One qubit is mapped to two level system in the standard model of quantum computation: a typical example is a spin-1/2 system. Further progress requires the use of network of multilevel quantum systems of different physical origin in solids: optical levels, large or strong interacting spins etc. We propose virtual qubit formalism and show how the quantum objects state space can be used as a basis for multiqubit information recording and processing in a quantum computer. It is suggested to map the qubits into solid state NMR spin system collective states instead of the state of the individual spin. Such an approach introduces the stable computational basis without any additional actions and allows to obtain the universal set of quantum gates, which operation time is determined only by the RF pulses.
Schemes of experimental realization of the main two-qubit processors for quantum computers and the Deutsch-Jozsa algorithm are derived in virtual spin representation. The results are applicable for every four quantum states allowing the required properties for quantum processor implementation if for qubit encoding, virtual spin representation is used. A four-dimensional Hilbert space of nuclear spin
3/2 is considered in detail for this aim.
A constructive scheme has been devised to enable mapping of any quantum computation into a spintronic circuit in which the computation is encoded in a basis that is, in principle, immune to quantum decoherence. The scheme is implemented by an algorithm that utilizes multiple physical spins to encode each logical bit in such a way that collective errors affecting all the physical spins do not disturb the logical bit. The scheme is expected to be of use to experimenters working on spintronic implementations of quantum logic. Spintronic computing devices use quantum-mechanical spins (typically, electron spins) to encode logical bits. Bits thus encoded (denoted qubits) are potentially susceptible to errors caused by noise and decoherence. The traditional model of quantum computation is based partly on the assumption that each qubit is implemented by use of a single two-state quantum system, such as an electron or other spin-1.2 particle. It can be surprisingly difficult to achieve certain gate operations . most notably, those of arbitrary 1-qubit gates . in spintronic hardware according to this model. However, ironically, certain 2-qubit interactions (in particular, spin-spin exchange interactions) can be achieved relatively easily in spintronic hardware. Therefore, it would be fortunate if it were possible to implement any 1-qubit gate by use of a spin-spin exchange interaction. While such a direct representation is not possible, it is possible to achieve an arbitrary 1-qubit gate indirectly by means of a sequence of four spin-spin exchange interactions, which could be implemented by use of four exchange gates. Accordingly, the present scheme provides for mapping any 1-qubit gate in the logical basis into an equivalent sequence of at most four spin-spin exchange interactions in the physical (encoded) basis. The complexity of the mathematical derivation of the scheme from basic quantum principles precludes a description within this article; it must suffice to report that the derivation provides explicit constructions for finding the exchange couplings in the physical basis needed to implement any arbitrary 1-qubit gate. These constructions lead to spintronic encodings of quantum logic that are more efficient than those of a previously published scheme that utilizes a universal but fixed set of gates.
The current proposals for the realization of quantum computer such as NMR, quantum dots and trapped ions are based on the using of an atom or an ion as one qubit. In these proposals a quantum computer consists from several atoms and the coupling between them provides the coupling between qubits necessary for a quantum gate. We discuss whether a {\it single} atom can be used as a quantum computer. Internal states of the atom serve to hold the quantum information and the spin-orbit and spin-spin interaction provides the coupling between qubits in the atomic quantum computer. In particular one can use the electron spin resonance (ESR) to process the information encoded in the hyperfine splitting of atomic energy levels. By using quantum state engineering one can manipulate the internal states of the natural or artificial (quantum dot) atom to make quantum computations.
In quantum error-correcting code (QECC), many quantum operations and measurements are necessary to correct errors in logical qubits. In the stabilizer formalism, which is widely used in QECC, generators
Gi(i=1,2,...) consist of multiples of Pauli matrices and perform encoding, decoding, and measurement. In order to maintain encoding states, the stabilizer Hamiltonian ${H}_{\mathrm{stab}}=\ensuremath{-}{\ensuremath{\sum}}_{i}{G}_{i}$ is suitable because its ground state corresponds to the code space. On the other hand, Hamiltonians of most solid-state qubits have two-body interactions and show their own dynamics. In addition solid-state qubits are fixed on substrate and qubit-qubit operation is restricted in their neighborhood. The main purpose of this paper is to show how to directly generate the stabilizer Hamiltonian
Hstab from conventional two-body Hamiltonians with Ising interaction and
XY interaction by applying a pulse control method such as an NMR technique. We show that generation times of
Hstab for nine-qubit code, five-qubit code, and Steane code are estimated to be less than 300 ns when typical experimental data of superconducting qubits are used, and sufficient pulse control is assumed. We also show how to prepare encoded states from an initial state $|0,...,0\ensuremath{\rangle}$. In addition, we discuss an appropriate arrangement of two- or three-dimensional arrayed qubits.