Quantum tomography for solid-state qubits

– We propose a method for the tomographic reconstruction of qubit states for a general class of solid-state systems in which the Hamiltonians are represented by spin operators, e.g., with Heisenberg-, XXZ-, or XY -type exchange interactions. We analyze the implementation of the projective operator measurements, or spin measurements, on qubit states. All the qubit states for the spin Hamiltonians can be reconstructed by using experimental data. Quantum information processing requires the effective measurement of quantum states. However, a single quantum measurement can only obtain partial information of a quantum state. The reconstruction of a quantum state requires measuring a complete set of observables on an ensemble of identically prepared copies of the system. This method called quantum state tomography [1], is very important because any unknown state can be ascertained by tomographic measurements. Moreover, the full description of qubit states can increase the accuracy of quantum operations. Tomographic measurements have been experimentally implemented for, e.g., the nuclear spin state of an NMR system [2], the electromagnetic field and photon state [3], the vibration state of molecules [4], the motional quantum state of a trapped atom [5], and atomic wave packets [6]. Experimental investigations on solid-state qubits are very promising, especially in superconducting [7,8] and quantum dot structures [9]. These recent achievements make it necessary to experimentally determine quantum states in solid-state systems. Although there are many theoretical studies on tomography (e.g., refs. [10] and references therein), to our knowledge, these are not specific to solid-state systems. Here, we focus on this question for quantum computing models using standard spin representations for solid-state qubits. Our proposal is related to tomographic measurements using NMR. The measurements of the density matrix in NMR experiments are obtained from the NMR spectrum of the linear combination of “product operators”, i.e. products of the angular-momentum operators [11]. However, experiments in solid-state systems usually involve the local single-qubit projective operator