Quantum-gate characterization in an extended Hilbert space

We describe an approach for characterizing the process performed by a quantum gate using quantum process tomography, by first modeling the gate in an extended Hilbert space, which includes nonqubit degrees of freedom. To prevent unphysical processes from being predicted, present quantum process tomography procedures incorporate mathematical constraints, which make no assumptions as to the actual physical nature of the system being described. By contrast, the procedure presented here assumes a particular class of physical processes, and enforces physicality by fitting the data to this model. This allows quantum process tomography to be performed using a smaller experimental data set, and produces parameters with a direct physical interpretation. The approach is demonstrated by example of mode matching in an all-optical controlled-NOT gate. The techniques described are general and could be applied to other optical circuits or quantum computing architectures.

Figure 1

Schematic of the CNOT gate using beam splitters with reflectivities $\eta$. $c$ and $t$ denote the control and target qubits, respectively. Modes labeled × are discarded and serve to balance the amplitudes in the different paths. The gate is postselected upon detection of exactly one photon between the $c$ modes and one between the $t$ modes. $\tau$ boxes represent mode-mismatch parameters, described in the text. We adopt the phase-asymmetric beam-splitter convention, where sign inversion takes place upon reflection from the gray beam-splitter surfaces.

Figure 2

Graphical representation of the mode-mismatch parameter space, in two degrees of freedom only ($k_1$ and $k_2$). (a) Mode mismatch in a single degree of freedom, represented by the point $\mathbf{P}$. (b) Equivalence of mode mismatch in any single degree of freedom, represented by the points $\mathbf{P}$ and $\mathbf{Q}$. (c) Equivalence of mode mismatch in multiples of degrees of freedom $\mathbf{M}$ and a single degree of freedom $\mathbf{N}$ through rotational invariance in the choice of axes.