j\leq l(P)}\left(1+\sigma_{\\{P^{j},P^{j+1}\\}}Z_{P^{j}}Z_{P^{j+1}}\right).$
(59)

The result will follow by summation over $P\in\mathcal{P}$. We first show:

$\displaystyle 1+(-1)^{l(P)+1}\left(\prod_{e\in
P}\sigma_{e}\right)Z_{P^{1}}Z_{P^{l(P)+1}}$
$\displaystyle=\left(1+\sigma_{\\{P^{1},P^{2}\\}}Z_{P^{1}}Z_{P^{2}}\right)+\sum_{2\leq
j\leq l(P)}(-1)^{j-1}\left(\prod_{1\leq
k<j}\sigma_{\\{P^{k-1},P^{k}\\}}\right)$ $\displaystyle\hskip
216.81ptZ_{P^{1}}Z_{P^{j}}\left(1+\sigma_{\\{P^{j},P^{j+1}\\}}Z_{P^{j}}Z_{P^{j+1}}\right).$
(60)

To show this, we can prove by recursion on $n\in[2,l(P)]$ that

$\displaystyle 1+(-1)^{n+1}\left(\prod_{1\leq j\leq
n}\sigma_{\\{P^{j},P^{j+1}\\}}\right)Z_{P^{1}}Z_{P^{n+1}}$
$\displaystyle=\left(1+\sigma_{\\{P^{1},P^{2}\\}}Z_{P^{1}}Z_{P^{2}}\right)+\sum_{2\leq
j\leq n}(-1)^{j-1}\left(\prod_{1\leq k<j}\sigma_{\\{P^{k-1},P^{k}\\}}\right)$
$\displaystyle\hskip
260.17464ptZ_{P^{1}}Z_{P^{j}}\left(1+\sigma_{\\{P^{j},P^{j+1}\\}}Z_{P^{j}}Z_{P^{j+1}}\right)$

For $n=2$,

$\displaystyle 1+(-1)^{n+1}\left(\prod_{1\leq j\leq
n}\sigma_{\\{P^{j},P^{j+1}\\}}\right)Z_{P^{1}}Z_{P^{n+1}}$
$\displaystyle=1-\sigma_{\\{P^{1},P^{2}\\}}\sigma_{\\{P^{2},P^{3}\\}}Z_{P^{1}}Z_{P^{3}}$
$\displaystyle=\left(1+\sigma_{\\{P^{1},P^{2}\\}}Z_{P^{1}}Z_{P^{2}}\right)+\left(-\sigma_{\\{P^{1},P^{2}\\}}\right)Z_{P^{1}}Z_{P^{2}}\left(1+\sigma_{\\{P^{2},P^{3}\\}}Z_{P^{2}}Z_{P^{3}}\right)$

which is the desired relation. Now, suppose the relation holds for
$n\in[2,l(P))$. Then

$\displaystyle 1+(-1)^{n+2}\left(\prod_{1\leq j\leq
n+1}\sigma_{\\{P^{j},P^{j+1}\\}}\right)Z_{P^{1}}Z_{P^{n+2}}$
$\displaystyle=1-\sigma_{\\{P^{n+1},P^{n+2}\\}}Z_{P^{n+1}}Z_{P^{n+2}}(-1)^{n+1}\left(\prod_{1\leq
j\leq n}\sigma_{\\{P^{j},P^{j+1}\\}}\right)Z_{P^{1}}Z_{P^{n+1}}$
$\displaystyle=1-\left(1+\sigma_{\\{P^{n+1},P^{n+2}\\}}Z_{P^{n+1}}Z_{P^{n+2}}-1\right)(-1)^{n+1}\left(\prod_{1\leq
j\leq n}\sigma_{\\{P^{j},P^{j+1}\\}}\right)Z_{P^{1}}Z_{P^{n+1}}$
$\displaystyle=1+(-1)^{n+1}\left(\prod_{1\leq j\leq
n}\sigma_{\\{P^{j},P^{j+1}\\}}\right)Z_{P^{1}}Z_{P^{n+1}}$
$\displaystyle\hskip 21.68231pt+(-1)^{n}\left(\prod_{1\leq j\leq
n}\sigma_{\\{P^{j},P^{j+1}\\}}\right)Z_{P^{1}}Z_{P^{n+1}}\left(1+\sigma_{\\{P^{n+1},P^{n+2}\\}}Z_{P^{n+1}}Z_{P^{n+2}}\right)$
$\displaystyle=\left(1+\sigma_{\\{P^{1},P^{2}\\}}Z_{P^{1}}Z_{P^{2}}\right)+\sum_{2\leq
j\leq n}(-1)^{j-1}\left(\prod_{1\leq k<j}\sigma_{\\{P^{k-1},P^{k}\\}}\right)$
$\displaystyle\hskip
238.49231ptZ_{P^{1}}Z_{P^{j}}\left(1+\sigma_{\\{P^{j},P^{j+1}\\}}Z_{P^{j}}Z_{P^{j+1}}\right)$
$\displaystyle\hskip 21.68231pt+(-1)^{n}\left(\prod_{1\leq j\leq
n}\sigma_{\\{P^{j},P^{j+1}\\}}\right)Z_{P^{1}}Z_{P^{n+1}}\left(1+\sigma_{\\{P^{n+1},P^{n+2}\\}}Z_{P^{n+1}}Z_{P^{n+2}}\right)$
$\displaystyle=\left(1+\sigma_{\\{P^{1},P^{2}\\}}Z_{P^{1}}Z_{P^{2}}\right)+\sum_{2\leq
j\leq n+1}(-1)^{j-1}\left(\prod_{1\leq
k<j}\sigma_{\\{P^{k-1},P^{k}\\}}\right)$ $\displaystyle\hskip
260.17464ptZ_{P^{1}}Z_{P^{j}}\left(1+\sigma_{\\{P^{j},P^{j+1}\\}}Z_{P^{j}}Z_{P^{j+1}}\right).$

This proves equation 33. Inequality 59 results from upper-bounding (in the
operator sense) the r.h.s. of equation 33:

$\displaystyle\sum_{2\leq j\leq l(P)}(-1)^{j-1}\left(\prod_{1\leq
k<j}\sigma_{\\{P^{k-1},P^{k}\\}}\right)Z_{P^{1}}Z_{P^{j}}\left(1+\sigma_{\\{P^{j},P^{j+1}\\}}Z_{P^{j}}Z_{P^{j+1}}\right)$
$\displaystyle=\sum_{2\leq j\leq l(P)}\left[1+(-1)^{j-1}\left(\prod_{1\leq
k<j}\sigma_{\\{P^{k-1},P^{k}\\}}\right)Z_{P^{1}}Z_{P^{j}}-1\right]\left(1+\sigma_{\\{P^{j},P^{j+1}\\}}Z_{P^{j}}Z_{P^{j+1}}\right)$
$\displaystyle=\sum_{2\leq j\leq
l(P)}\left(1+\sigma_{\\{P^{j},P^{j+1}\\}}Z_{P^{j}}Z_{P^{j+1}}\right)$
$\displaystyle\hskip 21.68231pt-\sum_{2\leq j\leq
l(P)}\left[1+(-1)^{j}\left(\prod_{1\leq
k<j}\sigma_{\\{P^{k-1},P^{k}\\}}\right)Z_{P^{1}}Z_{P^{j}}\right]\left(1+\sigma_{\\{P^{j},P^{j+1}\\}}Z_{P^{j}}Z_{P^{j+1}}\right)$
$\displaystyle\leq\sum_{2\leq j\leq
l(P)}\left(1+\sigma_{\\{P^{j},P^{j+1}\\}}Z_{P^{j}}Z_{P^{j+1}}\right).$

To obtain the last inequality, we used that the product of commuting positive
operators is positive. ∎

This result may straightaway be applied to generalize Bravyi et al.’s upper
bound (7, Theorem 2). For that purpose, we start by recalling some definitions
from (7) (see section 4.1 for graph-related notations):

######  Definition 34 ($\mathbf{Z}_{2}$-symmetric states and circuits).

A state $\ket{\psi}\in\mathbf{C}^{2^{n}}$ on $n$ qubits is called
$\mathbf{Z}_{2}$-symmetric if $X^{\otimes n}\ket{\psi}=\ket{\psi}$. A quantum
circuit $U\in\mathbf{U}(2^{n})$ acting on $n$ qubits is called
$\mathbf{Z}_{2}$-symmetric if $X^{\otimes n}UX^{\otimes n}=U$. A state will be
called “prepared by a $\mathbf{Z}_{2}$-symmetric circuit” if it can be
obtained by applying a $\mathbf{Z}_{2}$-symmetric circuit to a
$\mathbf{Z}_{2}$-symmetric product state.

######  Definition 35.

Let $G=(V,E)$ a graph. A circuit $U$ acting on qubits labelled by the vertices
of $V$ is said to have range $R$ if for all $v\in V$, all single-qubit
observable $\mathcal{O}_{v}$ supported on $v$, $U^{\dagger}\mathcal{O}_{v}U$
is supported on $B_{G}(v,R)$ (the $R$-neighbourhood of $v$ in $G$).

We are now ready to state and prove our generalization of (7, Theorem 2). The
main difference with the original theorem is that the new result not only
gives a lower bound on the expected energy of a ferromagnetic or
antiferromagnetic Ising Hamiltonian, but applies to Ising Hamiltonians with
arbitrary couplings in $\\{-1,1\\}$. Incidentally, this removes the need for
an extra step to relate the antiferromagnetic Ising Hamiltonian to the
ferromagnetic one, which is required by the original approach of (7) but does
not generalize well beyond ring graphs.

######  Proposition 36 (Generalization of (7, Theorem 2)).

Let $R\geq 1$ an integer, $k\geq 2$ an even integer, and $G$ a ring graph with
$n=(2R+1)k$ vertices. Then a state $\ket{\psi}$ prepared by a
$\mathbf{Z}_{2}$-symmetric range-$R$ quantum circuit satisfies for all choices
of signs $\left(\sigma_{e}\right)_{e\in E}\in\\{-1,1\\}^{E}$:

$\displaystyle\braket{\psi}{\sum_{e=\\{e_{0},e_{1}\\}\in
E}1+\sigma_{e}Z_{e_{0}}Z_{e_{1}}}{\psi}$ $\displaystyle\geq\frac{n}{2R+1}.$
(61)

In particular, for the MaxCut Hamiltonian $H_{\textnormal{MaxCut}}=\sum_{0\leq
i<k(2R+1)}\frac{1-Z_{i}Z_{i+1}}{2}$ associated to the ring graph,

$\displaystyle\braket{\psi}{H_{\textnormal{MaxCut}}}{\psi}$ $\displaystyle\leq
n-\frac{n}{2(2R+1)}.$ (62)

###### Proof.

Following the notation of lemma 33, consider the $k=\frac{n}{2R+1}$ edge-
disjoint length-$(2R+1)$ paths $P_{0},\ldots,P_{k-1}$ defined by
$P_{i}^{j}:=(2R+1)i+j$ ($0\leq j\leq 2R+1$). Applying lemma 33 then gives

$\displaystyle\braket{\psi}{\frac{n}{2R+1}-\sum_{0\leq
i<k}Z_{(2R+1)i}Z_{(2R+1)i+2R+1}}{\psi}$
$\displaystyle\leq\sum_{e=\\{e_{0},e_{1}\\}\in
E}\braket{\psi}{1+\sigma_{e}Z_{e_{0}}Z_{e_{1}}}{\psi}$

Now, by the range-$R$ assumptions,
$\braket{\psi}{Z_{(2R+1)i}Z_{(2R+1)i+2R+1}}{\psi}=0$, giving the result.
Finally, to upper bound the expectation of $H_{\textnormal{MaxCut}}$, it
suffices to lower-bound the expectation of $\sum_{e=\\{e_{0},e_{1}\\}\in
E}\frac{1+Z_{e_{0}}Z_{e_{1}}}{2}$ using the result just established. ∎

We now generalize this upper bound to grid graphs. In general, a grid graph is
a graph whose vertices and edges are defined by a lattice in $\mathbf{R}^{d}$.
Here, we will restrict to a square lattice. An example of such grid graph in
$d=2$ dimensions is given on figure 10.

Figure 10: A 2D grid graph

The following proposition applies to a two-dimensional grid graph:

######  Proposition 37.

Let $R\geq 1$ an integer and $G=(V,E)$ a 2D grid graph (with periodic boundary
conditions) of dimensions $(2R+1)k\times(2R+1)k$, $k\geq 1$ integer. Then a
state $\ket{\psi}$ prepared by a $\mathbf{Z}_{2}$-symmetric range-$R$ quantum
circuit satisfies for all choices of signs $\left(\sigma_{e}\right)_{e\in
E}\in\\{-1,1\\}^{E}$:

$\displaystyle\braket{\psi}{\sum_{e=\\{e_{0},e_{1}\\}\in
E}\left(1+\sigma_{e}Z_{e_{0}}Z_{e_{1}}\right)}{\psi}$
$\displaystyle\geq\frac{2n}{2R+1}$ (63)

where $n=(2R+1)^{2}k^{2}$ is the number of vertices in $G$. In particular, for
the MaxCut Hamiltonian777Note that the grid graph described here has $2n$
edges and that MaxCut is completely satisfiable for this graph.
$H_{\textrm{MaxCut}}$ of $G$, this implies

$\displaystyle\braket{\psi}{H_{\textnormal{MaxCut}}}{\psi}$ $\displaystyle\leq
2n-\frac{n}{2R+1}.$ (64)

###### Proof.

One may pack exactly $\underbrace{k}_{\begin{subarray}{c}\textrm{horizontal
paths}\\\ \textrm{per
row}\end{subarray}}\times\underbrace{k(2R+1)}_{\textrm{rows}}+\underbrace{k}_{\begin{subarray}{c}\textrm{vertical
paths}\\\ \textrm{per
column}\end{subarray}}\times\underbrace{k(2R+1)}_{\textrm{columns}}=2k^{2}(2R+1)=\frac{2n}{2R+1}$
horizonzal and vertical edge-disjoint paths of length $2R+1$ in $G$; an
example is given on figure 11 for $R=1$ (length $3$ paths).

Figure 11: Packing length-3 paths (colored either red, green, blue or purple)
in a 2D grid graph

By the range-$R$ property of the circuit, the vertices at the extremities of
each path are uncorrelated. Therefore, invoking lemma 33 and taking the
expectation of the operator inequality there on state $\ket{\psi}$ gives

$\displaystyle\frac{2n}{2R+1}$ $\displaystyle\leq\sum_{e=\\{e_{0},e_{1}\\}\in
E}\braket{\psi}{1+\sigma_{e}Z_{e_{0}}Z_{e_{1}}}{\psi}.$

∎

Note that the lower-bound $\frac{2n}{2R+1}$ for a two-dimensional grid graph
is twice the one for a ring graph (which is a one-dimensional grid graph). In
fact, by reviewing the last proof, it is not hard to see that this extra
factor comes from the number of dimensions. The previous proposition then
generalizes immediately:

######  Proposition 38.

Let $R\geq 1$ an integer and $G=(V,E)$ be a $d$-dimensional grid graph (with
periodic boundary conditions) of dimensions
$(2R+1)k\times\ldots\times(2R+1)k$, $k\geq 1$ integer. Then a state
$\ket{\psi}$ prepared by a $\mathbf{Z}_{2}$-symmetric range-$R$ quantum
circuit satisfies for all choices of signs $\left(\sigma_{e}\right)_{e\in
E}\in\\{-1,1\\}^{E}$:

$\displaystyle\braket{\psi}{\sum_{e=\\{e_{0},e_{1}\\}\in
E}\left(1+\sigma_{e}Z_{e_{0}}Z_{e_{1}}\right)}{\psi}$
$\displaystyle\geq\frac{dn}{2R+1}$ (65)

where $n=(2R+1)^{2}k^{2}$ is the number of vertices in $G$. In particular, for
the MaxCut Hamiltonian888Note that the grid graph described here has $2n$
edges and that MaxCut is completely satisfiable for this graph.
$H_{\textnormal{MaxCut}}$ of $G$, this implies

$\displaystyle\braket{\psi}{H_{\textnormal{MaxCut}}}{\psi}$ $\displaystyle\leq
dn-\frac{dn}{2R+1}.$ (66)

One may now wonder to what extent bounds 62, 64, 66 are tight. For 62, which
applies to a ring graph, (7, lemma B.2) shows a converse bound. A similar
result can proved for two-dimensional graph:

######  Proposition 39.

For a 2D grid graph $G$ with periodic boundary conditions of size
$k(2R^{2}+2R+1)\times k(2R^{2}+2R+1)$, with $k\geq 2$ an even integer, exists
a $\mathbf{Z}_{2}$-symmetric range-$R$ circuit such that

$\displaystyle\braket{\psi}{H_{\textrm{MaxCut}}}{\psi}$
$\displaystyle=2n-\frac{2R+1}{2R^{2}+2R+1}n.$ (67)

###### Proof.

Let us fix $k,R$ in the rest of the proof. We follow the strategy of proof of
(7, Lemma B.2), which consists to pack the graph with GHZ states “cells” as
illustrated on figure 12.

Figure 12: A $13\times 13$ vertices grid graph (periodic boundary conditions)
with packed GHZ states. Qubits forming a single GHZ state are delimited by a
gray cell.

For this packing to be exactly possible, each dimension of the lattice has to
be a multiple of $2R^{2}+2R+1$, hence the hypothesis of the proposition. A
single cell comprises $2R^{2}+2R+1$ qubits —the case $R=2$ is shown on figure
13.

Figure 13: A GHZ state cell

Now, a GHZ state can be prepared in a cell starting with all qubits in state
$\ket{0}$ and applying the circuit on figure 14.

Figure 14: Circuit preparing a GHZ on the cell qubits.

This circuit can be converted to a $\mathbf{Z}_{2}$-symmetric one (with range
$R$) acting on the $\mathbf{Z}_{2}$-symmetric state
$\ket{+}^{\otimes(2R^{2}+2R+1)}$ (instead of $\ket{0}^{\otimes(2R^{2}+2R+1)}$)
using the same circuit identities as in the proof of (7, Lemma B.1). The next
step consists to apply to each GHZ cell a series of alternated bitflips as
represented on figure 15(a). After performing these bit flips, every edge
$\\{i,j\\}$ connecting vertices in the same GHZ cell will be satisfied
—meaning the quantum state $\ket{\psi}$ will satisfy
$Z_{i}Z_{j}\ket{\psi}=-\ket{\psi}$ (illustration figure 15(b)).

(a) Alternated bit flip pattern

(b) Resulting satisfied edges

Figure 15: Applying bit flips to GHZ cell

However, every edge $\\{i,j\\}$ connecting vertices not in the same GHZ cell
will verify $\braket{\psi}{Z_{i}Z_{j}}{\psi}=0$. For each GHZ cell, there are
$4R^{2}$ edges belonging to the first category; since there are
$\frac{n}{2R^{2}+2R+1}$ GHZ cells, there are $\frac{4R^{2}}{2R^{2}+2R+1}n$
edges belonging to the first category. There are then
$2n-\frac{4R^{2}}{2R^{2}+2R+1}n=\frac{4R+2}{2R^{2}+2R+1}n$ edges in the second
category. This counting yields

$\displaystyle\braket{\psi}{H_{\textrm{MaxCut}}}{\psi}$
$\displaystyle=\sum_{\textrm{edges
}\\{i,j\\}}\braket{\psi}{\frac{1-Z_{i}Z_{j}}{2}}{\psi}$
$\displaystyle=\frac{4R^{2}}{2R^{2}+2R+1}n\times
1+\frac{4R+2}{2R^{2}+2R+1}n\times\frac{1}{2}$
$\displaystyle=\frac{4R^{2}+2R+1}{2R^{2}+2R+1}n$
$\displaystyle=2n-\frac{2R+1}{2R^{2}+2R+1}n.$

∎

We now apply lemma 33 to random $d$-regular graphs. In previous cases: ring
and grid graphs, the graphs were bipartite and the method bounded the cut
achievable by the algorithm away from $|E|$, the number of edges in a maximum
cut. This stands in contrast to random regular graph, as such a graph is, with
high probability, not bipartite (see e.g. (10) and remarks after proposition
40). Unfortunately, Bravyi et al.’s method can only bound the achievable cut
away from $|E|$ in this case; this means that the upper bound becomes trivial
for large enough $R$, but may still be relevant for low $R$. The bound is
precisely stated here:

######  Proposition 40.

Let $d\geq 3$ and $G=(V,E)$ be a random $d$-regular graph. For any
$\varepsilon>0$, with probability $\geq 1-\varepsilon$, for all choices of
signs $\left(\sigma_{e}\right)_{e\in E}\in\\{-1,1\\}^{E}$,

$\displaystyle(1-\varepsilon)\frac{dn}{2R+1}$
$\displaystyle\leq\sum_{e=\\{e_{0},e_{1}\\}\in
E}\braket{\psi}{1+\sigma_{e}Z_{e_{0}}Z_{e_{1}}}{\psi}$ (68)

holds for $n$ large enough for all state $\ket{\psi}$ prepared by a
$\mathbf{Z}_{2}$-symmetric range-$R$ circuit. In particular, for the MaxCut
Hamiltonian $H_{\textnormal{MaxCut}}$ associated to $G$,

$\displaystyle\braket{\psi}{H_{\textnormal{MaxCut}}}{\psi}$
$\displaystyle\leq\frac{nd}{2}-(1-\varepsilon)\frac{nd}{2(2R+1)}.$ (69)

###### Proof.

Contrary to the proof for ring and grid graphs, we will choose here a set of
paths $\mathcal{P}$ that overlap. Let us then consider the set of directed
paths starting at any vertex of $G$ and having length $2R+1$. (The
directedness means that two paths consisting of the same edges but walked in
opposite order are regarded as distinct.) Using the tree neighbourhood
property of random regular graphs stated in proposition 7, for large enough
$n$, with probability $1-\varepsilon$ at least $(1-\varepsilon)n$ vertices in
$G$ have trees as $(4R+1)$-neighbourhoods. For each such vertex, there are
$d(d-1)^{2R}$ distinct paths starting from this vertex. Therefore, these
vertices give rise to at least $\frac{1}{2}(1-\varepsilon)nd(d-1)^{2R}$
distinct paths, as illustrated on figure 16.

Figure 16: Directed paths arising from the central vertex for $d=3$ and $R=1$.
There are 12 such paths, corresponding to the leaves of the tree.

Each of these paths has their extremities $(2R+1)$ apart in the graph
—otherwise, the $(4R+1)$-neighbourhood of the vertex from which this path
starts would not be a tree. This lower-bounds the left-hand-side of equation
58 from lemma 33 by $\frac{1}{2}(1-\varepsilon)nd(d-1)^{2R}$. To upper-bound
the right-hand side, it remains to upper-bound the number of paths an edge may
appear in. Let then fix an edge $e\in E$. $e$ may then be the first edge of at
most $(d-1)^{2R}$ paths, the second edge of the same number of paths, and so
on until the $(2R+1)^{\textrm{th}}$ edge. This gives the upper-bound
$(2R+1)(d-1)^{2R}\sum_{e=\\{e_{0},e_{1}\\}\in
E}\left(1+\sigma_{e}Z_{e_{0}}Z_{e_{1}}\right)$. We then finally obtain:

$\displaystyle(1-\varepsilon)\frac{dn}{2R+1}$
$\displaystyle\leq\sum_{e=\\{e_{0},e_{1}\\}}\braket{\psi}{1+\sigma_{e}Z_{e_{0}}Z_{e_{1}}}{\psi}.$

∎

The bound 69 on MaxCut is unfortunately less satisfying than in the case of
ring and grid graphs. Indeed, for these graphs, MaxCut was completely
satisfiable and the right-hand side of the inequality was always smaller than
the maximum cut. On the contrary, MaxCut is usually not completely satisfiable
for a random regular graph. For instance, (10) shows that with high
probability, the MaxCut of a random $d$-regular graph $G=(V,E)$ includes
$|E|\left(\frac{1}{2}+\frac{P_{*}}{\sqrt{d}}(1+o_{d}(1))\right)$ edges, where
$P_{*}\approx 0.7632\ldots$ is the Parisi constant, which is a constant factor
below $|E|$. However, bound 69 merely guarantees
$\braket{\psi}{H_{\textnormal{MaxCut}}}{\psi}\leq|E|-(1-\varepsilon)\frac{|E|}{2R+1}$
and therefore becomes trivial for large enough $R$.

In particular, it is hopeless to construct a circuit achieving the bound.
However, one may still ask about the existence of a circuit similar to the
ones constructed in (7, lemma B.2) or proposition 39 achieving some lower
bound. Qualitatively, such a circuit is $\mathbf{Z}_{2}$-symmetric, has range
$R$ and perfectly anticorrelates some edges $\\{i,j\\}$:
$\braket{\psi}{Z_{i}Z_{j}}{\psi}=-1$ while perfectly decorrelating others:
$\braket{\psi}{Z_{i}Z_{j}}{\psi}=0$. We now demonstrate that for a random
regular graph, a cut sampled from a circuit verifying these assumptions would
necessarily have a poor approximation ratio. Therefore, maybe surprisingly,
Bravyi et al.’s construction is inoperational in the case of random regular
graphs. In fact, it will be interesting to start with a more general
proposition:

######  Proposition 41.

Let $G=(V,E)$ a graph with $n$ vertices. Assume that $\ket{\psi}$ is a
$\mathbf{Z}_{2}$-symmetric state with range $R$ such that for all
$e=\\{e_{0},e_{1}\\}\in E$,
$\braket{\psi}{Z_{e_{0}}Z_{e_{1}}}{\psi}\in\\{0,-1\\}$. Denote by $C_{l}$ the
number of cycles in the graph with length $\leq l$.

$\displaystyle\braket{\psi}{\sum_{e\in
E}\frac{1-Z_{e_{0}}Z_{e_{1}}}{2}}{\psi}$
$\displaystyle\leq\frac{1}{2}\left(|E|+n+(4R+1)C_{4R+1}\right)$ (70)

###### Proof.

Consider the edges $e$ such that $\braket{\psi}{Z_{e_{0}}Z_{e_{1}}}{\psi}=-1$;
call these edges $E^{\prime}$. Next, define $E^{\prime\prime}$ the set of
edges in $E^{\prime}$ which do not lie in a cycle of length $\leq 4R+1$;
therefore, $|E^{\prime\prime}|\geq|E^{\prime}|-(4R+1)C_{4R+1}$. Now, we show
that all paths with edges lying in $E^{\prime\prime}$ have length at most
$2R$. Assume, for the sake of contradiction, that there exists a path of
length $\geq 2R+1$ with edges in $E^{\prime\prime}$ and denote by
$v_{0},\ldots,v_{2R+1}$ the first $2R+2$ vertices of this path. Then
$v_{0}\ldots v_{2R+1}$ is a shortest path between $v_{0}$ and $v_{2R+1}$
—otherwise, some edges $v_{j}v_{j+1}$ would lie in a cycle of length $\leq
4R+1$. But then,

$\displaystyle 1$ $\displaystyle=\braket{\psi}{1+Z_{v_{0}}Z_{v_{2R+1}}}{\psi}$
$\displaystyle\leq\sum_{0\leq
j<2R+1}\braket{\psi}{1+Z_{v_{j}}Z_{v_{j+1}}}{\psi}$ $\displaystyle=0,$

a contradiction. Therefore, all paths in $E^{\prime\prime}$ have length at
most $2R$. Now, consider a connected component of the graph induced by the
edge set $E^{\prime\prime}$. Such a component then has diameter at most $2R$.
Besides, it has no cycle —since such a cycle would be of length $\leq 4R$,
contradicting the definition of $E^{\prime\prime}$. Therefore, the connected
component must be a tree and its number of edges is then bounded by the number
of vertices minus $1$. Summing over all connected components, one must then
have $|E^{\prime\prime}|\leq n$, hence $|E^{\prime}|\leq n+(4R+1)C_{4R+1}$.
The bound follows by recalling that edges $\\{i,j\\}$ not in $E^{\prime}$
satisfy $\braket{\psi}{Z_{i}Z_{j}}{\psi}=0$. ∎

As previously hinted, this proposition implies that for a random $d$-regular
graph, a circuit similar to the one proposed by Bravyi et al. produces a cut
with at most $|E|\left(\frac{1}{2}+\frac{1}{d}\right)$ satisfied edges, which
is bounded away from the typical value
$|E|\left(\frac{1}{2}+\frac{0.7632\ldots}{\sqrt{d}}(1+o_{d}(1))\right)$.

######  Corollary 42.

Let $G=(V,E)$ a random $d$-regular graph with $n$ vertices. Assume that
$\ket{\psi}$ is prepared by a $\mathbf{Z}_{2}$-symmetric range $R$ circuit
such that for all $e=\\{e_{0},e_{1}\\}\in E$,
$\braket{\psi}{Z_{e_{0}}Z_{e_{1}}}{\psi}\in\\{0,-1\\}$. Then for all
$\varepsilon>0$, with probability exponentially close to $1$ as $n\to\infty$,

$\displaystyle\braket{\psi}{\sum_{e=\\{e_{0},e_{1}\\}\in
E}\frac{1-Z_{e_{0}}Z_{e_{1}}}{2}}{\psi}$
$\displaystyle\leq|E|\left(\frac{1}{2}+\frac{1}{d}+\varepsilon\right)$ (71)

for any state $\ket{\psi}$ prepared by a $\mathbf{Z}_{2}$-symmetric range-$R$
quantum circuit.

###### Proof.

Using for instance (36, Theorem 1), for all $\varepsilon>0$, a random
$d$-regular graph $G$ has less than $\frac{\varepsilon}{4R+1}n$ cycles of
length less than $4R+1$ with probability exponentially close to $1$ as
$n\to\infty$. The result then follows from proposition 41. ∎

Coming back to grid graphs, it is worth noting that proposition 41 does not
contradict the achievability result stated in proposition 39. The reason is,
the two-dimensional grid graph considered in the latter theorem has many short
cycles compared to a random regular graph. For instance, for $R=1$,
$C_{4R+1}=C_{5}=n$, so proposition 41 gives a loose $\frac{13}{2}n$ upper
bound on the number of satisfied edges (while the grid graph only has $2n$
edges).

To conclude on the generalization of Bravyi et al.’s method, we finally show
that proposition 40 can be applied to derive upper bounds on the overlap of a
typical cut sampled from a shallow circuit with a maximum cut. Contrary to
previous upper bounds, these new bounds have the interest of being nontrivial
both for grid graphs and random regular graphs. They will result from the
following lemma:

######  Lemma 43.

Let $G=(V,E)$ a graph. Assume that for some state $\ket{\psi}$ and constant
$\alpha>0$,

$\displaystyle\alpha|E|$ $\displaystyle\leq\sum_{e=\\{e_{0},e_{1}\\}\in
E}\braket{\psi}{1+\sigma_{e}Z_{e_{0}}Z_{e_{1}}}{\psi}$ (72)

for all choices of signs $\left(\sigma_{e}\right)_{e\in E}\in\\{-1,1\\}^{E}$.
Denote by $\textnormal{MaxCut}(G)$ the number of satisfied edges in a maximum
cut of $G$. Then for all partition $E=E_{1}\sqcup E_{2}$ of the edges of $G$,

$\displaystyle\braket{\psi}{\sum_{e=\\{e_{0},e_{1}\\}\in
E_{1}}\frac{1-Z_{e_{0}}Z_{e_{1}}}{2}}{\psi}$
$\displaystyle\leq\frac{|E_{1}|+\textnormal{MaxCut}(G)}{2}-\frac{\alpha}{4}|E|.$
(73)

###### Proof.

Fix a partition $E=E_{1}\sqcup E_{2}$ of the edges of $G$. Define signs
$\left(\sigma_{e}\right)_{e\in E}$ by letting $\sigma_{e}=1$ if $e\in E_{1}$
and $\sigma_{e}=-1$ if $e\in E_{2}$. Then, by the assumption of the
proposition,

$\displaystyle\alpha|E|$
$\displaystyle\leq\braket{\psi}{\sum_{e=\\{e_{0},e_{1}\\}\in
E_{1}}(1+\sigma_{e}Z_{e_{0}}Z_{e_{1}})}{\psi}+\braket{\psi}{\sum_{e=\\{e_{0},e_{1}\\}\in
E_{2}}(1-\sigma_{e}Z_{e_{0}}Z_{e_{1}})}{\psi}.$

This implies successively:

$\displaystyle\alpha|E|\leq
2|E_{1}|-\braket{\psi}{\sum_{e=\\{e_{0},e_{1}\\}\in
E_{1}}(1-\sigma_{e}Z_{e_{0}}Z_{e_{1}})}{\psi}+\braket{\psi}{\sum_{e=\\{e_{0},e_{1}\\}\in
E_{2}}(1-\sigma_{e}Z_{e_{0}}Z_{e_{1}})}{\psi}$
$\displaystyle\alpha|E|-2|E_{1}|+2\braket{\psi}{\sum_{e=\\{e_{0},e_{1}\\}\in
E_{1}}(1-\sigma_{e}Z_{e_{0}}Z_{e_{1}})}{\psi}\leq\braket{\psi}{\sum_{e\in
E}\left(1-Z_{e_{0}}Z_{e_{1}}\right)}{\psi}.$

But $\braket{\psi}{\sum_{e\in E}\left(1-Z_{e_{0}}Z_{e_{1}}\right)}{\psi}\leq
2\textnormal{MaxCut}(G)$. The result follows. ∎

This lemma can right away be applied to a random regular graph:

######  Proposition 44.

Let $G=(V,E)$ a random $d$-regular graph. Let $E_{0}$ the set of satisfied
edges of a maximum cut. Then for all $\epsilon>0$, with probability $\geq
1-\varepsilon$ on the choice of $G$, for any state $\ket{\psi}$ prepared by a
$\mathbf{Z}_{2}$-symmetric range-$R$ circuit, the overlap between $E_{0}$ and
the satisfied edges of a cut sampled from $\ket{\psi}$ is bounded away from
$|E_{0}|$:

$\displaystyle\braket{\psi}{\sum_{e=\\{e_{0},e_{1}\\}\in
E_{0}}\frac{1-Z_{e_{0}}Z_{e_{1}}}{2}}{\psi}$
$\displaystyle\leq|E_{0}|-\frac{(1-\varepsilon)}{2(2R+1)}|E|.$ (74)

###### Proof.

This results from the previous lemma, applied to
$\alpha=\frac{2(1-\varepsilon)}{2R+1}$ (according to proposition 40),
$E_{1}=E_{0}$ and $E_{2}=E-E_{0}$. ∎

##  Appendix D Formulae for $p=1$ QAOA

###  D.1 Ring graph

Consider a vertex $0$ of a ring graph (with at least 3 vertices), having
neighbours $1$ and $2$.
$U_{C}(\gamma)^{\dagger}U_{B}(\beta)^{\dagger}Z_{0}U_{B}(\beta)U_{C}(\gamma)$
can be expressed as the sum of Pauli tensors represented on the next figure:

Figure 17: Terms of
$U_{C}(\gamma)^{\dagger}U_{B}(\beta)^{\dagger}Z_{0}U_{B}(\beta)U_{C}(\gamma)$
for $p=1$ QAOA on ring

###  D.2 3-regular graph

Consider a vertex $0$ of a 3-regular graph, having neighbours $1,2,3$.
$U_{C}(\gamma)^{\dagger}U_{B}(\beta)^{\dagger}Z_{0}U_{B}(\beta)U_{C}(\gamma)$
can be expressed as the sum of Pauli tensors represented on the next figure:

Figure 18: Terms of
$U_{C}(\gamma)^{\dagger}U_{B}(\beta)^{\dagger}Z_{0}U_{B}(\beta)U_{C}(\gamma)$
for $p=1$ QAOA on 3-regular graph


# Chirality Induced Spin Coherence in Electron Transfer Reactions

Thomas P. Fay<EMAIL_ADDRESS>Department of Chemistry, University
of Oxford, Physical and Theoretical Chemistry Laboratory, South Parks Road,
Oxford, OX1 3QZ, UK

###### Abstract

Recently there has been much interest in the chirality induced spin
selectivity effect, whereby electron spin polarisation, which is dependent on
the molecular chirality, is produced in electrode-molecule electron transfer
processes. Naturally, one might consider if a similar effect can be observed
in simple molecular charge transfer reactions, for example in light-induced
electron transfer from an electron donor to an electron acceptor. In this work
we explore the effect of electron transfer on spins in chiral single radicals
and chiral radical pairs using Nakajima-Zwanzig theory. In these cases
chirality, in conjuction with spin-orbit coupling, does not lead to spin
polarisation, but instead the electron transfer generates quantum coherence
between spins states. In principle, this chirality induced spin coherence
could manifest in a range of experiments, and in particular we demonstrate
that the OOP-ESEEM pulse EPR experiment would be able to detect this effect in
oriented radical pairs.

Recently there has been significant progress in experiments1, 2, 3, 4, 5, 6,
7, 8 on and theoretical descriptions9, 10, 11 of the chirality induced spin
selectivity (CISS) effect. Much of this has been motivated by the potential
for this effect to be exploited in spintronic devices,12 in which spins can be
used to transfer and store information, with the potential for use in quantum
computation and quantum information transfer.12, 13 The CISS effect involves
the generation of electron spin polarisation by the transmission of an
electron through a chiral environment, and this effect is generally explained
by the presence of the spin-orbit interaction in these systems.8, 9, 10 The
majority of experimental and theoretical studies involve the transmission of
electrons between electrodes and chiral molecules,8, 12, 1 however there has
been relatively little investigation into the role of chirality on the spin in
molecular electron transfers, and so this is precisely what we explore here
for reactions in involving radicals and radical pair formation.

It is instructive to first consider a charge transfer reaction in an $S=1/2$
doublet system,

$\displaystyle\ce{{}^{2}[D^{$\bullet-$}A]<=>[$k_{\mathrm{f}}$][$k_{\mathrm{b}}$]{}^{2}[DA^{$\bullet-$}]}.$

A simple model for this is a system which can exist in one of two charge
transfer (CT) states: $\ket{0}=\ket{\ce{D^{$\bullet-$}A}}$ and
$\ket{1}=\ket{\ce{DA^{$\bullet-$}}}$.14, 15, 16, 17 Associated with each of
these diabatic states are two spin states $\ket{M_{S}=\pm 1/2}$, so overall
there are four electronic states $\ket{j,M_{S}}$. For this system we can write
the total Hamiltonian as17, 14, 15, 16

$\displaystyle\hat{H}=\hat{H}_{0}\hat{\Pi}_{0}+\hat{H}_{1}\hat{\Pi}_{1}+\hat{V}_{\mathrm{DC}}+\hat{V}_{\mathrm{SOC}},$
(1)

where $\hat{\Pi}_{j}=\sum_{M_{S}=\pm 1/2}\outerproduct{j,M_{S}}{j,M_{S}}$ is a
projection operator onto charge transfer state $j$, $\hat{H}_{j}$ is the
Hamiltonian for charge transfer state $j$, $\hat{V}_{\mathrm{DC}}$ is the
diabatic coupling Hamiltonian and $\hat{V}_{\mathrm{SOC}}$ is the spin-orbit
coupling Hamiltonian. The Hamiltonian for each charge transfer state
$\hat{H}_{j}$ can be divided into a nuclear part
$\hat{H}_{j\mathrm{n}}=\hat{T}_{\mathrm{n}}+V_{j}(\hat{\mathbf{Q}})$ which is
a sum of nuclear kinetic energy and potential energy terms, and a spin part
$\hat{H}_{j\mathrm{s}}$, and for simplicity we can assume the spin part is
independent of the nuclear coordinates $\mathbf{Q}$.14, 15

The diabatic coupling term $\hat{V}_{\mathrm{DC}}$ does not act on the spin of
the system, instead it simply couples the two charge transfer states, and
within the Condon approximation (coupling independent of $\mathbf{Q}$) this
is17, 14, 15

$\displaystyle\hat{V}_{\mathrm{DC}}=\Delta(\outerproduct{0}{1}+\outerproduct{1}{0})$
(2)

where $\Delta$ is a real-valued constant. The spin-orbit coupling also couples
the charge transfer states, but it also acts on the spin as follows18, 19, 20

$\displaystyle\hat{V}_{\mathrm{SOC}}=-i(\outerproduct{0}{1}\bm{\Lambda}\cdot\hat{\mathbf{S}}-\bm{\Lambda}\cdot\hat{\mathbf{S}}\outerproduct{1}{0}).$
(3)

$\hat{\mathbf{S}}$ here is the unitless electron spin operator and the vector
coupling $\bm{\Lambda}$ is real valued for the spin-orbit coupling between
bound electronic states (a detailed justification of this is given in the SI).
Overall this means we can write the total charge transfer state coupling
Hamiltonian $\hat{V}=\hat{V}_{\mathrm{DC}}+\hat{V}_{\mathrm{SOC}}$ as

$\displaystyle\hat{V}=\Gamma\left(\hat{U}^{\dagger}\outerproduct{0}{1}+\outerproduct{1}{0}\hat{U}\right),$
(4)

where $\hat{U}$ is a unitary spin operator given by
$\hat{U}=(\Delta+i\bm{\Lambda}\cdot\hat{\mathbf{S}})/\Gamma$, and
$\Gamma=\sqrt{\Delta^{2}+|\bm{\Lambda}|^{2}/4}$. This unitary operator rotates
the electron spin about the axis $\mathbf{n}=\bm{\Lambda}/|\Lambda|$ by an
angle $2\theta$, with $\theta$ given by

$\displaystyle\theta=\atan(|\bm{\Lambda}|/(2\Delta)).$ (5)

Coherences between the spin states of each charge transfer state will be long-
lived because the potential energy surfaces for different spin states
$V_{jM_{S}}(\mathbf{Q})=V_{j}(\mathbf{Q})$ are identical, but this is not true
for coherences between the different charge transfer states, and as such we
describe the system with two spin density operators, one for each charge
transfer state, $\hat{\sigma}_{j\mathrm{s}}(t)$ which are related to the full
density operator $\hat{\rho}(t)$ by17

$\displaystyle\hat{\sigma}_{j\mathrm{s}}(t)=\Tr_{\mathrm{n}}[\expectationvalue{\hat{\rho}(t)}{j}].$
(6)

Our aim is now to find a quantum master equation for these spin density
operators. This is done by employing the Nakajima-Zwanzig equation21, 22
following the same steps as outlined in the derivation of radical pair spin
master equations in Ref. 17, which we summarise in the SI. Assuming that the
charge transfer coupling $\hat{V}$ can be treated perturbatively, and that the
population transfer is described accurately as an incoherent kinetic process,
with rates independent of the spin interactions in $\hat{H}_{j\mathrm{s}}$,
the resulting quantum master equations at second order in $\Gamma$ for
$\hat{\sigma}_{j\mathrm{s}}(t)$ are,

$\displaystyle\derivative{t}\hat{\sigma}_{0\mathrm{s}}(t)$
$\displaystyle=-\frac{i}{\hbar}[\hat{H}_{0\mathrm{s}},\hat{\sigma}_{1\mathrm{s}}(t)]-k_{\mathrm{f}}\hat{\sigma}_{0\mathrm{s}}(t)+k_{\mathrm{b}}\hat{U}^{\dagger}\hat{\sigma}_{1\mathrm{s}}\hat{U}$
(7) $\displaystyle\derivative{t}\hat{\sigma}_{1\mathrm{s}}(t)$
$\displaystyle=-\frac{i}{\hbar}[\hat{H}_{1\mathrm{s}},\hat{\sigma}_{1\mathrm{s}}(t)]-k_{\mathrm{b}}\hat{\sigma}_{1\mathrm{s}}(t)+k_{\mathrm{f}}\hat{U}\hat{\sigma}_{0\mathrm{s}}\hat{U}^{\dagger},$
(8)

where $[\cdot,\cdot]$ denotes the commutator. We see that as an electron is
transferred, its spin is rotated about the vector $\bm{\Lambda}$, and as it is
transferred back it is rotated about the vector $-\bm{\Lambda}$.

Thus far we do not appear to have included the idea of chirality in any of the
above discussion, so we shall now discuss how this manifests in these
equations. Chirality of the system determines the relative sign of $\Delta$
and $\bm{\Lambda}$. On spatial reflection, the sign of the diabatic coupling
$\Delta$ in unchanged, but the sign of the spin-orbit coupling vector
$\bm{\Lambda}$ is changed because the spin-orbit coupling operator is an axial
vector operator.10 This means that for an initially spin polarised state of
$\ket{0}$, $\hat{\sigma}_{0s}(0)=\outerproduct{\alpha}{\alpha}$,111Here we use
the standard notation for the electron spin states
$\ket{\alpha}=\ket{M_{S}=+1/2}$ and $\ket{\beta}=\ket{M_{S}=-1/2}$. this
polarisation will be transformed into coherence between spin states
$\ket{\alpha}$ and $\ket{\beta}$ in CT state $\ket{1}$ if $\bm{\Lambda}$ has a
component perpendicular to the $z$ axis.222This rotation of the electron spin
is reminiscent of the $\vec{C}$ term appearing in the theory of Dalum &
Hedegård in Ref. 10, derived for spin transport through a molecular junction
from the non-equilibrium Green’s function approach. A spin polarisation term
analogous to the $\vec{D}$ term in in Ref. 10 does not appear in our theory
for spin transport between bound electronic states because the initial and
final electronic states have no linear or angular momentum, so their
wavefunctions are real valued and the spin-orbit coupling vector is purely
real, so no spin polarisation appears. Interestingly this means that photo-
initiated electron transfer in a spatially oriented chiral doublet system,
initially at thermal equilibrium in a magnetic field, would yield $x/y$
magnetisation, and therefore a free-induction decay signal, without a
microwave pulse. For an achiral system there are equal contributions from
configurations with $\pm\bm{\Lambda}$, which means that spin polarisation
perpendicular to the $\bm{\Lambda}$ axis is simply lost in the charge transfer
process, and as such microwave pulse-free free-induction decay could not be
observed.

We are now in a position to consider the effect of spin-orbit coupling and
chirality on the charge transfer process between a singlet photoexcited
precursor state $\ce{DA^{*}}$ and a charge separated radical pair state
$\ce{D^{$\bullet+$}A^{$\bullet-$}}$,

$\displaystyle\ce{{}^{1}[DA^{*}]<=>[$k_{\mathrm{f}}$][$k_{\mathrm{b}}$]{}^{$2S+1$}[D^{$\bullet+$}A^{$\bullet-$}]}.$

Analogous to the above situation, a simple model for this includes the excited
precursor diabatic state $\ket{0}=\ket{\ce{DA^{*}}}$ and a charge separated
radical pair state $\ket{1}=\ket{\ce{D^{$\bullet+$}A^{$\bullet-$}}}$. We
assume that the state $\ket{0}$ can only exist in a singlet spin state, so
only $\ket{0,S=0,M_{S}=0}$ exists, but the radical pair state $\ket{1}$ can
exist in both singlet and triplet states which lie very close in energy,
therefore there are four (near-)degenerate $\ket{1,S,M_{S}}$ states which
exist, with $S=0$ and $S=1$. The total Hamiltonian for this system has the
same form as Eq. (1),
$\hat{H}=\hat{H}_{0}\hat{\Pi}_{0}+\hat{H}_{1}\hat{\Pi}_{1}+\hat{V}_{\mathrm{DC}}+\hat{V}_{\mathrm{SOC}},$
where again $\hat{H}_{j}=\hat{H}_{j\mathrm{s}}+\hat{H}_{j\mathrm{n}}$ is the
Hamiltonian for state $j$ and $\hat{\Pi}_{j}$ is a projection operator onto
the $j$ states of the allowed spin multiplicity for that charge transfer
state, i.e. $\hat{\Pi}_{0}=\outerproduct{0}{0}\hat{P}_{\mathrm{S}}$ and
$\hat{\Pi}_{1}=\outerproduct{1}{1}(\hat{P}_{\mathrm{S}}+\hat{P}_{\mathrm{T}})$,
where $\hat{P}_{\mathrm{S}}$ and $\hat{P}_{\mathrm{T}}$ are singlet and
triplet spin state projection operators. As before

$\displaystyle\hat{V}_{\mathrm{DC}}=\Delta(\hat{P}_{\mathrm{S}}\outerproduct{0}{1}+\outerproduct{1}{0}\hat{P}_{\mathrm{S}})$
(9)

is the diabatic coupling Hamiltonian, which now comes with a singlet
projection operator $\hat{P}_{\mathrm{S}}$ because the diabatic coupling is
spin conserving.17 The electron transfer can be regarded as a one electron
transfer process, where only electron 1 transfers, and therefore the spin-
orbit coupling term can be written as

$\displaystyle\hat{V}_{\mathrm{SOC}}=-i(\hat{P}_{\mathrm{S}}\outerproduct{0}{1}\bm{\Lambda}\cdot\hat{\mathbf{S}}_{1}-\bm{\Lambda}\cdot\hat{\mathbf{S}}_{1}\outerproduct{1}{0}\hat{P}_{\mathrm{S}}).$
(10)

where $\hat{\mathbf{S}}_{1}$ is the spin operator for electron 1 (this is
justified in more detail in the SI). This means the total charge transfer
Hamiltonian can be written as

$\displaystyle\hat{V}=\Gamma(\hat{P}_{\mathrm{S}}\hat{U}^{\dagger}\outerproduct{0}{1}+\outerproduct{1}{0}\hat{U}\hat{P}_{\mathrm{S}})$
(11)

where now $\hat{U}=(\Delta+i\bm{\Lambda}\cdot\hat{\mathbf{S}}_{1})/\Gamma$
only acts on electron 1.

From this we can obtain the quantum master equations for the spin density
operators of the two charge transfer states, once again using the Nakajima-
Zwanzig equation as in Ref. 17, which gives

$\displaystyle\begin{split}\derivative{t}\hat{\sigma}_{0\mathrm{s}}(t)&=-\frac{i}{\hbar}[\hat{H}_{0\mathrm{s}},\hat{\sigma}_{0\mathrm{s}}(t)]-k_{\mathrm{f}}\hat{\sigma}_{0\mathrm{s}}(t)+k_{\mathrm{b}}\hat{P}_{\mathrm{S}}\hat{U}^{\dagger}\hat{\sigma}_{1\mathrm{s}}\hat{U}\hat{P}_{\mathrm{S}}\end{split}$
(12)
$\displaystyle\begin{split}\derivative{t}\hat{\sigma}_{1\mathrm{s}}(t)&=-\frac{i}{\hbar}[\hat{H}_{1\mathrm{s}},\hat{\sigma}_{1\mathrm{s}}(t)]-\left\\{\frac{k_{\mathrm{b}}}{2}\hat{U}\hat{P}_{\mathrm{S}}\hat{U}^{\dagger},\hat{\sigma}_{1\mathrm{s}}(t)\right\\}-\frac{i}{\hbar}\left[2\delta
J\hat{U}\hat{P}_{\mathrm{S}}\hat{U}^{\dagger},\hat{\sigma}_{1\mathrm{s}}(t)\right]+k_{\mathrm{f}}\hat{U}\hat{P}_{\mathrm{S}}\hat{\sigma}_{0\mathrm{s}}(t)\hat{P}_{\mathrm{S}}\hat{U}^{\dagger}.\end{split}$
(13)

Here $\\{\cdot,\cdot\\}$ denotes the anti-commutator, and $2\delta
J\hat{U}\hat{P}_{\mathrm{S}}\hat{U}^{\dagger}$ is an effective spin coupling
term in the radical pair state (expressions for the master equation parameters
$k_{\mathrm{f}}$, $k_{\mathrm{b}}$ and $\delta J$ are given in the SI). Now
assuming the radical pair, state $\ket{1}$, is formed very rapidly and
irreversibly from an excited singlet precursor $\ket{0}$, as is often the case
for radical pairs formed by photoexcitation of a singlet ground state, then we
can take $k_{\mathrm{b}}=0$, and the radical pair spin density operator as
being formed in the state

$\displaystyle\hat{\sigma}_{1\mathrm{s}}(0)=\hat{\sigma}_{\mathrm{n}\mathrm{s}}\hat{U}\hat{P}_{\mathrm{S}}\hat{U}^{\dagger}$
(14)

where $\hat{\sigma}_{\mathrm{n}\mathrm{s}}$ is the initial nuclear spin state
of the precursor state. The initial electron spin state can be written as
$\hat{U}\hat{P}_{\mathrm{S}}\hat{U}^{\dagger}=\outerproduct{\psi_{0}}{\psi_{0}},$
where the initial spin state $\ket{\psi_{0}}=\hat{U}\ket{\mathrm{S}}$ is

$\displaystyle\ket{\psi_{0}}=\cos\theta\ket{\mathrm{S}}+i\sin\theta\ket{\mathrm{T}_{0}(\mathbf{n})},$
(15)

in which $\ket{\mathrm{S}}$ is an electron spin singlet state and
$\ket{\mathrm{T}_{0}(\mathbf{n})}$ is a $\mathrm{T}_{0}$ triplet state defined
with respect to the axis $\mathbf{n}=\bm{\Lambda}/|\bm{\Lambda}|$, and
$\theta$ is given by Eq. (5). Once again for the opposite enantiomer the sign
of $\bm{\Lambda}$ changes. Denoting the different enantiomers $+$ and $-$,
this means that $\mathbf{n}_{+}=-\mathbf{n}_{-}$, or equivalently we can write
the initial state as

$\displaystyle\ket{\psi_{0}^{\pm}}=\cos\theta_{+}\ket{\mathrm{S}}\pm
i\sin\theta_{+}\ket{\mathrm{T}_{0}(\mathbf{n}_{+})}.$ (16)

This means that the phase of the initial coherence between singlet and triplet
states is changed by the chirality in the radical pair. In an achiral system,
there will be equal but opposite contributions to charge transfer from
configurations with opposite signs of $\mathbf{n}$, and therefore there will
be no initial coherence between singlet and triplet electron spin states. It
is well known that spin-orbit coupled charge transport (SOCT) in radical pair
reactions leads to the formation of triplet radical pairs,20, 23, 24, 25, 26,
27, 28 but this shows that in chiral systems spin coherence between singlet
and triplet states is also generated by SOCT.

Figure 1: (A) A schematic representation radical pair formation in a magnetic
field, with donor, D, and acceptor, A, covalently linked by a chiral bridge,
B^*. (B) A summary of the pulse sequence in the OOP-ESEEM experiment with
$t_{0}=0$.

We will now consider how chirality induced spin coherence could be detected
experimentally by EPR. Because the spin coherence generated is orientation
dependent, any experiment probing this would have to be performed on radical
pairs somehow fixed in a specific spatial orientation. Here we consider the
OOP-ESEEM (out of phase electron spin echo envelope modulation) EPR
experiment, which has been developed as a technique for probing zero-quantum
coherences in photo-generated radical pairs, with the applied field
$\mathbf{B}$ parallel to $\bm{\Lambda}$ for oriented radical pairs as
illustrated in Fig. 1 (A).29, 30, 31, 32, 33 The standard OOP-ESEEM experiment
consists a laser flash ($h\nu$), in which the radical pairs are generated,
followed by a waiting time of $t_{0}$, followed by a non-selective
$(\pi/4)_{x}$ microwave pulse, and another $(\pi)_{x}$ pulse after a time
$\tau$, and after another waiting time $\tau$ the free-induction decay spin
echo is recorded over times $t$. This $h\nu-
t_{0}-(\pi/4)_{x}-\tau-(\pi)_{x}-\tau-\mathrm{FID}$ pulse sequence is
illustrated schematically in Fig. 1 (B). This pulse sequence gives access to
information about the zero quantum coherences in the spin density operator,32
which evolve in the waiting times $t_{0}$ and $\tau$. If we are interested in
the initial state of the spin density operator, in particular the phases of
the initial zero-quantum coherences, then we should minimise the time for the
coherences to evolve on their own, as well as any decoherence processes, and
therefore here we only consider the case of $t_{0}=0$, as shown in Fig. 1 (B).

Using the standard high-field radical pair spin Hamiltonian in the rotating
frame,32 and invoking the Schulten-Wolynes semiclassical approximation for the
nuclear spins in the radical pair,34 we can obtain expressions for the
$x$-channel OOP-ESEEM signal
$f_{x}(t)=\Tr_{\mathrm{s}}[(\hat{S}_{1x}+\hat{S}_{2x})\hat{\sigma}_{1\mathrm{s}}(t)]$
(details of this are given in the SI). In the weak coupling limit
($|\Omega_{1}-\Omega_{2}|\gg|J|,|d|$) we find the OOP-ESEEM FID signal to be,

$\displaystyle\begin{split}&f_{x}(t)=\frac{1}{2}\sin\left({\theta}\right)\sin\left(\left(J-\frac{d}{3}\right)(t+2\tau)\right)\sum_{i=1,2}\bigg{(}e^{-t^{2}/(2\tau_{i}^{2})}\\!\\!\left(\sin\left({\theta}\right)\cos\left(\Omega_{i}t\right)\\!-\\!\sqrt{2}\cos\left({\theta}\right)\sin\left(\Omega_{i}t\right)\right)\bigg{)}.\end{split}$
(17)

$\Omega_{i}$ is the resonant frequency of electron spin $i$ (in the rotating
frame), $J$ is the scalar coupling constant and $d$ is the effective dipolar
coupling constant, and
$\tau_{i}^{-2}=\frac{1}{3}\sum_{k=1}^{N_{i}}a_{i,k}^{2}I_{i,k}(I_{i,k}+1)$,
where $a_{i,k}$ and $I_{i,k}$ are the hyperfine coupling constants and spin
quantum numbers for the hyperfine coupled nuclei in radical $i$. For
simplicity we have ignored spin relaxation and radical pair recombination
processes. Importantly here we see that changing the sign of $\theta$ changes
the sign of certain terms in the $x$ channel FID signal after the OOP-ESEEM
pulse sequence. This means the signal would be sensitive to the chirality of
the molecule. As described in the SI, we can also calculate the OOP-ESEEM
signal in the intermediate–strong coupling limit numerically. Often in OOP-
ESEEM experiments, the waiting time $\tau$ is varied, so we also consider the
integrated FID echo signal as a function of $\tau$,
$F_{x}(\tau)=\int_{0}^{\infty}f_{x}(t)\differential{t}$.

Figure 2: OOP-ESEEM signals for chiral radical pairs with
$\Omega_{1}=-\Omega_{2}=1.5\mathrm{\ mT}\ \gamma_{\mathrm{e}}$,
$J-d/3=0.2\mathrm{\ mT}\ \gamma_{\mathrm{e}}$ and
$\tau_{1}=\tau_{2}=2.0\mathrm{\ mT}^{-1}\ \gamma_{\mathrm{e}}^{-1}$, typical
parameters for a real organic radical ion pair
($\gamma_{\mathrm{e}}=\mathrm{e}g_{\mathrm{e}}/2m_{\mathrm{e}}$ is the modulus
of the gyromagnetic ratio of the free electron spin). Achiral signals are an
average of the $\theta=\pm\pi/16$ signals. Top panel: the FID echo for
$\tau=200\ \mathrm{ns}$ calculated using Eq. (17). Bottom panel: the
integrated echo as a function of $\tau$ calculated using Eq. (S.26).  Figure
3: OOP-ESEEM signals for chiral radical pairs with
$\Omega_{1}=-\Omega_{2}=1.5\mathrm{\ mT}\ \gamma_{\mathrm{e}}$,
$J-d/3=3\mathrm{\ mT}\ \gamma_{\mathrm{e}}$ and
$\tau_{1}=\tau_{2}=2.0\mathrm{\ mT}^{-1}\ \gamma_{\mathrm{e}}^{-1}$. Achiral
signals are an average of the $\theta=\pm\pi/16$ signals. Top panel: the FID
echo for $\tau=200\ \mathrm{ns}$ calculated using Eq. (S.25). Bottom panel:
the integrated echo as a function of $\tau$ calculated using Eq. (S.25).

As an example of this we plot the $t$ dependent OOP-ESEEM signal of two
enantiomers of a chiral radical pair with $\theta=\pm\pi/16$ in Fig. 2. This
corresponds to an initial triplet fraction of $\sim\\!4\%$ which is typical of
organic radical pairs in solution.35, 36, 37 We see that the two OOP-ESEEM
signals are very different even in the case of a modest SOCT contribution. The
signals are not the exact negative of each other because the spin Hamiltonian
is chirality independent but chirality does change the sign of the phase of
coherences in the initial state, leading to a phase shift in the OOP-ESEEM
signals. In Fig. 3 we show these signals for the same model in the
intermediate coupling regime, calculated by sampling $10^{6}$ realisations of
the hyperfine fields, where again we see that there is a phase shift between
the $F_{x}(\tau)$ signals.

In this work we have discussed how spin-orbit coupling can lead to chirality
induced spin coherence in radicals and radical pair systems, an effect which
has not before been explored in spin chemistry. In contrast to the case of
electrode-molecule electron transfers, where chirality induces spin
polarisation, chirality in intramolecular electron transfers generates
specific coherent superpositions of spin states. This effect should manifest
in certain EPR experiments and other experiments such as those that probe
magnetic field effects (which we describe briefly in the SI).

Chirality induced spin coherence could be important in a range of systems. For
example, it has been suggested that radical pair reactions in cryptochrome
proteins form the basis of the magnetic compass sense of migratory birds,38,
39, 40 and because chirality induced spin coherence is orientation dependent,
this effect could hypothetically play a role in avian magnetoreception.
Radical-based systems have also been proposed as potential molecular
qubits,41, 42, 43, 44, 45 and various quantum information theoretic ideas have
been explored using these molecules.46, 44, 45 Electron transport in chiral
molecular systems could provide a way of further manipulating spin coherences
in these systems. Other chiral open-shell molecular systems, aside from
radicals and radical pairs, could potentially display chirality induced spin
coherence effects. For example it has been demonstrated that spin-orbit
coupling can play a role in singlet fission47 and triplet-triplet up-
conversion,48 so it stands to reason that in chiral systems spin-orbit
coupling could lead to interesting effects on these processes. Furthermore
chirality induced spin coherence effects could manifest in the photophysics of
open-shell chiral transition metal complexes with $S>0$. Overall we hope that
this work will lay foundations for exploring chirality induced spin effects in
a wide variety of molecular systems, beyond what has already been studied in
electrode-molecule interfaces.

## Acknowledgements

I would like to thank Peter Hore and David Manolopoulos for their comments on
this manuscript. I would also like to thank both Peter Hore and Jiate Luo for
introducing me to the problem of spin effects in electron transfer in chiral
molecules. I would like to acknowledge financial support from the Clarendon
Scholarship from Oxford University, an E.A. Haigh Scholarship from Corpus
Christi College, Oxford, the EPRSC Centre for Doctoral Training in Theory and
Modelling in the Chemical Sciences, EPSRC Grant No. EP/L015722/1, and the Air
Force Office of Scientific Research (Air Force Materiel Command, USAF award
no. FA9550-14-1-0095).

## Supporting Information

Discussion and derivation of the spin-orbit coupling Hamiltonian in the
diabatic representation used here, an outline of the second order Nakajima-
Zwanzig master equation theory, expressions for OOP-ESEEM signals of chiral
radical pairs in the intermediate-strong coupling limit, and a brief
discussion of the effect of chirality induced spin coherence on quantum yields
of radical pair reactions.

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