$1.77\text{\,}\mathrm{T}$\approx$50\text{\,}\mathrm{GHz}$$,
the ratios of spin-up and spin-down tunnel rates correspond to a temperature
of roughly $600\text{\,}\mathrm{mK}$. This value is seemingly high compared to
the base temperature of dilution refrigerators, but consistent with effective
electron temperatures observed in recent experiments[39].

Fig. 3b shows the simulation of the QND protocol for this set of parameters,
an initial $\downarrow_{\rm D}$, and driving the CR$(\downarrow_{\rm D})$,
i.e. the same as Fig. 2e but with the parameters in Fig. 3a. We can see that
now the initial peak belonging to the 1P state is much smaller, i.e. the
system resides in a 1P state for a much shorter time, due to the fast tunnel-
in rates. Additionally, the smaller ratio of $\Gamma^{\rm
in}_{\downarrow}/\Gamma^{\rm in}_{\uparrow}=2$ decreases the probability to
successfully reinitialize a $\downarrow_{\rm A}$ spin. For the chosen
read/load window of $t_{\rm R/L}$1\text{\,}\mathrm{ms}$$, this probability is
0.82. Looking at the state distribution after the Nth QND repetition, we can
see that the probability of flipping the data from $\downarrow_{\rm D}$ to
$\uparrow_{\rm D}$ is significantly increased compared to Fig. 2e. This is due
to the fact that the higher probability of accidentally loading an
$\uparrow_{\rm A}$ spin increases the likelihood of the $\downarrow_{\rm
D}\rightarrow\widetilde{\downarrow_{\rm A}\uparrow_{\rm D}}$ process. In the
equilibrium state, the probability of finding the data in a $\downarrow_{\rm
D}$ state (see Eq. (17)) is 0.13.

Plotting this quantity on a linear scale for all six considered cases, i.e.
initial $\uparrow_{\rm D}$ or $\downarrow_{\rm D}$ with CR$(\uparrow_{\rm
D})$, CR$(\downarrow_{\rm D})$, or CR$(\updownarrow_{\rm D})$, in Fig. 3c, we
recognize the same features as in Fig. 2f. However now the equilibrium is
shifted more towards $\uparrow_{\rm D}$ states. The data qubit flipping rates
on right in Fig. 3c are extracted by fitting Eq. (46). In comparison to the
natural spin relaxation rate
$\Gamma^{T_{1}}_{\uparrow}\sim$1\text{\,}{\mathrm{s}}^{-1}$$[46, 47] of an
donor-electron spin, the imperfections of the QND measurements are the
dominant process. We verify that including the relaxation process in our
simulations (see Eq. (43)) has but the effect to slightly accelerate the
$\Gamma_{\downarrow}$ rates. The precise results are given in Tab. 1.

###  IV.2 Nuclear spin hyperfine-coupled to a bound electron

Figure 4: Error channels in the QND measurement of a nuclear spin via the
hyperfine coupling to an electron, based on the example of a 31P atom with
nuclear spin $I=1/2$. (a) The energy levels of Eq. (26) are reordered compared
to Fig. 2a, since the nuclear spin energy splitting is much smaller than that
of the electron: $\epsilon_{\rm e}\gg A_{\rm I}\gtrsim|\epsilon_{\rm n}|$,
with $\epsilon_{\rm n}<0$. This gives a clear energetic distinction between
allowed transitions (Eq. (10)) of an $\uparrow$ or $\downarrow$ electron
tunneling. (b) Transition energies with respect to an exemplary Fermi
distribution in the lead for the read/load tuning. The resulting rates in Eq.
(11) belonging to an $\uparrow$ or $\downarrow$ electron tunneling in or out
are indicated. (c) Simulations of the QND measurement for $T>0$ (Eq. (51) and
Eq. (52)) with the rates in Fig. 3a, $s^{2}\approx 10^{-6}$, and a read/load
time of $t_{\rm R/L}=$1\text{\,}\mathrm{ms}$$. Shown is the probability of
finding the nucleus in a $\Uparrow$ state (Eq. (17)) after repeated QND
measurements as a function of time. The cases of driving the CR$(\Downarrow)$,
CR$(\Uparrow)$ alternating between the two CR$(\Updownarrow)$ are considered
for an initial $\Uparrow$ and $\Downarrow$ state. The nuclear flipping rates
on the right are extracted by fitting Eq. (46). (d) To reproduce the nuclear
flipping rates in Ref. [9] the $T>0$ model in (c) is augmented by including a
flip-flop relaxation rate $\Gamma^{\rm
ff}=$53.3\text{\times}{10}^{-3}\text{\,}{\mathrm{s}}^{-1}$$ (Eq. (53)).

The case of two exchange-coupled electrons considered so far, where the
individual qubit energy splittings are only slightly different (the case
$\epsilon_{\rm A}=\epsilon_{\rm D}$ is discussed in App. D), $\epsilon_{\rm
A}\gtrsim\epsilon_{\rm D}$, makes the system prone to errors because (i) the
small $\Delta\epsilon$ leads to a substantial hybridization of the
$\uparrow_{\rm D}\downarrow_{\rm A},\downarrow_{\rm D}\uparrow_{\rm A}$
states, and (ii) flip-flop transitions between data and ancilla qubits are
almost energy-conserving. In this section, we discuss the case $\epsilon_{\rm
A}\gg\epsilon_{\rm D}$, which is representative of a data qubit encoded in a
donor nuclear spin[9], read out via an electron spin ancilla, coupled to the
nucleus via Fermi contact hyperfine interaction.

Mathematically, this system is similar to Heisenberg exchange-coupled spins
(Eq. (8)), since the isotropic Fermi contact hyperfine interaction takes the
same Hamiltonian form:

$\displaystyle\hat{H}=\epsilon_{\rm e}\hat{S}_{z}+\epsilon_{\rm
n}\hat{I}_{z}+A_{\rm I}\vec{S}\cdot\vec{I},$ (26)

where $A_{\rm I}$ is the hyperfine coupling strength and
$\vec{I}=(\hat{I}_{x},\hat{I}_{y},\hat{I}_{z})$ is the vector of nuclear spin
operators in the basis $\\{\ket{\Uparrow},\ket{\Downarrow}\\}$ (we treat here
the simple case where the donor is a 31P atom with nuclear spin $I=1/2$). The
crucial difference is that the energy splitting of the ancilla electron is
much larger than that of the data nucleus due to their vastly different
gyromagnetic ratios[9], $|\gamma_{\rm e}/\gamma_{\rm n}B_{0}|>10^{3}$. With
$\epsilon_{\rm e}\gg A_{\rm I}\gtrsim|\epsilon_{\rm n}|$ ($\epsilon_{\rm n}<0$
for a 31P atom), the $\uparrow$ states are energetically well separated from
the $\downarrow$ ones in Fig. 4a, while the nuclear splitting is two orders of
magnitude smaller. As a result, the transition energies in Fig. 4b are split
depending on whether a spin up ($\mu_{\uparrow}$) or spin down
($\mu_{\downarrow}$) electron tunnels, independent of the nuclear spin state.
This energy reordering now allows an initial $\Downarrow$ nucleus to be
flipped in the zero-temperature loading process:

$\displaystyle\begin{pmatrix}\dot{\rho}_{\widetilde{\downarrow\Uparrow}}\\\
\dot{\rho}_{\downarrow\Downarrow}\\\
\dot{\rho}_{\Downarrow}\end{pmatrix}=\begin{pmatrix}0&0&\Gamma^{\rm
in}_{\downarrow}s^{2}\\\ 0&0&\Gamma^{\rm in}_{\downarrow}\\\ 0&0&-\Gamma^{\rm
in}_{\downarrow}(1+s^{2})\end{pmatrix}\begin{pmatrix}\rho_{\widetilde{\downarrow\Uparrow}}\\\
\rho_{\downarrow\Downarrow}\\\ \rho_{\Downarrow}\end{pmatrix}.$ (27)

This is in contrast to Eq. (14), where only an $\uparrow_{\rm D}$ data could
be flipped since the energy needed to excite an initially $\downarrow_{\rm D}$
data spin could not be provided at $\mu_{\downarrow}$. Using the parameters in
the experiment of Ref. [9], $A_{\rm I}=$117.5\text{\,}\mathrm{MHz}$$ and
$\Delta\epsilon=(\gamma_{\rm e}-\gamma_{\rm
n})B_{0}\approx$50\text{\,}\mathrm{GHz}$$ in a field of $B_{0}=1.77$ T, the
resulting flip probability is

$\displaystyle\frac{s^{2}}{1+s^{2}}\approx\left(\frac{A_{\rm
I}}{2\Delta\epsilon}\right)^{2}=$1.4\text{\times}{10}^{-6}\text{\,}$,$ (28)

orders of magnitude smaller than in the exchange-coupled electron pair in Eq.
(16). The nuclear spin readout is thus almost perfectly QND, even in presence
of the noncommuting terms in Eq. (26). This is thanks to the hierarchy of
energy scales $|\epsilon_{\rm e}|\gg|A|\gtrsim|\epsilon_{\rm n}|$, which
ensures that the eigenstates of Eq. (26) are almost exactly the product states
${\uparrow,\downarrow}\otimes{\Uparrow,\Downarrow}$. Furthermore, the
isotropic nature of the interaction weakly hybridizes only the
$\ket{\uparrow\Downarrow},\ket{\downarrow\Uparrow}$ states, leaving
$\ket{\uparrow\Uparrow},\ket{\downarrow\Downarrow}$ as exact eigenstates
unlike an anisotropic coupling discussed in the following section. Note that a
positive $\epsilon_{\rm n}$ merely flips the order of the 1P states, while the
above discussion remains valid.

Fig. 4c shows simulations of the probabilities of finding the nucleus in the
$\Uparrow$ state (compare Eq. (17)) after repeated QND measurements of the
nucleus via the electron, using the combined Liouvillian of Eq. (51) and Eq.
(52) with the rates from Fig. 3a. The extracted flipping rates on the right
are now on the order of $10^{-3}\,{\rm s}^{-1}$, i.e. on the order of a
million QND measurements to flip the nucleus instead of a thousand for the
exchange-coupled electron example. In general, we can see that the nucleus
flips faster if the system is driven, i.e. if the electron frequently tunnels
off and on. Consequently, driving the off-resonant transition will best
preserve the nuclear spin state in a QND measurement. This is in contrast to
the exchange-coupled spins, where driving the CR$(\downarrow_{\rm D})$
resulted in faster flipping rates for both spin configurations. As expected
from Eq. (27), which describes the dominant error channel of a $\Downarrow$
nucleus being flipped as an electron is loaded, the fastest flipping rate is
$\Gamma_{\Downarrow}$ in the case CR$(\Downarrow)$. In fact, in the zero-
temperature limit, this would be the only mechanism causing nuclear spin
flips.

Ref. [9] performed nuclear QND measurements using the CR$(\Updownarrow)$
transitions, and found a rate
$\Gamma_{\Uparrow}\approx$0.6\text{\times}{10}^{-3}\text{\,}{\mathrm{s}}^{-1}$$,
consistent with our prediction. However, they found a much faster
$\Gamma_{\Uparrow}\approx$15\text{\times}{10}^{-3}\text{\,}{\mathrm{s}}^{-1}$$,
attributed to the electron-nuclear flip-flop relaxation process[48, 47]
$\widetilde{\uparrow\Downarrow}\rightarrow\widetilde{\downarrow\Uparrow}$.
Including in our model for a flip-flop relaxation rate $\Gamma^{\rm
ff}=$53.3\text{\times}{10}^{-3}\text{\,}{\mathrm{s}}^{-1}$$ in Eq. (52)
increases $\Gamma_{\Uparrow}$ to
$15.73\text{\times}{10}^{-3}\text{\,}{\mathrm{s}}^{-1}$, consistent with Ref.
[9]. The state evolutions of all cases
(CR$(\Uparrow)$,CR$(\Downarrow)$,CR$(\Updownarrow)$) are shown in Fig. 4d with
the extracted nulcear flipping rates on the right. The CR$(\Downarrow)$ case
constantly brings a $\Downarrow$ state to the $\widetilde{\uparrow\Downarrow}$
state prone to flip-flop relaxation, making $\Gamma_{\Downarrow}$ faster. The
CR$(\Uparrow)$ case mostly avoids the $\widetilde{\uparrow\Downarrow}$ state
and remains the preferred option. The intrinsic relaxation rate of a nuclear
spin is immeasurably small[48] and needs not be included in the model.

Ref. [9] also performed a resonant tunneling (RT) experiment described in
detail App. F, where in each QND iteration $\mu_{\downarrow}$ is tuned in
resonance with $\mu_{\rm L}$ for a $0.7\text{\,}\mathrm{ms}$ time period.
During this time $\downarrow$ electrons repeatedly tunnel in and out at a high
rate. The resulting nuclear flipping rates can be found in Fig. 7c. The
CR$(\Updownarrow)$ case rates obtained with our model are in good agreement
with the results in Ref. [9].

###  IV.3 Anisotropic hyperfine coupling

Figure 5: Error channels in the QND measurement of a nuclear spin in the
presence of anisotropic hyperfine coupling with an electron. (a) Energy levels
of Eq. (29) for $\epsilon_{\rm e}\gg\epsilon_{\rm n}>A_{\rm I}>D_{ij}$. The
anisotropic terms hybridize all spin states, such that Eq. (10) now allows all
possible transitions marked by the colored arrows between the loaded (top) and
ionized (bottom) energy states. (b) Transition energies with respect to an
exemplary Fermi distribution in the SET island for the read/load tuning. The
resulting rates in Eq. (11) belonging to an $\uparrow$ or $\downarrow$
electron tunneling in or out are indicated. (c)
$M_{\Uparrow,\widetilde{\downarrow\Downarrow}}$ (see Eq. (10)) for eigenstates
of Eq. (29) on a logarithmic scale as a function of $B_{0}$ and $D_{xz}$ with
all other $D_{ij}=0$, $A_{\rm I}/2=$2.254\text{\,}\mathrm{MHz}$$,
$\epsilon_{\rm n}=$8.458\text{\,}\mathrm{MHz}\text{\,}{\mathrm{T}}^{-1}$\cdot
B_{0}$, and $\epsilon_{\rm
e}=$27.97\text{\,}\mathrm{GHz}\text{\,}{\mathrm{T}}^{-1}$\cdot B_{0}$. The
dashed lines mark $D_{xz}=$106.2\text{\,}\mathrm{kHz}$$ and
$B_{0}=$1.77\text{\,}\mathrm{T}$$. (d) Simulations of QND measurements in the
presence of anistropic hyperfine (not including $\Gamma^{\rm ff}$ or resonant
tunneling) for the parameters above, the tunnel rates in Fig. 3a, and a
read/load time of $t_{\rm R/L}=$1\text{\,}\mathrm{ms}$$. Shown are the
probabilities of finding the nucleus in an $\Uparrow$ state (Eq. (17)) after
repeated QND measurements as a function of time. The cases of driving the
CR$(\Downarrow)$, CR$(\Uparrow)$ alternating between the two
CR$(\Updownarrow)$ are considered for an initial $\Uparrow$ and $\Downarrow$
state. The nuclear flipping rates on the right are extracted by fitting Eq.
(46).

In the previous section, we considered a coupling that could hybridize the
antiparallel spin states, i.e. $\ket{\uparrow\Downarrow}$ and
$\ket{\downarrow\Uparrow}$, which introduced the error channels shown in Fig.
3a. As these states are split by $\Delta\epsilon\approx\epsilon_{\rm e}$,
which is much bigger than $A_{\rm I}$, this hybridization is weak and nuclear
flipping rates slow.

In this section, we extend the discussion of nuclear readout by including an
anisotropic hyperfine interaction[49]. The electron-nuclear Hamiltonian
becomes

$\displaystyle\hat{H}=\epsilon_{\rm e}\hat{S}_{z}+\epsilon_{\rm
n}\hat{I}_{z}+A_{\rm
I}\vec{S}\cdot\vec{I}+\sum_{i,j}D_{ij}\hat{S}_{i}\hat{I}_{j},$ (29)

where $D_{ij}$ describes the anisotropic dipolar coupling between electron and
nucleus, and we sum over the Cartesian coordinates. In the case of donors in
bulk silicon, $D_{ij}=0$ due to the spherical symmetry of the electron ground-
state wavefunction. Breaking this symmetry, e.g. by an electric field or by
strain, typically results in anistropic terms much smaller than the isotropic
ones $A_{\rm I}\gg D_{ij}$[50, 51]. This Hamiltonian describes many electron-
nuclear spin systems, such as NV-diamond[26, 8, 27, 28, 29, 30], defects in
SiC[52], molecular magnets[53], atoms on surfaces[54], rare-earth ions[55],
and 29Si nuclei coupled to donors[56, 31, 57] or quantum dots[58] in silicon.
Eq. (29) hybridizes all spin states in Fig. 5a, such that Eq. (10) now allows
all possible transitions between 1P and 2P states indicated by the arrows. As
a result, the anisotropic hyperfine coupling term in Eq. (29) introduces new
error channels in the QND measurement.

Crucially, the $D_{xz}$ and $D_{yz}$ components hybridize the $\downarrow$
subspace, i.e. $\ket{\downarrow\Uparrow}$ and $\ket{\downarrow\Downarrow}$, as
well as the $\uparrow$ one. The corresponding transitions in Fig. 5b can flip
the nuclear spin as a $\downarrow$ electron tunnels in, or an $\uparrow$
tunnels out, which routinely happens during the measurements. The amplitude of
the transition matrix elements $M_{\rm 1P,2P}$ (Eq. (10)) of this detrimental
process depends on the ratio of diagonal vs off-diagonal elements in the
respective subspace. Restricting ourselves to the $D_{xz}$ tensor component,
Eq. (29) in the $\\{\ket{\downarrow\Uparrow},\ket{\downarrow\Downarrow}\\}$
subspace has the form

$\displaystyle H=\frac{1}{2}\begin{pmatrix}-\epsilon_{\rm e}+(\epsilon_{\rm
n}-A_{\rm I}/2)&D_{xz}/2\\\ D_{xz}/2&-\epsilon_{\rm e}-(\epsilon_{\rm
n}-A_{\rm I}/2)\end{pmatrix}.$ (30)

We can interpret $D_{xz}$ as an effective field that tilts the quantization
axis of the nuclear spin through the presence of the electron[59], effectively
hybridizing the nuclear spin basis states. The smaller the nuclear energy
splitting $\epsilon_{\rm n}-A_{\rm I}/2$, the stronger the hybridization.
While $A_{\rm I}$ is typically only weakly tunable[48], $\epsilon_{\rm
n}=\gamma_{\rm n}B_{0}$ is a function of the applied magnetic field, which
allows the degree of hybridization to be tuned in an experiment. Fig. 5c shows
calculations of $M_{\Uparrow,\widetilde{\downarrow\Downarrow}}$, i.e the
$\downarrow$ subspace hybridization, as a function of $\epsilon_{\rm n}$ ($B$)
and $D_{xz}$ on a logarithmic scale for the parameters of an exemplary 29Si
atom near a phosphorus donor in silicon[56, 31]: $A_{\rm
I}/2=$2.254\text{\,}\mathrm{MHz}$$, $\gamma_{\rm
n}=$8.458\text{\,}\mathrm{MHz}\text{\,}{\mathrm{T}}^{-1}$$, and $\gamma_{\rm
e}=$27.97\text{\,}\mathrm{GHz}\text{\,}{\mathrm{T}}^{-1}$$. At the point where
$\epsilon_{\rm n}=A_{\rm I}/2$ the hybridization is strong as expected. To
reach the same degree of hybridization at a bigger splitting in Eq. (30)
requires an increasingly stronger coupling $D_{xz}$. The $\uparrow$ subspace
hybridization would be the strongest when $\epsilon_{\rm n}=-A_{\rm I}/2$.
However, for a negative magnetic field everything is effectively flipped, in
other words $\downarrow$ becomes $\uparrow$ and the hybridization in the
$\uparrow$ subspace is always weaker for positive $\gamma_{\rm n}$. This model
thus suggests measuring the nuclear flipping rate – i.e. the deviation from
QND measurement – as a function of $B_{0}$ is an experimentally realizable way
to quantify the anisotropy of the hyperfine interaction. This method would be
particularly useful in cases where the anisotropy is very weak, and therefore
would be difficult to observe directly in the resonance spectra, since the
resonance frequency is only affected by off-diagonal elements to second
order[56].

For the chosen 29Si atom[56] with $D_{xz}=$106.2\text{\,}\mathrm{kHz}$$ and a
field of $B=$1.77\text{\,}\mathrm{T}$$ indicated in Fig. 5c , where
$M_{\Uparrow,\widetilde{\downarrow\Downarrow}}\approx 4\times 10^{-6}$ and the
hybridization in the $\uparrow$ subspace
$M_{\Uparrow,\widetilde{\uparrow\Downarrow}}\approx 2\times 10^{-6}$, we
simulate the QND measurements using the Liouvillian in App. F and the rates in
Fig. 3a. Fig. 5d shows the probability of finding the nucleus in the
$\Uparrow$ state (compare Eq. (17)) after repeated QND measurements (not
including $\Gamma^{\rm ff}$ or RT) with the extracted flipping rates on the
right. Although the isotropic component of the hyperfine interaction $A_{\rm
I}=$4.5\text{\,}\mathrm{MHz}$$ is much smaller than that of a 31P atom
($A_{\rm I}=$117.5\text{\,}\mathrm{MHz}$$), the anisotropic components lead to
faster flipping rates than in Tab. 4c. This is a direct consequence of the
fact that, for a $D_{xz}$-like coupling, the simple presence of the electron
is sufficient to hybridize the nuclear spin states. In other words, unlike the
case of an isotropic $A_{\rm I}$ coupling, here the hybridization does not
require a process that includes an electron spin flip.

##  V Conclusion

In this paper, we established a theoretical framework to understand and
quantify the error channels in QND measurements in realistic spin systems.
Errors are introduced when the interaction, needed to map the spin state of
the data qubit onto the ancilla, entangles the two systems
(exchange/hyperfine) or tilts the quantization axis of the data spin
(anisotropic hyperfine). This enables transitions which flip the data spin as
the ancilla tunnels off and on as part of the spin readout process. Their
rates depend on the degree of hybridization and which transitions are
energetically allowed. We analysed this dependency for three different cases.

In exchange-coupled spins, the fact that $\epsilon_{\rm A}\approx\epsilon_{\rm
D}$ leads to a strong hybridization and fast error rates. The dominant error
channel is the flip of an $\uparrow_{\rm D}$ as its energy can be transferred
on to a $\downarrow_{\rm A}$ in the tunnel process. We showed that for a
realistic set of parameters this process is an order of magnitude faster than
the natural spin relaxation rate.

For a nuclear spin hyperfine-coupled to an electron, $\epsilon_{\rm
e}\gg\epsilon_{\rm n}$ holds and the much weaker hybridization leads to lower
error rates. The energy of the tunnel process now almost exclusively depends
on the electron spin state. Using a realistic set of parameters for a 31P atom
and including a flip-flop relaxation process, allowed us to reproduce the
nuclear flipping rates measured in Ref.[9].

Finally, we showed that anisotropic hyperfine components can significantly
hybridize the nuclear spin states and lead to faster flipping rates. As the
degree of hybridization can be tuned by varying the nuclear splitting with a
magnetic field, measuring the nuclear flipping rates as a function of field
and its direction could be used to probe the anisotropy of the hyperfine
coupling.

## Acknowledgements

The research was supported by the Australian Research Council (Grants No.
CE170100012, DP210103769), the US Army Research Office (Contract no.
W911NF-17-1-0200), and the Australian Department of Industry, Innovation and
Science (Grant No. AUSMURI000002). H.G.S. and I.F.d.F. Acknowledge support
from the Sydney Quantum Academy. The views and conclusions contained in this
document are those of the authors and should not be interpreted as
representing the official policies, either expressed or implied, of the Army
Research Office or the U.S. Government. The U.S. Government is authorized to
reproduce and distribute reprints for Government purposes notwithstanding any
copyright notation herein.

##  Appendix A Explicit treatment of the lead

We treat the lead as a continuum of states described by the Hamiltonian

$\displaystyle\hat{H}_{\rm
L}=\sum_{k,\sigma}\epsilon_{k}\hat{c}^{\dagger}_{k\sigma}\hat{c}_{k\sigma},$
(31)

where $\hat{c}_{k\sigma}^{\dagger}$ creates a particle with momentum $k$, spin
$\sigma$, and energy $\epsilon_{k}$ on the lead. Then the tunneling between
the lead and the ancilla is described by

$\displaystyle\hat{H}_{\rm
T}=\sum_{k,\sigma}t_{0}(\hat{a}_{\sigma}\hat{c}^{\dagger}_{k\sigma}+\hat{a}^{\dagger}_{\sigma}\hat{c}_{k\sigma}),$
(32)

where we assume the tunnel coupling $t_{0}$ to be constant over the relevant
energy range, i.e. the tunnel barrier is not changing much with energy (E).

We are now interested in processes, where a particle tunnels to (from) the
lead causing transition between the one-particle (1P) and two-particle (2P)
states of the data-ancilla system. Assuming that the lead is initially in the
state $\ket{\rm L}$, the rate of the process $\ket{\rm L}\ket{\rm
1P}\rightarrow\hat{c}_{k\sigma}\ket{\rm L}\ket{\rm 2P}$, where a particle with
energy $\epsilon_{k}$ and spin $\sigma$ tunnels to the ancilla, is given by

$\displaystyle\Gamma^{k,\sigma}_{{\rm 1P}\rightarrow{\rm 2P}}=|\braket{{\rm
L,1P}}{\hat{H}_{\rm T}\hat{c}_{k\sigma}}{{\rm
L,2P}}|^{2}\delta(\epsilon_{k}-\mu_{{\rm 1P}\leftrightarrow{\rm 2P}})$ (33)
$\displaystyle=|\braket{{\rm
1P}}{t_{0}\hat{a}_{\sigma}}{{2P}}|^{2}|\braket{{\rm
L}}{\hat{c}^{\dagger}_{k\sigma}\hat{c}_{k\sigma}}{{\rm
L}}|^{2}\delta(\epsilon_{k}-\mu_{{\rm 1P}\leftrightarrow{\rm 2P}}).$ (34)

The last term describes the fact that the particle can only tunnel if it
carries the energy equal to the chemical potential of the transition, i.e.
$\epsilon_{k}=\mu_{{\rm 2P}\leftrightarrow{\rm 1P}}$. The term $|\braket{{\rm
L}}{\hat{c}^{\dagger}_{k\sigma}\hat{c}_{k\sigma}}{{\rm L}}|^{2}$ is 1 if the
state with energy $\epsilon_{k}$ and spin $\sigma$ was initially occupied and
0 otherwise. By taking the thermal average over all states $\ket{\rm L}$, we
replace this term by the Fermi distribution $f(\epsilon_{k})$, i.e. the
fraction of states with energy $\epsilon_{k}$ that is on average occupied at a
given temperature. We may do so as the energy $\epsilon_{k}$ does not depend
on the spin. Summing over all spins $\sigma$ and momenta $k$ we get

$\displaystyle\Gamma_{{\rm 1P}\rightarrow{\rm
2P}}=\sum_{k,\sigma}\Gamma^{k,\sigma}_{{\rm 1P}\rightarrow{\rm 2P}}$ (35)
$\displaystyle=\sum_{k,\sigma}|\braket{{\rm
1P}}{t_{0}\hat{a}_{\sigma}}{{2P}}|^{2}f(\epsilon_{k})\delta(\epsilon_{k}-\mu_{{\rm
1P}\leftrightarrow{\rm 2P}})$ (36) $\displaystyle=|\braket{{\rm
1P}}{\sum_{\sigma}t_{0}\hat{a}_{\sigma}}{{2P}}|^{2}f(\mu_{{\rm
1P}\leftrightarrow{\rm 2P}})n(\mu_{{\rm 1P}\leftrightarrow{\rm 2P}}),$ (37)

where $n(E)$ is the density of states counting the number of states in the
lead at energy $E$. We have arrived at Eq. (7a) of the main text.

To derive Eq. (7b), we start by looking at the tunnel process $\ket{\rm
L}\ket{\rm 2P}\rightarrow\hat{c}^{\dagger}_{k\sigma}\ket{\rm L}\ket{\rm 1P}$
with rate

$\displaystyle\Gamma^{k,\sigma}_{{\rm 2P}\rightarrow{\rm 1P}}=|\braket{{\rm
L,2P}}{\hat{H}_{\rm T}\hat{c}^{\dagger}_{k\sigma}}{{\rm
L,1P}}|^{2}\delta(\epsilon_{k}-\mu_{{\rm 1P}\leftrightarrow{\rm 2P}})$ (38)
$\displaystyle=|\braket{{\rm
2P}}{t_{0}\hat{a}^{\dagger}_{\sigma}}{{1P}}|^{2}|\braket{{\rm
L}}{\hat{c}_{k\sigma}\hat{c}^{\dagger}_{k\sigma}}{{\rm
L}}|^{2}\delta(\epsilon_{k}-\mu_{{\rm 1P}\leftrightarrow{\rm 2P}})$ (39)
$\displaystyle=|\braket{{\rm
2P}}{t_{0}\hat{a}^{\dagger}_{\sigma}}{{1P}}|^{2}|\braket{{\rm
L}}{1-\hat{c}^{\dagger}_{k\sigma}\hat{c}_{k\sigma}}{{\rm
L}}|^{2}\delta(\epsilon_{k}-\mu_{{\rm 1P}\leftrightarrow{\rm 2P}}).$ (40)

From here we can follow the same steps as above to arrive at Eq. (7b).

##  Appendix B Liouvillian of Heisenberg exchange coupled spins

In the basis $\\{\uparrow_{\rm A}\uparrow_{\rm D},\widetilde{\uparrow_{\rm
A}\downarrow_{\rm D}},\widetilde{\downarrow_{\rm A}\uparrow_{\rm
D}},\downarrow_{\rm A}\downarrow_{\rm D},\uparrow_{\rm D},\downarrow_{\rm
D}\\},$ the Liouvillian of two Heisenberg exchange-coupled spins in the zero-
temperature limit is given by

$\displaystyle\mathcal{L}_{T=0}=\begin{pmatrix}-\Gamma^{\rm
out}_{\uparrow}&0&0&0&0&0\\\ 0&-\Gamma^{\rm
out}_{\uparrow}c^{2}&0&0&\Gamma^{\rm in}_{\downarrow}s^{2}&0\\\
0&0&-\Gamma^{\rm out}_{\uparrow}s^{2}&0&\Gamma^{\rm in}_{\downarrow}c^{2}&0\\\
0&0&0&0&0&\Gamma^{\rm in}_{\downarrow}\\\ \Gamma^{\rm
out}_{\uparrow}&0&0&0&-\Gamma^{\rm in}_{\downarrow}&0\\\ 0&\Gamma^{\rm
out}_{\uparrow}c^{2}&\Gamma^{\rm out}_{\uparrow}s^{2}&0&0&-\Gamma^{\rm
in}_{\downarrow}\end{pmatrix}.$ (41)

Here, the second-last column describes the loading process in Eq. (14). In the
$T>$0\text{\,}\mathrm{K}$$ case, we get the following additional matrix
elements

$\displaystyle\begin{split}\mathcal{L}_{T>0}=\begin{pmatrix}0&0&0&0&\Gamma^{\rm
in}_{\uparrow}&0\\\ 0&-\Gamma^{\rm out}_{\downarrow}s^{2}&0&0&0&\Gamma^{\rm
in}_{\uparrow}c^{2}\\\ 0&0&-\Gamma^{\rm out}_{\downarrow}c^{2}&0&&\Gamma^{\rm
in}_{\uparrow}s^{2}\\\ 0&0&0&-\Gamma^{\rm out}_{\downarrow}&0&0\\\
0&\Gamma^{\rm out}_{\downarrow}s^{2}&\Gamma^{\rm
out}_{\downarrow}c^{2}&0&-\Gamma^{\rm in}_{\uparrow}&0\\\ 0&0&0&\Gamma^{\rm
out}_{\downarrow}&0&-\Gamma^{\rm in}_{\uparrow}\end{pmatrix}.\end{split}$ (42)

The $T_{1}$ spin relaxation process with rate $\Gamma^{T_{1}}_{\uparrow}$ is
described by

$\displaystyle\mathcal{L}_{T_{1}}=\begin{pmatrix}-2\Gamma^{T_{1}}_{\uparrow}&0&0&0&0&0\\\
\Gamma^{T_{1}}_{\uparrow}&-\Gamma^{T_{1}}_{\uparrow}&0&0&0&0\\\
\Gamma^{T_{1}}_{\uparrow}&0&-\Gamma^{T_{1}}_{\uparrow}&0&0&0\\\
0&\Gamma^{T_{1}}_{\uparrow}&\Gamma^{T_{1}}_{\uparrow}&0&0&0\\\
0&0&0&0&-\Gamma^{T_{1}}_{\uparrow}&0\\\
0&0&0&0&\Gamma^{T_{1}}_{\uparrow}&0\end{pmatrix}.$ (43)

##  Appendix C Flipping rates and equilibrium state

Given that an $\uparrow$ is flipped with rate $\Gamma_{\uparrow}$ and a
$\downarrow$ with $\Gamma_{\downarrow}$, their time evolution is described by
the following rate equation

$\displaystyle\begin{pmatrix}\dot{\rho}_{\uparrow}\\\
\dot{\rho}_{\downarrow}\end{pmatrix}=\begin{pmatrix}-\Gamma_{\uparrow}&\Gamma_{\downarrow}\\\
\Gamma_{\uparrow}&-\Gamma_{\downarrow}\\\
\end{pmatrix}\begin{pmatrix}\rho_{\uparrow}\\\
\rho_{\downarrow}\end{pmatrix}.$ (44)

For an initial $\rho_{\uparrow}(0)=1$ we get the solution

$\displaystyle\rho_{\uparrow}(t)=$
$\displaystyle\frac{\Gamma_{\downarrow}}{\Gamma_{\uparrow}+\Gamma_{\downarrow}}\left(1+\frac{\Gamma_{\uparrow}}{\Gamma_{\downarrow}}e^{-(\Gamma_{\uparrow}+\Gamma_{\downarrow})t}\right),$
(45) $\displaystyle\rho_{\downarrow}(t)=$
$\displaystyle\frac{\Gamma_{\uparrow}}{\Gamma_{\uparrow}+\Gamma_{\downarrow}}\left(1-e^{-(\Gamma_{\uparrow}+\Gamma_{\downarrow})t}\right),$
(46)

which reaches the equilibrium state

$\displaystyle\rho_{\uparrow}(t=\infty)=\frac{\Gamma_{\downarrow}}{\Gamma_{\uparrow}+\Gamma_{\downarrow}},$
(47)
$\displaystyle\rho_{\downarrow}(t=\infty)=\frac{\Gamma_{\uparrow}}{\Gamma_{\uparrow}+\Gamma_{\downarrow}}.$
(48)

This means that the ratio of state occupation in equilibrium equals the ratios
of the flipping rates.

##  Appendix D Exchange-coupled spins with $\epsilon_{\rm A}=\epsilon_{\rm D}$

In the case where the two coupled electrons have the same individual energy
splitting, $\epsilon_{\rm A}=\epsilon_{\rm D}$, the hybridized states in Eq.
(9b) and Eq. (9a) become

$\displaystyle\ket{S}$ $\displaystyle=\frac{1}{\sqrt{2}}(\ket{\uparrow_{\rm
A}\downarrow_{\rm D}}-\ket{\downarrow_{\rm A}\uparrow_{\rm D}})$ (49)
$\displaystyle\ket{T}$ $\displaystyle=\frac{1}{\sqrt{2}}(\ket{\downarrow_{\rm
A}\uparrow_{\rm D}}+\ket{\uparrow_{\rm A}\downarrow_{\rm D}}).$ (50)

This makes it unreasonable to attempt the QND measurement, since an
$\uparrow_{\rm D}$ can be flipped within a single load cycle as shown in Fig.
6a. Here we use the tunnel rates in Fig. 3a with $s=c=1/\sqrt{2}$. The initial
$\uparrow_{\rm D}$ is quickly loaded into either a $S$ or $T$ state. However,
these states still contain a $\mu_{\uparrow}$ energy quantum, which can be
released by one of the electrons tunneling out, leaving a flipped
$\downarrow_{\rm D}$ behind. Finally, the ancilla is reloaded into
$\downarrow_{\rm A}\downarrow_{\rm D}$. Note that the simulated load time was
doubled to compensate for the $S$ and $T$ tunnel times that are twice as long
for $c^{2}=s^{2}=1/2$.

This process could also explain the experimentally observed increase of the
visibility of the $\uparrow_{\rm A}\uparrow_{\rm D}$ state in exchange-coupled
phosphorus donors[36] when the nuclear spins are parallel, i.e. $\epsilon_{\rm
A}=\epsilon_{\rm D}$. During the read/load simulated in Fig. 6b the ancilla
will tunnel off and back on twice, first while $\uparrow_{\rm A}\uparrow_{\rm
D}\rightarrow\uparrow_{\rm D}\rightarrow S/T$ and secondly while
$S/T\rightarrow\downarrow_{\rm D}\rightarrow\downarrow_{\rm A}\downarrow_{\rm
D}$. This gives the charge sensor two chances to detect a signal instead of
just once, which can make a visible difference when the single-shot readout
contrast is not perfect.

Figure 6: Simulations of a load/read period in the $\epsilon_{\rm
A}=\epsilon_{\rm D}$ case, i.e. $s^{2}=c^{2}=1/2$, and the tunnel rates in
Fig. 3a. Shown is the state distribution as a function of time. In (a) the
initial state is $\uparrow_{\rm D}$, while in (b) it is $\uparrow_{\rm
A}\uparrow_{\rm D}$.

##  Appendix E Liouvillian of a nucleus hyperfine-coupled to an electron

In the basis
$\\{\uparrow\Uparrow,\widetilde{\uparrow\Downarrow},\widetilde{\downarrow\Uparrow},\downarrow\Downarrow,\Uparrow,\Downarrow\\}$,
the $T=0$ Liouvillian of a nucleus hyperfine-coupled to an electron is

$\displaystyle\mathcal{L}_{T=0}=\begin{pmatrix}-\Gamma^{\rm
out}_{\uparrow}&0&0&0&0&0\\\ 0&-\Gamma^{\rm out}_{\uparrow}&0&0&0&0\\\
0&0&0&0&\Gamma^{\rm in}_{\downarrow}c^{2}&\Gamma^{\rm in}_{\downarrow}s^{2}\\\
0&0&0&0&0&\Gamma^{\rm in}_{\downarrow}\\\ \Gamma^{\rm
out}_{\uparrow}&\Gamma^{\rm out}_{\uparrow}s^{2}&0&0&-\Gamma^{\rm
in}_{\downarrow}c^{2}&0\\\ 0&\Gamma^{\rm
out}_{\uparrow}c^{2}&0&0&0&-\Gamma^{\rm
in}_{\downarrow}(1+s^{2})\end{pmatrix}.$ (51)

In the $T>$0\text{\,}\mathrm{K}$$ case, we get the following additional matrix
elements

$\displaystyle\mathcal{L}_{T>0}=\begin{pmatrix}0&0&0&0&\Gamma^{\rm
in}_{\uparrow}&0\\\ &0&0&0&\Gamma^{\rm in}_{\uparrow}s^{2}&\Gamma^{\rm
in}_{\uparrow}c^{2}\\\ 0&0&-\Gamma^{\rm out}_{\downarrow}&0&0&0\\\
0&0&0&-\Gamma^{\rm out}_{\downarrow}&0&0\\\ 0&0&\Gamma^{\rm
out}_{\downarrow}c^{2}&0&-\Gamma^{\rm in}_{\uparrow}(1+s^{2})&0\\\
0&0&\Gamma^{\rm out}_{\downarrow}s^{2}&\Gamma^{\rm
out}_{\downarrow}&0&-\Gamma^{\rm in}_{\uparrow}c^{2}\end{pmatrix}.$ (52)

Relaxation processes are described by

$\displaystyle\mathcal{L}_{T_{1}}=\begin{pmatrix}-\Gamma^{T_{1}}_{\uparrow}&0&0&0&0&0\\\
\Gamma^{T_{1}}_{\uparrow}s^{2}&-\Gamma^{T_{1}}_{\uparrow}c^{2}-\Gamma^{\rm
ff}&0&0&0&0\\\ \Gamma^{T_{1}}_{\uparrow}c^{2}&\Gamma^{\rm
ff}&-\Gamma^{T_{1}}_{\uparrow}s^{2}&0&0&0\\\
0&\Gamma^{T_{1}}_{\uparrow}c^{2}&\Gamma^{T_{1}}_{\uparrow}s^{2}&0&0&0\\\
0&0&0&0&0&0\\\ 0&0&0&0&0&0\end{pmatrix}.$ (53)

where we also included the flip-flop relaxation process[9, 48] with rate
$\Gamma^{\rm ff}$.

##  Appendix F Nuclear flipping via resonant electron tunneling

Figure 7: Nuclear spin flipping during resonant electron tunneling in a donor
system. (a) Illustration of two consecutive resonant tunneling (RT) experiment
iterations. Left: tuning of the transition energies with respect to an
exemplary Fermi distribution of the SET island during the different stages of
the RT iterations. Driving pluses are followed by a $1\text{\,}\mathrm{ms}$
read/load window, a $0.7\text{\,}\mathrm{ms}$ RT stage, and a
$0.3\text{\,}\mathrm{ms}$ load time to reinitialize a $\downarrow$ electron. A
RT iteration has a total length of $t_{\rm RT}=$2\text{\,}\mathrm{ms}$$.
During read and load periods we have the rates in Fig. 3a, while during RT
periods, where $\mu_{\downarrow}$ is tuned in resonance with $\mu_{\rm L}$,
$\Gamma^{\rm in}_{\downarrow}=\Gamma^{\rm
out}_{\downarrow}=$28\text{\times}{10}^{3}\text{\,}{\mathrm{s}}^{-1}$$,
$\Gamma^{\rm in}_{\uparrow}=$140\text{\,}{\mathrm{s}}^{-1}$$, and $\Gamma^{\rm
out}_{\uparrow}=$56\text{\times}{10}^{3}\text{\,}{\mathrm{s}}^{-1}$$.
$s^{2}\approx 10^{-6}$ and $\Gamma^{\rm
ff}=$53.3\text{\times}{10}^{-3}\text{\,}{\mathrm{s}}^{-1}$$. (b) Resulting
state distribution as a function of time. In the RT period an equilibrium of
an equally likely neutral (electron loaded) or ionized (electron unloaded)
donor is quickly reached. (c) Probabilities of finding the nucleus in a
$\Uparrow$ state (Eq. (17)) after repeated RT iterations as a function of
time. The cases of driving the CR$(\Downarrow)$, CR$(\Uparrow)$ alternating
between the two CR$(\Updownarrow)$ are considered for an initial $\Uparrow$
and $\Downarrow$ state. The nuclear flipping rates on the right are extracted
fitting Eq. (46).

Ref. [9] performed a QND measurement on a 31P nuclear spin hyperfine-coupled
to an electron, while including a resonant tunneling (RT) window for the
electron, whereby the $\downarrow$ electron state was tuned in resonance with
the electrochemical potential of the lead, $\mu_{\downarrow}=\mu_{\rm L}$ (see
RT in Fig. 7a). This tuning yields $\Gamma^{\rm out}_{\uparrow}>\Gamma^{\rm
out}_{\downarrow}=\Gamma^{\rm in}_{\downarrow}\gg\Gamma^{\rm in}_{\uparrow}$,
and results in $\downarrow$ electrons randomly and frequently tunneling on and
off the donor (akin to random telegraph signals[60]). The process is described
by the following rate equation

$\displaystyle\begin{pmatrix}\dot{\rho}_{\widetilde{\downarrow\Uparrow}}\\\
\dot{\rho}_{\downarrow\Downarrow}\\\ \dot{\rho}_{\Uparrow}\\\
\dot{\rho}_{\Downarrow}\end{pmatrix}=\begin{pmatrix}-\Gamma^{\rm
t}_{\downarrow}&0&\Gamma^{\rm t}_{\downarrow}c^{2}&\Gamma^{\rm
rt}_{\downarrow}s^{2}\\\ 0&-\Gamma^{\rm rt}_{\downarrow}&0&\Gamma^{\rm
rt}_{\downarrow}\\\ \Gamma^{\rm rt}_{\downarrow}c^{2}&0&-\Gamma^{\rm
rt}_{\downarrow}c^{2}&0\\\ \Gamma^{\rm rt}_{\downarrow}s^{2}&\Gamma^{\rm
rt}_{\downarrow}&0&-\Gamma^{\rm rt}_{\downarrow}(1+s^{2})\\\
\end{pmatrix}\begin{pmatrix}\rho_{\widetilde{\downarrow\Uparrow}}\\\
\rho_{\downarrow\Downarrow}\\\ \rho_{\Uparrow}\\\
\rho_{\Downarrow}\end{pmatrix},$ (54)

where $\Gamma^{\rm rt}_{\downarrow}=\Gamma^{\rm in/out}_{\downarrow}$. Due to
$\Gamma^{\rm out}_{\downarrow}=\Gamma^{\rm in}_{\downarrow}$, the system will
quickly reach an equilibrium
$\rho_{\Uparrow}=\rho_{\widetilde{\downarrow\Uparrow}}$ and
$\rho_{\Downarrow}=\rho_{\downarrow\Downarrow}$, which allows us to simplify
Eq. (54) to

$\displaystyle\begin{pmatrix}\dot{\rho}_{\Uparrow}\\\
\dot{\rho}_{\Downarrow}\end{pmatrix}=\begin{pmatrix}-\Gamma^{\rm
rt}_{\downarrow}s^{2}&\Gamma^{\rm rt}_{\downarrow}s^{2}\\\ \Gamma^{\rm
rt}_{\downarrow}s^{2}&-\Gamma^{\rm rt}_{\downarrow}s^{2}\\\
\end{pmatrix}\begin{pmatrix}\rho_{\Uparrow}\\\
\rho_{\Downarrow}\end{pmatrix}.$ (55)

Eq. (55) means that during the RT period both nuclear spin configurations will
be flipped at a rate $\Gamma^{\rm rt}_{\downarrow}s^{2}$, i.e. a significantly
faster flipping of the $\Uparrow$ nucleus.

Fig. 7a illustrates the tuning of the electrochemical potentials during a
repetition of the RT experiment[9]. Driving pluses are followed by a
$1\text{\,}\mathrm{ms}$ read/load window as in the usual QND experiment. Here,
this is followed by a $0.7\text{\,}\mathrm{ms}$ RT window and a
$0.3\text{\,}\mathrm{ms}$ load time to reinitialize a $\downarrow$ electron.

Fig. 7b shows simulations for pulses alternating between case 1 and 2, an
initial $\Uparrow$ nucleus. In the RT window we assume $\Gamma^{\rm
rt}_{\downarrow}=$28\text{\times}{10}^{3}\text{\,}{\mathrm{s}}^{-1}$$,
$\Gamma^{\rm in}_{\uparrow}=$140\text{\,}{\mathrm{s}}^{-1}$$, and $\Gamma^{\rm
out}_{\uparrow}=$56\text{\times}{10}^{3}\text{\,}{\mathrm{s}}^{-1}$$. Before
the first shown read/load window, a CR$(\Downarrow)$ leaves the
$\widetilde{\downarrow\Uparrow}$ initialized in a previous iteration
unchanged. Reading a $\downarrow$ electron only improves the state
preparation, i.e. decreases the remaining $\uparrow$ probability, through a
longer wait time. At the following RT tuning, we quickly reach the
$\rho_{\Uparrow}=\rho_{\widetilde{\downarrow\Uparrow}}=0.5$ equilibrium
discussed above at which the nucleus is flipped at the rate
$\Gamma_{\downarrow}^{\rm rt}s^{2}$. Before applying a driving pulse, now
conditional on a $\Uparrow$ nucleus (CR$(\Uparrow)$), a short loading window
initializes a $\downarrow$ electron state. In the subsequent read/load window,
the $\uparrow$ electron tunnels out and the donor is quickly reloaded with a
$\downarrow$ followed by the same RT and reload processes. The probabilities
of finding the nucleus in an $\Uparrow$ after many repetitions of this
sequence for the cases of driving the CR$(\Downarrow)$, CR$(\Uparrow)$ or
alternating between the two CR$(\Updownarrow)$, for an initial $\Uparrow$ and
$\Downarrow$ state, are shown in Fig. 7c with the extracted flipping rates on
the right.

##  Appendix G Liouvillian for anisotropic hyperfine coupling

In the basis of the eigenstates of Eq. (29)
$\\{\widetilde{\uparrow\Uparrow},\widetilde{\uparrow\Downarrow},\widetilde{\downarrow\Uparrow},\widetilde{\downarrow\Downarrow},\Uparrow,\Downarrow\\}$
the $T=0$ Liouvillian of nucleus coupled to an electron via anisotropic
hyperfine is

$\displaystyle\mathcal{L}_{T=0}=\begin{pmatrix}-\Gamma^{\rm
out}_{\widetilde{\uparrow\Uparrow}\rightarrow\Uparrow}-\Gamma^{\rm
out}_{\widetilde{\uparrow\Uparrow}\rightarrow\Uparrow}&0&0&0&0&0\\\
0&-\Gamma^{\rm
out}_{\widetilde{\uparrow\Downarrow}\rightarrow\Uparrow}-\Gamma^{\rm
out}_{\widetilde{\uparrow\Downarrow}\rightarrow\Uparrow}&0&0&0&0\\\
0&0&0&0&\Gamma^{\rm
in}_{\Uparrow\rightarrow\widetilde{\downarrow\Uparrow}}&\Gamma^{\rm
in}_{\Downarrow\rightarrow\widetilde{\downarrow\Uparrow}}\\\
0&0&0&0&\Gamma^{\rm
in}_{\Uparrow\rightarrow\widetilde{\downarrow\Downarrow}}&\Gamma^{\rm
in}_{\Downarrow\rightarrow\widetilde{\downarrow\Downarrow}}\\\ \Gamma^{\rm
out}_{\widetilde{\uparrow\Uparrow}\rightarrow\Uparrow}&\Gamma^{\rm
out}_{\widetilde{\uparrow\Downarrow}\rightarrow\Uparrow}&0&0&-\Gamma^{\rm
in}_{\Uparrow\rightarrow\widetilde{\downarrow\Uparrow}}-\Gamma^{\rm
in}_{\Uparrow\rightarrow\widetilde{\downarrow\Downarrow}}&0\\\ \Gamma^{\rm
out}_{\widetilde{\uparrow\Uparrow}\rightarrow\Downarrow}&\Gamma^{\rm
out}_{\widetilde{\uparrow\Downarrow}\rightarrow\Downarrow}&0&0&0&-\Gamma^{\rm
in}_{\Downarrow\rightarrow\widetilde{\downarrow\Uparrow}}-\Gamma^{\rm
in}_{\Downarrow\rightarrow\widetilde{\downarrow\Downarrow}}\end{pmatrix},$
(56)

where the rates are given by

$\displaystyle\Gamma^{\rm out}_{\rm 2P\rightarrow 1P}=\Gamma^{\rm
out}_{\uparrow}M_{\rm 1P,2P},$ (57a) $\displaystyle\Gamma^{\rm in}_{\rm
1P\rightarrow 2P}=\Gamma^{\rm in}_{\downarrow}M_{\rm 1P,2P}$ (57b)

In the $T>$0\text{\,}\mathrm{K}$$ case, we get the following additional matrix
elements

$\displaystyle\mathcal{L}_{T>0}=\begin{pmatrix}0&0&0&0&\Gamma^{\rm
in}_{\Uparrow\rightarrow\widetilde{\uparrow\Uparrow}}&\Gamma^{\rm
in}_{\Downarrow\rightarrow\widetilde{\uparrow\Uparrow}}\\\ 0&0&0&0&\Gamma^{\rm
in}_{\Uparrow\rightarrow\widetilde{\uparrow\Downarrow}}&\Gamma^{\rm
in}_{\Downarrow\rightarrow\widetilde{\uparrow\Downarrow}}\\\ 0&0&-\Gamma^{\rm
out}_{\widetilde{\downarrow\Uparrow}\rightarrow\Uparrow}-\Gamma^{\rm
out}_{\widetilde{\downarrow\Uparrow}\rightarrow\Downarrow}&0&0&0\\\
0&0&0&-\Gamma^{\rm
out}_{\widetilde{\downarrow\Downarrow}\rightarrow\Uparrow}-\Gamma^{\rm
out}_{\widetilde{\downarrow\Downarrow}\rightarrow\Downarrow}&0&0\\\
0&0&\Gamma^{\rm
out}_{\widetilde{\downarrow\Uparrow}\rightarrow\Uparrow}&\Gamma^{\rm
out}_{\widetilde{\downarrow\Downarrow}\rightarrow\Uparrow}&-\Gamma^{\rm
in}_{\Uparrow\rightarrow\widetilde{\uparrow\Uparrow}}-\Gamma^{\rm
in}_{\Uparrow\rightarrow\widetilde{\uparrow\Downarrow}}&0\\\ 0&0&\Gamma^{\rm
out}_{\widetilde{\downarrow\Uparrow}\rightarrow\Downarrow}&\Gamma^{\rm
out}_{\widetilde{\downarrow\Downarrow}\rightarrow\Downarrow}&0&-\Gamma^{\rm
in}_{\Downarrow\rightarrow\widetilde{\uparrow\Uparrow}}-\Gamma^{\rm
in}_{\Downarrow\rightarrow\widetilde{\uparrow\Downarrow}}\end{pmatrix},$ (58)

where the rates are given by

$\displaystyle\Gamma^{\rm in}_{\rm 1P\rightarrow 2P}=\Gamma^{\rm
in}_{\uparrow}M_{\rm 1P,2P},$ (59a) $\displaystyle\Gamma^{\rm out}_{\rm
2P\rightarrow 1P}=\Gamma^{\rm out}_{\downarrow}M_{\rm 1P,2P}$ (59b)

## References

  * Cohen-Tannoudji  _et al._ [1977] C. Cohen-Tannoudji, B. Diu, and F. Laloe,  _Quantum Mechanics, Volume 1_ , Vol. 1 (John Wiley & Sons, 1977).
  * Elzerman  _et al._ [2004] J. Elzerman, R. Hanson, L. Willems van Beveren, B. Witkamp, L. Vandersypen, and L. P. Kouwenhoven, Single-shot read-out of an individual electron spin in a quantum dot, Nature 430, 431 (2004).
  * Xiao  _et al._ [2004] M. Xiao, I. Martin, E. Yablonovitch, and H. Jiang, Electrical detection of the spin resonance of a single electron in a silicon field-effect transistor, Nature 430, 435 (2004).
  * Morello  _et al._ [2010] A. Morello, J. J. Pla, F. A. Zwanenburg, K. W. Chan, K. Y. Tan, H. Huebl, M. Möttönen, C. D. Nugroho, C. Yang, J. A. Van Donkelaar, _et al._ , Single-shot readout of an electron spin in silicon, Nature 467, 687 (2010).
  * Braginsky and Khalili [1996] V. B. Braginsky and F. Y. Khalili, Quantum nondemolition measurements: the route from toys to tools, Reviews of Modern Physics 68, 1 (1996).
  * Wallraff  _et al._ [2005] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, J. Majer, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Approaching unit visibility for control of a superconducting qubit with dispersive readout, Physical Review Letters 95, 060501 (2005).
  * Vijay  _et al._ [2011] R. Vijay, D. Slichter, and I. Siddiqi, Observation of quantum jumps in a superconducting artificial atom, Physical Review Letters 106, 110502 (2011).
  * Neumann  _et al._ [2010] P. Neumann, J. Beck, M. Steiner, F. Rempp, H. Fedder, P. R. Hemmer, J. Wrachtrup, and F. Jelezko, Single-shot readout of a single nuclear spin, Science 329, 542 (2010).
  * Pla  _et al._ [2013] J. J. Pla, K. Y. Tan, J. P. Dehollain, W. H. Lim, J. J. Morton, F. A. Zwanenburg, D. N. Jamieson, A. S. Dzurak, and A. Morello, High-fidelity readout and control of a nuclear spin qubit in silicon, Nature 496, 334 (2013).
  * Muhonen  _et al._ [2014] J. T. Muhonen, J. P. Dehollain, A. Laucht, F. E. Hudson, R. Kalra, T. Sekiguchi, K. M. Itoh, D. N. Jamieson, J. C. McCallum, A. S. Dzurak, _et al._ , Storing quantum information for 30 seconds in a nanoelectronic device, Nature Nanotechnology 9, 986 (2014).
  * Philips  _et al._ [2022] S. G. Philips, M. T. Mądzik, S. V. Amitonov, S. L. de Snoo, M. Russ, N. Kalhor, C. Volk, W. I. Lawrie, D. Brousse, L. Tryputen, _et al._ , Universal control of a six-qubit quantum processor in silicon, Nature 609, 919 (2022).
  * Terhal [2015] B. M. Terhal, Quantum error correction for quantum memories, Reviews of Modern Physics 87, 307 (2015).
  * Ruskov and Korotkov [2003] R. Ruskov and A. N. Korotkov, Entanglement of solid-state qubits by measurement, Physical Review B 67, 241305 (2003).
  * Itano  _et al._ [1990] W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, Quantum zeno effect, Physical Review A 41, 2295 (1990).
  * Gambetta  _et al._ [2008] J. Gambetta, A. Blais, M. Boissonneault, A. A. Houck, D. Schuster, and S. M. Girvin, Quantum trajectory approach to circuit qed: Quantum jumps and the zeno effect, Physical Review A 77, 012112 (2008).
  * Mabuchi and Doherty [2002] H. Mabuchi and A. Doherty, Cavity quantum electrodynamics: coherence in context, Science 298, 1372 (2002).
  * Blais  _et al._ [2021] A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Circuit quantum electrodynamics, Reviews of Modern Physics 93, 025005 (2021).
  * Koch  _et al._ [2007] J. Koch, M. Y. Terri, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Charge-insensitive qubit design derived from the cooper pair box, Physical Review A 76, 042319 (2007).
  * Mi  _et al._ [2018] X. Mi, M. Benito, S. Putz, D. M. Zajac, J. M. Taylor, G. Burkard, and J. R. Petta, A coherent spin–photon interface in silicon, Nature 555, 599 (2018).
  * Mendes and Vandersypen [2018] U. Mendes and L. Vandersypen, Strong spin-photon coupling in silicon, Science 359, 1123 (2018).
  * Arute  _et al._ [2019] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. Brandao, D. A. Buell, _et al._ , Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019).
  * Wu  _et al._ [2021] Y. Wu, W.-S. Bao, S. Cao, F. Chen, M.-C. Chen, X. Chen, T.-H. Chung, H. Deng, Y. Du, D. Fan, M. Gong, C. Guo, C. Guo, S. Guo, L. Han, L. Hong, H.-L. Huang, Y.-H. Huo, L. Li, N. Li, S. Li, Y. Li, F. Liang, C. Lin, J. Lin, H. Qian, D. Qiao, H. Rong, H. Su, L. Sun, L. Wang, S. Wang, D. Wu, Y. Xu, K. Yan, W. Yang, Y. Yang, Y. Ye, J. Yin, C. Ying, J. Yu, C. Zha, C. Zhang, H. Zhang, K. Zhang, Y. Zhang, H. Zhao, Y. Zhao, L. Zhou, Q. Zhu, C.-Y. Lu, C.-Z. Peng, X. Zhu, and J.-W. Pan, Strong quantum computational advantage using a superconducting quantum processor, Physical Review Letters 127, 180501 (2021).
  * Vandersypen  _et al._ [2017] L. Vandersypen, H. Bluhm, J. Clarke, A. Dzurak, R. Ishihara, A. Morello, D. Reilly, L. Schreiber, and M. Veldhorst, Interfacing spin qubits in quantum dots and donors—hot, dense, and coherent, npj Quantum Information 3, 34 (2017).
  * Nakajima  _et al._ [2019] T. Nakajima, A. Noiri, J. Yoneda, M. R. Delbecq, P. Stano, T. Otsuka, K. Takeda, S. Amaha, G. Allison, K. Kawasaki, _et al._ , Quantum non-demolition measurement of an electron spin qubit, Nature Nanotechnology 14, 555 (2019).
  * Xue  _et al._ [2020] X. Xue, B. D’Anjou, T. F. Watson, D. R. Ward, D. E. Savage, M. G. Lagally, M. Friesen, S. N. Coppersmith, M. A. Eriksson, W. A. Coish, _et al._ , Repetitive quantum nondemolition measurement and soft decoding of a silicon spin qubit, Physical Review X 10, 021006 (2020).
  * Jiang  _et al._ [2009] L. Jiang, J. Hodges, J. Maze, P. Maurer, J. Taylor, D. Cory, P. Hemmer, R. L. Walsworth, A. Yacoby, A. S. Zibrov, _et al._ , Repetitive readout of a single electronic spin via quantum logic with nuclear spin ancillae, Science 326, 267 (2009).
  * Robledo  _et al._ [2011] L. Robledo, L. Childress, H. Bernien, B. Hensen, P. F. Alkemade, and R. Hanson, High-fidelity projective read-out of a solid-state spin quantum register, Nature 477, 574 (2011).
  * Waldherr  _et al._ [2011] G. Waldherr, J. Beck, M. Steiner, P. Neumann, A. Gali, T. Frauenheim, F. Jelezko, and J. Wrachtrup, Dark states of single nitrogen-vacancy centers in diamond unraveled by single shot nmr, Physical Review Letters 106, 157601 (2011).
  * Dréau  _et al._ [2013] A. Dréau, P. Spinicelli, J. Maze, J.-F. Roch, and V. Jacques, Single-shot readout of multiple nuclear spin qubits in diamond under ambient conditions, Physical Review Letters 110, 060502 (2013).
  * Lovchinsky  _et al._ [2016] I. Lovchinsky, A. Sushkov, E. Urbach, N. P. de Leon, S. Choi, K. De Greve, R. Evans, R. Gertner, E. Bersin, C. Müller, _et al._ , Nuclear magnetic resonance detection and spectroscopy of single proteins using quantum logic, Science 351, 836 (2016).
  * Pla  _et al._ [2014] J. J. Pla, F. A. Mohiyaddin, K. Y. Tan, J. P. Dehollain, R. Rahman, G. Klimeck, D. N. Jamieson, A. S. Dzurak, and A. Morello, Coherent control of a single si 29 nuclear spin qubit, Physical Review Letters 113, 246801 (2014).
  * Dyte  _et al._ [2023] H. E. Dyte, G. Gillard, S. Manna, S. F. C. da Silva, A. Rastelli, and E. A. Chekhovich, Quantum non-demolition measurement of an electron spin qubit through its low-energy many-body spin environment, arXiv preprint arXiv:2307.00308 10.48550/arXiv.2307.00308 (2023).
  * Zajac  _et al._ [2018] D. M. Zajac, A. J. Sigillito, M. Russ, F. Borjans, J. M. Taylor, G. Burkard, and J. R. Petta, Resonantly driven cnot gate for electron spins, Science 359, 439 (2018).
  * Huang  _et al._ [2019] W. Huang, C. Yang, K. Chan, T. Tanttu, B. Hensen, R. Leon, M. Fogarty, J. Hwang, F. Hudson, _et al._ , Fidelity benchmarks for two-qubit gates in silicon, Nature 569, 532 (2019).
  * Kalra  _et al._ [2014] R. Kalra, A. Laucht, C. D. Hill, and A. Morello, Robust two-qubit gates for donors in silicon controlled by hyperfine interactions, Physical Review X 4, 021044 (2014).
  * Madzik  _et al._ [2021] M. T. Madzik, A. Laucht, F. E. Hudson, A. M. Jakob, B. C. Johnson, D. N. Jamieson, K. M. Itoh, A. S. Dzurak, and A. Morello, Conditional quantum operation of two exchange-coupled single-donor spin qubits in a mos-compatible silicon device, Nature Communications 12, 181 (2021).
  * Cota  _et al._ [2003] E. Cota, R. Aguado, C. Creffield, and G. Platero, Spin-polarized pumping in a double quantum dot, Nanotechnology 14, 152 (2003).
  * Golovach and Loss [2004] V. N. Golovach and D. Loss, Transport through a double quantum dot in the sequential tunneling and cotunneling regimes, Physical Review B 69, 245327 (2004).
  * Johnson  _et al._ [2022] M. A. Johnson, M. T. Mądzik, F. E. Hudson, K. M. Itoh, A. M. Jakob, D. N. Jamieson, A. Dzurak, and A. Morello, Beating the thermal limit of qubit initialization with a bayesian maxwell’s demon, Physical Review X 12, 041008 (2022).
  * Osika  _et al._ [2022] E. N. Osika, S. K. Gorman, S. Monir, Y.-L. Hsueh, M. Borscz, H. Geng, B. Thorgrimsson, M. Y. Simmons, and R. Rahman, Shelving and latching spin readout in atom qubits in silicon, Physical Review B 106, 075418 (2022).
  * Dehollain  _et al._ [2012] J. Dehollain, J. Pla, E. Siew, K. Tan, A. Dzurak, and A. Morello, Nanoscale broadband transmission lines for spin qubit control, Nanotechnology 24, 015202 (2012).
  * Joecker  _et al._ [2021] B. Joecker, A. D. Baczewski, J. K. Gamble, J. J. Pla, A. Saraiva, and A. Morello, Full configuration interaction simulations of exchange-coupled donors in silicon using multi-valley effective mass theory, New Journal of Physics 23, 073007 (2021).
  * Morello  _et al._ [2020] A. Morello, J. J. Pla, P. Bertet, and D. N. Jamieson, Donor spins in silicon for quantum technologies, Advanced Quantum Technologies 3, 2000005 (2020).
  * Morello  _et al._ [2009] A. Morello, C. Escott, H. Huebl, L. W. Van Beveren, L. Hollenberg, D. Jamieson, A. Dzurak, and R. Clark, Architecture for high-sensitivity single-shot readout and control of the electron spin of individual donors in silicon, Physical Review B 80, 081307 (2009).
  * Pirkkalainen  _et al._ [2010] J.-M. Pirkkalainen _et al._ , Probe and control of the reservoir density of states in single-electron devices, Physical Review B 81, 161304 (2010).
  * Tenberg  _et al._ [2019] S. B. Tenberg, S. Asaad, M. T. Madzik, M. A. Johnson, B. Joecker, A. Laucht, F. E. Hudson, K. M. Itoh, A. M. Jakob, B. C. Johnson, _et al._ , Electron spin relaxation of single phosphorus donors in metal-oxide-semiconductor nanoscale devices, Physical Review B 99, 205306 (2019).
  * Hsueh  _et al._ [2023] Y.-L. Hsueh, L. Kranz, D. Keith, S. Monir, Y. Chung, S. K. Gorman, R. Rahman, and M. Y. Simmons, Hyperfine-mediated spin relaxation in donor-atom qubits in silicon, Physical Review Research 5, 023043 (2023).
  * Savytskyy  _et al._ [2023] R. Savytskyy, T. Botzem, I. Fernandez de Fuentes, B. Joecker, J. J. Pla, F. E. Hudson, K. M. Itoh, A. M. Jakob, B. C. Johnson, D. N. Jamieson, _et al._ , An electrically driven single-atom “flip-flop” qubit, Science Advances 9, eadd9408 (2023).
  * Slichter [2013] C. P. Slichter,  _Principles of magnetic resonance_ , Vol. 1 (Springer Science & Business Media, 2013).
  * Wang  _et al._ [2016] Y. Wang, C.-Y. Chen, G. Klimeck, M. Y. Simmons, and R. Rahman, Characterizing si: P quantum dot qubits with spin resonance techniques, Scientific Reports 6, 31830 (2016).
  * Hile  _et al._ [2018] S. J. Hile, L. Fricke, M. G. House, E. Peretz, C. Y. Chen, Y. Wang, M. Broome, S. K. Gorman, J. G. Keizer, R. Rahman, _et al._ , Addressable electron spin resonance using donors and donor molecules in silicon, Science Advances 4, eaaq1459 (2018).
  * Bourassa  _et al._ [2020] A. Bourassa, C. P. Anderson, K. C. Miao, M. Onizhuk, H. Ma, A. L. Crook, H. Abe, J. Ul-Hassan, T. Ohshima, N. T. Son, _et al._ , Entanglement and control of single nuclear spins in isotopically engineered silicon carbide, Nature Materials 19, 1319 (2020).
  * Thiele  _et al._ [2014] S. Thiele, F. Balestro, R. Ballou, S. Klyatskaya, M. Ruben, and W. Wernsdorfer, Electrically driven nuclear spin resonance in single-molecule magnets, Science 344, 1135 (2014).
  * Willke  _et al._ [2018] P. Willke, Y. Bae, K. Yang, J. L. Lado, A. Ferrón, T. Choi, A. Ardavan, J. Fernández-Rossier, A. J. Heinrich, and C. P. Lutz, Hyperfine interaction of individual atoms on a surface, Science 362, 336 (2018).
  * Yang  _et al._ [2022] J. Yang, W. Fan, Y. Zhang, C. Duan, G. G. de Boo, R. L. Ahlefeldt, J. J. Longdell, B. C. Johnson, J. C. McCallum, M. J. Sellars, S. Rogge, C. Yin, and J. Du, Zeeman and hyperfine interactions of a single ${}^{167}\mathrm{Er}^{3+}$ ion in si, Physical Review B 105, 235306 (2022).
  * Ivey and Mieher [1975] J. L. Ivey and R. L. Mieher, Ground-state wave function for shallow-donor electrons in silicon. ii. anisotropic electron-nuclear-double-resonance hyperfine interactions, Physical Review B 11, 849 (1975).
  * Mądzik  _et al._ [2020] M. T. Mądzik, T. D. Ladd, F. E. Hudson, K. M. Itoh, A. M. Jakob, B. C. Johnson, J. C. McCallum, D. N. Jamieson, A. S. Dzurak, A. Laucht, _et al._ , Controllable freezing of the nuclear spin bath in a single-atom spin qubit, Science Advances 6, eaba3442 (2020).
  * Hensen  _et al._ [2020] B. Hensen, W. Wei Huang, C.-H. Yang, K. Wai Chan, J. Yoneda, T. Tanttu, F. E. Hudson, A. Laucht, K. M. Itoh, T. D. Ladd, _et al._ , A silicon quantum-dot-coupled nuclear spin qubit, Nature Nanotechnology 15, 13 (2020).
  * Prokof’Ev and Stamp [2000] N. Prokof’Ev and P. Stamp, Theory of the spin bath, Reports on Progress in Physics 63, 669 (2000).
  * House  _et al._ [2013] M. G. House, M. Xiao, G. Guo, H. Li, G. Cao, M. M. Rosenthal, and H. Jiang, Detection and measurement of spin-dependent dynamics in random telegraph signals, Physical Review Letters 111, 126803 (2013).