[2] It is also called channel-state duality by some authors in the quantum information area,[3] but mathematically, this is a more general correspondence between positive operators and the complete positive superoperators.
, which is a trace-preserving completely positive map from operator spaces
It allows us to represent a gate's properties and behavior as a Choi state.
, we would apply the Bell scheme on sites A and D. However, this can introduce Pauli byproduct operators, such as
The indirect Bell measurement is performed by applying a gate
The outcome of the indirect Bell measurement corresponds to either the singlet or the triplet state.
On the other hand, if the outcome is the triplet, which has the full symmetry of the relevant unitary group, the gate V is modified by applying a rotation T on the triplet state, equivalent to the action of
can be realised in a heralded manner, depending on the outcome from the qubit ancilla measurement.
However, a scheme based on direct-sum dilation can be employed to overcome this obstacle.
can be simulated by using a random-unitary channel, where the controlled-unitary gate U̘ acts on the joint system of the input state ρ and an ancilla qubit.
The action E(ρ) is obtained by restricting the evolution to the system subspace.
Finally, the action of the channel E on the input state ρ is obtained by considering the evolution restricted to the system subspace.
This channel requires a qutrit ancilla, and when the outcome is 2, indicating the occurrence of
, which is equal to 1 due to the trace-preserving condition, the simulation has to be restarted.
For special types of channels, the scheme can be significantly simplified.
These channels and program states are trivial since there is no entanglement, and they can be simulated using a measurement-preparation scheme.
From Stinespring's dilation, we know that it requires the form of Kraus operators, which are not easy to find in general given a Choi state.
From Choi, a channel is extreme if there exists a Kraus representation
It has been conjectured and numerically supported that an arbitrary channel can be written as a convex sum of at most
For the worst case, the upper bound for such a convex sum is
from Carathéodory's theorem on convex sets, which merely costs more random dits.
, hence permitting a convex-sum decomposition, one needs to sample the composition of gen-extreme channels.
Compared with the standard (tensor-product) dilation method to simulate a general channel, which requires two qudit ancillas, the method above requires lower circuit cost since it only needs a single qudit ancilla instead of two.
While the convex-sum decomposition, which is a sort of generalised eigenvalue decomposition since a gen-extreme Choi state can be mapped to a pure state, is difficult to solve for large-dimensional channels, it shall be comparable with the eigen-decomposition of the Choi state to find the set of Kraus operators.
We assume fault-tolerant qubits, gates, and measurements, which can be achieved with quantum error-correcting codes.
Additionally, we highlight two intriguing issues that establish connections with standard frameworks and results.
Our method can be used to teleport unitary universal gate sets.
, CNOT, and CZ are (generalised) permutations since they are Clifford gates, which preserve the Pauli group.
Instead, the affine forms of them contain a Hadamard-like gate as a sub-matrix, which means, in the Heisenberg picture, they are able to generate superpositions of Pauli operators.
This fact also generalises to the qudit case, with Hadamard replaced by Fourier transform operators.