Deriving Optimal QPD for Wire Cuts

Here we will briefly go over where the decomposition used for wire cuts comes from and how that is transfromed to operations implementable on a quantum computer. The derivation is based on [1][2].

Decompose Single Qubit Density Matrix

Any single qubit density matrix can be decomposed as:

\[ \rho=\frac{I}{2}+\frac{Tr[X\rho]}{2}X+\frac{Tr[Y\rho]}{2}Y+\frac{Tr[Z\rho]}{2}Z \]

We now want to express this as a sum of measure-prepare channels (measure in a pauli basis, prepare a pauli basis state).

Start by ignoring the \(\frac{I}{2}\) term for now. Each term has the form \(\frac{1}{2}Tr[\sigma\rho]\sigma\). Now expand this using the spectral decomposition \(\sigma=P^+-P^-\) where \(P^\pm=\frac{I+\sigma}{2}\) are the projectors to the \(\pm 1\) eigenstates of \(\sigma\). Similarly \(Tr[P^+\rho]-Tr[P^-\rho]\).

Expanding the product:

\[\begin{split} \begin{aligned} Tr[\sigma\rho]\sigma &= (Tr[P^+\rho]-Tr[P^-\rho])(P^+-P^-) \\ &= (Tr[P^+\rho]-Tr[P^+\rho])P^+ - (Tr[P^-\rho]-Tr[P^-\rho])P^- \\ &= Tr[\sigma\rho]P^+ - Tr[\sigma\rho]P^- \end{aligned} \end{split}\]

We can now observe that the \(\frac{1}{2}Tr[\sigma\rho]\sigma\) gives two measure-prepare channels, one per eigenstate:

Measure	Prepare	Coef
\(\sigma\)	\(\frac{I+\sigma}{2}\)	1/2
\(\sigma\)	\(\frac{I-\sigma}{2}\)	-1/2

Now looking at the \(\frac{I}{2}\) term we can write it as:

\[ \frac{I}{2}=\frac{1}{2}Tr[I\rho]\ket{0}\bra{0}+\frac{1}{2}Tr[I\rho]\ket{1}\bra{1} \]

These are again two measure-prepare channels.

Now writing out all the measure prepare channels for I, X, Y, Z:

Measure	Prepare	Coef
I	\(\ket{0}\bra{0}\)	1/2
I	\(\ket{1}\bra{1}\)	1/2
Z	\(\ket{0}\bra{0}\)	1/2
Z	\(\ket{1}\bra{1}\)	-1/2
X	\(\ket{+}\bra{+}\)	1/2
X	\(\ket{-}\bra{-}\)	-1/2
Y	\(\ket{i}\bra{i}\)	1/2
Y	\(\ket{-i}\bra{-i}\)	-1/2

Note that here the identity basis measurement (I) stands for a measurement that always reutrns the \(\ket{0}\) state. Practically this means not measuring and leaving a classical bit in the initial 0 position.

We have now decomposed any single qubit density matrix \(\rho\) as \(\sum_{i=1}^8c_iTr[O_i\rho]\rho_i\).

Summing over the absolute coefficient we get \(\gamma=4\) which is optimal (without communication).

References

1. T. Peng, A. W. Harrow, M. Ozols, and X. Wu, ‘Simulating Large Quantum Circuits on a Small Quantum Computer’, Phys. Rev. Lett., vol. 125, no. 15, p. 150504, Oct. 2020, doi: 10.1103/PhysRevLett.125.150504.

2. H. Harada, K. Wada, and N. Yamamoto, ‘Doubly Optimal Parallel Wire Cutting without Ancilla Qubits’, PRX Quantum, vol. 5, no. 4, p. 040308, Oct. 2024, doi: 10.1103/PRXQuantum.5.040308.