# Quantum Gates Between Mesoscopic Spin Ensembles

Mohamad Niknam  Department of Chemistry and Biochemistry, University of
California Los Angeles, 607 Charles E. Young Drive East, Los Angeles,
California 90095-1059, USA  Center for Quantum Science and Engineering, UCLA
Robert N. Schwartz  Department of Electrical and Computer Engineering, UCLA
Louis-S. Bouchard  Department of Chemistry and Biochemistry, University of
California Los Angeles, 607 Charles E. Young Drive East, Los Angeles,
California 90095-1059, USA  Center for Quantum Science and Engineering, UCLA
California NanoSystems Institute, UCLA

###### Abstract

Quantum algorithmics with single spins poses serious technological challenges
such as precision fabrication, rapid decoherence, atomic-scale addressing and
readout. To circumvent atomic-scale challenges, we examine the case of fully
polarized mesoscopic spin ensembles (spin-coherent states) whose total angular
momenta states map to qudit submanifolds. We show that in the limit where the
size of the ensembles is small compared to their separation, it is possible to
treat them as qubits with an effective coupling strength that scales with the
number of spins. If the spins within each ensemble are decoupled (e.g., via
control fields, spinning or diffusional averaging or materials engineering),
one- and two-qubit gate operations can be implemented with high fidelities.

Introduction.– Large-scale quantum operations are difficult to develop,
requiring isolation of a large number of individual qubits, precise control of
their quantum states, error correction schemes and high-accuracy measurements.
Existing qubits are implemented in environments requiring high vacuums or low
temperatures. Several recent initiatives have been driving hardware
development for the realization of new quantum technologies that are more
scalable, robust, and less physically demanding. Qubits should have long
coherence times or be able to retain quantum information for times much longer
than the operation time of quantum gates. Arbitrary unitary transformations on
the set of all qubits can be constructed up to desired precision by composing
quantum gates chosen from a set of universal quantum gates. The latter
typically consists of single-qubit gates (one per qubit) and one two-qubit
gate (acting on pairs of qubits) [1]. Implementation of the two-qubit gate has
proven to be the most challenging, and even the leading platforms for quantum
computing, i.e., superconducting qubits [2], trapped ions [3], and Rydberg
atoms [4] have error rates exceeding 0.1% per gate, which is an order of
magnitude larger than the threshold required for fault-tolerant quantum
computing [5].

Ensemble quantum computing has the advantage of many replicas, eliminating the
need to repeat projective readouts thousands of times to obtain the
statistical weights in the many-qubit wavefunction. Ensembles can have
favorable properties such as ease of fabrication, built-in robustness, longer
storage times, more sensitive detection, etc. Liquid-state NMR is of
historical significance because it provided the platform for the first
experimental demonstrations of a working quantum algorithm [6, 7]. Nuclear
spins in the liquid-state feature long coherence times, typically in the order
of seconds. They interact weakly, meaning that they behave independently and
can be addressed individually. Qubits can be selectively addressed using
frequency-selective (soft) pulses. This allows coherent control of quantum
devices containing up to 12 qubits [8] and implementation of Shor’s quantum
algorithm [9]. However, nuclear spin qubits are not scalable due to the
difficulty of initializing pure quantum states (highly mixed pseudo-pure
states above $\approx 1$ mK), and synthesizing molecules with a large number
of individually addressable nuclear spins coupled to one another. Indeed,
molecules can only accommodate the selective addressing of a small number of
distinct nuclei due to the narrow range of chemical shifts. Therefore,
obtaining entanglement in liquid-state qubits for ensemble NMR quantum
computing is nearly impossible [10]. NMR ensemble computing involves
performing collective quantum operations on molecules, followed by averaging
the state of qubits over the ensemble. To address the important issues of
scalability and ease-of-fabrication, we revisit the idea of ensemble computing
and consider reversing the order of averaging: ensemble-average spins to
obtain large qubits, then perform operations among the ensembles. In doing so,
we use the rotational properties of large-$J$ angular momentum operators,
which leads to a realization of qubit transformations similar to the single
spin case [11]. In this paper we will assume the ability to fully polarize
spins and work with spin-coherent states that we term E-qubits. Practical
considerations are discussed.

For nuclear spins the notation $\mathbf{I}$ is used instead of $\mathbf{J}$ to
denote the spin angular momentum. Suppose we have an ensemble of $N$
molecules, each with $n$ nuclear spins
$\\{\mathbf{I}^{1},\dots,\mathbf{I}^{n}\\}$. The Hilbert space for the nuclear
spin degrees of freedom in a molecule has dimension
$\prod_{i=1}^{n}(2I^{i}+1)$. The state of each molecule’s nuclear spins is
represented by a projector $P_{\omega}(\mathbf{I}^{1},\dots,\mathbf{I}^{n})$,
$\omega=1,\dots,N$. Let $\ket{\varphi_{i}(\omega)}$ be the wavefunction
describing the state of spin $i$ on molecule $\omega$. Each projector is
initially a product state, i.e.,
$P_{\omega}=\ket{\psi_{\omega}}\bra{\psi_{\omega}}$, where
$\ket{\psi_{\omega}}=\ket{\varphi_{1}(\omega)}\otimes\dots\otimes\ket{\varphi_{n}(\omega)}$,
i.e.,

$P_{\omega}=\ket{\varphi_{1}(\omega)}\bra{\varphi_{1}(\omega)}\otimes\dots\otimes\ket{\varphi_{n}(\omega)}\bra{\varphi_{n}(\omega)}.$

Here, $\omega$ is standard notation in probability theory to denote the
elementary outcome of a random variable. The unitary propagator corresponding
to the quantum circuit is denoted $U(\mathbf{I}^{1},\dots,\mathbf{I}^{n})$.
The final projector is
$U(\mathbf{I}^{1},\dots,\mathbf{I}^{n})P_{\omega}(\mathbf{I}^{1},\dots,\mathbf{I}^{n})U(\mathbf{I}^{1},\dots,\mathbf{I}^{n})^{\dagger}$.
Averaging over $N$ molecules (denoted by overline):

$\overline{U(\mathbf{I}^{1},\dots,\mathbf{I}^{n})P_{\omega}(\mathbf{I}^{1},\dots,\mathbf{I}^{n})U(\mathbf{I}^{1},\dots,\mathbf{I}^{n})^{\dagger}}\\\
=U(\mathbf{I}^{1},\dots,\mathbf{I}^{n})\rho(\mathbf{I}^{1},\dots,\mathbf{I}^{n})U(\mathbf{I}^{1},\dots,\mathbf{I}^{n})^{\dagger}\equiv\rho^{\prime}$

where

$\rho(\mathbf{I}^{1},\dots,\mathbf{I}^{n})\equiv\overline{P_{\omega}(\mathbf{I}^{1},\dots,\mathbf{I}^{n})}=\sum_{\omega=1}^{N}p_{\omega}P_{\omega}(\mathbf{I}^{1},\dots,\mathbf{I}^{n})$

is a statistical (density) operator and $\sum_{\omega}p_{\omega}=1$. The
readout of observable operator $A$ is obtained as:

$A^{(e_{1})}\equiv\braket{A(\mathbf{I}^{1},\dots,\mathbf{I}^{n})}=\mbox{Tr}[\rho^{\prime}(\mathbf{I}^{1},\dots,\mathbf{I}^{n})A(\mathbf{I}^{1},\dots,\mathbf{I}^{n})]$

where $e_{1}$ distinguishes this form of ensemble averaging. On the other
hand, suppose we start with $n$ ensembles, each described by a statistical
operator, $\rho_{i}(\mathbf{I}^{i})$, describing this initial ensemble
averaging. Each ensemble contains $N$ spins. The initial state is a product
state $\rho_{1}(\mathbf{I}^{1})\otimes\dots\otimes\rho_{n}(\mathbf{I}^{n})$,
where

$\rho_{i}(\mathbf{I}_{i})=\sum_{\omega}p_{\omega,i}\ket{\varphi_{i}(\omega)}\bra{\varphi_{i}(\omega)},\quad
i=1,\dots,n.$

The propagator for the quantum circuit is denoted
$V(\mathbf{I}^{1},\dots,\mathbf{I}^{n})$. (The notation $V$ instead of $U$
indicates that the circuit implementations may be different, depending on the
particular physical implementation of the qubits.) Evolution of this product
state leads to:

$\rho^{\prime\prime}=V(\mathbf{I}^{1},\dots,\mathbf{I}^{n})\rho_{1}(\mathbf{I}^{1})\otimes\dots\otimes\rho_{n}(\mathbf{I}^{n})V(\mathbf{I}^{1},\dots,\mathbf{I}^{n})^{\dagger}$

The readout for this ensemble-averaging method, labeled $e_{2}$, leads to:

$A^{(e_{2})}\equiv\braket{A(\mathbf{I}^{1},\dots,\mathbf{I}^{n})}=\mbox{Tr}[\rho^{\prime\prime}(\mathbf{I}^{1},\dots,\mathbf{I}^{n})A(\mathbf{I}^{1},\dots,\mathbf{I}^{n})].$

As a consequence of the different orders of averaging, we note that even if
$U=V$, $A^{(e_{1})}$ and $A^{(e_{2})}$ do not generally give the same result.
The averaging to obtain $A^{(e_{1})}$ only requires specifying the probability
distribution $\\{p_{\omega}\\}_{\omega=1}^{N}$ whereas averaging for
$A^{(e_{2})}$ is described by a set $i=1,\dots,n$ of distributions
$\\{p_{\omega,i}\\}_{\omega=1}^{N}$, $\sum_{\omega}p_{\omega,i}=1$.

Averaging over molecules before any computation is expected to yield different
outcomes. Qubits occupying larger physical volumes would significantly relax
the manufacturing requirements, as the placement of single-atom qubits in a
solid lattice (or in a host molecule, for that matter) at precise locations
and spacings has been challenging due to the lack of methods to implant point
defects with good precision [12]. A second advantage is the robustness of the
quantum operations with respect to environmental noise. A single spin
undergoing decoherence leads to the loss of its quantum state, whereas the
quantum state stored in ensembles may survive longer due to the inherent
robustness of ensembles (see Appendix I.3).

Suppose we have a large ensemble ($N\cdot n\gg 1$) of nuclear spins
$\\{\mathbf{I}^{i}\\}$, where $N$ is the number of localized spin ensembles
and $n$ is the number of spins per ensemble. We will call these mesoscopic
spin ensembles E-qubits for reasons that will become clear shortly. To
simplify the presentation we assume that $I=1/2$ for nuclear spins; we will
also assume that all spins in each localized ensemble have the same
gyromagnetic ratio. This choice only affects some constants and scaling
factors below, and does not lead to a loss of generality. The magnetic dipole
interaction between $N\cdot n$ spins cannot be used for quantum computing
because of the impossibility of addressing each spin individually. Also, the
exact couplings in atomic ensembles are not known, making it impossible to
design precise gates. We consider: (1) two ($N=2$) localized ensembles $A$ and
$B$ each containing $n$ spins; (2) assume full polarization of all spins in
the ensembles; (3) spins within each ensemble are decoupled. Spin ensembles
are rarely discussed as candidate qubits because of their multilevel energy
structure and high degeneracies, which differs from the structure of a qubit.
If each ensemble admits a $\mathfrak{su}(2)$-like algebra, the ensembles can
be viewed as qubits, even though they are in reality multilevel qudits (Fig.
1). The magnetic interaction between all pairs of spins is

$\\!\\!\sum_{i\in A,\atop j\in
B}\mathbf{I}^{i}\cdot\mathtensor{D}_{ij}\cdot\mathbf{I}^{j},\;\;\mathtensor{D}_{ij}=-\frac{\mu_{0}}{4\pi}\frac{\gamma_{i}\gamma_{j}\hbar^{2}}{|\vec{r}_{ij}|^{3}}\left(3\frac{\vec{r}_{ij}\otimes\vec{r}_{ij}}{|\vec{r}_{ij}|^{2}}-\mathbb{1}_{3}\right)$
(1)

where $\vec{r}_{ij}$ is the vector connecting spin $i$ from $A$ to spin $j$
from $B$ (Fig. 1a). $\mu_{0}$ is the vacuum permeability and $\gamma$ is the
gyromagnetic ratio. If there are $N$ such localized spin ensembles, each
containing $n$ spin 1/2 particles the dimension of the spin algebra
$\mathfrak{su}(2^{Nn})$ is $2^{2Nn}-1$. Clean quantum gates are still not
possible due to the random distribution of $\vec{r}_{ij}$ values. Moreover,
this type of interaction over macroscopic distances is rarely considered due
to the rapid $1/r^{3}$ falloff of the dipolar interaction. However, in a
specific cooperative geometry, clean gates are possible via cooperative
couplings, as explained below.

Consider a geometry involving two spherical volumes $A$ and $B$ (Fig. 1a)
whose centers are separated by a constant vector $\vec{r}_{0}$ and the
distance of each spin from the center of each region is much smaller than
$|\vec{r}_{0}|$, i.e., $\delta_{A}\ll r_{0}$, $\delta_{B}\ll r_{0}$. The
internuclear vector $\vec{r}_{ij}$ is nearly equal to $\vec{r}_{0}$. Expanding
$|\vec{r}_{ij}|^{-3}$ in the small parameter
$\vec{\epsilon}=\vec{\delta}_{B}-\vec{\delta}_{A}$ about $\vec{r}_{0}$ gives

$\frac{1}{|\vec{r}_{0}+\vec{\epsilon}|^{3}}=\frac{1}{r_{0}^{3}}-3\frac{\vec{r}_{0}\cdot\vec{\epsilon}}{r_{0}^{5}}+O(|\epsilon|^{2}).$
(2)

The zeroth order term (i.e., neglecting terms of order $|\epsilon/r_{0}|$ and
higher) gives the effective Hamiltonian $\mathcal{H}_{D}$

$\sum_{i\in A,\atop j\in
B}\mathbf{I}^{i}\cdot\mathtensor{D}_{AB}\cdot\mathbf{I}^{j},\quad\mathtensor{D}_{AB}=-\frac{\mu_{0}\gamma_{A}\gamma_{B}\hbar^{2}}{4\pi
r_{0}^{3}}\left(3\frac{\vec{r}_{0}\otimes\vec{r}_{0}}{r_{0}^{2}}-\mathbb{1}_{3}\right),$

where the coupling tensor is now constant. It is therefore independent of $i$
and $j$, the indices of summation. The Hamiltonian can be written as:

$\mathcal{H}_{D}=\bigl{(}\sum_{i=1}^{n}\mathbf{I}^{A,i}\bigr{)}\cdot\mathtensor{D}_{AB}\cdot\bigl{(}\sum_{j=1}^{n}\mathbf{I}^{B,j}\bigr{)}=\mathbf{I}^{A}\cdot\mathtensor{D}_{AB}\cdot\mathbf{I}^{B},$
(3)

where $\mathbf{I}^{A}=\sum_{i=1}^{n}\mathbf{I}^{A,i}$ and
$\mathbf{I}^{B}=\sum_{j=1}^{n}\mathbf{I}^{B,j}$ are total angular momentum
operators. We thus have a bilinear coupling
$\mathbf{I}^{A}\cdot\mathtensor{D}_{AB}\cdot\mathbf{I}^{B}$ involving two
large spins $A$ and $B$.

Also, since
$[I_{\alpha}^{A},I_{\beta}^{A}]=i\epsilon_{\alpha\beta\gamma}I_{\gamma}^{A}$,
$[\mathbf{I}^{A},\mathbf{I}^{B}]=0$, the action of the total angular momentum
operator on the composite $\prod_{i=1}^{n}(2I^{i}+1)$ dimensional Hilbert
space of a localized ensemble constitutes a representation of the
$\mathfrak{su}(2)$ Lie algebra that is reducible. Due to this rotational
symmetry that maps to that of qubits, ensembles $A$ and $B$ are examples of
E-qubits. (The $\mathfrak{su}(2)$ symmetry of large spins has been known since
1932 [13].) If we have $N$ localized spin ensembles, each with $n$ spins, the
spin-Hamiltonian is the sum of Zeeman (static, control fields) and point
dipolar interaction between the E-qubits,

$\displaystyle\mathcal{H}_{Z}+\mathcal{H}_{D}=\sum_{Q=1}^{N}\mathbf{I}^{Q}\cdot\bigl{(}\vec{h}^{Q}(t)+\frac{1}{2}\sum_{P\neq
Q}^{N}\mathtensor{D}_{QP}\cdot\mathbf{I}^{P}\bigr{)}$ (4)

with $\vec{h}^{Q}=(h_{x}^{Q},h_{y}^{Q},h_{z}^{Q})$ is an external field. It
has $\otimes\mathfrak{su}(2)^{N}\simeq\mathfrak{su}(2^{N})$ symmetry. Even
though E-qubits are multilevel systems, the Hamiltonian (4) of the $nN$-spins
system is equivalent to an effective $N$-qubit Hamiltonian, which is
controllable. Controllability would not be possible with the lower symmetric
form (1). The spin operator algebra is also much larger, with dimension
$(2I+1)^{2Nn}-1$ for (1) vs $(2I+1)^{2N}-1$ for Eq. (4).

Figure 1: a) Schematic of two E-qubits $A$ and $B$ coupled over mesoscopic
distance. Here,
$\vec{r}_{ij}=\vec{r}_{0}+\vec{\delta}_{B}-\vec{\delta}_{A}\equiv\vec{r}_{0}+\vec{\epsilon}$.
For $\epsilon\ll 1$ the Hamiltonian of these two $n$-spins systems is
equivalent to a two qubit Hamiltonian. We denote magnitudes of vectors without
an arrow, i.e., $\delta_{A}\equiv|\vec{\delta}_{A}|$, etc. b) Energy levels
for an E-qubit. The single-quantum transition energy is denoted
$\hbar\omega_{Q}$. The states with all spins up and all spins down, constitute
the two level structure of a qubit. Control sequences are applied on
collective state of spins. The case with $I=1/2$ and single-spin basis states
$\ket{\uparrow}$ and $\ket{\downarrow}$ is shown.

Qubit state.– Each E-qubit occupies a mesoscopic volume and can therefore be
addressed selectively using nanomagnets [14], nanowires, or similar
technologies. Assuming $I=1/2$, the density matrix of subsystem $A$ containing
$n$ fully polarized spins $(\frac{1}{2}\mathbb{1}\pm I_{z})^{\otimes n}$,
corresponding to $\ket{I^{A},\pm I^{A}}\bra{I^{A},\pm I^{A}}$, in product
operator notation is

$\frac{1}{2^{n}}\sum_{m=0}^{n}2^{m-n}(\pm
1)^{m}\mathbb{1}_{2^{n}}^{A}\sum_{i_{1}<\dots<i_{m}}I_{z}^{A,i_{1}}I_{z}^{A,i_{2}}\dots
I_{z}^{A,i_{m}}$

where

$I_{z}^{A,i}=\overbrace{\mathbb{1}_{2}^{A}\otimes\mathbb{1}_{2}^{A}\otimes\dots\underbrace{\otimes(\sigma_{z}^{A}/2)\otimes}_{i-\mathrm{th~{}position}}\dots\otimes\mathbb{1}_{2}^{A}\otimes\mathbb{1}_{2}^{A}}^{n~{}\mathrm{factors}}.$

A collective $\pi$ rotation on all $n$ spins using the control field
$h_{x}^{A}(t)I_{x}^{A}\otimes\mathds{1}^{B}$, reverses the sign of all $m$-odd
terms, flipping $\ket{I^{A},I^{A}}\bra{I^{A},I^{A}}$ into
$\ket{I^{A},-I^{A}}\bra{I^{A},-I^{A}}$ and vice versa. These states map to the
usual computational basis $\ket{0}_{A}$ and $\ket{1}_{A}$. Here we do not have
selectivity to address intermediate transitions within each ensemble, so these
intermediate levels are not accessible through unitaries (Fig. 1) or
decoherence (see Appendix I.3 and I.4).

The microstates of an E-qubit can be viewed as a very large $d$-level system.
For a $d$-level system a natural quantum computational basis consists of the
top and bottom energy levels. Transitions between the top and bottom levels of
a $d$-level system are multiple-quantum transitions requiring frequencies
proportional to $d$. When $d$ is large, such energies become out of reach to
the experimentalist. On the other hand, we know from decades of NMR and EPR
experiments that very large $d$ (multi-quantum) transitions are never needed
if the spins are uncoupled or weakly coupled (i.e., interaction strength is
much smaller than the Zeeman energy). In the weak coupling limit, only single-
quantum excitations are needed, as each spin can be viewed as behaving
independently. In this case, unitary transformation rules of conventional
quantum computing with qubits (spin 1/2) or those of qudits (see Appendices
I.1 and I.2) for large spins apply, as far as the generation of single- and
two-qubit gates. If the spins were coupled sufficiently strongly as to create
a large spin, impractically high energy excitations may be needed to excite
magnetic transitions between the lowest and highest energy states. To reduce
the ion-ion couplings engineered materials may be required in which the sum of
all interactions (e.g., dipolar, indirect spin-spin, or exchange between
neighboring spins is weak. A notable approach has been magnon-cavity coupling
where introduction of strong exchange coupling results in narrow-linewidth
magnetostatic modes [15, 16].

Quantum gates.– To have universal quantum gates we need arbitrary single qubit
gates complemented by one entangling two-qubit gate. Suppose the initial state
of an E-qubit is fully polarized, e.g., $\ket{I^{A},\pm I^{A}}$. This fully
polarized state corresponds to a linear combination of fully polarized high-
rank axial tensor operators $\mathscr{Y}^{(k)0}(\mathbf{I}^{A})$, $0\leq k\leq
2I^{A}$, such that the single-qubit rotations are described by
$\sum_{q}\mathscr{D}^{(k)}_{q0}(\Omega)\mathscr{Y}^{(k)0}(\mathbf{I}^{A})$,
where $\mathscr{D}^{(k)}_{qq^{\prime}}(\Omega)$ are elements of Wigner $D$
matrices representing the rotation group in the subspace of E-qubit $A$ (see
Appendices I.1 and I.2). These collective rotations can be implemented using
unitary operators of the form
$\text{e}^{-i\theta\hat{n}\cdot\mathbf{I}^{A}}\otimes\mathbb{1}^{B}$, where
$\hat{n}$ is a rotation axis and $\theta$ is the angle of rotation.
Transformations in the full rotation group generate coherent rotations needed
for single-qubit gates. This will work as long as the spins within each
ensemble are decoupled from each other. The fidelity of quantum gates depends
on linewidth. The latter is influenced by couplings between spins in each
E-qubit. The fidelity of a quantum gate, to the first order, depends linearly
on relaxation rate and inverse of gate time [17]. We discuss decoupling
schemes below. Figure 2(a) demonstrates such rotations of spin-coherent states
describing the E-qubit with $n=5$ and $N=2$. Initially polarized states such
as $\ket{I^{A},I^{A}}$ in the coupled spin basis can easily be rotated into
(say) $\ket{I^{A},-I^{A}}$ using standard operations.

Figure 2: Demonstration of single- and two-qubit gates for E-qubits. Single
qubit gates are applied with standard rotation operators; the control
frequencies are independent of $n$. a) Rabi oscillations for the $n=5$ with
$R_{x}^{A}(\theta)$ applied to E-qubit A. The vertical axis shows the
amplitude of observed signal resulting from projection of the density matrix
on the $X$, $Y$, and $Z$ axes. Spins are chosen at random locations inside two
spheres with $r_{A}=r_{B}=30$ nm and $r_{0}=100$ nm. b) Evolution under the
Ising $zz$ interaction needed for the entangling CNOT gate, with $n=1-5$.
Black/blue dots project the evolution of the E-qubit B with full/effective
Hamiltonian, Eq. (1)/Eq. (3), on the vertical axis. c) Fourier transform of
signal from the E-qubit B evolving with the full Hamiltonian, blue points from
panel b, show a clear linear dependency of the coupling frequency to $n$. The
signal amplitude also grows with $n$ as expected.

The CNOT gate where the first E-qubit is the control and the second is the
target can be implemented with the following sequence of unitaries:

$\sqrt{i}R^{A}_{z}(\frac{\pi}{2})\cdot R^{B}_{z}(\frac{-\pi}{2})\cdot
R^{B}_{x}(\frac{\pi}{2})\cdot U_{zz}(\frac{1}{2\|\mathtensor{D}_{AB}\|})\cdot
R^{B}_{y}(\frac{\pi}{2}),$

where $\|\mathtensor{D}_{AB}\|$ is the norm of $\mathtensor{D}_{AB}$,
$U_{zz}(\frac{1}{2\|\mathtensor{D}_{AB}\|})=\exp(i\pi
I_{z}^{A}\tiny{\otimes}I_{z}^{B})$ describes evolution under an Ising
($zz$)-type interaction derived from Eq. (4). Figure 2(c) indicates simulated
time-evolution under the full $U_{zz}$ dynamics for two qudits $N=2$, with
$n=1,2,3,4,5$ spins, showing that the behavior is similar to the evolution of
qubits, except that the evolution frequency scales with the number of spins in
each ensemble. Figure 2 proves that the coupling between E-qubits scales
linearly with $n$. Although cases $n>5$ ($>$ 10 spins total) are difficult to
simulate on an average classical computer, this linear relationship holds for
any $n$. We discuss below specific examples for much larger $n$. With spin
ensembles, storage of quantum information is inherently robust to noise if the
spin-flips affect only a small fraction of the total moment. This is an
advantage over single-atom qubits. We now discuss decoupling schemes.

Solid-state nuclear spins.– In solids, the magnetic dipole interaction between
nearest neighbors is strong. In contrast to the liquid state interactions, the
spatial part of dipolar coupling, $(1-3\cos^{2}\theta_{ij})r_{ij}^{-3}$, is
not averaged to zero as the result of molecular motion. Spin evolution under
homonuclear dipolar interaction
$\sum_{i<j}D_{ij}(2I_{z}^{i}I_{z}^{j}-\frac{1}{2}(I_{+}^{i}I_{-}^{j}+I_{-}^{i}I_{+}^{j}))$
transforms uncorrelated spin terms such as $I_{x}^{i}\otimes\mathds{1}^{j}$
into correlated terms such as $I_{y}^{i}\otimes I_{z}^{j}$. By connecting
multiple spin pairs, a network of multispin correlated terms is generated that
grows in size exponentially fast. These correlated terms are not directly
observable and result in short $T_{2}$ times, on the order of microseconds
[18, 19, 20, 21].

To average out the dipolar interactions inside each E-qubit while preserving
the inter-qubit interactions, well-known techniques of dynamic decoupling can
be applied to average out unwanted dipolar interactions and obtain the desired
effective Hamiltonian. The idea is to either suppress unwanted interactions or
shift them to a different frequency band. One may leverage solid-echo cycles
such as MREV-8 [22], or sequences based on magic echo [23], or use magic angle
spin-locking such as Lee-Goldberg (LG) [24]. Consider the polarization-
inversion spin-exchange at the magic angle (PISEMA) experiment which is a
variant of LG decoupling sequence. The homonuclear dipolar interaction is
averaged to zero by locking the spins to an effective field
$\mathbf{B}_{\text{eff}}$ along the magic angle,
$\theta_{m}=\arccos(\sqrt{1/3})$. The spatial part of the dipolar interaction
averages to zero in each E-qubit, as long as
$\|\mathbf{B}_{\text{eff}}\|\gg\|\omega_{D}\|$ (Fig. 3a). To understand the
influence on the inter-qubit interaction, consider the magnetic moment of
spins in each qubit. As their magnetic moment vectors rotate around
$\mathbf{B}_{\text{eff}}$, the perpendicular component gains a time dependence
and averages out over the cycle, while the parallel component survives. For
all the spins with initial magnetic moment along the $z$ axis, a portion of
their initial magnetic moment remains after the application of the locking
field $\tilde{I}_{z}^{\|,i}=I_{z}^{i}\sin{\theta_{m}}=0.82\times I_{z}^{i}$
[25]. This is a large scaling factor in comparison to rival decoupling
sequences. Figure 3b shows the pulse sequence that combines these ideas for
quantum control of two E-qubits. By applying the PISEMA sequence on both
E-qubits simultaneously, we ensure that the intra-qubit interactions are
averaged to zero. Selectivity in applying $\mathbf{B}_{\text{eff}}$ (direction
and amplitude) means that the effective inter-qubit interactions can be
engineered. In particular, the effective magnetic moment in each qubit can be
oriented at a different direction and with different precession frequencies
resulting in the inter-qubit interaction of the form
$\sin{(\theta_{m})}^{2}\sum_{i}I_{z}^{A,i}\otimes\sum_{j}I_{z}^{B,j}$.

Figure 3: Quantum operations with solid state nuclear-spin ensemble qubits. a)
In the LG spin-locking experiment, an effective field along the magic angle is
applied to remove the dipolar interaction inside each E-qubit. b) Decoupling
sequence for removing the intra-qudit dipolar interaction in E-qubits $A$ and
$B$, while preserving the $zz$ interaction between them. c) Proposed
experimental setup for nuclear spins. The radius of each sphere is 30 nm and
they are 100 nm apart. d) Two-qubit entangling gate between two E-qubits. This
experiment is limited by the efficiency of control sequence in removing the
intra-qubit interaction. Residual interactions larger than $1\%$ have a
devastating effect on the fidelity of the two-qubit gate. Average gate
fidelity reduces linearly with the linewidth of spins in E-qubits.

To get an idea of the effective coupling strength, consider the example of
solid hydrogen ($T<14.01$ K, density $0.086$ g/cm3). The gyromagnetic ratio of
1H is $26.75\times 10^{7}$ rad/T/s. Consider an arrangement of spin ensembles
each confined to spherical volumes with radius of $r=30$ nm. The centers of
the ensembles are placed $r_{0}=100$ nm away from the centers of their nearest
neighbors. There are $5.9\times 10^{6}$ spins in a sphere of solid hydrogen,
leading to $D_{AB}=7.51\times 10^{-4}$ rad/s. The magnetic field seen by a
spin in sphere $A$ due to the field of spins in sphere $B$ is $104~{}\mu$T.
This corresponds to a precession frequency of 4.4 kHz. Entangling gates
between the two ensembles can therefore be executed in 227 $\mu$s. This
proposed implementation is limited by the efficiency of the control sequence
in removing intra-qubit interactions. The fidelity of two-qubit gate is mainly
determined by the robustness of two-qubit unitary interactions such as $zz$.
Figure 3d shows the effect of residual intra-qubit interaction in the
implementation of a macroscopic Ising $zz$ interaction.

Liquid state or gas phase.– In the gas or liquid state Eq. (4) automatically
takes the form of a mesoscopic Ising ($zz$)-type interaction. In high magnetic
fields, dipolar interactions between nearby molecules vanish due to diffusion,
i.e., $\braket{3\cos^{2}\theta-1}_{S^{2}}=0$, leading to intra-qubit
decoupling. However, beyond the diffusion length, the magnetic dipole
interaction does not vanish, leading to a nonzero mesoscopic inter-qubit
dipolar interaction. The end result is an Ising $zz$ form
$\mathcal{H}_{AB}\propto I^{A}_{z}I^{B}_{z}$ [26, 27, 28, 29], where the
proportionality constant depends only on the time-averaged internuclear vector
connecting pairs of spins between each ensemble. In spite of molecular motions
this effective mesoscopic long-range interaction is static in the geometry
described here. For spherical volumes ($r\sim 30$ nm) filled with liquid
hydrogen, $r$ is less than the diffusion length of H2 molecules during the
timescale of a NMR experiment, the dipolar interaction is averaged out within
each sphere. The effective coupling strength for inter-qubit interactions when
$r_{0}=100$ nm is 3.6 kHz. This is to be contrasted with the long relaxation
times ($>$ 1 s) of nuclear spins in liquids.

Nuclear-spin polarization.– In the initialization step all spins must be fully
polarized, which is a non-trivial task for nuclear spins. Depending on the
physical implementation of the E-qubits, hyperpolarization techniques have
been developed which may be applicable including dynamic nuclear polarization
[30], spin exchange optical pumping [31] and parahydrogen induced polarization
[32]. We also note that nuclear ferromagnetic ordering has been observed at
nanokelvin temperatures [33, 34, 35, 36, 37].

Molecular magnets.– Moving on from nuclear spins to discussing electronic
spins we switch our notation from $\mathbf{I}$ to $\mathbf{S}$. Molecular
magnets are high-spin clusters in a generally anisotropic crystalline
structure [38]. They are modeled as giant spins [39]; Mn25 has
$S=\tfrac{51}{2}$ [40] whereas Chen et al. [41] reported a giant spin ground
state of magnitude $S=91$. A $T_{2}$ time of 31 ms was recently demonstrated
for 171Yb3+ at 1.2 K [42]. They are possible building blocks for the physical
implementation of quantum processors. A realization of Grover’s quantum search
algorithm was achieved by coherent manipulation of a TbPc2 molecular magnet
[43]. Unlike nuclear spins, nearly full thermal polarization of electron
spins, $\langle
S_{z}\rangle=\mbox{Tr}(S_{z}^{\dagger}e^{-\beta\mathcal{H}})/\mbox{Tr}(e^{-\beta\mathcal{H}})\approx|S|$,
$\mathcal{H}=D[S_{z}^{2}-\tfrac{1}{3}S(S+1)]+E[S_{x}^{2}-S_{y}^{2}]-g\mu_{B}\mathbf{S\cdot
B}$, can be asymptotically approached at high magnetic fields and low
temperatures. As an example, Gd3+ ions in GdW30 complexes [44] ($D=1281$ MHz,
$E=294$ MHz, $S=\tfrac{7}{2}$, $g=2$) can reach 95% polarization at 3 T and 2
K. The crystal-field (CF) term is sufficiently weak (for GdW30, 3% of the
Zeeman term) and does not interfere significantly with the spin dynamics. Weak
CF interactions are a general feature of rare-earth ions, stemming from the
isolated nature of the $4f$ electrons, making it possible to treat them like
nearly free paramagnetic ions (unlike transition-metal ions whose CF
splittings are much larger [45]). Other methods could be considered, such as
chirality-induced spin selectivity (CISS), whose helical electrons are fully
polarized at high temperatures [46].

The methods of the previous section can be extended to ensembles of large
spins ($S>1/2$) provided that the algebra of large spins is used. Molecular
magnets are normally treated as multipole tensors,
$\mathscr{Y}^{(k)q}(\mathbf{S})$ where $0\leq k\leq 2S$, $-k\leq q\leq k$, in
the EPR literature [45]. A full quantum treatment of multipoles (e.g., Ref.
[11]) is presented in Appendices I.1 and I.2. There, we demonstrate the
creation of one- and two-qubit gates involving arbitrary spins. In this
context, it is worth reviewing the well-known relationship from angular
momentum theory between coupled and uncoupled representations. It is the basis
that gives rise to large composite electronic spins. A molecular magnet is
made up of $n$ magnetic atoms each with spin $\\{\mathbf{s}^{i}\\}$
($i=1,\dots,n$). The total spin is $\mathbf{S}=\sum_{i=1}^{n}\mathbf{s}^{i}$.
The Zeeman interaction with an applied magnetic field $\mathbf{B}(t)$ is
$\mu_{B}\mathbf{B}(t)\cdot\sum_{i=1}^{n}\mathtensor{g}_{i}\cdot\mathbf{s}^{i}$,
where $\mu_{B}$ is the Bohr magneton and $\\{\mathtensor{g}_{i}\\}$ are
$g$-tensors. Using the Wigner-Eckart theorem

$\braket{\alpha^{\prime},j^{\prime}m^{\prime}}{\mathscr{Y}^{(k)q}}{\alpha,jm}=\braket{jk;mq}{jk;j^{\prime}m^{\prime}}\frac{\braket{\alpha^{\prime}j^{\prime}}{\mathscr{Y}^{(k)}}{\alpha
j}}{\sqrt{2j^{\prime}+1}}$

applied to vectors:

$\braket{\alpha^{\prime},j^{\prime}m^{\prime}}{V^{(1)q}}{\alpha,jm}=\braket{jm^{\prime}}{J^{(1)q}}{jm}\frac{\braket{\alpha^{\prime}j^{\prime}}{\mathbf{J\cdot
V}}{\alpha j}}{\hbar^{2}j(j+1)},$

where $\mathscr{Y}^{(k)q}$ are components of an irreducible tensor
$\mathscr{Y}^{(k)}$, $\braket{jk;mq}{jk;j^{\prime}m^{\prime}}$ is a Clebsch-
Gordan coefficient and
$\braket{\alpha^{\prime}j^{\prime}}{\mathscr{Y}^{(k)}}{\alpha j}$ is a reduced
matrix element, it is customary to write this interaction as
$\mu_{B}\mathbf{B}(t)\cdot\mathtensor{g}\cdot\mathbf{S}$. Here, $\mathbf{S}$
is the total spin of the molecular magnet and
$\mathtensor{g}=\sum_{i}\mathtensor{g}_{i}c_{i}$, with
$c_{i}=\braket{\alpha^{\prime}S}{\mathbf{S\cdot s}^{i}}{\alpha
S}/\hbar^{2}S(S+1)$. This is certainly the case when only the multiplet with
maximum spin $S$ is populated. Inside this multiplet, application of this
theorem for $\mathbf{V}=\mathbf{s}^{i}$, $\mathbf{J}=\mathbf{S}$,
$\ket{jm}\equiv\ket{SM}$, enables replacing $\mathbf{s}^{i}$ by
$c_{i}\mathbf{S}$, where $c_{i}$ is a numerical constant, i.e.,
$\pi\mathbf{s}^{i}\pi=c_{i}\pi\mathbf{S}\pi$, where $\pi$ is the projection
operator onto the multiplet of highest spin allowed in the ground state of the
molecular magnet, i.e. $\pi=\sum_{M=-S}^{S}\ket{SM}\bra{SM}$. The numerical
constants $c_{i}$ are determined by the nature of the exchange interactions
(e.g., ferro- vs antiferromagnetic) between the constituent spins.

Solid State Electronic Spins.– Solid state systems such as rare-earth ion or
transition-metal-doped semiconductors could be a potential platform for
E-qubits. In the dilute limit, spins are weakly interacting and residual
intra-qubit couplings can be addressed using decoupling schemes as discussed
earlier. In the dense limit, however, ions interact strongly. Fortunately,
uncoupling may be possible with the use of co-dopants or materials
engineering. Long decoherence times of Er3+ ions in the presence of oxygen co-
dopant have been observed in the system Er-Si [47, 48, 49]; we note that the
system Er:YSO is also of interest. For Er3+ the ZFS is smaller than the Zeeman
splitting from Tesla fields, making it possible to polarize electron spins
above $>$ 90%. Weak ZFS are a general feature of rare-earth ions.

Conclusion.– The design requirements of quantum processors based on large
collective spins (ensembles, molecules) may be less stringent that those based
on individual atomic spins. A scalable implementation of the latter would
require precise placement of atomic defects in the lattice with atomic
resolution, something that is not currently feasible at this time using
foundry techniques. Also, once the single atom decays, the entire quantum
state is lost. That is not the case for ensembles (E-qubits), where survival
of the fittest prevails, so-to-speak. Moreover, E-qubits can be selectively
addressed through spatial degrees of freedom, rather than frequency (spectral)
selectivity, providing a sensible path towards scalability. To date, a
prescription of gate implementation between collective spins has been lacking.
Herein we examined the feasibility of using direct macroscopic magnetic
dipolar interactions for the implementation of universal gates between
E-qubits. We have shown how to create coherent evolution where each E-qubit
behaves isomorphically to $\mathfrak{su}(2)$. Under conditions of intra-qubit
decoupling and effective mesoscopic Ising $zz$ interactions, possible
implementations could include nuclear-spin ensembles in the solid, liquid and
gas phases, as well as giant electronic spins from molecular magnets
(including possibly, ensembles of molecular magnets or rare-earth ion
dopants). Previous implementations of ensemble qubits such as neutral atoms
rely on interactions with an external field to apply quantum gates, where the
coupling scales with $\sqrt{n}$ [50, 51] as compared to $n$, a clear advantage
in our proposal. High-fidelity control and readout methods are of course,
critical to the success of such a proposal. To this end, we note the
possibility of electric-field control [52], nanomagnet control [14], nanowire
control [53] and nanosquid readout [54].

##  I Appendix

###  I.1 Quantum Treatment of Single- and Two-Qubit Gates For Arbitrary Spin

Here we give a full quantum treatment of gates for maximally polarized states
involving single-spin states from arbitrary spins. The formalism of state
multipoles $\mathscr{Y}^{(k)q}(\mathbf{S})$ [11] is employed. State multipoles
are defined in terms of the angular momentum states $\ket{SM}$ by Eq. (11)
below. Sets of arbitrary spins are best treated using the multispin formalism
of Sanctuary [11]. The multispin tensors for $N$ spins,
$T^{(k)q}_{\\{K\\}}(k_{1},k_{2},\dots,k_{N})$, are constructed from the
$\mathscr{Y}$’s as follows:

$T^{(k)q}_{\\{K\\}}(k_{1},k_{2},\dots,k_{N})=\left[\prod_{i=1}^{N}(S^{i})^{-1/2}\right]\\\
\times\sum_{q_{1},\dots,q_{N}}\braket{k_{1}q_{1}k_{2}q_{2}\dots
k_{N}q_{N}}{(k_{1}k_{2}\dots k_{N})\\{K\\}kq}\\\
\times\mathscr{Y}^{(k_{1})q_{1}}(\mathbf{S}^{1})\otimes\mathscr{Y}^{(k_{2})q_{2}}(\mathbf{S}^{2})\otimes\dots\otimes\mathscr{Y}^{(k_{N})q_{N}}(\mathbf{S}^{N}),$
(5)

where $\braket{k_{1}q_{1}k_{2}q_{2}\dots k_{N}q_{N}}{(k_{1}k_{2}\dots
k_{N})\\{K\\}kq}$ is a generalized Clebsch-Gordan coefficient [11],
$(S^{i})=(2S^{i}+1)$ and $\\{K\\}=K_{1},K_{2},\dots,K_{N-2}$ is the angular
momentum coupling scheme (set of intermediate values). The adjoint is:

$T^{(k)q}_{\\{K\\}}(k_{1},k_{2},\dots,k_{N})^{\dagger}=(-1)^{k-q}T^{(k)-q}_{\\{K\\}}(k_{1},k_{2},\dots,k_{N}).$
(6)

These operators are orthonormal in the following sense:

$\mbox{Tr}\left[T^{(k)q}_{\\{K\\}}(k_{1},k_{2},\dots,k_{N})^{\dagger}T^{(k^{\prime})q^{\prime}}_{\\{K^{\prime}\\}}(k_{1}^{\prime},k_{2}^{\prime},\dots,k_{N}^{\prime})\right]\\\
=\delta_{kk^{\prime}}\delta_{qq^{\prime}}\delta_{\\{K\\},\\{K^{\prime}\\}}\prod_{i=1}^{N}\delta_{k_{i},k_{i}^{\prime}},$
(7)

where
$\delta_{\\{K\\},\\{K^{\prime}\\}}=\delta_{K_{1},K_{1}^{\prime}}\delta_{K_{2},K_{2}^{\prime}}\dots\delta_{K_{N},K_{N}^{\prime}}$.
In this formalism a scalar operator on the $n$ spins can be written
$\phi=\sum_{k,\\{V\\}}\phi^{(k)}_{\\{V\\}}\odot^{k}T^{(k)}_{\\{V\\}}$, where
$T^{(k)}_{\\{V\\}}$ forms a basis for the irreducible representation of the
rotation group, $\phi^{(k)}_{\\{V\\}}$ are $k$-th rank tensor coefficients and
$\odot^{k}$ indicates a $k$-th order contraction of the two tensors [11]. In
this formalism the rotational invariance can be exploited but not when the
expansion
$\phi=\sum_{\alpha,\beta}\ket{\alpha}\bra{\alpha}\phi\ket{\beta}\bra{\beta}$
is used. Here the basis states $\ket{\alpha}$, which form a complete set, are
usually taken to be the product state of the $n$ spins. Such a treatment with
product states can be quite involved because the whole series must be used
[11].

####  I.1.1 Single-Qubit Gates

It is clear by inspection of Eq. (5) that single-spin operators ($k=1$,
$k_{i}=1$, $k_{j}=0$, $j\neq i$ for some $i=1,\dots,N$), i.e.,

$T^{(1)q}_{\\{K\\}}(00\dots 1_{i}\dots
00)\propto\mathscr{Y}^{(1)q}(\mathbf{S}^{i})\propto\mathbf{S}^{i}$

in the spherical basis, since for single-spin operators, $q=q_{i}$, according
to the generalized Clebsch-Gordan coefficient
$\braket{k_{1}q_{1}k_{2}q_{2}\dots k_{N}q_{N}}{(k_{1}k_{2}\dots
k_{N})\\{K\\}kq}$. By construction, these single-spin operators obey the usual
angular momentum commutation relations:

$\displaystyle[S_{\pm},\mathscr{Y}^{(k)q}(\mathbf{S})]=$
$\displaystyle\sqrt{k(k+1)-q(q\pm 1)}\mathscr{Y}^{(k)q\pm 1}(\mathbf{S})$
$\displaystyle[S_{z},\mathscr{Y}^{(k)q}(\mathbf{S})]=$ $\displaystyle
q\mathscr{Y}^{(k)q}(\mathbf{S}).$ (8)

And since these commutation rules hold, as would for any angular momentum
operator, any Euler rotation can be applied using multispin tensors of the
form $T^{(1)q}_{\\{K\\}}(00\dots 1_{i}\dots 00)$, where $1_{i}$ indicates the
component $k_{i}=1$, and therefore, single-qubit gates can be readily
achieved.

The density matrix for $N$ spins can be expanded in the multispin basis:

$\rho(t)=\frac{1}{\prod_{i=1}^{N}(2S^{i}+1)}\Bigl{[}\sum_{kq\alpha}\phi^{k}_{q}(\alpha,t)T^{(k)q}(\alpha)\Bigr{]}$
(9)

where $\alpha$ is shorthand for all quantum numbers except $k,q$, i.e., the
coupling scheme $\\{K\\}$ as well as $k_{1},\dots,k_{N}$. In light of (6) and
(7) the functions $\phi^{k}_{q}(\alpha,t)$ are defined as:

$\phi^{k}_{q}(\alpha,t)=\mbox{Tr}[T^{(k)q}(\alpha)^{\dagger}\rho(t)]\prod_{i=1}^{N}(2S^{i}+1).$

Each term in the density matrix is of the form (5), which in turn is a
summation of products
$\mathscr{Y}^{(k_{1})q_{1}}(\mathbf{S}^{1})\otimes\mathscr{Y}^{(k_{2})q_{2}}(\mathbf{S}^{2})\otimes\dots\otimes\mathscr{Y}^{(k_{N})q_{N}}(\mathbf{S}^{N})$.
A single-qubit gate amounts to applying a unitary constructed from the
generator $T^{(1)q}_{\\{K\\}}(00\dots 1_{i}\dots 00)\propto\mathbf{S}^{i}$,
which commutes with all tensors $\mathscr{Y}^{(k_{j})q_{j}}(\mathbf{S}^{j})$
in the product except $j=i$. Such rotations of angular momenta
$\mathbf{S}^{i}$ are analogous to the spin 1/2 case.

####  I.1.2 Two-Qubit Gates

In the expression for the density matrix (9) two-qubit gates can affect any
term, like the case of single-qubit gates. However, in a product operator
$\mathscr{Y}^{(k_{1})q_{1}}(\mathbf{S}^{1})\otimes\mathscr{Y}^{(k_{2})q_{2}}(\mathbf{S}^{2})\otimes\dots\otimes\mathscr{Y}^{(k_{N})q_{N}}(\mathbf{S}^{N})$
we need only consider two operators (say 1 and 2), the ones affected by the
bilinear coupling Hamiltonian:

$S_{z}^{1}S_{z}^{2}=\left\\{\tfrac{S^{1}(S^{1}+1)S^{2}(S^{2}+1)}{3}\right\\}^{1/2}\mathscr{Y}^{(1)0}(\mathbf{S}^{1})\otimes\mathscr{Y}^{(1)0}(\mathbf{S}^{2}).$
(10)

This Hamiltonian will affect
$\mathscr{Y}^{(k_{1})q_{1}}(\mathbf{S}^{1})\otimes\mathscr{Y}^{(k_{2})q_{2}}(\mathbf{S}^{2})$
in the product operator. In the multipole formalism, each term in the product
has a matrix representation:

$\mathscr{Y}^{(k_{i})q_{i}}(\mathbf{S}^{i})=(i)^{k_{i}}[(S^{i})(k_{i})]^{1/2}\\\
\times\sum_{M_{i}M_{i}^{\prime}}(-1)^{S^{i}-M_{i}}\left(\begin{matrix}S^{i}&k_{i}&S^{i}\\\
-M_{i}&q_{i}&M_{i}^{\prime}\end{matrix}\right)\ket{S^{i}M_{i}}\bra{S^{i}M_{i}^{\prime}}$
(11)

where $(S^{i})\equiv(2S^{i}+1)$ and $(k_{i})=(2k_{i}+1)$. For the axial
component ($q_{i}=0$) the selection rule from the Wigner $3j$ symbol implies
that $m_{i}=m_{i}^{\prime}$ and the terms are all diagonal in the basis of
projection operators $\ket{S^{i}M_{i}}\bra{S^{i}M_{i}^{\prime}}$. By extension
to the tensor product space,
$\mathscr{Y}^{(1)0}(\mathbf{S}^{1})\otimes\mathscr{Y}^{(1)0}(\mathbf{S}^{2})$
is diagonal in the tensor product basis
$\ket{S^{i}M_{i}}\bra{S^{i}M_{i}^{\prime}}\otimes\ket{S^{j}M_{j}}\bra{S^{j}M_{j}^{\prime}}$.
We can therefore exponentiate the operator
$\mathscr{Y}^{(1)0}(\mathbf{S}^{1})\otimes\mathscr{Y}^{(1)0}(\mathbf{S}^{2})$
as follows:

$\exp\left(iDt\mathscr{Y}^{(1)0}(\mathbf{S}^{1})\otimes\mathscr{Y}^{(1)0}(\mathbf{S}^{2})\right)=\sum_{M_{1},M_{2}}e^{-iDt[(S^{1})(1)(S^{2})(1)]^{1/2}(-1)^{S^{1}-M_{1}+S^{2}-M_{2}}\left(\begin{smallmatrix}S^{1}&1&S^{1}\\\
-M_{1}&0&M_{1}\end{smallmatrix}\right)\left(\begin{smallmatrix}S^{2}&1&S^{2}\\\
-M_{2}&0&M_{2}\end{smallmatrix}\right)}\\\
\times\ket{S^{1}M_{1}}\bra{S^{1}M_{1}}\otimes\ket{S^{2}M_{2}}\bra{S^{2}M_{2}}$

where $D$ is a coupling strength that also absorbs the numerical coefficient
of Eq. (10) for convenience. Evolution of a product state such as
$\mathscr{Y}^{(k_{1})q_{1}}(\mathbf{S}^{1})\otimes\mathscr{Y}^{(k_{2})q_{2}}(\mathbf{S}^{2})$
gives:

$\exp\left(iDt\mathscr{Y}^{(1)0}(\mathbf{S}^{1})\otimes\mathscr{Y}^{(1)0}(\mathbf{S}^{2})\right)\mathscr{Y}^{(k_{1})q_{1}}(\mathbf{S}^{1})\otimes\mathscr{Y}^{(k_{2})q_{2}}(\mathbf{S}^{2})\exp\left(-iDt\mathscr{Y}^{(1)0}(\mathbf{S}^{1})\otimes\mathscr{Y}^{(1)0}(\mathbf{S}^{2})\right)\\\
=\sum_{M_{1},M_{2}}\sum_{M_{1}^{\prime},M_{2}^{\prime}}e^{-iDt[(S^{1})(1)(S^{2})(1)]^{1/2}(-1)^{S^{1}-M_{1}+S^{2}-M_{2}}\left(\begin{smallmatrix}S^{1}&1&S^{1}\\\
-M_{1}&0&M_{1}\end{smallmatrix}\right)\left(\begin{smallmatrix}S^{2}&1&S^{2}\\\
-M_{2}&0&M_{2}\end{smallmatrix}\right)}\\\ \times
e^{iDt[(S^{1})(1)(S^{2})(1)]^{1/2}(-1)^{S^{1}-M_{1}^{\prime}+S^{2}-M_{2}^{\prime}}\left(\begin{smallmatrix}S^{1}&1&S^{1}\\\
-M_{1}^{\prime}&0&M_{1}^{\prime}\end{smallmatrix}\right)\left(\begin{smallmatrix}S^{2}&1&S^{2}\\\
-M_{2}^{\prime}&0&M_{2}^{\prime}\end{smallmatrix}\right)}\\\
\times\ket{S^{1}M_{1}}\bra{S^{1}M_{1}}\mathscr{Y}^{(k_{1})q_{1}}(\mathbf{S}^{1})\ket{S^{1}M_{1}^{\prime}}\bra{S^{1}M_{1}^{\prime}}\otimes\ket{S^{2}M_{2}}\bra{S^{2}M_{2}}\mathscr{Y}^{(k_{2})q_{2}}(\mathbf{S}^{2})\ket{S^{2}M_{2}^{\prime}}\bra{S^{2}M_{2}^{\prime}}$
(12)

Expressing the tensors as projectors using Eq. (11) we get:

$\sigma(t)=(i)^{k_{1}+k_{2}}[(S^{1})(k_{1})(S^{2})(k_{2})]^{1/2}\sum_{M_{1},M_{2}}\sum_{M_{1}^{\prime},M_{2}^{\prime}}(-1)^{S^{1}-M_{1}+S^{2}-M_{2}}\\\
\times
e^{-iDt[(S^{1})(1)(S^{2})(1)]^{1/2}(-1)^{S^{1}-M_{1}+S^{2}-M_{2}}\left(\begin{smallmatrix}S^{1}&1&S^{1}\\\
-M_{1}&0&M_{1}\end{smallmatrix}\right)\left(\begin{smallmatrix}S^{2}&1&S^{2}\\\
-M_{2}&0&M_{2}\end{smallmatrix}\right)}\\\ \times
e^{iDt[(S^{1})(1)(S^{2})(1)]^{1/2}(-1)^{S^{1}-M_{1}^{\prime}+S^{2}-M_{2}^{\prime}}\left(\begin{smallmatrix}S^{1}&1&S^{1}\\\
-M_{1}^{\prime}&0&M_{1}^{\prime}\end{smallmatrix}\right)\left(\begin{smallmatrix}S^{2}&1&S^{2}\\\
-M_{2}^{\prime}&0&M_{2}^{\prime}\end{smallmatrix}\right)}\\\
\times\left(\begin{matrix}S^{1}&k_{1}&S^{1}\\\
-M_{1}&q_{1}&M_{1}^{\prime}\end{matrix}\right)\left(\begin{matrix}S^{2}&k_{2}&S^{2}\\\
-M_{2}&q_{2}&M_{2}^{\prime}\end{matrix}\right)\ket{S^{1}M_{1}}\bra{S^{1}M_{1}^{\prime}}\otimes\ket{S^{2}M_{2}}\bra{S^{2}M_{2}^{\prime}}$
(13)

Measurement of a bilinear spin operator
$\mathscr{Y}^{(k_{3})q_{3}}(\mathbf{S}^{1})\otimes\mathscr{Y}^{(k_{4})q_{4}}(\mathbf{S}^{2})$
is obtained by computing the trace
$\mbox{Tr}\left[(\mathscr{Y}^{(k_{3})q_{3}}(\mathbf{S}^{1})\otimes\mathscr{Y}^{(k_{4})q_{4}}(\mathbf{S}^{2}))^{\dagger}\rho(t)\right]$.
Expanding $\mathscr{Y}^{(k_{3})q_{3}}(\mathbf{S}^{1})$ and
$\mathscr{Y}^{(k_{4})q_{4}}(\mathbf{S}^{2})$ in projectors (using Eq. (11))
and computing the trace using
$\mbox{Tr}(\cdot)=\sum_{ij}\bra{S^{1}M_{i}}\otimes\bra{S^{2}M_{j}}(\cdot)\ket{S^{1}M_{i}}\otimes\ket{S^{2}M_{j}}$.
(We have omitted the summation over $S^{1}$ and $S^{2}$; all other terms
different from $S^{1},S^{2}$ would be zero since $\rho(t)$ only contains terms
with $S^{1},S^{2}$.)

$\phi(t)=[(S^{1})(k_{1})(S^{2})(k_{2})]\sum_{M_{i},M_{j}}\sum_{M_{1},M_{2}}(-1)^{S^{1}-M_{1}+S^{2}-M_{2}}e^{-iDt[(S^{1})(1)(S^{2})(1)]^{1/2}(-1)^{S^{1}-M_{1}+S^{2}-M_{2}}\left(\begin{smallmatrix}S^{1}&1&S^{1}\\\
-M_{1}&0&M_{1}\end{smallmatrix}\right)\left(\begin{smallmatrix}S^{2}&1&S^{2}\\\
-M_{2}&0&M_{2}\end{smallmatrix}\right)}\\\ \times
e^{iDt[(S^{1})(1)(S^{2})(1)]^{1/2}(-1)^{S^{1}-M_{i}+S^{2}-M_{j}}\left(\begin{smallmatrix}S^{1}&1&S^{1}\\\
-M_{i}&0&M_{i}\end{smallmatrix}\right)\left(\begin{smallmatrix}S^{2}&1&S^{2}\\\
-M_{j}&0&M_{j}\end{smallmatrix}\right)}\\\
\times\left(\begin{matrix}S^{1}&k_{1}&S^{1}\\\
-M_{1}&q_{1}&M_{i}\end{matrix}\right)\left(\begin{matrix}S^{2}&k_{2}&S^{2}\\\
-M_{2}&q_{2}&M_{j}\end{matrix}\right)\left(\begin{matrix}S^{1}&k_{3}&S^{1}\\\
-M_{1}&q_{3}&M_{i}\end{matrix}\right)\left(\begin{matrix}S^{2}&k_{4}&S^{2}\\\
-M_{2}&q_{4}&M_{j}\end{matrix}\right)$ (14)

This formula gives the time-evolution for any initial state described by the
quantum numbers $S_{1},S_{2},k_{1},q_{1},k_{2},q_{2}$. The special case of a
single-spin operator is obtained (for example) by setting $k_{2}=0,q_{2}=0$.
Suppose that the initial state is $\mathscr{Y}^{(1)1}(\mathbf{S}^{1})$
($k_{1}=1$, $q_{1}=1$, $k_{2}=0$, $q_{2}=0$) and we are interested in the
amount of $\mathscr{Y}^{(1)1}(\mathbf{S}^{1})$ ($k_{3}=1$, $q_{3}=1$,
$k_{4}=0$, $q_{4}=0$). The product of Wigner $3j$ symbols is nonzero provided
that the triangle rules are obeyed (i.e., $0\leq k_{1},k_{3}\leq 2S^{1}$,
$0\leq k_{2},k_{4}\leq 2S^{2}$). The Wigner $3j$ symbols lead to the following
selection rules: $M_{2}=M_{j}$ and $M_{1}=M_{i}+1$, causing the sums over
$M_{i}$ and $M_{j}$ to collapse. Inside the argument of the exponential, we
may simplify the term:

$\left\\{\left(\begin{smallmatrix}S^{1}&1&S^{1}\\\
-M_{1}&0&M_{1}\end{smallmatrix}\right)-\left(\begin{smallmatrix}S^{1}&1&S^{1}\\\
1-M_{1}&0&M_{1}-1\end{smallmatrix}\right)\right\\}\left(\begin{smallmatrix}S^{2}&1&S^{2}\\\
-M_{2}&0&M_{2}\end{smallmatrix}\right)\\\
=\frac{(-1)^{M_{1}+S^{1}+M_{2}+S^{2}}M_{2}}{\sqrt{[S^{1}(S^{1}+1)(2S^{1}+1)][S^{2}(S^{2}+1)(2S^{2}+1)]}}$
(15)

provided that $-S^{1}\leq M_{1}-1\leq S^{1}$ and $-S^{2}\leq M_{2}\leq S^{2}$.
We get:

$\mathcal{C}(t)=[(S^{1})(1)(S^{2})(0)]\sum_{M_{1},M_{2}}(-1)^{S^{1}-M_{1}+S^{2}-M_{2}}\\\
\times
e^{-iDt[(S^{1})(1)(S^{2})(1)]^{1/2}(-1)^{M_{1}+M_{2}+S^{1}+S^{2}}M_{2}[S^{1}(S^{1}+1)(2S^{1}+1)S^{2}(S^{2}+1)(2S^{2}+1)]^{-1/2}}\\\
\times\frac{(M_{1}-1-S^{1})(S^{1}-M_{1})!}{2S^{1}(1+3S^{1}+2[S^{1}]^{2})(1+2S^{2})(M_{1}-1+S^{1})!}$
(16)

where $[S^{1}]^{2}$ denotes the square of $S^{1}$ (spin 1 magnitude). For
fixed $S^{1},S^{2}$ this equation describes a sum of oscillatory functions,
each with a slightly different frequency ($M_{1},M_{2}$ dependence) and
amplitudes. The initial coherence ($q_{1}=1$) returns to its original state
periodically, at a frequency that can be read out from the argument of the
exponential; the factor $M_{2}$ determines the particular harmonic for each
term in the summation.

The limit of large spin is obtained by taking $S^{1}=S^{2}=S$ and letting
$S\rightarrow\infty$ (i.e., $2S+1\approx S$, etc) and recalling that $D$ is
proportional to $[S]^{2}$ (see Eq. 10). The argument of the exponential
(evolution frequency) is then seen to be proportional to $([S]^{2}\cdot
S/[S]^{3})M_{2}=M_{2}$ and the frequencies are bounded by $|M_{2}|\leq S$.

On the other hand, suppose that we are interested in the amount of
$\mathscr{Y}^{(1)1}(\mathbf{S}^{1})\otimes\mathscr{Y}^{(1)0}(\mathbf{S}^{2})$
($k_{3}=1$, $q_{3}=1$, $k_{4}=1$, $q_{4}=0$). The Wigner $3j$ symbols again
lead to the following selection rules: $M_{2}=M_{j}$ and $M_{1}=M_{i}+1$,
causing the sums over $M_{i}$ and $M_{j}$ to collapse. We get:

$\mathcal{S}(t)=[(S^{1})(1)(S^{2})(0)]\sum_{M_{1},M_{2}}(-1)^{S^{1}-M_{1}+S^{2}-M_{2}}\\\
\times
e^{-iDt[(S^{1})(1)(S^{2})(1)]^{1/2}(-1)^{M_{1}+M_{2}+S^{1}+S^{2}}M_{2}}\\\
\qquad{}^{\times[S^{1}(S^{1}+1)(2S^{1}+1)S_{2}(S^{2}+1)(2S^{2}+1)]^{-1/2}}\\\
\cdot\frac{\times
M_{2}(1-M_{1}+S^{1})(S^{1}+M_{1})!}{2S_{1}(1+3S^{1}+2[S^{1}]^{2})\sqrt{S^{2}(1+S^{2})}(1+2S^{2})(M_{1}-1+S^{1})!}\\\
$ (17)

The harmonic content is the same as in Eq. (16) whereas the amplitudes are
slightly different. This describes an evolution of the form:

$\mathscr{Y}^{(1)1}(\mathbf{S}^{1})\xlongrightarrow[]{D\mathscr{Y}^{(1)0}(\mathbf{S}^{1})\otimes\mathscr{Y}^{(1)0}(\mathbf{S}^{2})}\\\
\mathscr{Y}^{(1)1}(\mathbf{S}^{1})\mathcal{C}(t)+\mathscr{Y}^{(1)1}(\mathbf{S}^{1})\otimes\mathscr{Y}^{(1)0}(\mathbf{S}^{2})\mathcal{S}(t)$
(18)

where $\mathcal{C}(t)$ and $\mathcal{S}(t)$ are periodic oscillatory functions
whose frequency content consists of harmonics of the fundamental frequency,
which is proportional to $D$ (see Eqs. 16 and LABEL:eq:osc1). This rule is
analogous to the rule for $J$-coupling evolution in NMR,

$I_{\pm}^{k}\rightarrow I_{\pm}^{k}\cos\pi J_{kl}\tau\mp
i2I_{\pm}^{k}I_{z}^{l}\sin\pi J_{kl}\tau$

except that the periodic coefficients contain harmonics because of the
multipoles. Similarly, it can be shown that entangled states like Bell states
can be created analogously to the case of spin-1/2 systems. The important
point to realize is that while quantum information can “leak” into other
tensor components (not all combinations of possible initial and final states
are shown here), the periodic nature of the coefficients shows that 2-qubit
gates can be realized even amongst large spins.

###  I.2 Definition of Gates and Computational Basis for Molecular Magnets

In this section we define the computational basis and sequences of operations
that lead to a universal set of gates. The Liouville space of operators for
arbitrary spins $S$ is spanned by the basis of irreducible tensor operators
$\mathscr{Y}^{(k)q}(\mathbf{S})$, $0\leq k\leq 2S$. The space is also spanned
by the outer products $\ket{SM}\bra{SM^{\prime}}$. The relationship between
$\ket{SM}\bra{SM^{\prime}}$ and irreducible tensors
$\mathscr{Y}^{(k)q}(\mathbf{S})$ is:

$\ket{SM}\bra{SM^{\prime}}=(-1)^{S-M}(2S+1)^{-1/2}\\\
\times\sum_{k=0}^{2S}\sum_{q=-k}^{k}(-i)^{k}(2k+1)^{1/2}\left(\begin{matrix}S&k&S\\\
-M&q&M^{\prime}\end{matrix}\right)\mathscr{Y}^{(k)q}(\mathbf{S}).$ (19)

We define the computational basis for arbitrary spins in terms of the qubit
states shown in Table 1. The transformation rules of multipole tensors are
listed in Table 2 whereas a possible implementation of quantum gates is show
in Table I.2. In these tables the operator ket notation $\lvert
B\rangle\rangle$ denotes the operator itself ($B$), whereas the bra
$\langle\langle A\rvert$ denotes the adjoint $A^{\dagger}$ and $\langle\langle
A\rvert\lvert B\rangle\rangle$ forms an inner product, often taken to be
$\mbox{Tr}[A^{\dagger}B]$.

Single-Spin State |  Multipole Representation
---|---
$\lvert 0\rangle\rangle$, $\lvert z\rangle\rangle$ |  $\sum\limits_{k=0}^{2S}\phi_{0}^{k}\mathscr{Y}^{(k)0}(\mathbf{S})=\sum\limits_{k=0}^{2S}\phi_{0}^{k}\lvert k0\rangle\rangle$
$\lvert 1\rangle\rangle$, $\lvert-z\rangle\rangle$ |  $\sum\limits_{k=0}^{2S}(-1)^{k}\phi_{0}^{k}\mathscr{Y}^{(k)0}(\mathbf{S})$
$\lvert x\rangle\rangle$ |  $\sum\limits_{k=0}^{2S}\phi^{k}_{0}\sum\limits_{q=-k}^{k}\mathscr{D}^{(k)}_{q0}(0,-\tfrac{\pi}{2},0)\mathscr{Y}^{(k)q}(\mathbf{S})$
$\lvert y\rangle\rangle$ |  $\sum\limits_{k=0}^{2S}\phi^{k}_{0}\sum\limits_{q=-k}^{k}\mathscr{D}^{(k)}_{q0}(\tfrac{\pi}{2},\tfrac{\pi}{2},0)\mathscr{Y}^{(k)q}(\mathbf{S})$
Table 1: Computational basis for arbitrary spins in terms of multipoles. The numerical coefficients $\phi_{0}^{k}$ defining the fully polarized state are obtained from Eq. (19) by setting $M,M^{\prime}=S$. Here, $\mathscr{D}^{(k)}_{qq^{\prime}}(\Omega)$ are Wigner $D$-matrices. Alternative notation for $\mathscr{Y}^{(k)q}(\mathbf{S})$ is $\lvert kq\rangle\rangle$.  Action |  Operator |  Transformation Rule
---|---|---
$R_{z}^{A}(\phi)$ |  $e^{\frac{-i}{2}\phi\mathscr{Y}^{(1)0}(\mathbf{S}^{A})}$ |  $\lvert x\rangle\rangle\rightarrow\lvert x\rangle\rangle\widetilde{\cos}{\phi}-i\lvert y\rangle\rangle\widetilde{\sin}{\phi}$
$R_{x}^{A}(\phi)$ |  $e^{\frac{-i}{2}\phi(\mathscr{Y}^{(1)1}(\mathbf{S}^{A})-\mathscr{Y}^{(1)-1}(\mathbf{S}^{A}))}$ |  $\lvert y\rangle\rangle\rightarrow\lvert y\rangle\rangle\widetilde{\cos}{\phi}-i\lvert z\rangle\rangle\widetilde{\sin}{\phi}$
$R_{y}^{A}(\phi)$ |  $e^{\frac{-i}{2}\phi(\mathscr{Y}^{(1)1}(\mathbf{S}^{A})+\mathscr{Y}^{(1)-1}(\mathbf{S}^{A}))}$ |  $\lvert x\rangle\rangle\rightarrow\lvert x\rangle\rangle\widetilde{\cos}{\phi}+i\lvert z\rangle\rangle\widetilde{\sin}{\phi}$
$U_{zz}(\frac{\pi}{2})$ |  $e^{\frac{-i\pi}{4}\mathscr{Y}^{(1)0}(\mathbf{S}^{A})\scriptstyle{\otimes}\mathscr{Y}^{(1)0}(\mathbf{S}^{B})}$ |  $\lvert x\rangle\rangle^{A}\otimes\mathds{1}^{B}\rightarrow\lvert y\rangle\rangle^{A}\otimes\lvert z\rangle\rangle^{B}$
Table 2:  Transformation rules for multipole tensors representing the
computational basis states. The notation $\widetilde{\cos}$ and
$\widetilde{\sin}$ is a reminder that individual tensor components are evolved
according to the harmonics generated by the transformation rules of tensor
operators. For single-qubit gates, this function is obtained from the
commutation relations (12). For two-qubit gates the periodic functions can be
read off from the argument to the exponential function in Eq. (LABEL:eq:osc1)
and (16).

Gate |  Symbol |  Unitary operator
---|---|---
X bit-flip |  | $R_{x}(\pi)$
Z phase-flip |  | $R_{z}(\pi)$
Hadamard |  | $R_{z}(\frac{\pi}{2})\cdot R_{x}(\frac{\pi}{2})\cdot R_{z}(\frac{\pi}{2})$
CNOT |  @C=1em @R=.7em |  1 |
|  |  | $\sqrt{i}R^{A}_{z}(\frac{\pi}{2})\cdot R^{B}_{z}(\frac{-\pi}{2})\cdot R^{B}_{x}(\frac{\pi}{2})\cdot U_{zz}(\frac{1}{2\|\mathtensor{D}_{AB}\|})\cdot R^{B}_{y}(\frac{\pi}{2})$

Table 3:  Implementation of the elementary gates as sequences of selective
rotations. Each operator except $U$ acts on the subspace of a single qudit.

###  I.3 Extended Survival Times in Non-Interacting Spins

Entangled spins within a localized ensemble decay faster than uncoupled spins,
whereas uncoupled spin ensembles survive longer than individual spins. Indeed,
non-entangled spins decay independently and the quantum information stored
within the polarization of the ensemble remains encoded over longer periods.
Here we give a rough estimate of the lifetime enhancement provided by an
ensemble relative to individual spins based on a probabilistic argument. Let
$X$ be a random variable describing the survival time of the spin. A single
spin decays at a rate $\lambda$. Assuming a Poisson process with distribution
function of the arrival times $\mathbb{P}(X>x)=e^{-\lambda x}$. The
corresponding PDF is $\frac{d\mathbb{P}(X\leq x)}{dx}=\lambda e^{-\lambda x}$.
Thus, the average arrival time (lifetime) for this single spin is
$\mathbb{E}(X)=\int_{0}^{\infty}x\lambda e^{-\lambda x}dx=\lambda^{-1}$, as
expected. Once this spin relaxes, the quantum state is lost; this is the main
disadvantage of using single spins for quantum computing. When the state has
$n$ spins, each of which decays in the same way (but independently), the
probability of the quantum state getting lost can be obtained by considering
the random variable $Y=\max\left\\{X_{1},\dots,X_{n}\right\\}$, where $X_{i}$
are iid describing the survival time of each of the $n$ spins. The lifetime of
the $n$-spins state is determined by relaxation of the last spin, hence our
interest in the maximum of $X_{1},\dots,X_{n}$. Assume that the spins are non-
interacting, the joint probability distribution is $\mathbb{P}(Y\leq
x)=\mathbb{P}(X_{1}\leq x,\dots,X_{n}\leq
x)=\prod_{i=1}^{n}\mathbb{P}(X_{i}\leq x)$, where each $\mathbb{P}(X_{i}\leq
x)=1-\mathbb{P}(X_{i}>x)=1-e^{-\lambda x}$ and the product form assumes the
spins are independent (non-interacting). The lifetime of the uncoupled
$n$-spins system is: E(Y) = ∫_0^∞y P(Y ∈dy) = ∫_0^∞y n (1-e^-λy)^n-1 λe^-λy
dy, which approaches $n\lambda^{-1}$ as $n$ increases. Thus, the spin-coherent
state lifetime is $n$ times longer than in the case of a single spin. This is
an important advantage over single spins or entangled spins in an ensemble. We
note that the lifetime of the “quantum memory” does not equal $n\lambda^{-1}$
since its usefulness rapidly disappears below the threshold of spin noise
($\propto\sqrt{n}$).

###  I.4 Dissipation in Many-Body Coupled Spin Systems

We now discuss the dissipation rates ($\lambda$ in the previous section). The
E-qubit requires weak couplings among the spins of the ensemble. Although
challenging, residual couplings should be removed. Nuclear spins can be
decoupled efficiently using a combination of sample spinning and rotor-
synchronized RF pulses. Narrowing of lines down to the sub-hertz range has
been demonstrated [23, 55]. For electrons, however, sample spinning is not an
option, whereas pulsed decoupling has limited value. Material engineering may
be the best option. Sellars et al. [48] demonstrated that co-doping with
oxygen can reduce line widths down to 10 Hz. If the spins are decoupled, the
lifetime of a quantum state encoded in the E-qubit will be extended relative
to that of a single spin.

In the presence of residual pairwise couplings a one-body state coherently
evolves into many-body states. The latter decay much faster than single-body
states. Pulsed decoupling can help prevent this coherent evolution. In this
section we show the detrimental effects of decoherence on many-body states.
The conclusion we draw is one that strongly advises against allowing the
formation of correlated many-body states, either intentionally via quantum
state preparation, or unintentionally via residual spin-spin couplings.

Consider a general Hamiltonian of the form
$\mathcal{H}=\mathcal{H}_{Z}+\mathcal{H}_{SB}+\mathcal{H}_{B}$, where
$\mathcal{H}_{Z}$ is the unperturbed spin part (e.g. Zeeman interaction),
$\mathcal{H}_{SB}$ is the spin-bath interaction including any spin-spin
interactions and $\mathcal{H}_{B}$ is the bath Hamiltonian. We denote the
corresponding Liouvillian superoperator as
$\mathscr{L}=\mathscr{L}_{Z}+\mathscr{L}_{SB}+\mathscr{L}_{B}$. Consider a set
of operators $\\{F_{k}\\}_{k=1}^{m}$ and an inner product
$(A,B)=\mbox{Tr}(A^{\dagger}B)$, often denoted $\langle\langle
A|B\rangle\rangle$. The operators $F_{k}$ are orthogonal, $\langle\langle
F_{i}|F_{j}\rangle\rangle=0$ for $i\neq j$. A projection operator is defined
as $\pi=\sum_{k=1}^{m}\frac{|F_{k}\rangle\rangle\langle\langle
F_{k}|}{\langle\langle F_{k}|F_{k}\rangle\rangle}$. It is customary to
transform the Liouvillian to the interaction representation generated by
$\mathscr{L}_{Z}$ and write the resulting Liouvillian as
$\mathscr{L}^{*}(t)\equiv\mathscr{L}_{B}+\mathscr{L}_{SB}(t)$, where $*$
denotes the interaction representation and $\mathscr{L}_{SB}(t)\equiv
e^{it\mathscr{L}_{Z}}\mathscr{L}_{SB}$. To study irreversible processes,
projection operator methods yield a quantum master equation for an observable
$F_{k}$:

$\displaystyle\frac{d}{dt}\langle\langle F_{k}|\rho(t)\rangle\rangle=$
$\displaystyle-i\langle\langle
F_{k}|\mathscr{L}^{*}(t)\pi|\rho(t)\rangle\rangle-i\langle\langle
F_{k}|\mathscr{L}^{*}(t)Te^{-i\int_{0}^{t}d\tau(1-\pi)\mathscr{L}^{*}(\tau)}(1-\pi)|\rho(0)\rangle\rangle$
$\displaystyle-\sum_{j=1}^{m}\int_{0}^{t}dt^{\prime}\frac{\langle\langle
F_{k}|\mathscr{L}^{*}(t)Te^{-i\int_{t^{\prime}}^{t}d\tau(1-\pi)\mathscr{L}^{*}(\tau)}(1-\pi)\mathscr{L}^{*}(t^{\prime})|F_{j}\rangle\rangle}{\langle\langle
F_{j}|F_{j}\rangle\rangle}\langle\langle F_{j}|\rho(t^{\prime})\rangle\rangle$

where $T$ denotes Dyson time-ordering of the exponential. Insertion of the
expression for the projection operator $\pi$ into the first term shows that it
causes coherent evolution (Bloch term). The second term can be made to vanish
with a proper choice of initial conditions. The last term is the dissipative
term we are interested in. Under some assumptions (weak collision limit, short
correlation time limit, average Liouvillian, stationarity of the memory
kernel), $Te^{-i\int_{t^{\prime}}^{t}d\tau(1-\pi)\mathscr{L}^{*}(\tau)}$ is
approximated by $e^{-i(t-t^{\prime})(1-\pi)\mathscr{L}^{*}}$, where
$\mathscr{L}^{*}$ is a time-averaged Liouvillian in the interaction
representation. It is customary to drop the Zeeman and spin-bath parts
($\|\mathscr{L}_{B}\|\gg\|\mathscr{L}_{Z}+\mathscr{L}_{SB}\|$), leaving only
the bath part, i.e. $\mathscr{L}^{*}\approx\mathscr{L}_{B}$, in the
exponential. Since there is no spin part left, the projector term is dropped
leaving $e^{-i(t-t^{\prime})\mathscr{L}_{B}}$. Finally, the term
$(1-\pi)\mathscr{L}^{*}(t^{\prime})|F_{j}\rangle\rangle$ gives two terms:
$\mathscr{L}^{*}(t^{\prime})|F_{j}\rangle\rangle$ and
$-\pi\mathscr{L}^{*}(t^{\prime})|F_{j}\rangle\rangle$. The second term is not
of interest in the derivation of dissipation rates because it describes a
product of average frequencies; we are instead interested in deviations from
averages. We therefore consider the first term. The dissipation term in this
wide-sense stationary Redfield limit reads:

$\displaystyle\sum_{j=1}^{m}\int_{0}^{\infty}d\tau\frac{\langle\langle
F_{k}|\mathscr{L}^{*}(0)e^{i\tau\mathscr{L}_{B}}\mathscr{L}^{*}(\tau)|F_{j}\rangle\rangle}{\langle\langle
F_{j}|F_{j}\rangle\rangle}\langle\langle F_{j}|\rho(t)\rangle\rangle$
$\displaystyle\qquad\equiv\sum_{j=1}^{m}\langle\langle
F_{j}|\rho(t)\rangle\rangle\int_{0}^{\infty}\mathcal{K}_{kj}(\tau)d\tau,$

where $W(F_{k})\equiv\int_{0}^{\infty}\mathcal{K}_{kj}(\tau)d\tau$ is a
dissipation rate for the state $F_{k}$. The spin-bath interaction is taken to
be a sum of pairwise couplings. In the interaction representation, it acquires
a phase factor $e^{-i\omega q\tau}$:

$\displaystyle\mathcal{H}_{SB}^{*}(\tau)=$
$\displaystyle\sum_{i<j}\sum_{k=0,1,2}\sum_{q=-2}^{2}(-1)^{q}A_{kq}(\mathbf{S}^{i},\mathbf{S}^{j})T_{k,-q}(\mathbf{S}^{i},\mathbf{S}^{j})$
$\displaystyle\qquad\times e^{-i\omega q\tau}$

where we used the shorthand notation $T_{k,-q}(\mathbf{S}^{i},\mathbf{S}^{j})$
for the multispin tensor $T_{k,-q}(000\underbrace{\dots
1\dots}_{i\mathrm{th~{}pos.}}\underbrace{\dots
1\dots}_{j\mathrm{th~{}pos.}}000)$, which we will denote as
$T^{kq}(k_{1}k_{2})$ with $k_{1}=k_{2}=1$.

####  I.4.1 One-Body States

The one-body state describes a spin among an ensemble of uncoupled spins
(E-qubit), or weakly coupled spins undergoing decoupling. It also describes an
initial state ($t=0$) where the spins did not undergo coherent evolution. It
is described by a vector (rank 1) tensor operator corresponding to a spin of
interest (denoted $m$ henceforth): —F_m ⟩⟩= Y^(1)q(S^m) where $q=-1,0,1$.
Spin-lattice ($T_{1}$) relaxation corresponds to the case $q=0$ whereas spin-
spin relaxation corresponds to $q=\pm 1$. We now compute the numerator of the
memory function, $\langle\langle
F_{k}|\mathscr{L}^{*}(0)e^{i\tau\mathscr{L}_{B}}\mathscr{L}^{*}(\tau)|F_{j}\rangle\rangle$.
We start with the commutator:

$\displaystyle\mathscr{L}^{*}(\tau)F_{m}$
$\displaystyle\equiv[\mathcal{H}_{SB}^{*}(\tau),F_{m}]$
$\displaystyle=\sum_{i<j}\sum_{k=0,1,2}\sum_{q=-2}^{2}(-1)^{q}A_{kq}(\mathbf{S}^{i},\mathbf{S}^{j})$
$\displaystyle\quad\times[T_{2,-q}(\mathbf{S}^{i},\mathbf{S}^{j}),\mathscr{Y}^{(1)q}(\mathbf{S}^{m})]e^{-i\omega
q\tau}.$

Substituting

$\displaystyle T^{kq}(k_{1}k_{2})$
$\displaystyle=[(2S^{1}+1)(2S^{2}+1)]^{-1/2}$
$\displaystyle\times\sum_{q_{1}q_{2}}(-1)^{k_{1}-k_{2}+k}\sqrt{(2k+1)}$
$\displaystyle\times(-1)^{k-q}\left(\begin{matrix}k&k_{1}&k_{2}\\\
-q&q_{1}&q_{2}\end{matrix}\right)\mathscr{Y}^{(k_{1})q_{1}}(\mathbf{S}^{1})\mathscr{Y}^{(k_{2})q_{2}}(\mathbf{S}^{2})$

we get: L^*(τ) F_m = 12 ∑_i,j ∑_k=0,1,2 ∑_q=-2^2 (-1)^q A_kq( S^i, S^j) ×∑_q_1
q_2 (-1)^k_1 - k_2 + k (2k+1) [(2S^i+1)(2S^j+1)]^-1/2 (-1)^k-q ×( k k1k2-q
q1q2 ) [ Y^(k_1)q_1 (S^i) Y^(k_2)q_2 (S^j) , Y^(1)q(S^m) ] e^- i ωq τ.
Substitution of the commutator

$\displaystyle[\mathscr{Y}^{(l)m}(\mathbf{S}),$
$\displaystyle\mathscr{Y}^{(k)q}(\mathbf{S})]=2\sum_{k^{\prime}q^{\prime}}\frac{\phi(klk^{\prime})}{(2S+1)}$
$\displaystyle\times\langle\langle\mathscr{Y}^{(k^{\prime})q^{\prime}}(\mathbf{S})|\mathscr{Y}^{(l)m}(\mathbf{S})|\mathscr{Y}^{(k)q}(\mathbf{S})\rangle\rangle\mathscr{Y}^{(k^{\prime})q^{\prime}}(\mathbf{S})$

where $\phi(klm^{\prime})$ is $1$ if $k+k^{\prime}+l$ is odd and zero
otherwise,

$\displaystyle\langle\langle\mathscr{Y}^{(k^{\prime})q^{\prime}}(\mathbf{S})$
$\displaystyle|\mathscr{Y}^{(l)m}(\mathbf{S})|\mathscr{Y}^{(k)q}(\mathbf{S})\rangle\rangle$
$\displaystyle=$
$\displaystyle\mbox{Tr}\left\\{\mathscr{Y}^{(k^{\prime})q^{\prime}}(\mathbf{S})^{\dagger}\mathscr{Y}^{(l)m}(\mathbf{S})\mathscr{Y}^{(k)q}(\mathbf{S})\right\\}$
$\displaystyle=$
$\displaystyle(-1)^{k^{\prime}-q^{\prime}}\left(\begin{matrix}k^{\prime}&l&k\\\
-q^{\prime}&m&q\end{matrix}\right)\langle\langle
k^{\prime}\|\mathscr{Y}^{(l)}\|k\rangle\rangle$

(Wigner-Eckart theorem) and

$\displaystyle\langle\langle$ $\displaystyle
k^{\prime}\|\mathscr{Y}^{(l)}\|k\rangle\rangle=(-1)^{l+k+k^{\prime}+2S}(i)^{k+k^{\prime}+l}$
$\displaystyle\times[(2l+1)(2k+1)(2S+1)(2k^{\prime}+1)]^{1/2}\left\\{\begin{matrix}k^{\prime}&l&k\\\
S&S&S\end{matrix}\right\\},$

the reduced matrix element, in terms of Wigner $6-j$ symbols, yields:

$\displaystyle\mathscr{L}^{*}(\tau)F_{m}=\sum_{i<j}\sum_{k=0,1,2}\sum_{q=-2}^{2}(-1)^{q}A_{kq}(\mathbf{S}^{i},\mathbf{S}^{j})(-1)^{k-q}$
$\displaystyle\times\sum_{q_{1}q_{2}}(-1)^{k_{1}-k_{2}+k}\sqrt{(2k+1)}[(2S^{i}+1)(2S^{j}+1)]^{-1/2}$
$\displaystyle\times\left(\begin{matrix}k&k_{1}&k_{2}\\\
-q&q_{1}&q_{2}\end{matrix}\right)[\mathscr{Y}^{(k_{1})q_{1}}(\mathbf{S}^{i})\mathscr{Y}^{(k_{2})q_{2}}(\mathbf{S}^{j}),\mathscr{Y}^{(1)q}(\mathbf{S}^{m})]e^{-i\omega
q\tau}.$

Acting on $\mathscr{L}^{*}(\tau)F$ with the bath propagator
$e^{i\mathscr{L}_{B}\tau}$ introduces the time dependence
$A_{kq}(\mathbf{S}^{i},\mathbf{S}^{j})\rightarrow
A_{kq}(\mathbf{S}^{i},\mathbf{S}^{j})(\tau)$. Now,

$\displaystyle[$
$\displaystyle\mathscr{Y}^{(k_{1})q_{1}}(\mathbf{S}^{i})\mathscr{Y}^{(k_{2})q_{2}}(\mathbf{S}^{j}),\mathscr{Y}^{(1)q}(\mathbf{S}^{m})]=\frac{2}{(2S^{m}+1)}\sum_{k^{\prime}q^{\prime}}$
$\displaystyle\times\Bigl{\\{}\delta_{im}\phi(1k_{1}k^{\prime})\langle\langle\mathscr{Y}^{(k^{\prime})q^{\prime}}(\mathbf{S}^{m})|\mathscr{Y}^{(k_{1})q_{1}}(\mathbf{S}^{m})|\mathscr{Y}^{(1)q}(\mathbf{S}^{m})\rangle\rangle$
$\displaystyle\times\mathscr{Y}^{(k^{\prime})q^{\prime}}(\mathbf{S}^{m})\mathscr{Y}^{(k_{2})q_{2}}(\mathbf{S}^{j})$
$\displaystyle+\delta_{jm}\phi(1k_{2}k^{\prime})\langle\langle\mathscr{Y}^{(k^{\prime})q^{\prime}}(\mathbf{S}^{m})|\mathscr{Y}^{(k_{2})q_{2}}(\mathbf{S}^{m})|\mathscr{Y}^{(1)q}(\mathbf{S}^{m})\rangle\rangle$
$\displaystyle\times\mathscr{Y}^{(k^{\prime})q^{\prime}}(\mathbf{S}^{m})\mathscr{Y}^{(k_{1})q_{1}}(\mathbf{S}^{i})\Bigr{\\}},$

where $i\neq j$ (LHS) implying that $j\neq m$ in the first term and $i\neq m$
in the second term (RHS). The final expression is:

$\displaystyle
e^{\mathscr{L}_{B}\tau}\mathscr{L}^{*}(\tau)F_{m}=\sum_{i<j}\sum_{k=0,1,2}\sum_{q=-2}^{2}(-1)^{q}A_{kq}(\mathbf{S}^{i},\mathbf{S}^{j})(\tau)$
$\displaystyle\sum_{q_{1}q_{2}}(-1)^{k_{1}-k_{2}+k}\sqrt{(2k+1)}[(2S^{i}+1)(2S^{j}+1)]^{-1/2}(-1)^{k-q}$
$\displaystyle\times\left(\begin{matrix}k&k_{1}&k_{2}\\\
-q&q_{1}&q_{2}\end{matrix}\right)\frac{2e^{-i\omega
q\tau}}{(2S^{m}+1)}\sum_{k^{\prime}q^{\prime}}\Bigl{\\{}\delta_{im}\phi(1k_{1}k^{\prime})$
$\displaystyle\times\langle\langle\mathscr{Y}^{(k^{\prime})q^{\prime}}(\mathbf{S}^{m})|\mathscr{Y}^{(k_{1})q_{1}}(\mathbf{S}^{m})|\mathscr{Y}^{(1)q}(\mathbf{S}^{m})\rangle\rangle$
$\displaystyle\times\mathscr{Y}^{(k^{\prime})q^{\prime}}(\mathbf{S}^{m})\mathscr{Y}^{(k_{2})q_{2}}(\mathbf{S}^{j})$
$\displaystyle+\delta_{jm}\phi(1k_{2}k^{\prime})\langle\langle\mathscr{Y}^{(k^{\prime})q^{\prime}}(\mathbf{S}^{m})|\mathscr{Y}^{(k_{2})q_{2}}(\mathbf{S}^{m})|\mathscr{Y}^{(1)q}(\mathbf{S}^{m})\rangle\rangle$
$\displaystyle\times\mathscr{Y}^{(k^{\prime})q^{\prime}}(\mathbf{S}^{m})\mathscr{Y}^{(k_{1})q_{1}}(\mathbf{S}^{i})\Bigr{\\}}$
(20)

where $k_{1}=k_{2}=1$. We also compute the quantity
$(\mathscr{L}^{*}(0)F_{\tilde{m}})^{\dagger}$ by conjugation using
$\mathscr{Y}^{(k)q}{}^{\dagger}(\mathbf{S})=(-1)^{k-q}\mathscr{Y}^{(k)-q}(\mathbf{S})$.
Thus,

$\displaystyle(\mathscr{L}^{*}(0)F_{m})^{\dagger}=\sum_{\tilde{i}<\tilde{j}}\sum_{\tilde{k}=0,1,2}\sum_{\tilde{q}=-2}^{2}(-1)^{\tilde{q}}A_{\tilde{k}\tilde{q}}(\mathbf{S}^{\tilde{i}},\mathbf{S}^{\tilde{j}})(0)^{\dagger}$
$\displaystyle\times\sum_{\tilde{q}_{1}\tilde{q}_{2}}(-1)^{k_{1}-k_{2}+\tilde{k}}\sqrt{(2\tilde{k}+1)}[(2S^{\tilde{i}}+1)(2S^{\tilde{j}}+1)]^{-1/2}$
$\displaystyle\times(-1)^{\tilde{k}-\tilde{q}}\left(\begin{matrix}\tilde{k}&k_{1}&k_{2}\\\
-\tilde{q}&\tilde{q}_{1}&\tilde{q}_{2}\end{matrix}\right)\frac{2}{(2S^{\tilde{m}}+1)}\sum_{\tilde{k}^{\prime}\tilde{q}^{\prime}}\Bigl{\\{}\delta_{\tilde{i}\tilde{m}}\phi(1k_{1}\tilde{k}^{\prime})$
$\displaystyle\times\langle\langle\mathscr{Y}^{(\tilde{k}^{\prime})\tilde{q}^{\prime}}(\mathbf{S}^{\tilde{m}})|\mathscr{Y}^{(k_{1})\tilde{q}_{1}}(\mathbf{S}^{\tilde{m}})|\mathscr{Y}^{(1)\tilde{q}}(\mathbf{S}^{\tilde{m}})\rangle\rangle$
$\displaystyle\times\mathscr{Y}^{(\tilde{k}^{\prime})\tilde{q}^{\prime}}(\mathbf{S}^{\tilde{m}})^{\dagger}\mathscr{Y}^{(k_{2})\tilde{q}_{2}}(\mathbf{S}^{\tilde{j}})^{\dagger}$
$\displaystyle+\delta_{\tilde{j}\tilde{m}}\phi(1k_{2}\tilde{k}^{\prime})\langle\langle\mathscr{Y}^{(\tilde{k}^{\prime})\tilde{q}^{\prime}}(\mathbf{S}^{\tilde{m}})|\mathscr{Y}^{(k_{2})\tilde{q}_{2}}(\mathbf{S}^{\tilde{m}})|\mathscr{Y}^{(1)\tilde{q}}(\mathbf{S}^{\tilde{m}})\rangle\rangle$
$\displaystyle\times\mathscr{Y}^{(\tilde{k}^{\prime})\tilde{q}^{\prime}}(\mathbf{S}^{\tilde{m}})^{\dagger}\mathscr{Y}^{(k_{1})\tilde{q}_{1}}(\mathbf{S}^{\tilde{i}})^{\dagger}\Bigr{\\}},$
(21)

where $k_{1}=k_{2}=1$. We have relabeled the index $m$ as $\tilde{m}$ to allow
for the possibility of cross-relaxation. Next, we compute the inner product:

$\displaystyle\bigl{(}\mathscr{L}^{*}(0)F_{\tilde{m}},$ $\displaystyle
e^{i\mathscr{L}_{B}\tau}\mathscr{L}^{*}(\tau)F_{m}\bigr{)}=$
$\displaystyle\mbox{Tr}\bigl{[}(\mathscr{L}^{*}(0)F_{\tilde{m}})^{\dagger}e^{i\mathscr{L}_{B}\tau}\mathscr{L}^{*}(\tau)F_{m}\bigr{]}.$
(22)

To this end, we use the trace formula:  Tr { Y^(k)q(S)^†Y^(k’)q’(S) } = (2S+1)
δ_kk’ δ_qq’. Upon multiplying $(\mathscr{L}^{*}(0)F_{\tilde{m}})^{\dagger}$
and $e^{i\mathscr{L}_{B}\tau}\mathscr{L}^{*}(\tau)F_{m}$ we get four terms.
The operator part gives:

$\displaystyle\mbox{Tr}\left[\mathscr{Y}^{(\tilde{k}^{\prime})\tilde{q}^{\prime}}(\mathbf{S}^{\tilde{m}})^{\dagger}\mathscr{Y}^{(k_{2})\tilde{q}_{2}}(\mathbf{S}^{\tilde{j}})^{\dagger}\mathscr{Y}^{(k^{\prime})q^{\prime}}(\mathbf{S}^{m})\mathscr{Y}^{(k_{2})q_{2}}(\mathbf{S}^{j})\right]$
$\displaystyle=(2S^{m}+1)(2S^{j}+1)\bigl{[}\delta_{\tilde{m}m}\delta_{\tilde{j}j}\delta_{\tilde{k}^{\prime}k^{\prime}}\delta_{\tilde{q}^{\prime}q^{\prime}}\delta_{\tilde{q}_{2}q_{2}}$
$\displaystyle\quad+\delta_{\tilde{j}m}\delta_{\tilde{m}j}\delta_{1k^{\prime}}\delta_{\tilde{q}_{2}q^{\prime}}\delta_{\tilde{k}^{\prime}1}\delta_{\tilde{q}^{\prime}q_{2}}\bigr{]}$
(23)
$\displaystyle\mbox{Tr}\left[\mathscr{Y}^{(\tilde{k}^{\prime})\tilde{q}^{\prime}}(\mathbf{S}^{\tilde{m}})^{\dagger}\mathscr{Y}^{(k_{2})\tilde{q}_{2}}(\mathbf{S}^{\tilde{j}})^{\dagger}\mathscr{Y}^{(k^{\prime})q^{\prime}}(\mathbf{S}^{m})\mathscr{Y}^{(k_{1})q_{1}}(\mathbf{S}^{i})\right]$
$\displaystyle=(2S^{m}+1)(2S^{i}+1)\bigl{[}\delta_{\tilde{m}m}\delta_{\tilde{j}i}\delta_{\tilde{k}^{\prime}k^{\prime}}\delta_{\tilde{q}^{\prime}q^{\prime}}\delta_{\tilde{q}_{2}q_{1}}$
$\displaystyle\quad+\delta_{\tilde{j}m}\delta_{\tilde{m}i}\delta_{1k^{\prime}}\delta_{\tilde{q}_{2}q^{\prime}}\delta_{\tilde{k}^{\prime}1}\delta_{\tilde{q}^{\prime}q_{1}}\bigr{]}$
(24)
$\displaystyle\mbox{Tr}\left[\mathscr{Y}^{(\tilde{k}^{\prime})\tilde{q}^{\prime}}(\mathbf{S}^{\tilde{m}})^{\dagger}\mathscr{Y}^{(k_{1})\tilde{q}_{1}}(\mathbf{S}^{\tilde{i}})^{\dagger}\mathscr{Y}^{(k^{\prime})q^{\prime}}(\mathbf{S}^{m})\mathscr{Y}^{(k_{2})q_{2}}(\mathbf{S}^{j})\right]$
$\displaystyle=(2S^{m}+1)(2S^{j}+1)\bigl{[}\delta_{\tilde{m}m}\delta_{\tilde{i}j}\delta_{\tilde{k}^{\prime}k^{\prime}}\delta_{\tilde{q}^{\prime}q^{\prime}}\delta_{\tilde{q}_{1}q_{2}}$
$\displaystyle\quad+\delta_{\tilde{i}m}\delta_{\tilde{m}j}\delta_{1k^{\prime}}\delta_{\tilde{q}_{1}q^{\prime}}\delta_{\tilde{k}^{\prime}1}\delta_{\tilde{q}^{\prime}q_{2}}\bigr{]}$
(25)
$\displaystyle\mbox{Tr}\left[\mathscr{Y}^{(\tilde{k}^{\prime})\tilde{q}^{\prime}}(\mathbf{S}^{\tilde{m}})^{\dagger}\mathscr{Y}^{(k_{1})\tilde{q}_{1}}(\mathbf{S}^{\tilde{i}})^{\dagger}\mathscr{Y}^{(k^{\prime})q^{\prime}}(\mathbf{S}^{m})\mathscr{Y}^{(k_{1})q_{1}}(\mathbf{S}^{i})\right]$
$\displaystyle=(2S^{m}+1)(2S^{i}+1)\bigl{[}\delta_{\tilde{m}m}\delta_{\tilde{i}i}\delta_{\tilde{k}^{\prime}k^{\prime}}\delta_{\tilde{q}^{\prime}q^{\prime}}\delta_{\tilde{q}_{1}q_{1}}$
$\displaystyle\quad+\delta_{\tilde{i}m}\delta_{\tilde{m}i}\delta_{1k^{\prime}}\delta_{\tilde{q}_{1}q^{\prime}}\delta_{\tilde{k}^{\prime}1}\delta_{\tilde{q}^{\prime}q_{1}}\bigr{]}.$
(26)

Collecting all the terms, and restricting the various summations according to
the various Kronecker delta functions, we get the overall dissipation rate.
This multi-index expression, which involves substituting Eqs. (23), (24),
(25), (26), (21), (20) into (22), will not be provided here, as it is neither
informative, nor particularly illuminating. We only need to highlight some key
features. We see that the dissipation rate of the one-body term,
$W(\mathscr{Y}^{(1)q}(\mathbf{S}^{m}))$, is proportional to bath
autocorrelation functions of the form $\langle
A_{\tilde{k}\tilde{q}}(\mathbf{S}^{\tilde{i}},\mathbf{S}^{\tilde{j}})(0)^{\dagger},A_{kq}(\mathbf{S}^{i},\mathbf{S}^{j})(\tau)\rangle_{B}$,
which according to Abragam [56] must be taken as thermal averages, i.e.
$\langle AA(t)\rangle_{B}\equiv\mbox{tr}[\rho AA(t)]$, where
$\rho(\mathcal{H}_{B})=\exp(-\beta_{L}\mathcal{H}_{B})/\mbox{Tr}[\exp(-\beta_{L}\mathcal{H}_{B})]$
is the Boltzmann density matrix and tr denotes the partial trace over the bath
degrees of freedom.

Assuming that spins are all of the same type ($S^{i}=S$), two types of terms
arise. The first is:

$\displaystyle\Gamma_{1}^{kq\tilde{k}\tilde{q}}(q\omega)=$
$\displaystyle\int_{0}^{\infty}d\tau\sum_{j}\sum_{\tilde{i}<j}\sum_{i<j}\langle
A_{\tilde{k}\tilde{q}}(\mathbf{S}^{\tilde{i}},\mathbf{S}^{j})^{\dagger}(0)A_{kq}(\mathbf{S}^{i},\mathbf{S}^{j})(\tau)\rangle_{B}e^{-i\omega
q\tau}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}d\tau\int_{\mathbb{R}^{3}}d^{3}\mathbf{r}_{1}\int_{|\mathbf{r}_{3}-\mathbf{r_{1}}|>\epsilon}d^{3}\mathbf{r}_{2}\int_{|\mathbf{r}_{3}-\mathbf{r_{1}}|>\epsilon}d^{3}\mathbf{r}_{3}\langle
A_{\tilde{k}\tilde{q}}(\mathbf{r}_{2},\mathbf{r}_{1})^{\dagger}(0)A_{kq}(\mathbf{r}_{3},\mathbf{r}_{1})(\tau)\rangle_{B}e^{-i\omega
q\tau}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}d\tau\int_{|\mathbf{r}_{31}|>\epsilon}d^{3}\mathbf{r}_{31}\int_{|\mathbf{r}_{21}|>\epsilon}d^{3}\mathbf{r}_{21}\int_{\mathbb{R}^{3}}d^{3}\mathbf{r}_{1}\langle
A_{\tilde{k}\tilde{q}}(\mathbf{r}_{21})^{\dagger}(0)A_{kq}(\mathbf{r}_{31})(\tau)\rangle_{B}e^{-i\omega
q\tau}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}d\tau\int_{\mathbb{R}^{3}}d^{3}\mathbf{r}_{1}\langle\overline{A_{\tilde{k}\tilde{q}}(\mathbf{r}_{21})^{\dagger}(0)}\cdot\overline{A_{kq}(\mathbf{r}_{31})(\tau)}\rangle_{B}e^{-i\omega
q\tau}$

where $\epsilon>0$, $\mathbf{r}_{21}\equiv\mathbf{r}_{2}-\mathbf{r}_{1}$,
$\mathbf{r}_{31}\equiv\mathbf{r}_{3}-\mathbf{r}_{1}$ and the overline denotes
a volume integral over $\mathbf{r}_{21}$ or $\mathbf{r}_{31}$ (as seen from
the point $\mathbf{r}_{1}$, but excluding it). All integrals, including the
one over $\mathbb{R}^{3}$, cover only the volume of interest (E-qubit). In the
third equality we assumed spatial homogeneity (distribution of spins is a
stationary random field). This expression is the single-sided temporal Fourier
transform of the volume averaged two-point spatio-temporal autocorrelation
function of the spin-bath coupling.

Assuming spatial homogeneity the second type of term is of the form:

$\displaystyle\Gamma^{kq\tilde{k}\tilde{q}}_{2,m,\tilde{m}}(q\omega)=$
$\displaystyle\int_{0}^{\infty}d\tau\sum_{\tilde{i}<m}\sum_{i<\tilde{m}}\langle
A_{\tilde{k}\tilde{q}}(\mathbf{S}^{\tilde{i}},\mathbf{S}^{m})^{\dagger}(0)A_{kq}(\mathbf{S}^{i},\mathbf{S}^{\tilde{m}})(\tau)\rangle_{B}e^{-i\omega
q\tau}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}d\tau\int_{|\mathbf{r}_{1}-\mathbf{r}_{m}|>\epsilon}d^{3}\mathbf{r}_{1}\int_{|\mathbf{r}_{2}-\mathbf{r}_{\tilde{m}}|>\epsilon}d^{3}\mathbf{r}_{2}\langle
A_{\tilde{k}\tilde{q}}(\mathbf{r}_{1},\mathbf{r}_{m})^{\dagger}(0)A_{kq}(\mathbf{r}_{2},\mathbf{r}_{\tilde{m}})(\tau)\rangle_{B}e^{-i\omega
q\tau}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}d\tau\int_{|\mathbf{r}_{1m}|>\epsilon}d^{3}\mathbf{r}_{1m}\int_{|\mathbf{r}_{2\tilde{m}}|>\epsilon}d^{3}\mathbf{r}_{2\tilde{m}}\langle
A_{\tilde{k}\tilde{q}}(\mathbf{r}_{1m})^{\dagger}(0)A_{kq}(\mathbf{r}_{2\tilde{m}})(\tau)\rangle_{B}e^{-i\omega
q\tau}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}d\tau\langle\overline{A_{\tilde{k}\tilde{q}}(\mathbf{r}_{1m})^{\dagger}(0)}\cdot\overline{A_{kq}(\mathbf{r}_{2\tilde{m}})(\tau)}\rangle_{B}e^{-i\omega
q\tau},$

where $\epsilon>0$, $\mathbf{r}_{1m}\equiv\mathbf{r}_{1}-\mathbf{r}_{m}$,
$\mathbf{r}_{2\tilde{m}}\equiv\mathbf{r}_{2}-\mathbf{r}_{\tilde{m}}$ and the
overline denotes a spatial autocorrelation function for the spin-bath
interaction (as seen from $\mathbf{r}_{1m}$ and $\mathbf{r}_{2\tilde{m}}$ but
excluding those points). This dissipation rate describes how spatial
autocorrelations functions of the microscopic field are temporally correlated.
The spectral density functions $\Gamma^{kq\tilde{k}\tilde{q}}_{1}(q\omega)$
and $\Gamma^{kq\tilde{k}\tilde{q}}_{2,m,\tilde{m}}(q\omega)$ combine to give
the overall dissipation rate involving the states $|F_{m}\rangle\rangle$ and
$|F_{\tilde{m}}\rangle\rangle$. The remaining summations and indices are
geometrical factors related to the selection rules for coupling of angular
momenta.

####  I.4.2 Many-Body States

Correlated many-body states can arise due to quantum state preparation or via
coherence evolution in the presence of residual couplings. Their dissipation
rate is much faster. Consider the $n$-spin state F = Y^(1)q_1(S^1)
Y^(1)q_2(S^2) …Y^(1)q_n(S^n) where $q_{i}\in\\{-1,0,1\\}$. The idea here is
that when computing $\mathscr{L}F=[\mathcal{H},F]$, and making use of the
commutator rule [ H, A_1 …A_n] = [ H, A_1] A_2 …A_n \+ A_1 [ H, A_2] A_3 …A_n
+ …+ A_1 …A_n-1 [ H, A_n], which yields $n$ terms, there will be $n^{2}$ terms
in total when computing the inner product
$\left(\mathscr{L}^{*}(0)F,e^{\mathscr{L}_{B}\tau}\mathscr{L}^{*}(\tau)F\right)$
since $\mathscr{L}F$ appears twice. And because each term contributes a
dissipation rate of the same magnitude as the ones computed in the previous
section (for single-spin term), the overall dissipation rate,
$W(\mathscr{Y}^{(1)q_{1}}(\mathbf{S}^{1})\mathscr{Y}^{(1)q_{2}}(\mathbf{S}^{2})\dots\mathscr{Y}^{(1)q_{n}}(\mathbf{S}^{n}))$,
which is a sum of $n^{2}$ such terms, decays $n^{2}$ times faster than in the
case of the single-body state.

From this we conclude that entangled many-body states are more vulnerable to
decoherence than uncoupled single-spin states. Their dissipation rates can in
principle be computed exactly from the above analysis, if the bath two-body
spatio-temporal autocorrelation functions are known.

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