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arXiv:quant-ph/9610008 · International Journal of Modern Physics B

Design of Gates for Quantum Computation: The Three-Spin XOR Gate in Terms of Two-Spin Interactions

Dima Mozyrsky, Vladimir Privman A, Steven P. Hotaling

Published Feb 20, 1998• 6 citations
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Abstract

We propose to design multispin quantum gates in which the input and output two-state systems (spins) are not necessarily identical. We describe the motivations for such studies and then derive an explicit general two-spin interaction Hamiltonian which accomplishes the quantum XOR gate function for a system of three spins: two input and one output.

DOI
10.1142/s0217979298000351

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We consider interactions needed to operate the quantum-mechanical NOT gate in the conventional formulation when the evolution is in time only and also in the case of spatially separated Input and Output two-state systems. Explicit expressions for the Hamiltonian are derived for the interaction which is time-independent for the duration of the gate operation. We develop a general approach which can be used to obtain Hamiltonians of this sort for quantum computer gates. We also discuss extensions to the case of time-dependent interactions.
We derive an explicit Hamiltonian for copying the basis up and down states of a quantum two-state system—a qubit—onto n “copy ” qubits (n ≥ 1) initially all prepared in the down state. In terms of spin components, for spin- 1 2 particle spin states, the resulting Hamiltonian involves n- and (n + 1)-spin interactions. The case n = 1 also corresponds to a quantum-computing controlled-NOT gate. PACS numbers: 03.65.Bz, 85.30.St. – 1 – Interest in quantum computing [1-27] has boosted studies of quantum mechanics of two-state systems such as the spin states of spin- 1 2 particles. We will use “spin ” to indicate a two-state system in this context. The “binary ” up and down spin states are of particular significance and
Quantum computation offers the potential for solving specific problems exponentially faster than the best classical machines, by virtue of its non-classical dynamics and information representation. Quantum mechanical bits (called 'qubits') can be both 0 and 1 at the same time, in which case the qubit is said to be in a superposition state. A quantum gate with input 0 or 1 gives the same output as a classical gate, but if the input is 0 and 1 simultaneously, the quantum gate performs the two corresponding operations simultaneously. Similarly, a 2-qubit gate with an input state prepared as a superposition of the 4 combinations 00, 01, 10 and 11, performs four simultaneous computations. In general, an N-qubit gate could perform 2/sup N/ computations at the same time. So as we increase the size of the system, the computational capacity of the apparatus goes up exponentially, compared to linearly for a classical computer. Using commercial NMR-spectrometers, simple quantum circuits are experimentally implemented consisting of multiple cascaded logic gates.