Quantum circuit cutting is a method to partition a large quantum circuit into smaller, more manageable parts. In particular, during the NISQ era of quantum computing the execution of quantum circuits is limited by the size, i.e., number of qubits, as well as their high susceptibility to noise. Therefore it is desirable to partition a quantum circuit into smaller chunks, run them on possibly smaller quantum devices and then to perform postprocessing to retrieve results which are close to the results one would expect from the uncut circuit. It has been shown in experiments, that the application of circuit cutting can indeed reduce the impact of noise.
Quasi probability decomposition
In order to perform wire or gate cutting, one can decompose the quantum channel V {\displaystyle {\mathcal {V}}} of the operation to cut as quasi probability decomposition
V ( ρ ) = ∑ i a i Φ i ( ρ ) {\displaystyle {\mathcal {V}}(\rho )=\sum _{i}a_{i}\Phi _{i}(\rho )}
with coefficients a i ∈ R {\displaystyle a_{i}\in \mathbb {R} } such that ∑ i a i = 1 {\displaystyle \sum _{i}a_{i}=1} . Those coefficients can be either positive or negative which is why this is coined as quasiprobability distribution. The Φ i {\displaystyle \Phi _{i}} 's are quantum channels which can act on disjoint qubits in the case of gate cutting, i.e., Φ i ( ρ A ⊗ ρ B ) = Φ i A ( ρ A ) ⊗ Φ i B ( ρ B ) {\displaystyle \Phi _{i}(\rho ^{A}\otimes \rho ^{B})=\Phi _{i}^{A}(\rho ^{A})\otimes \Phi _{i}^{B}(\rho ^{B})} , where A , B {\displaystyle A,B} denote the partitions resulting from the cut. For the case of wire cutting, the Φ i {\displaystyle \Phi _{i}} 's are measure-and-prepare-channels. Therefore, in the case of gate cutting the channel V {\displaystyle {\mathcal {V}}} is the channel of some quantum gate and for wire cutting it is the identity channel, as quantum wires act as identities on the qubits.
Sampling overhead Cutting a circuit comes with the price of a sampling overhead, scaling exponentially in the number of qubits involved in the cut. The sampling overhead depends on the L1 norm of the coefficients, i.e., κ = ∑ i | a i | {\displaystyle \kappa =\sum _{i}|a_{i}|} . From Hoeffding's inequality one can derive that the number of samples needed to retrieve an expectation value of the cut circuit which is close to the result expected from the uncut circuit scales as O ( κ 2 ) {\displaystyle {\mathcal {O}}(\kappa ^{2})} . The value κ {\displaystyle \kappa } scales exponentially with the number of cuts and therefore imposes a serious limitation on circuit cutting techniques. Thus, it is desired to reduce this value as much as possible. Considering the cut of n {\displaystyle n} parallel qubit wires, the method proposed by Peng et al. requires κ 2 = 16 n {\displaystyle \kappa ^{2}=16^{n}} . It was argued that the usage of classical communication between the different partitions can reduce the sampling overhead. The cutting technique introduced by Harada et al. employs classical communication and requires κ 2 = ( 2 n + 1 − 1 ) 2 {\displaystyle \kappa ^{2}=(2^{n+1}-1)^{2}} which is the lowest possible if LOCC is used.
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