*[enumerate]label=(0)

# Optimal joint cutting of two-qubit rotation gates

Christian Ufrecht<EMAIL_ADDRESS>Fraunhofer IIS,
Fraunhofer Institute for Integrated Circuits IIS, Division Positioning and
Networks, Nuremberg, Germany  Laura S. Herzog  Fraunhofer IIS, Fraunhofer
Institute for Integrated Circuits IIS, Division Positioning and Networks,
Nuremberg, Germany  Daniel D. Scherer  Fraunhofer IIS, Fraunhofer Institute
for Integrated Circuits IIS, Division Positioning and Networks, Nuremberg,
Germany  Maniraman Periyasamy  Fraunhofer IIS, Fraunhofer Institute for
Integrated Circuits IIS, Division Positioning and Networks, Nuremberg, Germany
Sebastian Rietsch  Fraunhofer IIS, Fraunhofer Institute for Integrated
Circuits IIS, Division Positioning and Networks, Nuremberg, Germany  Axel
Plinge  Fraunhofer IIS, Fraunhofer Institute for Integrated Circuits IIS,
Division Positioning and Networks, Nuremberg, Germany  Christopher Mutschler
Fraunhofer IIS, Fraunhofer Institute for Integrated Circuits IIS, Division
Positioning and Networks, Nuremberg, Germany

###### Abstract

Circuit cutting, the partitioning of quantum circuits into smaller independent
fragments, has become a promising avenue for scaling up current quantum-
computing experiments. Here, we introduce a scheme for joint cutting of two-
qubit rotation gates based on a virtual gate-teleportation protocol. By that,
we significantly lower the previous upper bounds on the sampling overhead and
prove optimality of the scheme. Furthermore, we show that no classical
communication between the circuit partitions is required. For parallel two-
qubit rotation gates we derive an optimal ancilla-free decomposition, which
include CNOT gates as a special case.

##  I Introduction

Current quantum computing hardware faces serious limitations, such as low
qubit numbers and high susceptibility to noise. As a result, quantum hardware
will likely be unable to execute algorithms with provable speedup like Shor’s
[1] or Grover’s [2] algorithm in the near future. On the other hand,
experiments with heuristic quantum algorithms have been conducted on a small
number of qubits, primarily applied to toy problems from the fields of, e.g.
optimization [3, 4], machine learning [5, 6], and chemical simulations [7, 8].
However, scaling up both problem sizes and the number of qubits is crucial for
assessing the potential usefulness of quantum computers. Even though recent
progress [9] suggests that quantum hardware based on superconducting qubits
might be on the verge of entering a so-called utility regime where the first
practically interesting problems might be approached, to achieve relevant
hardware size, modular quantum computing has been proposed [10, 11]. In this
paradigm, multiple quantum computing platforms are interconnected through
quantum links possibly enhanced by virtual entanglement distillation [12, 13].
Until reaching a higher technology readiness level, circuit cutting could
serve as a useful approach, replacing quantum links with classical links and a
post-processing step.

Given a unitary quantum channel $\mathcal{V}$, circuit cutting describes the
method of decomposing the channel as

$\mathcal{V}=\sum_{i}a_{i}\mathcal{F}_{i}\,,$ (1)

with the real coefficients $a_{i}$, to reduce the circuit size or the impact
of noise. Circuit cutting can be categorized into two methods. Wire cutting
[14, 15, 16, 17, 18, 19] describes the decomposition of the identity channel
corresponding to empty qubit wires. Here, $\mathcal{F}_{i}$ are measure-and-
prepare channels. The second method is known as gate cutting [20, 21, 22, 23,
24] where $\mathcal{F}_{i}$ are elements of LOCC($A$,$B$), that is local
operations on two partitions $A$ and $B$ of the quantum circuit and classical
communication between them. If all wires or gates that connect the partitions
$A$ and $B$ are cut, the sub-circuits become independent and can be run on two
quantum computers only connected by classical communication links. To evaluate
Eq. 1, it can be rewritten as a quasi-probability decomposition (QPD) and
evaluated via Monte-Carlo sampling [25]. As a result, all circuit-cutting
methods incur a sampling overhead, the factor of more samples required to
estimate the expectation value of an observable with the same accuracy as with
respect to the original uncut circuit. This sampling overhead has been shown
to be $\kappa^{2}$ where $\kappa=\sum_{i}|a_{i}|$ [25, 26]. The task is,
therefore, to find a decomposition of $\mathcal{V}$ with minimal $\kappa$,
which we will refer to as $\gamma$.

Figure 1: Cutting of parallel CNOT gates. Consider two partitions $A$ and $B$
of a quantum circuit connected by $n$ CNOT gates. In the case of gates that
can be executed at the same time slice of the circuit (parallel gates), we
present a joint optimal ancilla-free decomposition with coefficients $a_{i}$
and unitary or measurement operations $\mathcal{F}^{A}_{i}$ and
$\mathcal{F}^{B}_{i}$. The sampling overhead is $\mathcal{O}(\gamma^{2})$
where $\gamma=\mathcal{O}(2^{n})$. This is a strong improvement compared to
cutting $n$ CNOT gates independently for which $\gamma=\mathcal{O}(3^{n})$.
Note that the equality in the figure has to be understood on the superoperator
rather than on the gate level.

Note the similarity of the approach to probabilistic error cancellation where
the (possibly unphysical) inverse of a noisy quantum channel is decomposed
into physical operations [27]. Recent work [23] employed CNOT-gate
teleportation [28, 29, 30] consuming one Bell state initially connecting the
two partitions per CNOT gate. The optimal QPD with $\gamma=3$ of the Bell-
state density matrix then translates into the optimal decomposition of the
gate. The crucial insight of Ref. [23] was that the joint QPD of $n$ Bell
states required for the teleportation of $n$ CNOT gates can be constructed
more efficiently compared to the individual decomposition of the gates. As a
result, the $\gamma$ parameter is reduced from $\mathcal{O}(3^{n})$ to
$\mathcal{O}(2^{n})$, however at the cost of one ancilla qubit per partition
and gate. In this cutting scheme, $\mathcal{F}_{i}\in\mathrm{LOCC}(A,B)$ and
two-way classical information is shared between the partitions. The protocol
was extended to general Clifford gates and later also to wire cutting [15].
Recently, it was shown that optimal cutting of parallel wires is possible
without ancilla qubits [16].

In this work, we extend previous proposals to the joint cutting of non-
Clifford two-qubit rotation gates. We significantly simplify existing methods
and, by that, enable implementation on current NISQ hardware. Two-qubit
rotation gates play a crucial role for example in the QAOA algorithm [4] or
for the simulation of spin systems. In Sec. II, we introduce a virtual
teleportation scheme that effectively consumes less than one entanglement bit
(ebit). After explaining the protocol for a single gate instance, we
generalize the scheme to the joint cutting of $n$ two-qubit rotation gates and
demonstrate the optimality of the derived circuit-cutting method. With
optimality, we refer to the circuit cutting scheme with the minimal possible
sampling overhead. The $\gamma$ parameter for a two-qubit rotation gate with
parameter $\theta$ is [21, 23]

$\displaystyle\gamma$ $\displaystyle=1+2|\sin(\theta)|\,.$ (2) As we will
show, when $n$ instances of this gate are jointly cut, we will find the
reduced effective $\gamma$ parameter per gate $\displaystyle\gamma$
$\displaystyle=1+|\sin(\theta)|$ (3)

asymptotically as $n$ approaches infinity, which matches the lower bound. Our
scheme offers several improvements over previous work. First, it eliminates
the need for classical communication between the partitions, and consequently,
no real-time feedback is required. This contrasts the gate-teleportation
protocol used in Ref. [23] where local correction operators conditioned on
intermediate measurement results are necessary. Moreover, our protocol can be
executed sequentially on the same quantum computer, unlike schemes where two-
way classical communication exchange necessitates the use of two separate
quantum computers in parallel. Second, it has recently been emphasized that
the number of operations $\mathcal{F}_{i}$ can be the limiting factor for the
execution of circuit-cutting protocols. We derive a quasi-probability
decomposition of the entanglement resource states with an exponentially
reduced number of elements. Third, in Sec. III, we show that parallel gates
can be cut without the need for ancilla qubits. Parallel gates refer to gates
that can be executed within the same time slice of a circuit as shown in Fig.
1. Note that controlled rotation gates and CNOT gates (for $\theta=\pi/2$) are
equivalent to two-qubit rotation gates up to local operations. Thus, all
results of our work also apply to these types of gates.

##  II Virtual gate teleportation

In this section, we aim to find an optimal joint decomposition of two-qubit
rotation gates based on gate teleportation. A teleportation scheme for a two-
qubit rotation gate $R_{zz}(\theta)$ [30] is shown in Fig. 2. It consumes one
initial Bell state (wiggly line in the figure) prepared on two ancilla qubits.
These qubits are subsequently measured in the computational basis, and the
outcomes, $k,l\in\\{0,1\\}$, are communicated between the partitions $A$ and
$B$ via classical communication to select the local correction operations
$X^{k}$ and $Z^{l}$. For $\theta=\pi/2$ the two-qubit rotation gate is
equivalent to a CNOT gate up to local operations.

Figure 2: Teleportation protocol for a two-qubit rotation gate
$R_{zz}(\theta)$ [30]. The scheme consumes an initial Bell state (wiggly line)
as entanglement resource between the partitions $A$ and $B$ and requires two-
way classical communication visualized by the dashed arrows. The measurements
are performed in the computational basis with the outcomes $l,k\in\\{0,1\\}$.

In this case, simulating the initial Bell state by a quasi-probability
distribution with $\gamma=3$ translates into an optimal cutting scheme for a
CNOT gate (also $\gamma=3$) [23]. When employing the teleportation scheme for
$n$ CNOT gates together with a joint quasi-probability decomposition of the
$n$ Bell states, the $\gamma$ parameter behaves sub-multiplicatively reducing
from $\mathcal{O}(3^{n})$ to $\mathcal{O}(2^{n})$.

When attempting to achieve a similar reduction for joint cutting of two-qubit
rotation gates, we encounter the following difficulty: A decomposition of a
two-qubit rotation gate based on the teleportation scheme would be sub-optimal
since it always consumes one Bell state resulting in $\gamma=3$ while the
$\gamma$ parameter of a two-qubit rotation gate in Eq. 2 is smaller for most
values of $\theta$. On the other hand, a deterministic gate teleportation
protocol consuming less than one ebit does not exist [31, 32] but only
probabilistic implementations [33, 34, 35], which would lead to suboptimal
sampling overhead. As a result, a naive generalization of Ref. [23] is
impossible.

###  II.1 Single gate instance

In the following, we introduce the alternative scheme depicted in Fig. 3 for
which we prepare the initial state $|\Psi_{kl}\rangle$, to be specified below,
on the ancilla qubits.

Figure 3: Virtual gate teleportation. As explained in the main text, for the
state shown in Eq. 4, the correct gate is only teleported for the measurement
results $k=l$, resulting in suboptimal sampling overhead when discarding other
outcomes. Since we are only interested in a teleportation scheme together with
a quasi-probability simulation of the initial state, we show that in this
case, the result $k\neq l$ can be corrected in the final classical post-
processing step. This effectively corresponds to the preparation of the
initial state $|\Psi_{kl}\rangle$ defined in Eq. 10, which depends on the
outcome of the measurement performed later in time, therefore the term virtual
teleportation.

As a first step, we choose the initial state

$|\Psi\rangle=\mathrm{cos}(\theta/2)|00\rangle+\mathrm{sin}(\theta/2)|11\rangle\,.$
(4)

As we detail in Appendix A, an optimal QPD of a pure-state density matrix can
be readily calculated based on the robustness of entanglement measure [36,
23]. The $\gamma$ parameter is calculated as

$\gamma=2\Big{(}\sum_{j}c_{j}\Big{)}^{2}-1$ (5)

where $c_{j}$ are the Schmidt-coefficients of the state. Consequently, for the
state shown in Eq. 4, we obtain
$\gamma=2(|\cos(\theta/2)|+|\sin(\theta/2)|)^{2}-1$ matching Eq. 2. However,
it is straightforward to convince ourselves that the correct gate is
teleported only for $k=l$, and its Hermitian conjugate for $k\neq l$ [34]. On
the one hand, local operators on the partitions $A$ and $B$ conditioned on the
measurement results to achieve a deterministic protocol do not exist [31, 32].
On the other hand, a repeat-until-success version that is disregarding all
measurement outcomes with $k\neq l$ would lead to the suboptimal sampling
overhead $4\gamma^{2}$. Note, however, that we are not interested in deriving
an actual deterministic gate teleportation protocol but in a QPD to simulate
the two-qubit rotation gate with optimal sampling overhead.

In Appendix A, we derive an alternative optimal QPD for a pure-state density
matrix compared to the one presented in Ref. [36]. Applied to Eq. 4, we find

$\displaystyle|\Psi\rangle\langle\Psi|$
$\displaystyle=\mathrm{cos}^{2}(\theta/2)|00\rangle\langle
00|+\mathrm{sin}^{2}(\theta/2)|11\rangle\langle 11|$ (6)
$\displaystyle+\mathrm{sin}(\theta)(\sigma^{+}-\sigma^{-})\,.$ (7)

The explicit form of the separable states $\sigma^{+}$ and $\sigma^{-}$ is
specified in Appendix A. To see that the decomposition is optimal, we sum over
the absolute of the prefactors, finding again Eq. 2. The states $\sigma^{+}$
and $\sigma^{-}$ are independent of $\theta$, which is the crucial property to
perform our protocol as described in the following: As mentioned above, for
$k\neq l$, the Hermitian conjugate of the two-qubit rotation gate is
teleported, which corresponds to $\theta\rightarrow-\,\theta$ in Eq. 6 and Eq.
7. Because $\sin(-\,\theta)=-\sin(\theta)$, we could equally choose a minus
sign in front of Eq. 7, keep results with $k\neq l$, and disregard results
with $k=l$.

Figure 4: Generalization of the virtual teleportation protocol from the single
gate instance shown in Fig. 3 to $n$ two-qubit rotation gates. Each gate
requires one ancilla qubit per partition on which we effectively prepare the
states $|\Psi_{k_{s}l_{s}}\rangle$ for $s=1,...,n$ by a quasi-probability
decomposition. The result of the measurements on the ancilla qubits determines
the sign of the quasi-probabilities in the classical post-processing step.

To make use of both situations, we define the QPD

$\displaystyle|\Psi_{kl}\rangle\langle\Psi_{kl}|$
$\displaystyle=\mathrm{cos}^{2}(\theta/2)|00\rangle\langle
00|+\mathrm{sin}^{2}(\theta/2)|11\rangle\langle 11|$ (8)
$\displaystyle+(-1)^{k+l}\mathrm{sin}(\theta)(\sigma^{+}-\sigma^{-})$ (9)

in which the signs of the quasi-probabilities depend on the measurement
outcomes in the form of the factor $(-1)^{k+l}$. To execute the protocol, we
proceed as detailed in the following: To estimate an expectation value
experimentally, one typically defines a post-processing function
$f(s)\in[-1,1]$ [14] as a function of the bitstring $s$ observed at each
experimental run and then takes the sample mean over many runs. For our
protocol, we perform Monte-Carlo sampling as, for example, explained in Ref.
[24]. For each run, we prepare one of the states in Eq. 8 and Eq. 9 sampled
according to the probability given by the absolute of the prefactor divided by
$\gamma$. In case we initialize the ancilla qubits with one of the states
constituting $\sigma^{+}$ or $\sigma^{-}$, we accumulate $\pm(-1)^{k+l}f(s)$
in the sample mean where the sign depends on the outcome of the measurements
on the ancilla qubits. We will refer to our protocol as virtual gate
teleportation since an actual entanglement resource state is not physically
prepared but only simulated by a quasi-probability decomposition. In this
respect, the protocol effectively makes use of the state

$|\Psi_{kl}\rangle=\mathrm{cos}(\theta/2)|00\rangle+(-1)^{k+l}\mathrm{sin}(\theta/2)|11\rangle$
(10)

which depends on the outcomes of the measurement performed later in time.

###  II.2 Joint cutting of multiple gates

In this subsection, we generalize the virtual protocol to the joint cutting of
$n$ two-qubit rotation gates at arbitrary positions in the circuit. When
cutting $n$ gate instances independently, the overall $\gamma$ parameter
defined here as $\gamma^{(n)}_{\mathrm{ind.}}$ behaves multiplicatively, that
is

$\gamma^{(n)}_{\mathrm{ind.}}=\Pi_{s=1}^{n}(1+2|\mathrm{sin}(\theta_{s})|)$
(11)

where we used Eq. 2. In order to investigate if joint decomposition can be
done more efficiently, the single-instance virtual teleportation protocol is
generalized in the following to $n$ gates as shown in Fig. 4. In the figure we
view the $n$ two-qubit states $|\Psi_{k_{s}l_{s}}\rangle$ for $s=1,...,n$ on
the ancilla qubits as a $2n$-qubit state
$|\Psi_{k_{1}l_{1}...k_{n}l_{n}}\rangle$. Again, a specific choice of
$k_{s},l_{s}$ for $s=1,...,n$ defining the resource state only translates into
the signs of the quasi probabilities of its QPD. In turn, we can choose
$k_{s}=l_{s}=0$ for all $s$ of the resource state but correct unwanted
measurement results in the post-processing step.

Using Eq. 10 and the qubit arrangement as shown in Fig. 4, we find

$|\Psi_{k_{1}l_{1}...k_{n}l_{n}}\rangle=\sum_{j\in\\{0,1\\}^{n}}c_{j}|j_{1}...j_{n}\rangle|j_{n}...j_{1}\rangle$
(12)

with $c_{j}=\Pi_{s=1}^{n}\mathscr{c}_{j_{s}}$ and

$\mathscr{c}_{j_{s}}=\left\\{\begin{array}[]{lr}\mathrm{cos}(\theta_{s}/2)&\quad\text{for
}j_{s}=0\\\ (-1)^{k_{s}+l_{s}}\,\mathrm{sin}(\theta_{s}/2)&\quad\text{for
}j_{s}=1\end{array}\right.\,.$ (13)

In Eq. 12, the first (second) ket describes the state on the ancilla qubits of
partition $A$ ($B$). If we redefine, e.g., the first ket on the right-hand
side of Eq. 12 by absorbing the sign of $c_{j}$, this equation is the Schmidt
decomposition of $|\Psi_{k_{1}l_{1}...k_{n}l_{n}}\rangle$ allowing us to
calculate $\gamma_{\mathrm{joint}}^{(n)}$ for the QPD as

$\displaystyle\gamma_{\mathrm{joint}}^{(n)}$
$\displaystyle=2\Big{(}\sum_{j\in\\{0,1\\}^{n}}|c_{j}|\Big{)}^{2}-1$ (14)
$\displaystyle=2\,\Pi_{s=1}^{n}(1+|\mathrm{sin}(\theta_{s})|)-1$ (15)
$\displaystyle<\gamma_{\mathrm{ind.}}^{(n)}\,.$ (16)

In Appendix A, we explicitly state an optimal quasi-probability decomposition
for any pure-state density matrix. Applied to Eq. 12, the measurement-
dependent signs in Eq. 13 only appear as the signs of the quasi probabilities,
again allowing correction in the classical post-processing step. The $\gamma$
parameter in Eq. 15 is, in general, significantly smaller than the $\gamma$
parameter for independent cuts. Therefore, we have shown that there exists a
decomposition for multiple instances of two-qubit rotation gates with sub-
multiplicative behavior of the $\gamma$ parameter. In Appendix B, we prove
that $\gamma_{\mathrm{joint}}^{(n)}$ also is a lower bound for the optimal
decomposition of $n$ two-qubit rotation gates. Consequently, joint optimal
virtual gate teleportation also translates into optimality of the gate
decomposition.

Note that no classical communication is required between the partitions 111It
can be shown that all operations in the virtual teleportation protocol are
elements of the extended definition of local operations used in Ref. [23,
15].. Joint gate cutting without the need for classical communication
significantly alleviates implementation on current hardware for two reasons.
First, no real-time feedback for local operations conditioned on measurement
results is needed, distinct from conventional gate teleportation protocols as,
for example, shown in Fig. 2. Second, if all gates connecting the two
partitions are cut, the severed sub-circuits can be executed sequentially on
the same hardware. This stands in stark contrast to protocols employing two-
way classical communications, where two quantum devices connected by classical
links must operate in parallel. Lastly, we mention again that our protocol
contains the joint cutting of $n$ CNOT gates as a special case with optimal
$\gamma_{\mathrm{joint}}^{(n)}=2^{n+1}-1$ [23].

###  II.3 Multi-qubit rotation gates

Finally, we consider multi-qubit $Z$ rotation gates. A multi-qubit rotation
gate can be written as a two-qubit rotation gate sandwiched by ladders of CNOT
gates [38] as shown in Fig. 5. The task of cutting a multi-qubit $Z$ rotation
gate then reduces to the cutting of a two-qubit rotation gate. The resulting
decomposition is optimal. This can be seen by noting that if the $\gamma$
parameter of the multi-qubit rotation gate was smaller than that of a two-
qubit rotation gate, we could use the former to construct a decomposition of
the latter with smaller $\gamma$, leading to a contradiction.

Figure 5: A multi-qubit rotation gate is equivalent to a two-qubit rotation
gate up to local operations (with respect to the two-partitions $A$ and $B$).
Consequently, the $\gamma$ parameter of this gate equals the one of the two-
qubit rotation gate.

##  III Parallel gates can be cut without ancilla qubits

Interestingly, as we will show in the following, there is an ancilla-free
joint decomposition of parallel gates. With parallel gates, we refer to gates
that can be executed at the same time slice of the circuit, for example, as
shown in Fig. 1 for the case of CNOT gates. In Appendix C, we start from the
protocol in Fig. 4 and subsequently show that it can be reduced to an ancilla-
free decomposition in terms of local operations on the partition $A$ and $B$.
Remarkably, again, no classical communication between the partitions is
required, and optimality is preserved. The explicit formula for the
decomposition is shown in Eq. 42. It contains single-qubit $Z$ gates, as well
as multi-qubit operations on all subsets of the $n$ qubits on each partition
$A$ and $B$. The operations consist of two different types. The first are
multi-qubit rotation gates by angles $\pm\pi/2$. They can be implemented by
single-qubit rotation gates sandwiched by ladders of CNOT gates [38]. The
second are non-unitary operations, where the single-qubit rotation gates
within the CNOT ladders are replaced by computational-basis measurements; see
a detailed discussion in Appendix C. As a consequence, each term in the
decomposition can be implemented with $\mathcal{O}(n)$ CNOT gates. It has
recently been emphasized that the sampling overhead is not the only relevant
metric measuring the quality of a circuit-cutting scheme [16] but also the
number $L$ of different channels to be applied. In principle, the number of
experimental shots required to evaluate an expectation value to additive error
$\epsilon$ scales as $\mathcal{O}(\gamma^{2}/\epsilon^{2})$ which is
independent of $L$. In practice, however, the latency introduced by the
additional compilation time needed for each of the $L$ sub-circuits
corresponding to the elements $\mathcal{F}_{i}$ for $i=1,...,L$ on
topologically constrained hardware can be the dominant factor in the runtime
of the algorithm [16]. As discussed in Appendix A, our decomposition
guarantees $L=\mathcal{O}(4^{n})$ for the case of parallel as well as non-
parallel gates, which is an exponential improvement over previous results. It
should be noted that we could trade compilation time versus additional circuit
depth. Indeed, a multi-qubit rotation gate can be trivially routed to
topologically constrained hardware by SWAP insertion. This decreases
compilation latency to almost zero but, in turn, might drastically increase
the CNOT-gate count and the additional circuit depth $d$, resulting in a more
severe impact of noise. Note that a multi-qubit rotation gate can be
implemented on fully connected hardware with depth
$d=\mathcal{O}(\lceil\mathrm{log}(n)\rceil)$ when replacing each CNOT gate
ladder by a balanced tree [38]. For hardware with constrained connections,
efficient compilation methods exist for multi-qubit rotation gates based on
minimum spanning trees [39] and CNOT gate re-synthesis [40]. In what sense the
low depth of the circuits can be retained will be the focus of further
investigations.

##  IV Conclusion

In this work, we introduced joint cutting of non-Clifford two-qubit rotation
gates based on virtual teleportation and proved optimality. In the case of
parallel gates, we further derived an ancilla-free optimal decomposition. We
conclude by stressing the following aspects: As mentioned before, controlled
rotation gates are equivalent to two-qubit rotation gates up to local unitary
operations. In addition, CNOT gates are special instances of this gate class.
Consequently, the joint virtual teleportation protocol proposed in this work
can also be used for optimal joint cutting of CNOT gates with several
improvements over previous work [23]. (1) In our virtual teleportation
protocol, no classical communication needs to be shared. As a result, no real-
time feedback during the quantum computation in the form of local correction
operations conditioned on the measurement outcomes on the ancilla qubits is
required. (2) As a consequence, the cut circuits can be run in sequence on the
same hardware as opposed to parallel execution, which was necessary for
previous methods. (3) By our alternative quasi-probability decomposition of
pure-state density matrices, we reach an exponential reduction of the number
of terms required compared to Refs. [23, 36], directly translating into an
exponential reduction of the number of channels in the cutting scheme. (4) Our
decomposition method allows us to derive ancilla-free optimal joint cutting
schemes for CNOT gates, two-qubit rotation gates, and controlled rotation
gates, which has not yet been considered in the literature.

###### Acknowledgements.

C.U. would like to thank M. Bechtold, D. Sutter, and F. Wagner for helpful
discussions. The research was funded by the project QuaST, supported by the
Federal Ministry for Economic Affairs and Climate Action on the basis of a
decision by the German Bundestag.

##  Appendix A Optimal QPD for a pure-state density matrix

In this appendix, we derive an alternative optimal QPD for arbitrary pure-
state density matrices. In Ref. [36], Vidal and Tarrach consider the following
problem: Given two partitions and a bipartite state $\rho$. What is the
minimal mixing of the state with a separable state such that the mixed state
becomes separable as well? This problem can be rephrased as follows [23]: What
is the minimal $R$, called robustness of entanglement, for which

$\rho=(1+R)\eta_{1}-R\eta_{2}$ (17)

where $\eta_{1}$ and $\eta_{2}$ are separable. While such decomposition is
hard to find in general, for a pure state $\rho=|\psi\rangle\langle\psi|$ the
robustness can be calculated via the $m$ Schmidt coefficients $c_{j}$ with
$j=0,...,m-1$ of the $|\psi\rangle$ state as

$R=\left(\sum_{j=0}^{m-1}c_{j}\right)^{2}-1\,.$ (18)

Consequently,

$\gamma=1+2R\,.$ (19)

In Ref. [36], the authors also provide an explicit representation for
$\eta_{1}$ and $\eta_{2}$ in which, however, the separable states themselves
are functions of the Schmidt coefficients. In contrast, the alternative QPD
derived in the following only contains the Schmidt coefficients in the quasi-
probabilities. As discussed in the main text, this guarantees that, when
applied to the protocol in Fig. 3 and Fig. 4, we can correct the unwanted
measurement results in the classical post-processing step. A similar
representation has been derived in Refs. [41, 42]. In the following, we first
derive the general form of the decomposition and subsequently show optimality:

Consider a bipartite state $|\psi\rangle$ on the partitions $A$ and $B$ and an
expansion of the form

$|\psi\rangle=\sum_{j=0}^{m-1}c_{j}|\varphi_{j}\rangle\otimes|\varphi_{j}^{\prime}\rangle$
(20)

with $c_{j}$ real but not necessarily positive. Then

$\displaystyle|\psi\rangle\langle\psi|$
$\displaystyle=\sum_{j=0}^{m-1}c_{j}^{2}|\varphi_{j}\rangle\langle\varphi_{j}|\otimes|\varphi_{j}^{\prime}\rangle\langle\varphi_{j}^{\prime}|$
$\displaystyle+2\sum_{i>j}c_{i}c_{j}\left(\sigma_{ij}^{+}-\sigma_{ij}^{-}\right)$
(21)

holds. In this expression

$\sigma_{ij}^{\pm}=\frac{1}{\alpha}\sum_{r=1}^{\alpha}|\xi^{\pm}_{rij}\rangle\langle\xi^{\pm}_{rij}|\otimes|\tau_{rij}\rangle\langle\tau_{rij}|\,.$
(22)

are separable density matrices with respect to the partitions $A$ and $B$. The
states take the form

$\displaystyle|\xi^{\pm}_{rij}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|\varphi_{i}\rangle\pm\mathrm{e}^{i\phi_{r}}|\varphi_{j}\rangle\right)$
(23) and $\displaystyle|\tau_{rij}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|\varphi_{i}^{\prime}\rangle+\mathrm{e}^{-i\phi_{r}}|\varphi^{\prime}_{j}\rangle\right)\,.$
(24)

Finally, the phases $\phi_{r}$ can be chosen such that

$\sum_{r=1}^{\alpha}\mathrm{e}^{i\phi_{r}}=\sum_{r=1}^{\alpha}\mathrm{e}^{i2\phi_{r}}=0\,.$
(25)

Furthermore, if Eq. 20 is the Schmidt decomposition of $|\psi\rangle$ \- in
this case $c_{j}\geq 0$ \- then Eq. 21 is optimal in that it achieves the
minimal possible $\gamma$.

Proof: The statement is most easily proven by substituting Eqs. (22)-(25) into
Eq. 21. It remains to show optimality in case Eq. 20 is the Schmidt
decomposition of $|\psi\rangle$. To this end, we sum over the absolute of the
coefficients for all terms with the result stated in Eq. 18 and Eq. 19.
Therefore, the decomposition is optimal.

We choose $\phi_{r}=2\pi r/\alpha$ for integer $\alpha\geq 3$. The choice
$\alpha=3$ involves the least amount of terms, but we observe in Appendix C
that $\alpha=4$ allows removing ancilla qubits for parallel gates. Eq. 21
involves $\mathcal{O}(m^{2})$ terms. This is an exponential improvement to the
decomposition used in Ref. [36] with $\mathcal{O}(2^{m})$ terms.

##  Appendix B Lower bounds

In this appendix we prove a lower bound on $\gamma_{\mathrm{joint}}^{(n)}$,
the $\gamma$ parameter for jointly cutting $n$ two-qubit rotation gates. This
bound will agree with the upper bound established in Eq. 15 by the joint
virtual teleportation scheme, thereby proving optimality. A lower bound on the
$\gamma$ parameter of a quantum gate can be obtained from the QPD of the Choi
state of the gate’s unitary [23]. The argument is the following: Since the
Choi state of a gate’s unitary can be formed by local (with respect to the
partitions $A$ and $B$) operations, the $\gamma$ parameter of the gate cannot
possibly be smaller than that of the Choi state. If it were, construction of
the Choi state with smaller $\gamma$ would be possible, leading to a
contradiction. Thus, Eq. 5 provides

$\gamma\geq 2\Big{(}\sum_{j}c_{j}\Big{)}^{2}-1$ (26)

where $c_{j}$ are the Schmidt coefficients of the Choi state of the gate.
Further note that the Choi state of a gate whose matrix representation is
diagonal in the computational basis has the same Schmidt coefficients as the
normalized vector containing the diagonal elements. In our case, this vector
is just $|\Psi_{k_{1}l_{1}...k_{n}l_{n}}\rangle$ for $k_{s}=l_{s}=0$ for all
$s$ up to local operations. Consequently, the lower bound matches the upper,
proving the optimality of the joint virtual teleportation approach for joint
cutting of two-qubit rotation gates.

Ref. [23] proved optimality for a large class of gate decomposition derived in
Ref. [22]. Note, however, that the lower bound obtained in this way can be
relatively loose for some gates. E.g., in case of a $K$-qubit version of a
Toffoli gate, it can be shown that the right-hand side of Eq. 26 for a cut
after one qubit tends to one as $K\rightarrow\infty$. Since, however, this
gate can be used to create a Bell state between two partitions, $\gamma\geq 3$
must be true.

##  Appendix C Parallel gates

In this appendix, we derive the ancilla-free decomposition of $n$ parallel
two-qubit rotation gates for which we start from the protocol in Fig. 4.
First, we define the map

$\Lambda(|\psi\rangle\langle\psi|)[\cdot]=2^{n}\sum_{i,j}\langle
i|\psi\rangle\langle\psi|j\rangle P_{i}\,\cdot\,P_{j}$ (27)

where $P_{i}=|i\rangle\langle i|$ and $\\{|i\rangle\\}_{i=0}^{2^{n}-1}$ is the
computational basis. This map corresponds to the transformation $U\cdot
U^{\dagger}$ where $U$ contains the computational basis elements of
$|\psi\rangle$ multiplied by $\sqrt{2^{n}}$ on its diagonal. Thus, $U$ is
unitary if $|\langle i|\psi\rangle|=1/\sqrt{2^{n}}$ for all $i$. Further, note
that $\Lambda$ is linear in its first argument, and if $|\psi\rangle$
factorizes over two partitions, then the map does as well.

Let us define as $\mathcal{R}_{zz}^{(n)}$ the $2n$-qubit unitary channel
corresponding to the $n$ two-qubit rotation gates we want to cut and a state
$|\psi\rangle$ such that

$\Lambda(|\psi\rangle\langle\psi|)[\cdot]=\mathcal{R}_{zz}^{(n)}\,.$ (28)

This state is just

$|\psi\rangle=H^{\otimes n}\otimes(HS^{\dagger})^{\otimes n}|\Psi\rangle$ (29)

with $|\Psi\rangle$ as in Eq. 12 but with $k_{s}=l_{s}=0$ for all $s$, that is

$|\Psi\rangle=\sum_{j\in\\{0,1\\}^{n}}c_{j}|j_{1}...j_{n}\rangle|j_{n}...j_{1}\rangle$
(30)

with $c_{j}=\Pi_{s=1}^{n}\mathscr{c}_{j_{s}}$ and

$\mathscr{c}_{j_{s}}=\left\\{\begin{array}[]{lr}\mathrm{cos}(\theta_{s}/2)&\quad\text{for
}j_{s}=0\\\ \mathrm{sin}(\theta_{s}/2)&\quad\text{for
}j_{s}=1\end{array}\right.\,.$ (31)

It is left to insert a QPD of $|\psi\rangle\langle\psi|$ calculated with
Appendix A into Eq. 27 and then use linearity and factorization properties of
$\Lambda$. Comparison with Eq. 20 shows $m=2^{n}$,
$|\varphi_{j}\rangle=H^{\otimes n}|j_{1},...,j_{n}\rangle$, and
$|\varphi_{j}^{\prime}\rangle=(HS^{\dagger})^{\otimes
n}|j_{n},...,j_{1}\rangle$. To calculate the map $\Lambda$, we first make use
of linearity and calculate the different terms individually. When we define
the unitary channel

$\mathcal{Z}_{j}[\cdot]=Z^{j_{1}}\otimes...\otimes Z^{j_{n}}\cdot
Z^{j_{1}}\otimes...\otimes Z^{j_{n}}$ (32)

corresponding to the application of single qubit $Z$ gates, we find

$\Lambda(|\varphi_{j}\rangle\langle\varphi_{j}|)[\cdot]=\mathcal{Z}_{j}[\cdot]$
(33)

where we used

$\sqrt{2}\sum_{k_{s}\in\\{0,1\\}}\langle k_{s}|H|j_{s}\rangle
P_{k_{s}}=\sum_{k_{s}\in\\{0,1\\}}(-1)^{j_{s}k_{s}}P_{k_{s}}=Z^{j_{s}}\,.$
(34)

Equally, we find the same result for $|\varphi_{j}^{\prime}\rangle$ for $j$
replaced by $\tilde{j}:=(j_{n},...,j_{1})$. The evaluation of the states
$|\xi^{\pm}_{rij}\rangle$ and $|\tau_{rij}\rangle$ requires a little bit more
work. First, we insert $|\xi^{\pm}_{rij}\rangle$, defined in Eq. 23, with the
result

$\displaystyle\sqrt{2}^{n}\sum_{k}\langle k$
$\displaystyle|\xi^{\pm}_{rij}\rangle P_{k}=Z^{i_{1}}\otimes...\otimes
Z^{i_{n}}\times$ $\displaystyle\left[(\mathbb{I}^{\otimes
n}\pm\mathrm{e}^{i\phi_{r}}Z^{j_{1}-i_{1}}\otimes...\otimes
Z^{j_{n}-i_{n}})/\sqrt{2}\right]\,.$ (35)

For $\phi_{r}=2\pi r/\alpha$ with $\alpha=4$, we first consider
$\mathrm{exp}(i\phi_{1})=i$ and $\mathrm{exp}(i\phi_{3})=-i$. In this case,
the expression in the brackets is unitary. Indeed,

$R_{ij}(\pm\pi/2)=\frac{\mathbb{I}^{\otimes n}\mp
iZ^{j_{1}-i_{1}}\otimes...\otimes Z^{j_{n}-i_{n}}}{\sqrt{2}}$ (36)

is a multi-qubit rotation gate on the qubits with $|i_{s}-j_{s}|=1$. Again, we
define the corresponding channel $\mathcal{R}_{ij}(\pm\pi/2)$ using
calligraphic notation. Furthermore, we introduce the abbreviation

$\mathcal{R}_{ij}=\frac{\mathcal{R}_{ij}(\pi/2)-\mathcal{R}_{ij}(-\pi/2)}{2}$
(37)

of the unitary channels. We now turn to Eq. 35 with
$\mathrm{exp}(i\phi_{2})=-1$ and $\mathrm{exp}(i\phi_{4})=1$. In this case, we
define

$P^{k}_{ij}=\frac{\mathbb{I}^{\otimes
n}+(-1)^{k}Z^{j_{1}-i_{1}}\otimes...\otimes Z^{j_{n}-i_{n}}}{2}$ (38)

for $k=0,1$, which is non-unitary. The channel $\mathcal{P}_{ij}^{k}$ can be
evaluated on a quantum computer using a ladder of CNOT gates [43] and a
projector on $|0\rangle$ or $|1\rangle$, e.g.

$\frac{\mathbb{I}^{\otimes t}+(-1)^{k}Z^{\otimes
t}}{2}\quad=\quad\vbox{\hbox{\includegraphics[width=107.0pt]{CNOT_chain.pdf}}}$
(39)

for integer $t>0$. We then define

$\mathcal{P}_{ij}=\mathcal{P}_{ij}^{0}-\mathcal{P}_{ij}^{1}$ (40)

which is neither trace-preserving nor positive. However, $\mathcal{P}_{ij}$ is
the difference of two completely positive trace non-increasing (CPTN) maps,
whose sum is completely positive and trace preserving (CPTP). Such maps can be
simulated on a quantum device without extra sampling overhead [24, 23, 21,
22]. Thus, both $\mathcal{R}_{ij}$ and $\mathcal{P}_{ij}$ can be simulated
with $\gamma=1$. The evaluation of Eq. 27 for $|\tau_{rij}\rangle$ proceeds
along the same lines. We only need to replace
$i,j\rightarrow\tilde{i},\tilde{j}$, choose the plus sign in Eq. 35, and
replace $\phi_{r}\rightarrow-\phi_{r}$. Due to the extra $S^{\dagger}$ gate in
the definition of $|\varphi^{\prime}\rangle$, we additionally have to replace
$Z\rightarrow-iZ$, introducing a factor $(-i)^{\nu_{ij}}$ with

$\nu_{ij}=\sum_{s=1}^{n}(j_{s}-i_{s})\,.$ (41)

Collecting all the terms, we finally arrive at

$\mathcal{R}_{zz}^{(n)}=\sum_{j\in\\{0,1\\}^{n}}\\!c_{j}^{2}\,\mathcal{Z}_{j}\otimes\mathcal{Z}_{\tilde{j}}+2\sum_{i>j}c_{i}c_{j}[\mathcal{Z}_{i}\otimes\mathcal{Z}_{\tilde{i}}]\circ\begin{cases}\;\,(-1)^{\nu_{ij}/2}\,\Big{[}\mathcal{P}_{ij}\otimes\mathcal{P}_{\tilde{i}\tilde{j}}-\mathcal{R}_{ij}\otimes\mathcal{R}_{\tilde{i}\tilde{j}}\Big{]}\quad&\text{for
$\nu_{ij}$ even}\\\
\;\,(-1)^{(\nu_{ij}-1)/2}\Big{[}\mathcal{R}_{ij}\otimes\mathcal{P}_{\tilde{i}\tilde{j}}+\mathcal{P}_{ij}\otimes\mathcal{R}_{\tilde{i}\tilde{j}}\Big{]}\quad&\text{for
$\nu_{ij}$ odd}\end{cases}\,.$ (42)

This decomposition is optimal with

$\gamma=2\,\Pi_{s=1}^{n}(1+|\mathrm{sin}(\theta_{s})|)-1\,.$ (43)

It consists of single-qubit $Z$ gates, multi-qubit $Z$ rotations by the angles
$\pm\pi/2$ and measurements within CNOT-gate ladders. Due to the double sum,
the number of terms in Eq. 42 grows as $\mathcal{O}(4^{n})$. Interestingly,
the decomposition achieves the optimal $\gamma$ parameter even though no
exchange of classical information is required between the partitions.

## References

  * Shor [1994] P. Shor, Algorithms for quantum computation: discrete logarithms and factoring, in  _Proceedings 35th Annual Symposium on Foundations of Computer Science_ (IEEE Comput. Soc. Press, 1994) p. 124.
  * Grover [1996] L. K. Grover, A fast quantum mechanical algorithm for database search, in  _Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing_ (Association for Computing Machinery, New York, NY, USA, 1996) p. 212.
  * Harrigan  _et al._ [2021] M. P. Harrigan, K. J. Sung, M. Neeley, K. J. Satzinger, F. Arute, K. Arya, J. Atalaya, J. C. Bardin, R. Barends, S. Boixo, M. Broughton, B. B. Buckley, D. A. Buell, B. Burkett, N. Bushnell, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, S. Demura, A. Dunsworth, D. Eppens, A. Fowler, B. Foxen, C. Gidney, M. Giustina, R. Graff, S. Habegger, A. Ho, S. Hong, T. Huang, L. B. Ioffe, S. V. Isakov, E. Jeffrey, Z. Jiang, C. Jones, D. Kafri, K. Kechedzhi, J. Kelly, S. Kim, P. V. Klimov, A. N. Korotkov, F. Kostritsa, D. Landhuis, P. Laptev, M. Lindmark, M. Leib, O. Martin, J. M. Martinis, J. R. McClean, M. McEwen, A. Megrant, X. Mi, M. Mohseni, W. Mruczkiewicz, J. Mutus, O. Naaman, C. Neill, F. Neukart, M. Y. Niu, T. E. O’Brien, B. O’Gorman, E. Ostby, A. Petukhov, H. Putterman, C. Quintana, P. Roushan, N. C. Rubin, D. Sank, A. Skolik, V. Smelyanskiy, D. Strain, M. Streif, M. Szalay, A. Vainsencher, T. White, Z. J. Yao, P. Yeh, A. Zalcman, L. Zhou, H. Neven, D. Bacon, E. Lucero, E. Farhi, and R. Babbush, Quantum approximate optimization of non-planar graph problems on a planar superconducting processor, Nat. Phys. 17, 332 (2021).
  * Farhi  _et al._ [2014] E. Farhi, J. Goldstone, and S. Gutmann, A quantum approximate optimization algorithm (2014), arXiv:1411.4028 .
  * Havlíček  _et al._ [2019] V. Havlíček, A. D. Córcoles, K. Temme, A. W. Harrow, A. Kandala, J. M. Chow, and J. M. Gambetta, Supervised learning with quantum-enhanced feature spaces, Nature 567, 209 (2019).
  * Biamonte  _et al._ [2017] J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Quantum machine learning, Nature 549, 195 (2017).
  * Lanyon  _et al._ [2010] B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, Towards quantum chemistry on a quantum computer, Nat. Chem. 2, 106 (2010).
  * Kandala  _et al._ [2017] A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets, Nature 549, 242 (2017).
  * Kim  _et al._ [2023] Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. van den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zaletel, K. Temme, and A. Kandala, Evidence for the utility of quantum computing before fault tolerance, Nature 618, 500 (2023).
  * Devoret and Schoelkopf [2013] M. H. Devoret and R. J. Schoelkopf, Superconducting circuits for quantum information: An outlook, Science 339, 1169 (2013).
  * Monroe and Kim [2013] C. Monroe and J. Kim, Scaling the ion trap quantum processor, Science 339, 1164 (2013).
  * Yuan  _et al._ [2023] X. Yuan, B. Regula, R. Takagi, and M. Gu, Virtual quantum resource distillation (2023), arXiv:2303.00955 .
  * Bechtold  _et al._ [2023] M. Bechtold, J. Barzen, F. Leymann, and A. Mandl, Circuit cutting with non-maximally entangled states (2023), arXiv:2306.12084 .
  * Peng  _et al._ [2020] T. Peng, A. W. Harrow, M. Ozols, and X. Wu, Simulating large quantum circuits on a small quantum computer, Phys. Rev. Lett. 125, 150504 (2020).
  * Brenner  _et al._ [2023] L. Brenner, C. Piveteau, and D. Sutter, Optimal wire cutting with classical communication (2023), arXiv:2302.03366 .
  * Harada  _et al._ [2023] H. Harada, K. Wada, and N. Yamamoto, Doubly optimal parallel wire cutting without ancilla qubits (2023), arXiv:2303.07340 .
  * Lowe  _et al._ [2023] A. Lowe, M. Medvidović, A. Hayes, L. J. O’Riordan, T. R. Bromley, J. M. Arrazola, and N. Killoran, Fast quantum circuit cutting with randomized measurements, Quantum 7, 934 (2023).
  * Uchehara  _et al._ [2022] G. Uchehara, T. M. Aamodt, and O. D. Matteo, Rotation-inspired circuit cut optimization, in  _2022 IEEE/ACM Third International Workshop on Quantum Computing Software (QCS)_ (IEEE Computer Society, Los Alamitos, CA, USA, 2022) p. 50.
  * Pednault [2023] P. Pednault, An alternative approach to optimal wire cutting without ancilla qubits (2023), arXiv:2303.08287 .
  * Hofmann [2009] H. F. Hofmann, How to simulate a universal quantum computer using negative probabilities, J. Phys. A: Math. Theor. 42, 275304 (2009).
  * Mitarai and Fujii [2021a] K. Mitarai and K. Fujii, Constructing a virtual two-qubit gate by sampling single-qubit operations, New J. Phys. 23, 023021 (2021a).
  * Mitarai and Fujii [2021b] K. Mitarai and K. Fujii, Overhead for simulating a non-local channel with local channels by quasiprobability sampling, Quantum 5, 388 (2021b).
  * Piveteau and Sutter [2023] C. Piveteau and D. Sutter, Circuit knitting with classical communication, IEEE Trans. Inf. Theory , 1 (2023).
  * Ufrecht  _et al._ [2023] C. Ufrecht, M. Periyasamy, S. Rietsch, D. D. Scherer, A. Plinge, and C. Mutschler, Cutting multi-control quantum gates with ZX calculus, Quantum 7, 1147 (2023).
  * Pashayan  _et al._ [2015] H. Pashayan, J. J. Wallman, and S. D. Bartlett, Estimating outcome probabilities of quantum circuits using quasiprobabilities, Phys. Rev. Lett. 115, 070501 (2015).
  * Piveteau  _et al._ [2022] C. Piveteau, D. Sutter, and S. Woerner, Quasiprobability decompositions with reduced sampling overhead, NPJ Quantum Inf. 8, 12 (2022).
  * Temme  _et al._ [2017] K. Temme, S. Bravyi, and J. M. Gambetta, Error mitigation for short-depth quantum circuits, Phys. Rev. Lett. 119, 180509 (2017).
  * Collins  _et al._ [2001] D. Collins, N. Linden, and S. Popescu, Nonlocal content of quantum operations, Phys. Rev. A 64, 032302 (2001).
  * Gottesman and Chuang [1999] D. Gottesman and I. L. Chuang, Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations, Nature 402, 390 (1999).
  * Eisert  _et al._ [2000] J. Eisert, K. Jacobs, P. Papadopoulos, and M. B. Plenio, Optimal local implementation of nonlocal quantum gates, Phys. Rev. A 62, 052317 (2000).
  * Stahlke and Griffiths [2011] D. Stahlke and R. B. Griffiths, Entanglement requirements for implementing bipartite unitary operations, Phys. Rev. A 84, 032316 (2011).
  * Soeda  _et al._ [2011] A. Soeda, P. S. Turner, and M. Murao, Entanglement cost of implementing controlled-unitary operations, Phys. Rev. Lett. 107, 180501 (2011).
  * Cirac  _et al._ [2001] J. I. Cirac, W. Dür, B. Kraus, and M. Lewenstein, Entangling operations and their implementation using a small amount of entanglement, Phys. Rev. Lett. 86, 544 (2001).
  * Dür and Cirac [2001] W. Dür and J. I. Cirac, Nonlocal operations: Purification, storage, compression, tomography, and probabilistic implementation, Phys. Rev. A 64, 012317 (2001).
  * Groisman and Reznik [2005] B. Groisman and B. Reznik, Implementing nonlocal gates with nonmaximally entangled states, Phys. Rev. A 71, 032322 (2005).
  * Vidal and Tarrach [1999] G. Vidal and R. Tarrach, Robustness of entanglement, Phys. Rev. A 59, 141 (1999).
  * Note [1] It can be shown that all operations in the virtual teleportation protocol are elements of the extended definition of local operations used in Ref. [23, 15].
  * Cowtan  _et al._ [2020] A. Cowtan, S. Dilkes, R. Duncan, W. Simmons, and S. Sivarajah, Phase gadget synthesis for shallow circuits, EPTCS 318, 213 (2020).
  * Gogioso and Yeung [2023] S. Gogioso and R. Yeung, Annealing optimisation of mixed ZX phase circuits, in  _Proceedings 19th International Conference on Quantum Physics and Logic , Wolfson College, Oxford, UK_, EPTCS, Vol. 394, edited by S. Gogioso and M. Hoban (2023) pp. 415–431.
  * Nash  _et al._ [2020] B. Nash, V. Gheorghiu, and M. Mosca, Quantum circuit optimizations for nisq architectures, Quantum Sci. Technol. 5, 025010 (2020).
  * Sperling and Vogel [2009a] J. Sperling and W. Vogel, Representation of entanglement by negative quasiprobabilities, Phys. Rev. A 79, 042337 (2009a).
  * Sperling and Vogel [2009b] J. Sperling and W. Vogel, Erratum: Representation of entanglement by negative quasiprobabilities [Phys. Rev. A 79, 042337 (2009)], Phys. Rev. A 80, 029905 (2009b).
  * Mitarai and Fujii [2019] K. Mitarai and K. Fujii, Methodology for replacing indirect measurements with direct measurements, Phys. Rev. Research 1, 013006 (2019).